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Mathematical Discourse: Language, Symbolism and Visual Images

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Page 1: Mathematical Discourse: Language, Symbolism and Visual Images
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Mathematical Discourse

Language, Symbolism and Visual Images

Kay L. O'Halloran

continuumL O N D O N • N E W Y O R K

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ContinuumThe Tower Building, 15 East 26th Street,11 York Road, New York, NY 10010London SE1 7NX

© Kay L. O'Halloran 2005

All rights reserved. No part of this publication may be reproduced or transmitted in any formor by any means, electronic or mechanical, including photocopying, recording or anyinformation storage or retrieval system, without permission in writing from the publishers.

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

ISBN 0-8264-6857-8 (hardback)

Library of Congress Cataloging-in-Publication DataA catalogue record for this book is available from the Library of Congress

Typeset by RefineCatch Ltd, Bungay, SuffolkPrinted and bound in Great Britain by Cromwell Press Ltd, Trowbridge, Wilts

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Contents

Acknowledgements viii

Copyright Permission Acknowledgements ix

1 Mathematics as a Multisemiotic Discourse 11.1 The Creation of Order 11.2 Halliday's Social Semiotic Approach 61.3 Mathematics as Multisemiotic 101.4 Implications of a Multisemiotic View 131.5 Tracing the Semiotics of Mathematics 171.6 Systemic Functional Research in Multimodality 19

2 Evolution of the Semiotics of Mathematics 222.1 Historical Development of Mathematical Discourse 222.2 Early Printed Mathematics Books 242.3 Mathematics in the Early Renaissance 332.4 Beginnings of Modern Mathematics: Descartes and Newton 382.5 Descartes' Philosophy and Semiotic Representations 462.6 A New World Order 57

3 Systemic Functional Linguistics (SFL) and Mathematical Language 603.1 The Systemic Functional Model of Language 603.2 Interpersonal Meaning in Mathematics 673.3 Mathematics and the Language of Experience 753.4 The Construction of Logical Meaning 783.5 The Textual Organization of Language 813.6 Grammatical Metaphor and Mathematical Language 833.7 Language, Context and Ideology 88

4 The Grammar of Mathematical Symbolism 944.1 Mathematical Symbolism 944.2 Language-Based Approach to Mathematical Symbolism 964.3 SF Framework for Mathematical Symbolism 974.4 Contraction and Expansion of Experiential Meaning 1034.5 Contraction of Interpersonal Meaning 1144.6 A Resource for Logical Reasoning 1184.7 Specification of Textual Meaning 1214.8 Discourse, Grammar and Display 1254.9 Concluding Comments 128

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VI CONTENTS

5 The Grammar of Mathematical Visual Images 1295.1 The Role of Visualization in Mathematics 1295.2 SF Framework for Mathematical Visual Images 1335.3 Interpersonally Orientating the Viewer 1395.4 Visual Construction of Experiential Meaning 1425.5 Reasoning through Mathematical Visual Images 1455.6 Compositional Meaning and Conventionalized Styles of

Organization 1465.7 Computer Graphics and the New Image of Mathematics 148

6 Intersemiosis: Meaning Across Language, Visual Images and Symbolism 1586.1 The Semantic Circuit in Mathematics 1586.2 Intersemiosis: Mechanisms, Systems and Semantics 1636.3 Analysing Intersemiosis in Mathematical Texts 1716.4 Intersemiotic Re-Contexualization in Newton's Writings 1776.5 Semiotic Metaphor and Metaphorical Expansions of Meaning 1796.6 Reconceptualizing Grammatical Metaphor 184

7 Mathematical Constructions of Reality 1897.1 Multisemiotic Analysis of a Contemporary Mathematics Problem 1897.2 Educational Implications of a Multisemiotic Approach to

Mathematics 1997.3 Pedagogical Discourse in Mathematics Classrooms 2057.4 The Nature and Use of Mathematical Constructions 208

References 211

Index 223

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In memory of my father, Jim O'Halloran.For my brother Greg and his family.

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Acknowledgements

This study of mathematical discourse is based on Michael Halliday'ssystemic functional model of language and Jim Martin's extensive contri-butions to systemic theory. Michael O'Toole's application of systemicfunctional theory to displayed art provides the inspiration for the modelsfor mathematical symbolism and visual image presented here. Jay Lemkepioneered the application of systemic functional theory to science andmathematics as multisemiotic discourses. This work would not be possiblewithout these founding contributions.

My special thanks to the director and librarians from the John RylandsUniversity Library of Manchester (JRULM) for making so readily availablethe mathematics manuscripts in the Mathematical Printed Collection. Ithank Linda Thompson (Director of the Language and Literacy StudiesResearch Group, Faculty of Education, University of Manchester) for sup-porting my visit to JRULM. My special thanks to Philip J. Davis (EmeritusProfessor, Applied Mathematics Division, Brown University) for his interestin this project. Our lively correspondence has contributed to the contentsof this book.

I thank Michael O'Toole and Frances Christie for their friendship andsupport, and I thank my past and present friends and colleagues at theNational University of Singapore - most notably Joe Foley, Chris Stroud,Linda Thompson, Desmond Allison, Ed McDonald, Umberto Ansaldo andLisa Lim.

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Copyright Permission Acknowledgements

The author is grateful to the following organizations for the right to repro-duce the images which appear in this book. Every effort has been made tocontact copyright holders of material produced in this book. The pub-lishers apologize for any omissions and will be pleased to rectify them at theearliest opportunity.

Chapter 1

Plate 1.1(1) Photographs from Beevor (2002: Chapter 24)The photographs are reprinted with kind permission from:Photograph 43: Bildarchiv PreuBischer Kulturbesitz, BerlinPhotograph 44: Ullstein Bild, BerlinPhotograph 45: Jiirgen Stumpff/Bildarchiv PreuBischerKulturbesitz, Berlin

Plate 1.3(1) Language, Visual Images and Symbolism (Kockelkoren etal, 2003: 173)Reprinted with kind permission from Elsevier

Chapter 2

Plate 2.2(1) The Treviso Arithmetic (reproduced from Swetz, 1987: 140)Reprinted by permission from Open Court PublishingCompany, a division of Carus Publishing Company, Peru, ILfrom Capitalism and Arithmetic by F. Swetz, © 1987 by OpenCourt Publishing Company

The following have been reprinted by courtesy of the director andlibrarian, the John Rylands University Library of Manchester:

Plate 2.2(2) The Hindu-Arabic system versus counters and lines (Reisch,1535: 267)

Plate 2.2(3) Printing counter and line calculations (Reisch, 1535: 326)

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X COPYRIGHT PERMISSION ACKNOWLEDGEMENTS

Plate 2.2(4a) Euclid's Elements: Venice 1482, Erhard Ratdolt (Thomas-Stanford, 1926: Illustration II)

Plate 2.2(4b) Euclid's Elements: Venice 1505, J. Tacuinus (Thomas-Stanford, 1926: Illustration IV)

Plate 2.2(6) QuadraturaParaboles (Archimedes, 1615: 437)Plate 2.3(2) Hitting a target (Tartaglia, 1546: 7)Plate 2.3(3) Arithmetic calculations to hit a target (Tartaglia, 1546:

106)Plate 2.3(4) Positioning a target (Galileo, 1638: 67)Plate 2.4(la) Removing the human body (Descartes, 1682: 111)Plate 2.4(2a) Movement in space and time: the stone (Descartes, 1682:

217)Plate 2.4(2b) Movement in space and time: the model (Descartes, 1682:

217)Plate 2.4(4a) Context, circles and lines (Descartes, 1682: 226)Plate 2.4(4b) Circles and lines (Descartes, 1682: 228)Plate 2.4(5a) Descartes' semiotic compass (1683: 54) (Book Two)Plate 2.4(5b) Drawing the curves (Descartes, 1683: 20) (Book Two)Plate 2.5(1) Illustration from Newton's (1729) The Mathematical Principles

of Natural Philosophy

My thanks to the Bodleian Library, Oxford for access to the microfilm ofthe following manuscript held by the British Library, London:

Plate 2.4(8a) Newton's (1736: 80-81) Method of Fluxions and Infinite SeriesPlate 2.4(8b) Newton's (1736: 100) Method of Fluxions and Infinite Series

The following have been reprinted by courtesy of Dover Publications:

Plate 2.2(5) Translation of Euclid (reproduced from Euclid, 1956: 283)Plate 2.4(lb) Removing the human body: Newton (1952: 9)Plate 2.4(6a) Descartes' description of curves (1954: 234)Plate 2.4(6b) Descartes' use of symbolism (1954: 186)Plate 2.4(7) Newton's algebraic notes on Euclid (reproduced from

Cajori, 1993: 209) Latin edition (1655) of Barrow'sEuclid (Taken from Isaac Newton: A Memorial Volume[ed. WJ. Greenstreet: London, 1927], p. 168)

Chapter 4

Plate 4.3(1) Mathematical Symbolic Text (Stewart, 1999: 139)From Calculus: Combined Single and Multivariable 4th editionby Stewart. © 1999. Reprinted with permission of Brooks/Cole, a division of Thompson Learning:www.thompsonrights.com. Fax: 800 730-2215

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COPYRIGHT P E R M I S S I O N ACKNOWLEDGEMENTS xi

The following are reprinted with permission from Elsevier:

Plate 4.5(1) Mathematical Symbolic Text (Wei and Winter, 2003: 159)Plate 4.7(1) Textual Organization of Mathematical Symbolism (Clerc,

2003: 117)

Chapter 5

Plate 5.2(1) Interpretation of the Derivative as the Slope of a Tangent(Stewart, 1999: 130)From Calculus: Combined Single and Multivariable 4th editionby Stewart. © 1999. Reprinted with permission of Brooks/Cole, a division of Thompson Learning:www.thompsonrights.com. Fax: 800 730-2215

Plate 5.7(1) Evolving Images of Computer Graphics(a) Figure 5 Stills from a computer-made movie: wrapping arectangle to form a torus (Courtesy T. Banchoff and C. M.Strauss) (Davis, 1974: 126)Reprinted by kind permission of T. Banchoff and C. M.Strauss through Philip J. Davis (Emeritus Professor, AppliedMathematics Division, Brown University Providence, RI,USA)(b) MATLAB graphics, circa 1985 (courtesy of Philip J.Davis)Reprinted with kind permission Philip J. Davis (EmeritusProfessor, Applied Mathematics Division, Brown UniversityProvidence, RI, USA)(d) Graphical and Diagrammatic Display of Patterns (Bergeet al., 2003: 194)Reprinted with kind permission from Elsevier

Chapter 6

Plate 6.31 Newton's (1736: 46) Procedure for Drawing TangentsReprinted from microfilm in the Bodleian Library, Oxford.Reproduced with permission from the British Library(London) which holds the original manuscript.

Plate 6.32 The Derivate as the Instantaneous Rate of Change (Stewart,1999: 132)From Calculus: Combined Single and Multivariable 4th editionby Stewart. © 1999. Reprinted with permission of Brooks/Cole, a division of Thompson Learning:www.thompsonrights.com. Fax: 800 730-2215

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Chapter 7

Plate 7.1 (1) Mathematics Example 2.24 (Burgmeier et al, 1990: 76-77)From Burgmeier, J. W., Boisen, M. B. and Larsen, M. D.(1990) Brief Calculus with Applications. New York:McGraw-Hill. © 1990. Reprinted with kind permission fromThe McGraw-Hill Publishing Company, New York

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1 Mathematics as a Multisemiotic Discourse

1.1 The Creation of Order

Success is right. What does not succeed is wrong. It was, for example, wrong to per-secute the Jews before the war since that set the Anglo-Americans against Germany.It would have been right to postpone the anti-Jewish campaign and begin it afterGermany had won the war. It was wrong to bomb England in 1940. If they hadrefrained, Great Britain, so they believe, would have joined Hitler in the war againstRussia. It was wrong to treat Russian and Polish [prisoners of war] like cattle since nowthey will treat Germans in the same way. It was wrong to declare war against the USAand Russia because they were together stronger than Germany.

In this extract from Berlin: The Downfall 1945, Beevor (2002:429) summarizesthe views of over three hundred pro-Nazi generals after Germany's defeat inthe Second World War, based on a report of interviews by the SupremeHeadquarters Allied Expeditionary Force in Europe (SHAEF). The Germangenerals are seen to possess a view of events; one they envisaged would haveworked towards victory rather than defeat. Their guiding principle, asexpressed by Beevor (2002: 429), is 'Success is right. What does not succeedis wrong.' Many millions participated in the enactment of those views, andthe familiar question arises as to how this could be possible. How could somany people be persuaded to take part in the events which unfolded duringthe course of the Second World War? There have been a variety of responsesto this question. Goldhagen (1996), for instance, suggests that most of theordinary Germans involved in the holocaust were 'willing executioners' whoactually believed in the events that took place. No doubt a variety of meanswere used incrementally over a long period of time in order to mobilize thepopulation in the war effort. In the past century, such massive mobilizationshave not been confined to Germany. Weitz (2003), for example, documentsthe unprecedented programmes of genocide which have taken place in thetwentieth century, including Stalin's Soviet Union, Cambodia under theKhmer Rouge and the former Yugoslavia. In these cases and many others,significant portions of the population take part in the war effort. But howcan so many people be convinced of the necessity of such programmes, theimpact of which lasts for generations?

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In attempting to answer such a question, it is worthwhile to consider asimple reformulation of the German generals' guiding principles 'Successis right. What does not succeed is wrong' (Beevor, 2002: 429). That is, if thephrase for the Nazi party is inserted, the statement becomes 'Success [for theNazi party] is right. What does not succeed [for the Nazi party] is wrong.'Such a reformulation introduces in unequivocal terms the basis uponwhich the guiding principles are constructed. The simple inclusion of thebeneficiary 'the Nazi party' makes clear the premise underlying the lin-guistic statement, and the specific interests which are being served. Such aninclusion also provides room for argumentation and negation, whereas thefinality accompanying the original cliched statement 'Success is right' ismuch more difficult to counteract. In a similar manner, the import of lin-guistic choices may be seen in George W. Bush's statement to the worldafter 11 September 2001 attacks on the United States: 'Either you are withus, or you are with the terrorists' (CNN.com/US 20 September 2001).Expressed in simple terms of a relational set of circumstances, the dichot-omy is based on pro-American interests ('with us') versus anti-Americaninterests ('against us'). Such a simple division of the world into two oppos-ing sets of relations leaves few options for a negotiated peace settlementalong other possible lines of interest. Language functions in this way tostructure the world largely in terms of categories, the nature of whichdepends upon the choices which are made.

The value of using language and other systems of meaning to createa world view conducive to the war effort was well recognized in NaziGermany. These strategies included the use of the media for news reportsand documentaries (involving language, visual images and music), politicalspeeches and rallies (for example, language, visual images, embodiedaction, music, and architectural features of the platform and seatingarrangements), and particular styles of dress and the distinctive salute ofthe Nazi party (for example, the uniforms, insignias, actions and gestures).These strategies have direct parallels in existence today, where choicesfrom the different resources combine to create particular meanings to theexclusion of others. However, the contexts which give rise to the orderingof reality are not confined to those which are specifically designed formass consumption in the form of 'propaganda' programmes. Order ismaintained, negotiated and challenged in every situation which involveschoices from language, visual images, gesture, styles of dress and so forth.

Page (2001: 10) comments: 'There is a privilege in being raised in a timeof peace. A luxury that your life is not under immediate threat. Warbecomes something labelled as heroic, often patriotic, nationalistic. Thereis a cause, it is just and right, and it somehow excuses all the pain and all theloss.' The use of language and other sign systems for the structuring ofthought and reality in the ways described by Page is the subject of this study.This approach is not intended to downplay strategies of physical and men-tal coercion and abuse. However, violence commences somewhere, and inmany cases, for ordinary citizens at least, the starting point is the ordering

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of reality along certain lines through semiosis; that is, acts of meaningthrough choices from language and other sign systems. The major aim ofthis study is to introduce a theory and approach for examining the natureand impact of semiosis in contexts which span the supposedly inane tothe discourses of immense influence, which include the subject matter ofthis investigation; namely, mathematics and science. War is chosen as thetopic to introduce this approach.

The role of language for structuring thought and reality is well recog-nized today within a wide range of disciplines which include socio-linguistics, critical discourse theory, communication studies, psychologyand sociology (for example, Berger and Luckmann, 1991; Bourdieu, 1991;Fairclough, 1989; Gumperz, 1982; Halliday, 1978; Herman and Chomsky,1988; Vygotsky, 1986). In addition, the functions of visual images areincreasingly taken into account (for example, Barthes, 1972; Lynch andWoolgar, 1990; Mirzoeff, 1998; van Leeuwen andjewitt, 2001). This is espe-cially important in the electronic age where the ease with which pictorialrepresentations may be reproduced is expanding. Beevor (2002), forexample, includes visual images in the form of black and white photo-graphs and maps to depict the advance of the Red Army and the finalcollapse of the Third Reich. Berlin: The Downfall 1945 is a text or discourseconstructed through choices from the English language, photographs andmaps. These choices work together to create Beevor's account of the hor-ror of the final months of the Second World War in Germany. In whatfollows, the types of meanings afforded by Beevor's (2002) photographsare investigated and compared to meanings which are made usinglanguage.

Photographs 43-45 displayed in Plate 1.1(1) appear in Chapter 24 inBeevor (2002: 354—369). These photographs appear among a group ofinserted photographs which are numbered 30-49. As seen in Plate 1.1(1),Photograph 43 is a picture of a German teenage conscript at the end of thewar, Photograph 44 shows a Russian female medical assistant attending to awounded Russian soldier, and the official signing of the final surrender byGeneral Stumpff, Field Marshal Keitel and Admiral von Friedburg in May1945 is shown in Photograph 45. In Beevor (2002), these pictures are pre-ceded by photographs of Russians engaged in street fighting in Berlin,scenes outside the Reich Chancellery, convoys of Russian-controlled armedforces, German soldiers surrendering in Berlin, Russian soldiers washingand civilians cooking in the streets of Berlin, victory celebrations betweendelegates from the Red Army and the US Army, and German civiliansescaping across the Elbe River to American territory. Immediately followingPlate 1.1(1), there are further photographs of soldiers in the streets ofBerlin, the Russian victory parade, and a full-page photograph of Red Armyofficials visiting the battleground inside the Reichstag. The photographsdisplayed in Plate 1.1(1) have contextual meaning within this sequenceof photographs. Beevor's (2002) linguistic account of the fall of Berlinsimilarly unfolds as a staged text consisting of sentences, paragraphs, pages

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44

45

Plate 1.1(1) Photographs from Beevor (2002: Chapter 24)

and chapters which have contextual meaning within the sequence of thenarrative. However, there are differences in the types of meaning affordedby Beevor's linguistic and photographic account of the fall of Berlin. These

43

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differences relate to the meaning potential of language and visual images.This point is developed below.

From existing photographs of the fighting and aftermath in Berlin in1945, a selection of photographs has been chosen to be included in Beevor(2002). In turn, each photograph in the sequence represents a set ofchoices made by the photographer, which, in the case of war, most likelyhappen more by chance rather than design. The photographer captures aninstance of time according to the camera angle, the camera distance, theperspective and light conditions, for example. Certain scenes are frozenwithin the frame, and within those frames human figures are engaged insome form of action in a setting. Further to this, the photographs aredeveloped and reproduced under certain conditions which include choicesin terms of paper quality, darkroom techniques, and the possibility forvarious forms of editing, including cropping and erasure. Putting aside themateriality of the medium and the production process, following O'Toole's(1994) framework for the analysis of paintings, each photograph representschoices at the rank of the whole frame or the Work (in terms of the setting,actions and circumstance), the Episodes in each frame (the activities whichare captured), the Figures (the individual people and other objects) andtheir Members (in terms of body parts and parts of the objects). The impactof these choices in the photographs displayed in Plate 1.1(1) merit closeattention.

The settings, physical actions, gestures, facial expressions and the natureof the averted gazes of the human figures in the photographs are juxta-posed in what is a grotesque opposition between the devastation faced bythose involved in the fighting (Photographs 43-44) and the well-fed andwell-attired defiance of those taking part in the official surrender (Photo-graph 45). This opposition is marked at each rank of the Work, Episode,Figure and Member. For example, the contrast between the physical andemotional state of the soldiers, the medical attendant and the Germangenerals becomes evident in a glance. The quality, style and condition oftheir respective uniforms at the rank of Figure and Member are similarlydiametrically opposed. Compositionally, even the grainy quality of thestreet scene where the Russian medical assistant attends to the injuriessuffered by a soldier (Photograph 44) is placed in stark opposition to thesmooth textual quality of the photograph of the official German surrender(Photograph 45).

The situational contexts, actions, experiences and the emotional andphysical states of the participants in the fall of Berlin according to circum-stance, nationality, age, gender and position are thus constructed by thephotographs. Even if Beevor had the space to describe these dimensions,the meanings of these black and white photographs are impossible toexactly reproduce in narrative form. A linguistic description cannot makethe same meanings as Photographs 43-45. The scenes, the interplay ofEpisodes, the actions and events, the mood of the Figures realized throughtheir embodied actions and appearance cannot be captured using words.

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In the same manner, 'Success [for the Nazi party] is right. What does notsucceed [for the Nazi party] is wrong' cannot be captured pictorially.Different resources such as language and visual images have differentpotentials to create meaning. In simplest terms, language tends to orderthe world in terms of categorical-type distinctions, while visual images suchas photographs create order in a manner which to varying degrees accordswith our dynamic perceptual experience of the world. The two types ofmeanings afforded by language and visual images combine in Beevor'saccount of the fall of Berlin and the collapse of the Third Reich.

The semantic realm explored in this study is not war, rather it is the worldoffered by mathematics, the discourse which underlies the scientific view ofthe world. This world came into being largely through the developmentand refinement of a new sign system, namely mathematical symbolism,which was designed to function in co-operation with language and special-ized forms of visual images. The mathematical and the scientific ways ofordering the world permeate our everyday existence, and thus the aimof this study is to understand the nature and the implications of sucha view. Before moving to the field of mathematics, Michael Halliday'ssocial-semiotic approach which informs this study is introduced.

1.2 Halliday's Social Semiotic Approach

We impose order on the world, and that order is expressed semioticallythrough choices from a variety of sign systems. These semiotic resources, orsign systems, include language, paintings and other forms of visual images,music, embodied systems of meaning such as gesture, action and stance,and three-dimensional man-made items and objects such as clothes, sculp-tures and buildings. A culture may be understood as typical configurationsof choices from a variety of semiotic resources. The lecture, the pop song,the political speech, the news report and the textbook are to a large extentpredictable configurations of semiotic choices. In a general sense, thisunderstanding of semiotics pertains to 'the specificity of human semiosis'(Cobley, 2001: 260) where 'Semiosis is the name given to the action ofsigns. Semiotics might therefore be understood as the study of semiosisor even as a "metasemiosis", producing "signs about signs" '(Cobley, 2001:259). As Cobley (2001: 259) claims, 'Behind this simple definition [ofsemiotics] lies a universe of complexity.'

Noth (1990) describes the diversity in theoretical and appliedapproaches to study of semiotics and Chandler (2002: 207) sees semioticsas 'a relatively loosely defined critical practice rather than a unified, fully-fledged analytical method or theory'. There are many schools andbranches of theoretical and applied semiotics, with various definitions andmeanings. Noth (1990), for instance, categorizes semiotics as being con-cerned with the study of language and language-based codes, text (forexample, rhetoric and stylistics, poetry, theatre and drama, narrative, myth,ideology and theology), non-verbal communication, aesthetics and visual

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communication. Noth (1990: 5-6) provides alternative subdivisions whichinclude the semiotics of culture, multimedia communication, popularculture, anthropology, ethnosemiotics, and other topics such as psycho-semiotics, socio-semiotics and semiotic sociology, together with thesemiotics of disciplines such as mathematics, psychiatry, history and soforth.

Michael Halliday's (1978, 1994, 2004) social-semiotic theory of languageknown as Systemic Functional Linguistics (SFL) is located within the theor-etical realm of what Noth (1990: 6) terms 'socio-semiotics'. Halliday is con-cerned with the social interpretation of the meaning of language, and thisview is extended to include other semiotic resources such as the maps andphotographs found in Beevor (2002) and the mathematical symbolism anddiagrams found in the discourse of mathematics. While the basic tenetsof the Hallidayan approach to language are introduced below, morecomprehensive accounts may be found elsewhere (for example, Bloorand Bloor, 1995; Eggins, 1994; Martin, 1992; Martin and Rose, 2003;Thompson, 1996).

Halliday (1978, 1994) sees language as a tool, where the means throughwhich language is used to achieve the desired results are located within thegrammar. The grammar is theorized according to the functions language isrequired to serve. Halliday (1994) identifies the 'metafunctions of lan-guage' as (i) the experiential - the construction of our experience of theworld, (ii) the logical - the construction of logical relations in that world,(iii) the interpersonal - the enactment of social relations, and (iv) thetextual - the means for organizing the message. The grammatical systemsthrough which these four metafunctions of language are realized aredescribed in Chapter 3.

From the Hallidayan perspective, meaning is thus made through choicesfrom the metafunctionally based grammatical systems. The meaning of achoice (the sign or the syntagm) is understood in relation to the otherpossible choices within the system networks (the paradigmatic options).Halliday uses the term 'social semiotic' to explain that the meanings of thesigns (the semiotic choices) depend on the context of use (the social). Themeanings arising from choices from the system networks are negotiatedwithin the social and cultural context in which those choices are made. Forexample, a linguistic statement such as 'Success is right' does not exist as anabstract independent entity. Rather, the statement means within a contextof use, in this case in Beevor's (2002) account of the fall of Berlin. In thesame fashion, contexts are established semiotically. For example, the fall ofBerlin is constructed by Beevor (2002) and other historians throughchoices from the semiotic resources of language, maps and photographs.Similarly, the academic lecture is a typical configuration of semioticchoices from the resources of language, visual images, dress, gesture,objects, architecture, seating, lighting and so forth. The configuration ofthe academic lecture is recognizable by members of a culture, even thoughthe form varies according to discipline and institution.

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In order to communicate, members of a culture, or groups within thatculture, must possess some sense of shared contextual meaning. Beingpart of a culture means learning, using and experimenting with themeaning potential of the semiotic systems to create, maintain and nego-tiate the reality which is socially constructed. Semiotic activity is also usedfor acts of resistance, which may materialize, for example, in the form ofemail messages or websites where 'standard' linguistic practices are sub-verted from the point of view of grammar, lexical choice, text colour andgraphics. The dynamic nature of the electronic medium is such that thedistinction between the spoken and written modes becomes increasinglyblurred with the variations in genre configurations, language choices andgraphical representations. However, these new practices eventually becomein themselves standardized in much the same way that video texts in themusic industry become predictable. The resistance which some discoursesinitially appear to offer (for example, in the music and film industry, sportand the internet) typically become absorbed into mainstream culture,often in the form of re-packaged commercial products.

The contextual values attached to different choices or combinations ofchoices from semiotic resources are socially and culturally determined.Members of a culture recognize and maintain or resist those values. Com-panies such as McDonalds, Nike and Coca Cola, for example, invest largeamounts of money in advertising to ensure that their brands andaccompanying icons maintain 'the right' social value among the otherproducts on offer. In this way they seek to create and maintain a market forgroups of consumers. In one study, Cheong (2004) found that apart fromthe interpersonally salient component of an advertisement designed toattract the attention of the reader (in many cases a visual image), the onlyobligatory item in a print advertisement is the company logo. Presumablyif the logo was missing, the intertextual relations with other texts in theadvertising campaign would ensure that the brand is easily identifiable.Advertising as such means creating an image so that the product or serviceis viewed as desirable by groups of members of a community. Buying theproduct thus means acquiring the social and cultural connotative value ofthat product (Barthes, 1972, 1974).

Human life is negotiated through semiotic exchange within the realmsof situational and cultural contexts. Certain combinations of selectionsfunction more prominently to structure reality to the exclusion of others.Studies in Systemic Functional Linguistics (SFL) attempt to document andaccount for the typical linguistic patterns in different types of social inter-action or genres; for example, casual conversation (Eggins and Slade,1997), service encounters (Ventola, 1987), pedagogical discourse (Christie,1999; Christie and Martin, 1997; O'Halloran, 2000, forthcoming b;Unsworth, 2000) and scientific writing (Martin, 1993b; Martin and Veel,1998). Other studies of language look at typical patterns along contextualparameters such as gender (for example, Tannen, 1995) and sexuality (forexample, Cameron and Kulick, 2003). Forensic linguistics, on the other

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MATHEMATICS AS A MULTISEMIOTIC DISCOURSE 9

hand, is concerned with identifying typical language patterns of theindividual (for example, Coulthard, 1993).

Bourdieu's (1991) notion of symbolic and cultural capital of the 'habi-tus', which is the set of acquired dispositions of an individual or group ofindividuals, may be conceptualized as semiotic capital; that is, the ability toconstruct, interpret and reconstruct the world in contextually specific ways.However, the ability to make appropriate meanings in a range of contextsthrough the use of semiotic resources is unevenly distributed across sec-tions of any community or culture. The reason for this unequal distributionof semiotic capital is related to the educational, economic, social and cul-tural background of individuals and groups within any community. Forexample, Bernstein (1977,1990) identifies the disadvantages students fromlower social class backgrounds face in participating in the linguistic prac-tices rewarded in educational institutions. In a sense, being 'educated'means being able to participate in certain types of 'valued' semioticexchange; for instance, the discourses of medicine, science, business, law,music and art. Certain groups within a society, typically those with wealthand connections, are relatively well placed within the semantic domainswhich are rewarded (usually by members of that same group). Othergroups to varying degrees are marginalized. Increasingly the market-drivenpractices adopted in schools and universities, such as making entrancedependent on money rather than merit, function to reinforce thesedivisions of inclusion and exclusion.

Participation in everyday discourse includes semiotic exchange in termsof performative action; that is, selections in the form of gesture, stance,proxemics and dress. Whether delivering a conference paper or giving apolitical speech, the speaker needs to talk the talk (using appropriate lin-guistic and phonological choices), walk the walk (in terms of non-verbalbehaviour and action), and increasingly look the look (in terms of clothing,hairstyle, make-up, body size, body shape, height, and skin and hair colour,for example) according to the parameters established as desirable in thatculture. More generally men and women are urged to identify their'unique selling point' (USP), be it the talk, the walk or the look. Increas-ingly acts of meaning inscribed on and through the human body (forexample, physical appearance which is increasingly the product of medicalprocedures and other forms of practices involving drugs, chemicals and soforth) often outweigh the import of other acts of meaning (the talk). Inthe electronic medium, the performative action and physical creation ofidentity becomes a textual act. One is no longer constrained by semiosisemanating from the body and the immediate material context. Multipleidentities can be established according to the limits of the electronicmedium and platforms that are offered, and the user's ability to makeuse of different semiotic resources, including language, visual images,music and so forth. Semiotic capital comes into play in new ways throughcomputer technology.

The social-semiotic construction of reality (Berger and Luckmann, 1991)

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is determined as much by what is included as to what is excluded. As seen inthe example of the German generals' 'guiding principles', (i) there arelimits to what options are selected, and (ii) there are limits to what can beselected from the existing systems. In the first case, semiotic selectionsfunction as meaning through choice, and so some options (for example,'success') are chosen to the exclusion of others (for example, 'justice' or'freedom'), while other possible options are left out (for example, 'for theNazi party'). In the second case, although systems are dynamic and con-stantly changing with each contextual instantiation, there are nonethelessat any one time a limited number of options available. We are containedwithin particular semantic domains according to the limitations of thesystems which are available. These systems, however, constantly evolve sothat meaning making is a dynamic practice in which change is possible.

Realms of meaning do not exist until they become semiotic choices;for example, the concepts of women's rights, gender and Freud's (forexample, 1952, 1954) concept of psychoanalysis are comparatively newlinguistic choices. Although perhaps pre-existing as disparate practices, theintroduction of these options in language led to radically new ways ofconceptualizing women, women's roles and what has become the innerpsychosexual self. Similarly, the scientific revolution in the seventeenthcentury introduced radically new ways of conceptualizing the physicalworld. The basis for this scientific re-ordering of reality was the develop-ment of mathematics which offered new resources in the form of the sym-bolism and visual display. These semiotic resources combine in significantways with language to create a new world order. The nature of that order isinvestigated in this study.

1.3 Mathematics as Multisemiotic

Mathematics and science are considered as 'multisemiotic' constructions;that is, discourses formed through choices from the functional sign systemsof language, mathematical symbolism and visual display. These discoursesare commonly constituted as written texts, although mathematical and sci-entific practices are not confined to these forms of semiotic activity. Thereare many different 'multimodal' genres constituting mathematical andscientific practices; for example, lectures, conference papers, softwareprograms and laboratory investigations. In addition to the written mode,these types of semiotic activity involve spoken discourse, physical actionand gesture in environments, which include digital media and day-to-daythree-dimensional material reality. The major line of enquiry in thisstudy, however, is directed towards multisemiosis in printed discoursesof mathematics, largely because modern mathematical symbolism is asemiotic resource which developed in written format. In order to developtheoretical frameworks for mathematical symbolism and visual display, theprint medium has been chosen for investigation. In addition, the effects ofcomputer technology on the nature of mathematical discourse are also

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considered in this study. With the exception of the systemic functional (SF)approach to mathematics (Lemke, 1998b; O'Halloran, 1996, 1999b, 2000,2003a, 2003b, forthcoming a; Veel, 1999), few studies exist in the field ofthe semiotics of mathematics (for example, Anderson et al, 2003; Rotman,1987,1988,1993,2000).

Mathematical discourse involves language, mathematical symbolism andvisual images as displayed in Plate 1.3(1), a page reproduced from PhysicaD, a journal for research in dynamical systems theory. Plate 1.3(1) containsequations (11), (12) and (13), which are mathematical symbolic statementsspatially separated from the main body of the linguistic text. Symbolicstatements and elements are also embedded within the linguistic text. Forexample, symbolic elements function as elements within the linguisticstatements in the text located between equations (11) and (13). Inaddition, there are visual images in the form of mathematical graphs in thethree panels labelled Fig. 2 in Plate 1.3(1). Mathematical written discoursemay also contain tables which are forms of textual organization where thereader may access information quickly and efficiently (Baldry, 2000a;Lemke, 1998b). As seen in Plate 1.3(1), mathematical printed texts aretypically organized in very specific ways which simultaneously permitsegregation and integration of the three semiotic resources.

An SF approach to mathematics as social-multisemiotic discoursemeans that each of the three semiotic resources - language, visual imagesand mathematical symbolism - is perceived to be organized according tounique discourse and grammatical systems through which meaning is real-ized. That is, each semiotic resource is considered to be a functional signsystem which is organized grammatically. Mathematical texts such as thosedisplayed in Plate 1.3(1) represent specific semiotic choices from the avail-able grammatical systems in each of the three resources. As seen in thegraphs and linguistic and symbolic components of the mathematics text inPlate 1.3(1), choices from the three semiotic resources function integra-tively. That is, the linguistic text and the graphs contain symbolic elementsand the symbolic text contains linguistic elements. This feature of math-ematical discourse means that the grammars of each resource must beconsidered in relation to each other.

The similarities and differences in the organizing principles of the threesemiotic resources are considered intra-semiotically in terms of the grammarsand functions of each resource. In addition, mathematical discourse is con-sidered inter-semiotically; that is, in terms of the meaning which arises fromthe relations and shifts between the three semiotic resources. Royce(1998a, 1998b, 1999) refers to intersemiotic semantic relations betweenlinguistic and visual components of a text as 'intersemiotic complementar-ity', and ledema (2003: 30) calls the process of semiotic shift as 'resemioti-cization', which he defines as 'the analytical means for . . . tracing howsemiotics are translated from one into another as social processes unfold'.In mathematics, intersemiotic shifts take place on a macro-scale acrossstretches of text, and they also take place on a micro-scale within stretches

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J. Kockelkoren et al./Physica D 174 (2003) 16X-175

of the wavelength Xc of the patterns is at criticalityabout 13% off from the theoretical value; however, weare not interested here in the absolute value, but in therelative variation of XC/X.

The difficulty of comparing theory and experimenton the variation of the wavelength is that the only the-oretically sharply defined quantity is the wavelength

sufficiently far behind the front, A.^, and that one hasto go beyond the lowest order Ginzburg-Landau treat-ment to be able to study the pattern wavelength leftbehind. For example, if we use a Swift-Hohenbergequation for a system with critical wavenumber kc andbare correlation length £o.

then a node counting argument [4,6] yields for theasymptotic wavelength A.as far behind the front [6]:

In the Rayleigh-Benard experiments, kc « 2.75/d,where d is the cell height; the theoretical value is £o =0.385rf, so if our conjecture that the value is some 15%larger is correct, we get £Q ̂ 0.4<W. This then gives

(13)

As we stressed already above Xas is the wavelength farbehind the front; for a propagating pulled front, thereis another important quantity which one can calculateanalytically, the local wavelength A* measured in theleading edge of the front. For the Swift-Hohenberg

Fig. 2. Top panel: shadowgraph trace of a propagating front in theexperiments of FS for f = 0.012 [16]. The time difference betweensuccessive traces is 0.42fv, where fv is the vertical diffusion timein the experiments, and the distances are measured in units d(the cell height) (from [9]). Middle panel: similar data obtainedfrom numerical integration of the Swift-Hohenberg equation alsoat e = 0.012 starting with a localized initial condition. The timedifference between successive traces corresponds to 0.42/v. Bottompanel: velocity versus time in the experiment, as obtained byinterpolating the maxima of the traces in the top panel, as explainedin the text. The dashed line shows the analytical result (8) andthe dotted curve the result of the amplitude equation simulationof Fig. 1 with nln" = 1.2. Note that the curves are not fitted,only the absolute scale is affected by adjusting £o

Plate 1.3(1) Language, visual images and symbolism (Kockelkoren et al,2003: 173)

of text. The potential of intersemiotic processes to produce metaphoricalconstruals is formulated through the notion of 'semiotic metaphor'.

Through close examination of the meaning realized within and acrossthe three semiotic resources, the functions and the semantic realm of

173

(11)

(12)

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mathematics as a discourse are tentatively formulated in this study. It mustbe stressed that this is not an account of the entire field of mathematics.Rather it is an account of the semiotic processes and the discourse andgrammatical strategies through which mathematics operates to structurethe world. From this position, the semantic realm with which mathematicsis concerned may be appreciated. This is in part achieved through a com-parison of the functions of mathematics with those of language. However,mathematics evolved as a discourse capable of creating a world view whichextends beyond that possible using linguistic resources alone. The resultof that re-ordering in what is viewed as the scientific revolution is alsoconsidered in this study. The implications of viewing mathematicaldiscourse as a multisemiotic construction are considered below.

1.4 Implications of a Multisemiotic View

The multisemiotic approach, where language, visual images and mathemat-ical symbolism are considered as semiotic resources (O'Halloran, 1996),originally stems from O'Toole's (1994, 1995, 1999) extensions of Halliday's(1978, 1994) SF approach to displayed art, and Lemke's (1998b, 2000,2003) early work in mathematical and scientific discourse. The SFapproach to mathematics is welcomed by Rotman (2000: 42) who explainsthat such an approach offers 'a linguistic/semiotic framework wellgrounded in natural language that . . . [is] abstract enough to include themaking of meaning in mathematics'.

Halliday's (1994) Systemic Functional Grammar (SFG) includes docu-mentation of the metafunctionally based systems which are the grammat-ical resources through which meaning is made. Halliday's account of theabstract language systems includes statements of how these choices arerealized in text. SFG is essentially a 'natural' grammar as it explains howlanguage is organized to fulfil the metafunctions of language: the experien-tial, logical, interpersonal and textual. Halliday's (1994) model of languagedescribed in Chapter 3 provides the basis for the Systemic FunctionalGrammars (SFG) presented for mathematical symbolism and visual imagesin Chapters 4 and 5 respectively. These grammars and a framework withsystems for intersemiosis are used for discourse analyses of mathematicaltexts in Chapters 6 and 7. The discussion includes an account of the edu-cational implications of a multisemiotic view of mathematics and the natureof pedagogical discourses in mathematics classrooms.

The SFGs for mathematical symbolism and visual images are inspired byO'Toole's (1994, 1995) systemic frameworks for the analysis of semiosis inpaintings, architecture and sculpture. O'Toole (1994) demonstrates howthe SF frameworks may be used so that the viewer can learn to engagedirectly with instantiations of displayed art rather than depending on the'knowledge' handed down by art historians and other accredited experts.Bourdieu (1989) further explains that aesthetics and art appreciationare discourses which function covertly to maintain existing social class

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distinctions. In this view, 'taste' is a social and cultural product throughwhich group and individual identities are indexed and, as with all symbolicinvestments, different values are placed on those indices. Needless to say,the highest values are accorded to those who constitute the powerful insociety.

the main reason for this close [semiotic] engagement with the details before our eyesis that it enables everyone to sharpen their perceptions and join in the discussion assoon as they begin to recognize the systems at work in the painting. And everyone cansay something new and insightful about the work in front of them. Art history, on theother hand, requires a long apprenticeship . . . before they are expected to be ableto contribute any new information to a discussion of the work in question. Andwhat kind of information might this be? . . . Don't they in fact 'mystify' the paintingand make us feel we have nothing to contribute? . . . the result is to build aninsurmountable wall around this precious property.

(O'Toole, 1994: 171)

Following O'Toole's (1994) example, rather than producing a discursivecommentary about the nature of mathematics and its intellectual achieve-ments, the intention behind the SF approach in this study is activeengagement with mathematical text in order to understand the strategiesthrough which the presented reality is structured, the content of that realityand the nature of the social relations which are subsequently established.The result is an appreciation and understanding of the functions of math-ematical discourse and the strategies through which this is achieved. This isessentially a new approach to mathematics for practising mathematicians,and teachers and students of mathematics. This approach also offersinsights for outsiders who typically possess a limited understanding andknowledge of mathematics. The implications of an SF approach to math-ematics as a multisemiotic discourse are outlined below in relation to thekey ideas and formulations developed in this study. These ideas arerevisited in Chapter 7 after the theory and approach have been developedin Chapters 2-6.

Mathematical and Scientific Language

The view of mathematics as multisemiotic has implications about the waysmathematical and scientific language are understood. Traditionally, thenature of scientific language has been viewed in isolation rather than as asemiotic resource which has been shaped through the use of mathematicalsymbolism and visual display. Scientific language developed in certain waysas a response to the functions which were fulfilled symbolically and visually.On a more global scale, our entire linguistic repertoire has been shaped bythe use of other semiotic resources, with the result that many of our con-temporary linguistic constructions are metaphorical in nature. Forexample, certain views become common sense under the guise of meta-phorical labels such as 'economic rationalism', 'entrepreneurship' and

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'freedom and democracy'. Despite their grammatical instantiation asnouns, these are not concrete or material objects. On the contrary, rathercomplex and dynamic sets of practices are subsumed under such labels. Anunderstanding of the functions of mathematical symbolism, visual imagesand other semiotic resources permits a re-evaluation of the role of languagein constructing such a naturalized view of the world. As with the vestedinterest behind the guiding principle, 'Success is right', metaphorical termsneed to be critically understood in a historical and contextual manner inorder to appreciate the premises behind their construction.

The Grammar of Mathematical Symbolism

An SF framework for mathematical symbolism is presented so that thegrammatical strategies through which meaning is encoded symbolicallycan be documented. This is significant because the grammatical strategiesfor organizing meaning in symbolic statements differ from those found inlanguage. While members of a culture are capable of using language as afunctional resource in various ways, typically the use of mathematical sym-bolism is restricted to certain groups. One reason for this limited access isthat the grammar of mathematical symbolism is not generally well under-stood. It is important to demonstrate how mathematical meaning is organ-ized, and how the unique grammatical strategies specifically developed inmathematical symbolism so that this semiotic could be used for the solutionof mathematics problems. The underlying premise is that mathematicalsymbolism developed as a semiotic resource with a grammar which had thecapacity to solve problems in a manner that is not possible with othersemiotic resources. The SFG of mathematical symbolism presented inChapter 4 explains how this functionality is achieved.

Grammar of Visual Images in Mathematics

Visual images in mathematics are specialized types of visual representation,most typically in the form of abstract graphs, statistical graphs and dia-grams. The systemic functional framework for abstract graphs is used toexplain how the systems are organized to make very specific meaningswhich provide a link between the linguistic description of a problem andthe symbolic solution. Once again, the functions fulfilled by mathematicalvisual images are different to those achieved linguistically and symbolically.The systems through which the functions of abstract graphs are achievedare discussed in Chapter 5. This discussion includes insights into thechanging roles of visual images in mathematics due to the impact of com-puter technology. Visualization is undergoing a rapid resurgence due tothe increasing sophistication of computer graphics which display numer-ical solutions generated by the computer. The new ways of manipulatingand viewing data through computers are discussed in Chapter 5.

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Intrasemiosis and Intersemiosis

While the three semiotic resources in mathematics fulfil individual func-tions which are not replicable across the other resources (Lemke, 1998b,2003; O'Halloran, 1996), the success of mathematics depends on utilizingand combining the unique meaning potentials of language, symbolism andvisual display in such a way that the semantic expansion is greater than thesum of meanings derived from each of the three resources. Lemke (1998b)refers to this expansion of meaning as the multiplicative aspect of multi-semiosis. Mathematical discourse thus depends on intrasemiotic activity,or semiosis through choices from the grammatical systems within eachresource, and intersemiotic activity, or semiosis through grammatical systemswhich function across the three resources. Intersemiosis involves recon-strual of particular elements in a second or third resource throughintersemiotic shifts or 'code-switching'. Intrasemiosis, or meaning withinone semiotic resource, is important because the types of meaning madeby each semiotic are fundamentally different. Intersemiosis, however,is equally important because not only is the new meaning potential ofanother resource accessed, but also metaphorical expressions can arisewith such shifts. This important process, which may arise in any multi-semiotic discourse, is developed in this study through the notion of semi-otic metaphor. The functions of mathematics are therefore achievedthrough intrasemiosis and intersemiosis; that is, meaning through eachsemiotic resource, and meaning across the three semiotic resources wheremetaphor plays an important role in the expansion of meaning.

Intersemiotic Mechanisms, Systems and Semiotic Metaphor

Intersemiotic mechanisms provide a description of the ways in whichintersemiosis takes place across language, visual images and mathematicalsymbolism. The intersemiotic mechanisms take place through metafunc-tionally based systems which are documented in Chapter 6. Semiotic meta-phor refers to the phenomenon of metaphorical construals which arisefrom such shifts across semiotic resources. This process means that expan-sions in meaning can occur when a functional element is reconstrued in adifferent resource. For instance, an action realized through a verb in lan-guage (for example, 'measuring') may be reconstrued as an entity in asecond semiotic resource (for example, a visual line segment or a symbol-ical distance). Such reconstruals permit expansions of meaning on a scalewhich is not possible within a single semiotic resource. As explained inChapter 6, one of the key elements in the success of mathematics is the meta-phorical reformulation of elements across the three semiotic resources.

Mathematics Education

The view of mathematics as a multisemiotic discourse is significant in apedagogical context as often teachers and students do not seem to be aware

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of the grammatical systems for mathematical symbolism and visual display,and the types of metaphorical construals which take place in mathematicstexts and in the classroom. The ways in which a social-semiotic perspectivecan inform mathematics teaching and learning are described in Chapter 7.This discussion is based on the functions of language, visual images and thesymbolism, their respective grammatical systems and the nature of theintersemiotic activity. Chapter 7 includes a discussion of the nature ofpedagogical discourse in mathematics.

1.5 Tracing the Semiotics of Mathematics

In order to introduce the types of meaning found in modern mathematics,a historical perspective is adopted in Chapter 2 to examine the semioticunfolding of mathematics from the period of the early Renaissance tomodern contemporary mathematics. The nature of the projects of earlymodern mathematics, as exemplified by Descartes and Newton, is seen tolead to the creation of a mathematical and scientific reality which is locatedwithin a limited semantic domain. However, at the same time, the semanticexpansions afforded by the visual images and mathematical symbolismpermitted expansions in the form of scientific description, prediction andprescription. Contemporary thought in mathematics, for example, chaosand dynamical systems theory, also reveals the changes in mathematicaltheorizations of reality.

Significantly, the mathematical practices advocated by Descartes andNewton have been re-inscripted into new contexts in contemporary times.The beginnings of modern mathematics and science developed in whatwas originally conceived as a transcendental realm which necessitated theexistence of God, as seen in the discussion of Cartesian and Newtonianphilosophy in Chapter 2. The re-inscription of the supposedly 'value free'discourses of mathematics and science as universal truth into new realms ofhuman endeavour such as the social sciences, education, business, econom-ics and politics is questioned from the relatively fresh perspective of thesocio-semiotics of mathematics in Chapter 7. This discussion also contrib-utes to an appreciation of the metaphorical nature of our semiotic con-structions and the limitations of the contexts in which mathematics may beusefully applied. Mathematics is thus first viewed in a historical contextso the functions for which mathematics was originally designed and thecontext of that development may be appreciated. From this point, the newcontexts in which mathematics is re-inscribed are critically examined.Although mathematics has expanded into new fields, the semioticresources nonetheless essentially remain linguistic, visual and symbolic.Computation is considered a symbolic undertaking which is instantiated inan electronic medium.

An understanding of the scientific view of the world made possiblethrough mathematics is an overriding theme because such a view isvital for an understanding of contemporary Western culture which

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materializes as a technological project shaped by the discourse of math-ematics and science. Looking back, the rationalist project of the eighteenthcentury and the consequent mathematical, scientific and technologicalachievements of the modern period appeared to hold much promise forthe world. As Horkheimer and Ardorno (1972) claim, the much-toutedaim of progress was the improvement of the human condition accom-panied by freedom, equality and justice. In retrospect, however, suchprogress seems to have been made for the advantage of the relativelyprivileged few. In addition to providing the infrastructure for unequallydistributed goods and services such as healthcare and education,advances in mathematical and scientific knowledge appear to have pri-marily provided the means for technological development which is dir-ected and controlled by military, business and political interests. As Davis(2000: 291) claims: 'Through advanced science and technology, warfareutilizes many mathematical ideas and techniques. The creation of vastnumbers of new mathematical theories over the past fifty years was due inconsiderable measure to the pressures and the financial support of themilitary.'

The self-evident deliverables of the scientific project were underscoredin the aftermath of the Second World War and, in a more recent case, theUS-led war in Iraq in 2003 where the destructive power of military techno-logical innovation was widely televised. As Horkheimer (1972: 3) claims: Tnthe most general sense of progressive thought, the Enlightenment hasalways aimed at liberating men from fear and establishing their sovereignty.Yet the fully enlightened earth radiates disaster triumphant.' Today theextent to which the military, business and the political institutions can bedifferentiated as separate functioning bodies becomes increasingly difficultto ascertain. One could include universities on the list of institutions whichincreasingly function pragmatically along the lines of business-orientatedcommercial interests.

The soundness of reason depends on the explicit or implicit premisesupon which that reasoning is based. The view that mathematical and scien-tific reasoning is constructed to order the world along certain principleswhich change is not new (for example, Derrida, 1978; Foucault, 1970,1972;Kuhn, 1970). However, the approach adopted in this study is to understandthe systems and strategies through which that ordering takes place. In thisway, the functions of these discourses may be understood, and throughsuch awareness we can understand our own positions and explore possi-bilities other than those directly offered. This is an exploration of the worldview offered by mathematics and science, a view which dominates oureveryday thinking. It is also a critique of that world view which is so oftenmisunderstood as universal truth.

The path is developed through an excursion through early printedmathematical texts to understand the context behind modern mathemat-ics. SFGs are used to critically interpret the nature of meanings made incontemporary mathematics. Through an understanding of the discourse,

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we may start to count the gains and costs of the mathematical and scientificview of the world. In the view of Davis (2000: 291):

The mathematical spirit both solves problems and creates other problems. What is themathematical spirit? It is the spirit of abstraction, of objectification, of generalization,of rational or 'logical' deduction, of universal quantization, of computational recipes.It claims universality and indubitability. I have the conviction . . . that this spirit isnow . . . pushing us too hard, pushing us to the edge of dehumanization.

The ways in which 'mathematics is pushing us too hard' are investigatedthrough an understanding of mathematics as a multisemiotic resource.Only then can we begin to appreciate the ways in which this discourse andscientific order function to shape our view of ourselves, and our relations toothers and the world around us.

1.6 Systemic Functional Research in Multimodality

This study of mathematics represents part of a growing movement in SFL(see ledema, 2003) where language is conceptualized as one resourcewhich functions alongside other semiotic resources. This research field iscommonly called 'multimodality', or the study of 'multimodal discourse'(for example, Baldry, 2000b; Baldry and Thibault, forthcoming a; Kress,2000, 2003; Kress et al., 2001; Kress and van Leeuwen, 1996, 2001; Levineand Scollon, 2004; O'Halloran, 2004a; Unsworth, 2001; Ventola et al.,forthcoming). Apart from the research in mathematics (Lemke, 2003;O'Halloran, 1996, 1999b, 2003b, forthcoming a), studies have been com-pleted in a wide range of fields including science (Baldry, 2000a; Kress etal, 2001; Lemke, 1998b, 2000, 2002), biology (Guo, 2004b; Thibault,2001), multiliteracy (Lemke, 1998a; Unsworth, 2001), film and television(ledema, 2001; O'Halloran, 2004b; Thibault, 2000), music (Callaghan andMcDonald, 2002), museum exhibitions (Pang, 2004), shopping displays(Ravelli, 2000), TESOL (Royce, 2002), hypertext and the electronicmedium (Jewitt, 2002; Kok, 2004; Lemke, 2002) and advertising (forexample, Cheong, 2004). Research in the field of multimodality alsoincludes the development by Anthony Baldry et al (Baldry, 2004, forth-coming; Baldry and Thibault, 2001, forthcoming a, forthcoming b) of anon-line multimodal concordancer, the Multimodal Corpus Authoring (MCA)system, which is web-based software for the analysis of phase and transitionsin dynamic texts such as television advertisements, film and web pages.

There have been attempts to construct grammatical frameworks fordifferent semiotic resources (see Kress and van Leeuwen, 1996, 2002;O'Halloran, 2004a). However, with the exception of Thibault's (2001)approach to the theory and practice of multimodal transcription andBaldry and Thibault's notion of phase for the analysis of dynamic texts(Baldry, 2004, forthcoming; Baldry and Thibault, 2001, forthcoming b),few comprehensive theoretical and practical approaches have beendeveloped in the field of multimodality. Consequently, a meta-language for

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an overarching model for the theory and practice of multimodal discourseanalysis remains at a preliminary stage. Partly as a consequence of this lackof a meta-theory, there exist problems of terminology in studies of multi-modality, as noted by ledema (2003: 50). For example, there is confusionover the use of the terms 'mode' versus 'semiotic', and, consequently, 'multi-modal' versus 'multisemiotic'. Given that this field represents a relativelynew area of research, this is to be expected as the much needed frameworksundergo development.

As an example of mixed terminology, Kress and van Leeuwen (2001:21-22) define 'mode' as the 'semiotic resources which allow the simul-taneous realization of discourses and types of (inter) action. Designs thenuse these resources, combining semiotic modes, and selecting from theoptions which they make available according to the interests of a particularcommunication situation.' From this position, Kress and van Leeuwen(2001: 22) see Narrative, for example, as a mode. In this study, however, theterm 'semiotic' is used to refer to semiotic resources such as language,visual images and mathematical symbolism. These semiotic resources haveunique grammatical systems through which they are organized. Any dis-course that involves more than one semiotic resource is therefore termed'multisemiotic' rather than 'multimodal'. The use of the term 'multimodal'is explained below.

The term 'mode' in SFL, following Halliday and Hasan (1985), typicallymeans the role language is playing (spoken or written) in an interaction.This sense, adopted in this study, is concerned with the nature of the actionof semiosis; that is, whether it is auditory, visual or tactile, for example. Itfollows that different semiotic resources are constrained in terms of pos-sible modes through which the semiotic activity can take place. Forexample, language may be instantiated orally or visually, but visual imagesare instantiated through the visual mode in different media such as print,electronic media and three-dimensional space. On the other hand, Kressand van Leeuwen (2001: 22) use the term 'medium' to refer to the'[m]aterial resources used in the production of semiotic products andevents, including both the tools and the materials used (for example, themusical instrument and air; the chisel and the block of wood. They areusually specially produced for this purpose, not only in culture (ink, paint,cameras, computers), but also in nature.'

In order to maintain existing systemic terminology, in this study the termmode is used to refer to the channel (auditory, visual or tactile, for example)through which semiotic activity takes place, medium for the materialresources of the channel, and genre for text types such as the Narrative(which is realized through language in the spoken or written form). Theterm multisemiotic is used for texts which are constructed from more thanone semiotic resource and multimodality is used for discourses which involvemore than one mode of semiosis. A radio play featuring speech, music anddiegetic sound is therefore multisemiotic rather than multimodal as itinvolves multiple semiotic resources realized through the auditory mode of

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sound through the medium of the radio. However, a website which containswritten linguistic text and a music video clip is multisemiotic (involvinglanguage, visual images, music) and multimodal (visual and auditory). Thepractices adopted here do not attempt to solve the problems of mixedusages of terminology, rather they seek to clarify the use of the termsadopted in this study. In this respect, mathematics is referred to as multi-semiotic as it consists of three semiotic resources, language, visual imagesand mathematical symbolism. Mathematics is considered to be primarily awritten discourse produced in printed and electronic media. There are alsomultimodal genres in the field of mathematics, such as the academic lec-ture, which involves spoken discourse and other semiotic resources. Themultimodal nature of mathematical pedagogical discourse is discussed inChapter 7.

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2 Evolution of the Semiotics of Mathematics

2.1 Historical Development of Mathematical Discourse

A historical view of the changing nature of multisemiosis in mathematicaldiscourse from the early Renaissance to the present is a useful method forintroducing in general terms the development of the semantic realmof mathematics (O'Halloran, 2003b, forthcoming a). Such an examinationof visual images, symbolism and language in mathematical texts demon-strates how particular dimensions of meaning are incorporated to theexclusion of others. This excursion includes a discussion of the first knownprinted mathematics book, the Treviso Arithmetic 1478, and an examinationof early mathematical and scientific printed texts from the sixteenth to theeighteenth centuries. In particular, Descartes' shift of emphasis from per-ception to what he called 'the intellect' and Newton's reformulation ofnature in mathematical terms are investigated.

Descartes (1596-1650) and Newton (1642-1727) are seen to provideimportant points of departure in the seventeenth century for what was tobecome the contemporary mathematical and scientific project. In the firstcase, Descartes successfully used mathematical symbolism to describe anddifferentiate between curves. It appears that this success with symbolic andvisual semiotic tools was incorporated into an approach upon which Des-cartes could base his philosophical method, a method aimed at securing'true knowledge'. This method involved dispensing with the 'secondary'qualities of matter, such as colour, odour and taste perceived by the bodilysenses, and accepting only 'primary' qualities which could be dealt withthrough 'the mind'. Newton developed Descartes' mathematical semiotictools to provide a symbolic description of physical reality. In doing so,Newton in fact re-admitted sense experience to the philosophical andscientific realm in such a way that made the invisible (for example, force andattraction) visible through mathematicized symbolic description (Barry,1996). The scientific age and the use of experimentation and technologybegan in earnest with Newton. In this movement, much effort wasexpended in developing mathematical symbolism as a semiotic resourcewith a grammar that could directly interact with the grammars of graphs,

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diagrams and language. In this chapter, some necessarily fragmentedglimpses of these early events are traced in order to introduce the nature ofmeanings found in contemporary mathematics.

To make this undertaking more accessible, the historical investigationinto the semiotic realm of mathematics and science takes the form of adiscussion of the illustrations and diagrams which appear in early printedmathematical texts. In addition, observations concerning the differentforms of symbolism in these texts are made. As Cajori (1993) explains, thehistory of the development of mathematical symbolism is complex andinvolves rivalry among mathematicians. Cajori's (1993) detailed account ofthe history of mathematical notation reveals that the majority of forms ofmathematical symbolism became obsolete with only a few forms survivingto the present day. These developments are not included in the accountpresented here, nor is it possible to include a discussion of the origins ofalgebra documented by mathematical historians (for example, Klein,1968). The broader view presented in this study is that algebra developed inthree stages (see Joseph, 1991; Swetz, 1987). First there existed rhetoricalalgebra which involved linguistic descriptions and solutions to problems.The second stage was syncopated algebra where quantities and operationswhich were used frequently were symbolized. The last stage of the devel-opment was symbolic algebra where the mathematical symbolism developedas a semiotic resource in its own right. Rather than providing a completedescription of the three stages, the changing nature of multisemiosis inmathematics as the symbolism developed is explored.

This discussion of the history of mathematics differs from most accountsin that the view is essentially semiotic. In other words, out of the possibleoptions within the different sign systems for language, visual images andsymbolism, it may be seen that only certain ranges of choice were incorpor-ated in mathematics during different time periods. The shifting nature ofthose choices becomes evident as the printed classical mathematical textsof antiquity and early practical arithmetic books were replaced with newforms of semiosis in the mathematics of the early Renaissance. Descartesand Newton reformulated the mathematical realm in what marks thebeginning of modern science during the seventeenth century. However,today that mathematical realm functions in different contexts from thatwhich supported the original mathematical formulations. The implicationsof this re-contextualization of mathematical and scientific formulations infields which include the arts, social sciences and humanities are addressedin Chapter 7.

The reasons for the changes in the nature of mathematics are traced tothe cultural, intellectual and economic climate of the different timeperiods, the functions which the mathematics was designed to serve and theavailable technology. This remains true today where economic, com-mercial, political and private interests combine with advances in computertechnology to determine the type of mathematics developed and thenature of scientific projects which are undertaken. As Wilder (1986) claims,

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mathematics is a cultural practice, and so, like other forms of discourse, it ispolitically motivated. Consequently, following Koestler (1959) and laterphilosophers of science such as Kuhn (1970), the development of math-ematics and science has not been orderly: 'The progress of science [andmathematics] is generally regarded as a clean, rational advance along astraight descending line; in fact it has followed a zigzag course, more bewil-dering than the evolution of political thought' (Koestler, 1959: 11). In thefollowing discussion, views of the changing nature of the semiotics of math-ematics are oudined in relation to the cultural and situational contextswhich gave rise to those discourses.

In explaining the effectiveness of mathematics, Hamming (1980)claims that what is seen is what is looked for, that the kind of mathemat-ics used is selected from a range of possible choices and that in thisprocess very few problems are answered. Following this line of argument,mathematics is seen to deal with a limited semantic field in limited ways,but in doing so has the potential to solve problems which would beimpossible to solve using other semiotic resources. Seen in this light, thebreakthrough which led to the scientific revolution was a new way ofconceptualizing the world using new forms of semiosis. This is basicallythe position developed in the following discussion of the evolution ofmodern mathematics.

2.2 Early Printed Mathematics Books

The first known printed Western mathematical book is the Treviso Arithmetic1478. Partially translated from Italian into English by David Eugene Smithin the 1920s, the first complete translation appears in Frank Swetz's (1987)Capitalism and Arithmetic. The author of the original manuscript is unknownand the title arises from the date and place of publication, the Italian townof Treviso. The book is concerned with practical arithmetic for calculationsin trade and commerce. Swetz (1987) explains that the content is typical ofthe earliest known mathematics books in Europe. The majority of thebooks were written by masters in reckoning schools and guilds which flour-ished in Italy during the fifteenth century. These institutions were popularplaces to learn the mathematics necessary for the expanding merchanttrade.

As seen in Plate 2.2(1), the Hindu-Arabic system is used for the calcula-tions in the Treviso Arithmetic, and there are drawings to demonstrate howthe calculations are performed. Although Swetz (1987) is an English trans-lation of an Italian text, it can be seen that the original text is constructedsemiotically through the use of language, numerical symbols and particularforms of drawings. The style is rhetorical algebra where unknown quantitiesare realized as entities such as 'profit' rather than symbolic quantities suchas 'P'. According to Swetz (1987), in the late fifteenth century some Italianwriters were starting to use syncopated algebra with forms of abbreviationsfor recurring terms and mathematical operations.

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EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 25

Thus the problem is solved, and theanswer is that there falls to Piero as profit 138

ducats, 21 grossi, n pizoli and

Plate 2.2(1) The Treviso Arithmetic (reproduced from Swetz, 1987: 140)

At the time of the Treviso Arithmetic, there were controversies over the bestmethod for performing arithmetical calculations as pictured in Plate2.2(2). The controversy concerned the abacists, who manually used coun-ters and ruled lines to perform the calculations and recorded the result inRoman numerals, and the algorists, who used the Hindu-Arabic numer-ation system and algorithms to calculate and record. Computation at thistime centred around prestigious counting tables, and the proposed shift toalgorists' Hindu-Arabic system represented a threat to those who hadvested interests in maintaining the tables. Despite the obvious benefits ofthe new system, the shift to the Hindu-Arabic numeration system, firstintroduced in Europe as early as AD 1000 was slow because of the resis-tance exerted by those who controlled the tables. By the fifteenth centuryItaly, however, was ahead of other European countries in using the

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Plate 2.2(2) The Hindu-Arabic system versus counters and lines (Reisch,1535: 267)

Hindu-Arabic system as the means for performing arithmetic calculations.The Treviso Arithmetic demonstrates how the nature of mathematics is influ-enced by cultural and economic concerns of the time (in this case, mer-chant trade and commerce) and pressure from special interest groups (forexample, those supporting the Hindu-Arabic number system). In addition,technology plays an important role as seen below in the discussion of the

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impact of the printing press on mathematics. More generally, the nature ofthe development of mathematics is a convergence of these factors.

The use of printing press technology explains the increased popularity ofthe Hindu-Arabic system which lent itself to this type of reproduction (forexample, Dantzig, 1954; Eisenstein, 1979; Swetz, 1987). The calculationscompleted through lines and counters on the counting tables were clumsyto reproduce and required special printing techniques. As may be seen inPlate 2.2(3), the diagrams on the left-hand side (which include small pic-tures of hands) demonstrate how the calculations are performed using thelines and counters. As mathematical texts increasingly appeared in print,this form of representation could not compete with the more efficientsemiotic form of the Hindu-Arabic system of computation. The expansionof the use of the Hindu-Arabic system is significant for two reasons. First, ascommercial mathematics increasingly became semiotic instantiations inthe written mode, the algorithms for the calculations became more widelydisseminated and commercial arithmetic moved from the hands and coun-ters of the few to a wide audience. Second, Swetz (1987: 32) explains thatthe increased focus, attention and recording of the mathematical tech-niques in the Hindu-Arabic practical arithmetical texts in effect paved theway for the development of symbolic algebra. The printed text permittedclose examination and development of arithmetical algorithms, and thestandardization of mathematical procedures, techniques and symbolswhich led to the range of mathematical notations documented by Cajori(1993).

Under the economic and intellectual impetus of this time, not only were math-ematical techniques being more widely learned but they were, in many cases, newtechniques based on the use of Hindu-Arabic numerals and their accompanyingalgorithms. From this period onward, computation involving numbers would be moreeasily executed and efficiently recorded. The visual stimulus of a mathematical pro-cess written out allowed for a re-examination and questioning of the process; patternscould be noted and mathematical structure discerned. Printing also forced a stand-ardization of mathematical terms, symbols, and concepts. The way was now openedfor even greater computational advances and the movement from a rhetorical algebrato a symbolic one.

(Swetz, 1987: 284)

Swetz (1987) explains that at the time the Treviso Arithmetic was published,the Classicist mathematicians demanded a printed edition of Euclid. How-ever, it appears that this priority was placed second to arithmetic where'Practical necessity was the motivating force in this printing decision'(Swetz, 1987: 25). 'Perhaps the typographical problems inherent in settingtype for geometrical figures were responsible for the delay, but more likelyit was due to the economic and intellectual demands of the marketplace'(ibid.). In Thomas-Stanford's (1926: 3) view, the early Venetian printingpresses published few mathematics books due to the problems of printingthe diagrams. Two examples of early printed editions of Euclid's Elementsare displayed in Plates 2.2(4a-b). Thomas-Stanford (1926: 3) states that the

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first printed edition, which appeared in 1482, was 'an epoch-making' bookin many respects: 'It was [one of] the first attempt[s] - and a highly success-ful one - to produce a long mathematical book illustrated by diagrams.' Inthis version of Euclid displayed in Plate 2.2(4a), the running linguistic textin the style of rhetorical algebra is ornately decorated and the diagrams areneatly offset to one side. Plate 2.2(4b) shows the richness of the borderpatterns and text which appears in the early editions of Euclid. In the

Plate 2.2(3) Printing counter and line calculations (Reisch, 1535: 326)

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Plate 2.2(4a) Euclid's Elements: Venice 1482, Erhard Ratdolt (Thomas-Stanford, 1926: Illustration II)

original version of Plate 2.2(4b) reproduced in Thomas-Stanford (1926:Illustration IV), parts of the text and the decoration are coloured red.

Thomas-Stanford (1926: 4) observes that' [it] would almost seem that atVenice especially the printers sought by a refinement of ornamentation torelieve the austerity of the subject-matter', the nature of which may beappreciated from the modern translation of Euclid which is displayed inPlate 2.2(5). The mathematics appears as objective statements which areaccompanied by perfect geometrical shapes. Apart from the statement, 'Isay that. . .' in Euclid's discourse, the author is absent. In Euclid's writings

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Plate 2.2(4b) Euclid's Elements: Venice 1505, J. Tacuinus (Thomas-Stanford, 1926: Illustration IV)

displayed in Plates 2.2(4a-b) and 2.2(5), there is also a noticeable lackof symbolism in the text, which appears only in the form of a, b, c and d andA, B, C and D to refer to the points, sides, angles and triangles in the

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Plate 2.2(5) Translation of Euclid (reproduced from Euclid, 1956: 283)

mathematical diagrams. As we shall soon see, Newton rewrote Euclid'sgeometry in symbolic form.

Needless to say, the quality of the production of the mathematics printedtexts was not always consistent. For example, as displayed in Plate 2.2(6),the translated version of Archimedes (1615: 437) printed in Paris containsmathematical diagrams which are crude especially with respect to linewidth and horizontal alignment on the page. In addition, typesetting linesseparate the Greek and Latin versions of the text, and the headings andmargins. Presumably early printing presses possessed to different degreesthe technology, expertise and finance to produce printed mathematicalbooks. While the absence of mathematical symbolism in Archimedes'(287-212 BC) rhetorical-style text is not surprising, the textual layoutincludes spatial separation of the text and diagrams, which is a feature ofcontemporary mathematical texts.

Febvre and Martin (1976: 259) claim that 'printing does not seem to haveplayed much part in developing scientific theory at the start'. This view isbased on an observation that some influential works in arithmetic and

PROPOSITION 18.

In any triangle the greater side subtends the greater angle.For let ABC be a triangle having the side AC greater

than AB;I say that the angle ABC is also greater than the angle

BCA.For, since AC is greater than AB, let AD be made equal

toAB [i. 3], and let BD be joined.Then, since the angle ADB

is an exterior angle of the triangleBCD,

it is greater than the interiorand opposite angle DCB. [i. 16]

But the angle ADB is equalto the angle ABD,

since the side AB is equal to AD;therefore the angle ABD is also greater than the angle

ACB;therefore the angle ABC is much greater than the angleACS.

Therefore etc.Q. E. D.

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Plate 2.2(6) Quadratura Paraboles (Archimedes, 1615: 437)

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algebra, such as Nicolas Chuquet's (1484) The Triparty, remained in manu-script form in the late fifteenth century. However, early printing pressesmust have taken time to become established and to be economically viable,the printers must have made careful choices as to what they published.Given the economic demands of the time and the pressure from certainestablished circles, it is not surprising that the first books in the late fif-teenth century were concerned with practical arithmetic and classical textsrather than new works in algebra. When the printed texts on algebra didappear, they were influential. Whitrow (1988), for example, states that thekey to the mathematical revolution in the sixteenth century was the begin-nings of the development of algebra, and the first book on the subject wasLuca Pacioli's Summa de Arithmetica (1494). Whitrow (1988: 267) notes:'[this book] was extremely influential, presumably because it was printed'.The printing of mathematical texts had an immense impact on the math-ematics that was subsequently developed, for as Eisenstein (1979: 467)claims '[cjounting on one's fingers or even using an abacus did notencourage the invention of Cartesian coordinates'. Eisenstein (1979) fur-ther explains that Newton mastered the classical works of the ancients andcontemporary mathematicians such as Descartes from the books heobtained from libraries and book fairs. Newton was self-taught, and thisdiffered greatly from previous practices where learning took place in anoral tradition which involved the elder masters. Likewise, Leibniz hadread most of the important mathematical texts of his time before he wastwenty years of age (Smith, 1951). Mathematics became widely accessible,and in some sense standardized, through the medium of the printingpress.

Before moving beyond the times of the Treviso Arithmetic, it is importantto note that commercial arithmetic was not the only concern of this timesimply for the reason that commerce and trade do not only involve count-ing. As Swetz (1987: 25) explains, the reckoning masters were the forerun-ners of applied mathematicians, and their concerns spanned commercialarithmetic to land surveying, construction of calendars, and cask gauging.As trade and colonialization expanded, there was a need to refine navi-gational techniques and increase military strength. As becomes evident inSection 2.3, mathematical and scientific descriptions at the beginning ofthe Renaissance included the study of warfare. At this time, mathematicsbecame a recognized profession which was freed from the mysticism of theMiddle Ages, and it developed into a discipline that ranked alongside orabove other more established fields of study. In this climate, studies inmathematics expanded rapidly into new ways of thinking about old ideas,and new ways of thinking about new ideas.

2.3 Mathematics in the Early Renaissance

The new movements in sixteenth-century Europe were fuelled throughthe decline of feudalism and the growth of cities and towns in which the

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wealth created through manufacture, commerce and trade meant theintroduction of a new power base. In the climate of relative social stabilityin the West, a re-examination of ideas occurred in what is generally termedthe Renaissance. Swetz (1987: 5) explains that 'Intellectual humanism waspatronised by capitalism and secularism, which broadened man's horizonsof inquiry and innovation.' The arts flourished and the nature of math-ematical and scientific thinking also changed. The nature of the change inmathematics is explored below through the examination of several printedmathematical texts of that time.

Niccolo Fontana, known as Niccolo Tartaglia, was a pupil of the Italianreckoning schools and later became a prominent mathematician in his ownright. Tartaglia wrote La Nova Scientia (The New Science) in 1537, and thefrontispiece to this book, displayed in Plate 2.3(1), contains a picture of atower with different academic fields represented by human figures. Thenew place of mathematics in what marks the beginning of the Renaissanceis made clear in this scene. After being admitted by Euclid and passing bythe two firing cannons with their attendants, the two female figures stand-ing by Tartaglia to greet the visitor are labelled Arithmetic and Geometry.The other female figures are labelled Astronomy, Music, Poetry, andAstrology. The figures of Plato and Aristotle stand at the entrance to thesecond level and the female figure of Philosophy is located at the top level.The banner that Plato holds reads 'hue geometriae expers ingrediatur' or'Nobody enters who is not expert of mathematics' (translated by DavidPingree in Davis, 2000: 293). As Davis (2000: 293) explains, 'What we havehere is the hierarchy of knowledge as set out by St. Thomas Aquinas . . . thatlacks its topmost thomist level: theology!' The message is that those seekingwisdom must know mathematics, and it appears that an integral part of thatknowledge is somehow associated with cannons, a topic which is furtherinvestigated below.

In the new spirit of the early Renaissance, mathematical discourse beganto appear in a very different form from the earlier classics which were stillin place as the authorative texts. The circumstantial context of the math-ematical problem was often made explicit, and, in addition, the humanrealm was depicted. For instance, the concept of volume is illustratedthrough spears which pierce the body of a naked man standing on a moundof earth in Reisch (1535: 424). This conception of volume differs dramatic-ally from that found in Euclid, for example. Tartaglia's (1546: 7) researchinto the trajectory of cannonballs displayed in Plate 2.3(2) clearly showsthe circumstantial context of the problem including the target which is tobe hit. The accompanying mathematical symbolic text includes extendedarithmetical calculations as seen in Plate 2.3(3). The arithmetic is diffi-cult to read as it is embedded within the linguistic text, and the notationdoes not include the shorthand forms which are found today; forexample, 1020 for 100,000,000,000,000,000,000. This number appears as'100000000000000000000' in Tartaglia (1546). Apart from Tartaglia, mili-tary concerns in the form of hitting targets are reflected in other studies;

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Plate 2.3(1) Tartaglia's (1537) La Nova Scientia Frontispiece

for example, Galileo (1638: 67) who attempts to calculate the angle ofelevation of a building on a hill, this time from different positions on theground as seen in Plate 2.3(4).

Many mathematical texts in the early Renaissance appear to involvehuman figures participating in some form of physical or perceptual activity

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Plate 2.3(2) Hitting a target (Tartaglia, 1546: 7)

where the circumstantial context of the exercise is included. For example,the context surrounding Tartaglia's (1537) concern with predicting thepath of a cannonball is explicit in Plate 2.3(5). A man is engaged in firing acannon to hit a target which appears to be the building on the other sideof the river or lake. There is a human actor engaged in a material activitywhich is presented as the problem to be solved. The problem is approachedthrough geometrical constructions involving lines and a circle (which waslater proved to be incorrect through the work of Galileo). While Tartaglialater regretted his work on cannon fire as a contribution to the art ofwarfare (see Davis, 2000: 293), these texts were nonetheless explicit asto the purpose of the mathematical exercise. Contemporary textbooksin mathematics also have images where the context of the problem isvisualized; for example, introductory exercises pose problems in order tointroduce the mathematical theory which is to be developed, and practiceexercises apply that theory. However, reasons for the theory such as warfareare not typically depicted, at least in the public eye. Rather, the math-ematical theory is developed in the abstract, and retrospectively shown tohave applications, or alternatively, a suitable example of a problem is posedso that the reader can appreciate the usefulness of the mathematics whichis subsequently developed. Mathematical theory is, however, often pre-sented non-contextually in contemporary discourse. However, in the

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Plate 2.3(3) Arithmetic calculations to hit a target (Tartaglia, 1546: 106)

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Plate 2.3(4) Positioning a target (Galileo, 1638: 67)

Renaissance texts viewed here, the realm of human activity is an integralpart of the mathematics texts.

The semiotic visual rendition of the mathematics problem permits are-organization of perceptual reality. For example, Tartaglia's (1537) draw-ing of the men firing the cannon in Plate 2.3(5) demonstrates how thevisualization of the problem allows new objects or entities to be introducedsemiotically. That is, the line segments, circles and arcs and the resultanttriangles are entities which only exist in the semiotic construal of thematerial context of the problem. Significantly, these new entities becomethe focus of attention for Descartes who attempts to construct differentcurves and in doing so discovers that they may be described algebraically.From this point, Descartes discards the human realm of sense perception.In what follows, an examination of Descartes' mathematical and philo-sophical writings reveals that mathematical symbolism developed as a semi-otic system to form 'a semantic circuit' with the visual images and language.That is, the visual images of the curves, the symbolic description of thosecurves and the use of language function hand-in-hand to create a newversion of reality.

2.4 Beginnings of Modern Mathematics: Descartes and Newton

There was a shift in the nature of the semiotic construction of mathematicalproblems in the seventeenth to eighteenth centuries where the circles,curves and line segments increasingly became the major focus of attentionrather than the depiction of the material context of the problem that fea-tured so prominently in the early Renaissance texts. For example, thehuman body gradually disappears, or alternatively is replaced by a part of

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Plate 2.3(5) Predicting the path of cannon fire (Tartaglia, 1537)

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the body, as we may see in the case of Descartes' (1682: 111) eyes in Plate2.4(la) and Newton's drawings of the path of light in Plate 2.4(lb). Whilethe mental process of perception is still construed, the body part of 'theeye' now acts as the sensor. In these new diagrams, the focus shifts awayfrom the human actor and the context of the problem to the semioticentities of lines, curves and triangles. With the decline of the human agent,the line segments, circles and arcs and their accompanying spatial andtemporal relations take centre stage. In Newton's diagram in Plate 2.4(lb),

Plate 2.4(la) Removing the human body: Descartes (1682: 111)

Plate 2.4(lb) Removing the human body: Newton (1952: 9)

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for example, the dynamic process of the refraction of light is constructedthrough intersecting line segments through a lens. The drawing of the eyecontextualizes the scene, but this eye possesses secondary importancecompared to the lens and the path of light depicted by the line segments.This secondary importance is signalled by the size and position of the eyecompared to size, centrality and labelling of the line segments and thelens.

The human figure disappears in visual representations of materialactions as well as acts of perception. For example, in Descartes' (1985c:259) drawing in Plate 2.4(2a), a hand rather than a complete human figureis drawn swinging a stone from point A to point F. The situational contextof the problem is absent and the stone is not swung for any conceivablepurpose other than to trace the movement of the stone. Descartes labelsthe path A, B and F at different points which adds a temporal dimension tothe visual semiotic representation. Descartes is concerned with spatial andtemporal dimensions of the path in the visual semiotic construction of theproblem. Although the path of the cannonball is drawn in Plate 2.3(5),Tartaglia does not attempt to mark so explicitly the unfolding temporaldynamics at particular points of time, but rather the more dynamic aspect

Plate 2.4(2a) Movement in space and time: the stone (Descartes, 1682:217)

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of his drawing appears in the firing of the cannon. Descartes' attention,however, lies with the path of the stone at different times which areexplicitly drawn, labelled and marked spatially with line segments. Des-cartes draws a model of a compass to trace the movement of the stone inPlate 2.4(2b). This drawing depicts a material compass that swings on apivot at point E. As we shall soon see, the semiotic compass plays a majorrole in the development of Descartes' geometry.

Newton's (1953: 31) construction of the path of two swinging pendulums

Plate 2.4(2b) Movement in space and time: the model (Descartes, 1682:217)

Plate 2.4(3) Movement in space and time: the pendulum (Newton, 1953:31)

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is displayed in Plate 2.4(3). In this diagram the human figure and thecontext have completely disappeared, and the material everyday objectsuch as a stone is replaced with a pendulum, which is a piece of scientificequipment like the lens in Plate 2.4(lb). Newton marks the relative pos-ition of the pendulums in more detail than Descartes' path of the stone inPlate 2.4(2a). One can see the development of the semiotic construction ofthe prediction of the path of objects as a continuous mapping of spatial andtemporal dimensions.

The shift from the realm of the material, everyday world of human actionand perception to de-contextualized visual images in the beginnings ofmodern science is apparent in Descartes' drawings in Plates 2.4(4a-b).These illustrations display the path of a ball through water. In Plate 2.4(4a),a man is shown hitting the ball downwards into what looks like a lake or apond. We have, in a manner similar to Tartaglia's drawings, features of thecontext of the situation which include a complete human figure involved insome material action in a setting. The path of the ball is constructed as aseries of line segments in relation to a circle. However, in Plate 2.4(4b)Descartes shifts his attention to the line segments and curves. Once again,the human figure and the context are eliminated and the major partici-pants are the new semiotic entities of line segments and circles which aresituated in specific relations to each other.

The appearance of de-contextualized visual images where the majorprocesses are spatial, temporal and relational with entities in the form of

Plate 2.4(4a) Context, circles and lines (Descartes, 1682: 226)

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44 MATHEMATICAL DISCOURSE

the line segments, circles and curves requires explanation. One reasonis the growing significance of the role of the mathematical symbolism.Descartes discovered that curves could be described using mathematicalsymbolism, and therefore he moved (a) from the semiotic construction ofthe material context (b) to the semiotic construction of the lines and curves(c) which were described symbolically (d) to solve the problem. Newtonand other mathematicians used this path to lay the foundations of modernscience. This path is explored in the remainder of this chapter.

Descartes was interested in constructing curves using a semioticallygrounded compass which was conceived from the material compass andruler used by the Greeks. From the material actions depicted in Plate2.4(5a), Descartes devised a method of semiotically constructing curvesbased on proportionality as displayed in Plate 2.4(5b). For Descartes, 'Thisnew [semiotic] instrument does not have to be physically applied; it isenough to be able to visualise it and use it as a computing device. In otherwords, pen and paper is all that is required, since the nature of the curve isrevealed in its tracing' (Shea, 1991: 45). The shifts in the nature of thesemiotic construals of Descartes' geometry seem to occur in stages. In theinitial stages, the semiotic rendition included drawing the material actionof tracing the curve as displayed in Plate 2.4(5a). However, this materialdrawing of the curve (which includes the actions of the hand) developedinto a semiotic rendition of the material compass to trace curves as dis-played in Figure 2.4(5b). Descartes' main concern was the proportionalrelations which he mapped visually and spatially as curves using his semioticcompass.

Descartes discovered that mathematical symbolism could be used todifferentiate between the curves he constructed. Although this was a major

Plate 2.4(4b) Circles and lines (Descartes, 1682: 228)

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EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 45

Plate 2.4(5a) Descartes' semiotic compass (1683: 54) (Book Two)

Plate 2.4(5b) Drawing the curves (Descartes, 1683: 20) (Book Two)

breakthrough, Descartes' interest remained in the construction of thecurves and he did not realize, as later mathematicians such Newtonand Leibniz did, that the symbolism provided a complete description ofthe curves rather than a means for construction. While the methodamounted to an algebraization of ruler-and-compass constructions (Davisand Hersh, 1986), Descartes nonetheless 'simplified algebraic notation and

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46 MATHEMATICAL DISCOURSE

set geometry on a new course by his discovery that algebraic equations wereuseful not only in classifying geometrical curves, but in actually devising thesimplest possible construction' (Shea, 1991: 67). Davis and Hersh (1986: 5)comment: 'In its current form, Cartesian geometry is due as much toDescartes' own contemporaries and successors as to himself.' Despite this,Descartes' increasing reliance on the symbolism is evident in his geometryas displayed in Plates 2.4(6a-b). The symbolism features as an integral partof Descartes' geometry, one that now depends on language, visual imagesand mathematical symbolism.

The significance of Descartes' algebraic descriptions for curves cannotbe underestimated because this is the point from which modern mathemat-ics and science developed as an integrated multisemiotic discourse inwhich a central role was assigned to the symbolism. For example, Newton'sreliance on mathematical symbolic descriptions is seen in Plates 2.4(7) and2.4(8a-b). Newton rewrites Euclid's geometry in algebraic terms in Plate2.4(7), and in printed versions of Newton's work in Plates 2.4(8a-b) thereliance on algebraic symbolism as a method for reasoning is evident. New-ton proceeds symbolically step-by-step in Plate 2.4(8a), and efficientlyorganizes these symbolic descriptions into table format in Plate 2.4(8b).Newton's semiotic is the symbolism which functions in conjunction withlanguage and the mathematical graphs.

The implications and circumstances surrounding Descartes' move to thesymbolic are worthy of further investigation, not only because this providedNewton with the tools to rewrite nature, but also because it appears that thenewly de-contextualized and algebraicized geometry provided Descarteswith the foundations for his influential philosophy. Descartes' method isconcerned with establishing what is 'true knowledge' through the path ofintelligibility rather than sensory experience. This method appears tooperate within the boundaries of Descartes' new form of symbolic semiosis,which offers much more at the price of admitting substantially less. Theprogramme of objectivity and truth in the mathematical descriptionsinherited from Descartes exists today.

2.5 Descartes' Philosophy and Semiotic Representations

While Descartes did not fully utilize the potential of the mathematical sym-bolic descriptions, he certainly appreciated the power of this form of semi-osis. Descartes repeatedly insists, for instance, that language is inadequatefor his purpose of achieving certainty of knowledge beyond the common-sense kind. Descartes' distrust of the linguistic semiotic is openly expressedin the Second Meditation in 'Meditations on First Philosophy' (Descartes,1952, 1985b) where he attempts to describe what can be known with cer-tainty through a discussion of a ball of wax. Descartes explains that know-ledge achieved through the senses (for example, colour, flavour, smell,shape and size) is unreliable because these properties change as thewax is heated. Descartes concludes that mental facilities alone permit

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EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 47

*to LA GEOMETRIE .

Plate 2.4(6a) Descartes' description of curves (1954: 234)

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Plate 2.4(6b) Descartes' use of symbolism (1954: 186)

48 MATHEMATICAL DISCOURSE

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Plate 2.4(7) Newton's algebraic notes on Euclid

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Plate 2.4 (8a) Newton's (1736: 80-81) Method of Fluxions and Infinite Series

50 MATHEMATICAL DISCOURSE

The Method ..of FLUXIONS,llli To determine -a Conic SeSiimt, at any.Point of which, the Cur-

vntitre and Pofitian of the Tangent, (inrefpeftofthe AxhJ) may be liketo the Curvature and' Pofition of the. tangent, at a Point aj/igiid ofany other Curve.

"21. The ufe of which Problem is. this* that inftead of Ellipfes ofthe fecond kind, whole Properties- of refradling Light are explain'dby DCS Cartes in his Geometry, Conic Sections may be fubftituted,which Hull perform the fame | thing, very nearly, as to their Re-irasftions. And the fame may be underftooci of other Curves.

P R O B. VII.20 find as .many Curves as you pleafe^ wbofe Areas may

be exhibited by finite. Equations.

E X A M P L E S .

80

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EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 51

their Abfcifles or Bafes AB and AC. Then the Increments or Fluxionso( the Areas which they defcribe, will be as thofe Ordinates drawn

M into

Plate 2.4(8a) - cont

P R O B. VIII.

70 find as many Curves as you pleafe^ wbofe Areas Jhallhave a relation to the Area of any given Curve> ajjlgn-able by finite Equations.i. Let FDH be a given Curve, and GEI the Curve required, and

conceive their Ordinates DB and EC to move, at right Angles upon

10. And thus from the Areas, however they may be feign'd, youmay always determine the Ordinates to which they belong.

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52 MATHEMATICAL D I S C O U R S E

1OO fhe Method < ? / F L U X I O N S ,

Plate 2.4(8b) Newton's (1736: 100) Method of Fluxions and Infinite Series

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EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 53

examination of the reliable essence of matter which he conceptualizes asmotion and extension in the form of length, breadth and depth. Two trans-lations where Descartes explicitly criticizes the use of language are repro-duced below: the first is an English translation of Descartes' original 1642Latin text and the second is a translation of the 1647 French version of thatLatin text.

But as I reach this conclusion I am amazed at how [weak and] prone to error my mindis. For although I am thinking about these matters within myself, silently and withoutspeaking, nonetheless the actual words bring me up short, and I am almost tricked byordinary ways of talking.

(Descartes, 1985b: 21)

I am indeed amazed when I consider how weak my mind is and how prone to error.For although I can, dispensing with words, [directly] apprehend all this in myself,none the less words have a hampering hold upon me, and the accepted usages ofordinary speech tend to mislead me.

(Descartes, 1952: 209)

The two translations express Descartes' criticism of the use of language todescribe knowledge which he claims is certain. His intent is clear: 'Butaiming as I do at knowledge superior to the common, I should be ashamedto draw grounds for doubt from the forms and terms of ordinary speech'(ibid.: 210). The semiotic Descartes installs as the one most appropriate forhis purposes is mathematical symbolism. In this process, the accompanyinggeometrical curves are only to be used as an aid to thought. Descartesdescribes his symbolic expressions which are to replace the linguisticdescriptions:

Whatever, therefore, is to be regarded as an item . . . we shall designate by a uniquesign, which can be freely chosen. For convenience sake, we employ the letters, a, b andc, etc., to express magnitudes already known, and A, B, C, etc. for unknown magni-tudes. To them we shall often prefix the signs 1, 2, 3, etc., to indicate their numericalquantity, and shall also append them to indicate the number of relations which are tobe recognised in them. Thus if I write 2a\ that will be as if I should write the double ofthe magnitude signified by the letter a, which contains three relations. By this devicenot only do we obtain a great economy in words, but also, what is more important, wepresent the terms of the difficulty so plain and unencumbered that, while omittingnothing which is needed, there is also nothing superfluous, nothing which engagesour mental powers to no purpose . . . Rule XVI

(ibid.: 101)

Descartes (1952: 101-102) claims that linguistic descriptions such as 'thesquare' or 'the cube' are confusing, and that they should be abandoned.

The first is entitled the root, the second the square, the third the cube, the fourth thebiquadratic, etc. These terms have, I confess, long misled me. For, after the line andsquare, nothing it seemed to me allowed of being more clearly exhibited to theimagination than the cube and other shapes; and with their aid I solved not a fewdifficulties. But at last after many trials I came to realise that by this way of conceivingthings I had discovered nothing which I could not have learnt much more easily and

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54 MATHEMATICAL DISCOURSE

distinctly without it, and that all such denominations should be entirely abandoned, asbeing likely to cause confusion in our thinking.

The underlying reason for Descartes' claim that the linguistic descriptionscause 'confusion in our thinking' is that the linguistic version means some-thing quite different to what is generally considered to be the symbolicequivalent. As explained in the discussion of language and photographs inChapter 1, different semiotic resources have the potential to mean differentthings. In Descartes' example, the linguistic term 'the cube' is semantically afixed entity through its grammatical instantiation as a noun. It is an object,a thing, a participant or entity. On the other hand, the symbolic ax1 is not afixed object, rather it is a complex of interactive participants which combinethrough the process of multiplication as seen in the expanded form ax xx xx x. The symbolic expression is not a stable fixed entity like the linguisticnominal group, rather it is a dynamic complex which may be reconfiguredin different ways. This type of symbolic expression offers countless alterna-tives to describe different curves; for example x3, 2X3, Sx3 and so forth. Giventhat Descartes' aim is to construct and differentiate between curves, it isnot surprising that the symbolic descriptions are preferred.

Descartes' (1985a: 9-78, 111-151) 'Rules for the Direction of the Mind'and 'Discourse on the Method for Rightly Conducting One's Reason andSeeking the Truth in the Sciences' explain his method for securing trueknowledge. Basically the method entails finding the simplest parts whichare known to be true, and ordering and enumerating these parts to under-stand the more complex whole. The results should be checked to makesure that there are no errors. Shea (1991: 131) explains more fully themethod:' (a) nothing is to be assented to unless evidently known to be true;(b) every subject-matter is to be divided into the smallest possible parts, andeach dealt with separately; (c) each part is to be considered in the rightorder, the simplest first; and (d) no part is to be omitted in reviewing thewhole'. This is Descartes' method. Construct the problem out of the sim-plest elements possible, and rearrange those elements to solve the problemof the more complex.

The symbolic descriptions fulfil Descartes' criterion for the rightphilosophical method upon which to proceed to secure true knowledgebecause the method appears to be based upon his success in algebraicizedgeometry. The relationship between Descartes' philosophical method andhis algebraicized geometry is quite apparent. First, the (algebraic) elementsare broken down into their simplest components to understand themore complex (symbolic) configuration. Second, the (symbolic) state-ments do not include any superfluous information which may function as adistraction. Third, the (symbolic) expressions may be checked to ensurethere is no error. Descartes' procedures in geometry match his method forsecuring true knowledge. There are several important implications of Des-cartes' mathematics and philosophy which shaped the course of modernmathematics and science.

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EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 55

Descartes' philosophy rested on mathematical formulations and pro-cedures which revolutionized the nature of semiosis through the promin-ence accorded to the symbolic and the secondary position accorded to thelinguistic. Language was considered inadequate for the knowledge towhich Descartes aspired. Language belonged to the common-sense worldof perceptions and sensory input which was deemed unreliable. The fam-ous Cartesian mind and body duality was attached to the semiotic formswhich were used to represent each realm. Symbolic and specialized math-ematical forms of visual semiosis were located in the realm of the mindwhich was, according to Descartes, the proper site for securing knowledge.Language belonged to the realm of the everyday and to the body and itssensory apparatus. This remarkable shift in emphasis resulted in a sharpdichotomy between the different forms of semiosis, and this included adifference in the values attached to each. The differing values accorded tothe sciences and the arts and social sciences continue today.

The focus of concern in mathematics became curves or patterns whichwere exactly describable through the symbolism. The types of processes inthese visual representations are spatial and temporal relations, and relativeproportional rates of change. The major visual participants are lines, linesegments, circles, arcs and curves and geometrical shapes which are thevisual representations of the relations. Those relations are described sym-bolically through mathematical processes such as multiplication, addition,subtraction and division between the symbolic participants. The continu-ous nature of these relations is depicted graphically and described exactlythrough the symbolism. Human actors participating in material, affective,perceptive, behavioural or verbal processes in the context of the everydaywere removed from the realm of mathematics and science. The new semi-otic tools are designed to work within particular semantic fields, and thesedo not include the human realm of the material, the emotive, and thesensory which were considered superfluous. The human realm was putaside in this major re-evaluation of knowledge.

The mystical claims of the Middle Ages were replaced with a new type ofknowledge and a new basis for legitimizing truth. In addition, the claimsupheld on the basis of mathematical descriptions were backed by experi-mental evidence. With the material success achieved through the math-ematicization of nature, gradually the God which was central to Descartesand Newton's formulations (see Plate 2.5(1)) was removed from modernscience. The significance of this re-contextualization in the modern math-ematical view of the world is explored in Chapter 7. As Koestler (1959: 11)comments 'all cosmological systems [visions of the universe] reflect theunconscious prejudices, the philosophical or even political biases of theirauthors; and from physics to physiology, no branch of Science, ancient ormodern, can boast freedom from metaphysical bias of one kind oranother'. The question remains as to the impact of our adoption of themathematical in a present-day context which differs from the cosmologywithin which it developed.

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Plate 2.5(1) Illustration from Newton's (1729) The Mathematical Principles ofNatural Philosophy

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EVOLUTION OF THE SEMIOTICS OF MATHEMATICS 57

Mathematical symbolism originated from rhetorical algebra in the formof linguistic descriptions and commands which explained the methodthrough which to proceed. Syncopated algebra saw abbreviations for recur-ring participants, but basically the linguistic grammar still provided thebasis for these discourses. However, through the work of Descartes, Newtonand other mathematicians, especially Leibniz who 'made a prolonged studyof matters of notation' (Book Two, Cajori, 1993: 180), the symbolismdeveloped as the semiotic tool which was central to the mathematics whichsubsequently developed. The symbolic grammar was based on economythrough the need to describe in the simplest and most condensed form thatwhich also needed to be rearranged to explain the more complex. Thismeant the development of new systems in the grammar of mathematicalsymbolism which did not exist in language. There was to be no confusion,no room for error and no superfluous information in the new formsof reasoning provided by the mathematical symbolism. This grammar ofmathematical symbolism is the focus of Chapter 4. From this point, Newtonstarted a trend which could only be called a new world order.

2.6 A New World Order

The new approach advocated by Descartes proved to be significant becauseNewton and others created a movement which involved a new representa-tion of the physical world using new semiotic tools. In this movement,matter and perceptual data were re-admitted by Newton, but in a newmathematicized form (Barry, 1996). As Sweet Stayer (1988: 3) claims, whileNewton explained the motion of bodies through his calculus and com-pleted research in the fields of optics, tides, thin films and gravitation, TheMathematical Principles of Natural Philosophy was the culmination of hiswork, and it 'profoundly changed the perspective with which we view theworld'. Newton's new semiotic constructions explained the visible worldthrough invisible properties which were made 'real' or 'concrete' throughmathematical symbolic description. One key to this success was that themathematical symbolism, the visual images and language worked together.Descartes discarded sense data and developed a method with a form ofsemiosis which could describe exactly relations. These entities could bevisualized and they could be described exactly in a symbolic form whichallowed the rearrangement of those relations to solve problems and concep-tualize the more complex phenomena. Newton dispensed with Descartes'position in that he accepted the world of perception, but at the same timehe reconstructed that world using Descartes' semiotic tools. Newton usedmathematical symbolism to create metaphorical entities which explainedthe everyday material world. Newton's new mathematical tools permittedexact description of that which was perceived in terms of the propertiesof matter. The physical world became the object of concern, and with thisnew engagement, the means for industrialization, colonialization andcommerce rapidly increased. The reason for the status of mathematics is

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precisely the goals and objectives which have been fulfilled through thisform of semiotic representation. Mathematics and science fulfil functionswhich have transformed the face of the world and life on earth.

Newton instigated a movement which increased control over the physicalworld because he included experimentation as an integral part of his scien-tific method. Galileo first established the ideas of experimentation: 'Gali-leo's work laid the foundations of the modern scientific method whichregards the collection of experimental evidence as the essential prelude tothe formulation of scientific laws and theories' (Hooper, 1949: 201). One ofNewton's major contributions was the use of technology and scientificequipment in a laboratory setting to test empirically his theories. The the-ories had to fit the empirical evidence, or at least are seen to fit. Sciencebecame a matter of description, prediction and prescription within theconfines of the practices established by the laboratory and the semiotictools which permitted those representations. Although matter may havebeen re-admitted by Newton, there were constraints on how the sensoryphenomena could be viewed and described, and those constraints wereestablished through his semiotic tools and the technology of his scientificequipment. As Descartes openly states, all 'superfluous' information wasremoved, and what remained was what was possible with the symbolism,visual images and language which formed the semantic circuit with whichNewton constructed the new world order.

New branches of mathematics have developed since Newton, and theidea of an ordered mechanical physical world has been largely abandonedwith the development of the notion of chaos and dynamical systems theory.This new view of the world is based on the idea of non-linearity where it isassumed that the behaviour of physical systems is in fact indeterminate; thatis, the behaviour of a system cannot be predicted exactly. Davies (1990)explains that the approach advocated by Descartes where systems arebroken down into constituent components to understand the complexwhole is reasonably successful because most physical systems behave in thislinear format up to a certain point. This method of analysis is only partiallysuccessful however: 'On the other hand, they [all physical systems] turn outto be nonlinear at some level. When nonlinearity becomes important, it isno longer possible to proceed by analysis, because the whole is now greaterthan the sum of the parts' (ibid.: 16). When this point is reached, theconstraints, boundary conditions and initial conditions of the system mustbe taken into account if the behaviour of the system is to be predicted withsome degree of success.

The new mathematics of non-linear dynamical systems theory is madepossible through computer technology. As the computing ability increases,together with the potential for highly sophisticated dynamic graphicalimages, so the nature of the mathematics changes; that is, mathematicsand science are intimately linked to the state of the art of computer tech-nology which affords new possibilities in what has literally become a virtualworld. Indeed, computer technology is such that visual images are now

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increasingly exploited as a semiotic resource that offers new possibilitiesfor modelling the world. It could be that the superfluous information oncediscarded by Descartes can now be incorporated in a new more fullyinclusive view of the world.

In conclusion, mathematics and science offer a particular representationof the world - one that is limited by the semiotic tools and the technologyemployed in its construction. In order to appreciate the nature of thatconstruction, texts and contexts must be analysed to understand the typesof meanings that are made, and the means through which this is achieved.For this reason, the grammatical systems for mathematical symbolism andvisual display are presented in Chapters 4—5. After discussing the uniquegrammars for each resource, the semantic circuit in mathematics involvingthe linguistic, the visual and the symbolic is discussed in Chapters 6-7.Discourse analyses of mathematical texts demonstrate that intersemiosisacross the three resources is critical for the semantic expansions thattake place in mathematics. The analyses represent a close engagementwith mathematics as a multisemiotic discourse in order to appreciate thepotential and the limitations of the meanings which are made.

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3.1 The Systemic Functional Model of Language

The investigation of multisemiosis in mathematics is based on MichaelHalliday's (1973, 1978, 1985, 1994) systemic functional (SF) approach tolanguage which has been extended by Jim Martin and others to incorporatediscourse systems (Martin, 1992; Martin and Rose, 2003), genre and ideol-ogy (for example, Christie, 1999; Christie and Martin, 1997; Hasan, 1996b;Martin, 1997). An outline of systemic functional (SF) theory and theaccompanying grammatical and discourse systems of language is providedin order to explain the conceptual apparatus underlying this study ofmathematics. The discussion is necessarily technical, but further explan-ations of SF theory are provided elsewhere (for example, Bloor and Bloor,1995; Eggins, 1994; Eggins and Slade, 1997; Halliday, 1994; Halliday andMatthiessen, 1999; Martin etal, 1997; Matthiessen, 1995; Thompson, 1996)including a collection of Halliday's writings (Webster, 2002-). The descrip-tion of Systemic Functional Linguistics (SFL) and the discussion of thenature of mathematical language in this chapter function to contextualizethe systemic frameworks for mathematical symbolism and visual displaydeveloped in Chapters 4-5. This leads to an investigation of the meaningarising from the integrated use of language, mathematical symbolism andvisual display in Chapters 6-7.

The description of mathematical and scientific language in this chapteris general (for a more detailed analysis, see Halliday and Martin, 1993;Halliday and Matthiessen, 1999; Martin and Veel, 1998) as the major con-cern of this study is the extension of SF theory to mathematical symbolismand visual display in order to investigate the multisemiotic nature of math-ematical discourse. In what follows, Halliday's SFL model and linguisticsystems at the rank of clause and clause complex and Martin's discoursesystems at the level of paragraph and text are used to examine the nature ofmathematical language. It becomes apparent in this discussion that thestudy of mathematical and scientific language needs to take into accountthe meaning arising from the symbolism and visual display. Martin's dis-course systems where meaning is made across stretches of text are therefore

3 Systemic Functional Linguistics (SFL) andMathematical Language

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extended in Chapter 6 to include meaning arising from intersemiosisacross linguistic, visual and symbolic components of the text. In addition,the description of grammatical metaphor in this chapter is furtherdeveloped through an examination of semiotic metaphor in Chapters 6-7.In a similar fashion, the discussion of register, genre and ideology is revisitedin relation to the multisemiotic nature of the discourse of mathematics inChapter 7.

Halliday 's SF Theory of Language

The fundamental assumption behind Halliday's SF social-semiotic theory isthat language is a resource for meaning through choice. Halliday (1994)comprehensively documents the grammatical systems through whichlanguage is used to achieve different functions. For, as Halliday explains,language has evolved to satisfy human needs and its grammatical organiza-tion is therefore functional with respect to those needs. Halliday (ibid.: xiii)states: 'A functional grammar is essentially a "natural" grammar, in thesense that everything in it can be explained, ultimately, by reference to howlanguage is used.' Any instance of written or spoken language does notunfold haphazardly as an abstract artefact as formal linguists would lead usto believe, but rather all texts are constructed in some context of use. Thechoices in the text's patternings reflect the uses that language is serving inthat particular instance.

The underlying assumptions of SFL may be contrasted to the positionadopted in formal linguistics where language is conceptualized as a systemof rules. Descriptions in these traditions show which sentences are accept-able and explanations reveal 'why the line between in and out falls where itdoes in terms of an innate neurological speech organism' (Martin, 1992:3). Rather than adopting an individual mentalist perspective, SFL viewslanguage as a resource consisting of a network of relationships. Descrip-tions show 'how these relationships are interrelated' and explanationsreveal 'the connections between these relations and the use to which lan-guage is put' (ibid.: 3). SFL is thus orientated to choice, 'what speakersmight and tend to do', as opposed to restriction, 'what speakers are neuro-logically required not to do' (ibid.: 3-4). The SFL approach is concernedwith the analysis of how language is used to achieve certain goals throughthe description of lexicogrammatical (that is, lexical and grammatical) anddiscourse systems, and the analysis of the choices that have been made inany instance of language use. SFL discourse analysis is a critical interpret-ation of how language choices function to construct a particular viewof reality, and the nature of social relations that are enacted in thatconstruction.

SFL evolved from Firthian linguistics and consequently is a type of sys-tem/structure theory where the key idea is meaning through choice fromthe available systems. Following Hjelmslev (1961), paradigmatic relationsare mapped onto the available options in the system network (the range of

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choices) and syntagmatic relations are mapped on to actual choices (theprocess which takes the form of a chain of words). The concept of 'realiz-ation' relates system and process in that realization statements specify thesystems in process and give the structural arrangement of the selectedoptions. Halliday does not privilege either system or process: 'I prefer tothink of these [system and process] as a single complex phenomenon:the "system" only exists as potential for the process, and the process is theactualisation of that potential. Since this is a language potential, the"process" takes the form of what we call a text' (Thibault, 1987: 603).

The systems in Halliday's grammar of English are organized accordingto meaning. Halliday separates the two main types of meaning into the'ideational' or the reflective, and the 'interpersonal' or the socially active.Halliday further separates ideational meaning into two components, the'experiential' and the 'logical', which are respectively concerned withthe construction of experience and logical relations in the world. Hallidayalso identifies the enabling function of language, the 'textual' componentwhich organizes language choices into coherent message forms. These fourtypes of meaning, the experiential, logical, interpersonal and textual, arecalled the metafunctions of language as they are manifestations of thegeneral purposes of language: ' (i) to understand the environment (idea-tional), and (ii) to act on the others in it (interpersonal)' (Halliday, 1994:xiii). The SF approach means that although the grammatical classes such asnouns, verbs, adjectives and so forth still have a place, for example, in thedescriptions of grammatical metaphor (Derewianka, 1995; Halliday andMartin, 1993; Martin et al., 1997; Simon-Vandenbergen et al, 2003), theelements of language are described by functional rather than word classlabels. Language is conceived as an 'organic configuration of functions'and 'each part is interpreted as functional with respect to the whole'(Halliday, 1994: xiv).

There is equal emphasis on the interpretation of the interpersonalmetafunction as well as experiential, logical and textual meanings in SFL.As Poynton (1990) explains, the focus on social relations and the expres-sion of personal attitudes and feelings has traditionally been marginalizedin the majority of linguistic theories. The focus on system and referentialmeaning in linguistics, perpetuated with Chomsky's reformulation ofSaussure's (1966) langue/parole (language system versus language use)distinction as competence/performance, was accompanied by an explicitemphasis on the cognitive domain (for example, Chomsky, 1965, 2000).Poynton (1990) explains that such cognitively orientated conceptions oflanguage support dichotomies such as objective versus subjective, andreason versus emotion. As Poynton claims, the higher values accorded toobjectivity and reason have obvious significance in the development ofmathematics and science, and they have also been invoked in areas of socialcontrol. These issues are explored through an interpretation of the con-cepts of reason, objectivity and truth based on the analysis of a mathematicstext in Chapter 7.

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Halliday (1994) is largely concerned with lexicogrammar at the ranks ofword, word group/phrase, clause and the clause complex. On the otherhand, Martin (1992; Martin and Rose, 2003) is concerned with metafunc-tionally based systems which operate across paragraphs and the whole text.Martin's work follows from Halliday and Hasan's (1976) systemic analysis oftextual cohesion where the basic opposition is between structural (gram-matical) and non-structural (cohesive) devices. Martin (1992: 1) organizeshis divisions stratally 'as an opposition between grammar and semantics(between clause orientated and text orientated resources for meaning)'.Martin thus establishes a separate discourse semantics stratum to comple-ment Halliday's lexicogrammar. Martin's proposals lead to a languageplane with two strata,1 discourse semantics and lexicogrammar, and anexpression plane which is concerned with phonology and graphology/typography (Eggins, 1994: 81-82). The resulting SF model of language,which also includes the communication planes of register, genre and ideol-ogy, is displayed in Table 3.1 (1).

Halliday's Lexicogrammatical Systems and Martin's Discourse Systems

The major systems in Halliday's lexicogrammar and Martin's (1992; Martinand Rose, 2003) metafunctionally based discourse systems are listed inTable 3.1(2). Following systemic conventions, the lexicogrammatical anddiscourse systems are capitalized. The major lexicogrammatical systems areMOOD for interpersonal meaning, THEME for textual meaning and

Table 3.1(1) Language, Expression and Communication Planes

IDEOLOGY

GENRE

REGISTER

LANGUAGE

EXPRESSION PhonologyGraphology/Typography

CONTENT Discourse semanticsParagraph and text

LexicogrammarClause complexClauseWord group and phraseWord

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64

Table 3.1(2) Metafunctional Organization of Halliday's (1994) LexicogrammaticalSystems and Martin's (1992; Martin and Rose, 2003) Discourse Systems

Metafunction Lexicogrammar Discourse Systems

interpersonal clause:MOOD; MODALIZATION;MODULATION; POLARITY;TAGGING; VOCATION;ELLIPSIS

word group:PERSON; ATTITUDE(attitudinal modifiers,intensifiers); COMMENT(comment adjuncts); LEXIS(expressive words, stylisticorganization of vocabulary)

NEGOTIATION(exchange rank includingSPEECH FUNCTION at themove rank)Structure: Exchange Structurelinking moves

APPRAISALStructure: surges, flows andfalls mapped through wordgroups, phrases and clausesin text

textual clause:THEME

clause and word group:SUBSTITUTION; ELLIPSISword group: DEIXIS (nominal)

IDENTIFICATION(phoricity, reference)Structure: reference chainslinking participants

logical clause complex:LOGICO-SEMANTICRELATIONS andINTERDEPENDENT

CONJUNCTION andCONTINUITY(based on classifications ofLOGICO-SEMANTICSRELATIONS and semanticrelations respectively)Structure: conjunctive reticulalinking messages

experiential clause:TRANSITIVITY; AGENCY

word group:TENSE; LEXIS (lexical'content'); collocation

IDEATION(lexical relations)Structure: lexical strings andnuclear relations linkingmessage parts

TRANSITIVITY for experiential meaning at the clause rank, which is thebasic unit in which the semantic features are represented. The elements inthe clause (word, word group/phrase) are explained by their functions ineach of the metafunctionally based systems. At the rank of clause complexor sentence, the systems for logical meaning are LOGICO-SEMANTICRELATIONS and INTERDEPENDENT. Halliday's systems are discussed in

MATHEMATICAL DISCOURSE

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(SFL) AND MATHEMATICAL LANGUAGE 65

relation to the types of selections found in mathematical discourse inSections 3.2-3.5. These systems are also considered in relation to thegrammatical organization of mathematical symbolism in Chapter 4.

The realization structures for Halliday's grammatical systems takedifferent forms. Textual meanings arising from the system of THEME andinterpersonal meanings through the system of MOOD are described byparadigmatic oppositions realized through syntagmatic structures of thefunctional categories. Experiential meanings from TRANSITIVITY choices,on the other hand, are represented as clusters of participant/process/circumstance rather than sequences of functional elements in the clause.In addition, the patterns of realization vary across the metafunctions.Following Halliday, the realizations of the metafunctions in discourse takethe forms of particle-like experiential meanings, irregular prosodic swellsof interpersonal meanings (where the concept of volume comes into play)and regular periodic wave-like textual meanings.

The realization of the logical metafunction at the rank of clausecomplex through the system of LOGICAL-SEMANTIC RELATIONS issomewhat different from the other grammatical systems as particularelements (for example, the structural conjunctions 'and' and 'or') may beselected more than once in a clause complex, while the other systems havemultiple variables which may be selected only once in the clause. Logicalmeaning is described by the types of INTERDEPENDENCY relations(dependent or independent) and by LOGICO-SEMANTIC relationsbetween clauses (relations of logical expansion or projection of speechand thought).

Martin's (1992; Martin and Rose, 2003) discourse systems includeNEGOTIATION, APPRAISAL, IDENTIFICATION, CONJUNCTION andCONTINUITY, and IDEATION. The metafunctional organization andstructure of the discourse systems are included in Table 3.1(2). NEGOTI-ATION is orientated towards spoken discourse, but Martin and Rose (2003)also include the system of APPRAISAL for capturing graduations of attitude(affect, judgement, appreciation) and engagement in written (and spoken)discourse. 'Appraisal is concerned with evaluation: the kinds of attitudesthat are negotiated in a text, the strength of those feelings involved and theways in which values are sourced and readers aligned' (ibid.: 22).

Martin's concept of discourse systems is useful for the analysis ofstretches of text which involve language, visual images and mathematicalsymbolism. Martin's frameworks, however, need reworking as they aredeveloped for the analysis of linguistic text (that is, intrasemiosis in lan-guage), rather than the analysis of meaning within and across differentsemiotic resources (that is, intrasemiosis in mathematical symbolism andvisual images, and intersemiosis across the three semiotic resources). Dis-course systems similar to those proposed by Martin are introduced in the SFframeworks for mathematical symbolism and visual images in Chapters 4and 5, and the framework for intersemiosis in Chapter 6. The analysis ofintra- and inter-semiosis in mathematical discourse includes the typography

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66 MATHEMATICAL DISCOURSE

of the text at the expression stratum, the importance of which becomesevident in Chapters 4-6. This stratum has not typically been included inSFL analysis.

Martin's discourse structures interact systematically with each other andthe lexicogrammatical structures (giving rise to incidences of grammaticalmetaphor, for example) resulting in the 'texture' of a text. The ways inwhich the discourse systems co-operate with each other to make a text isnot as well understood as the nature of interaction across the grammaticaland discourse strata (see for example, Halliday, 1994; Hasan, 1984). Martin(1992:392) refers to the systematic interaction between discoursal and gram-matical structures as modal responsibility, cohesive harmony and themethod of development of the text. While these formulations are notspecifically developed in this study, the grammatical density arising fromthe interactions between language, visual images and the symbolism acrossdifferent strata becomes apparent in the analyses of the mathematics textspresented in Chapters 6-7. The texture of discourse in this case involvesthe dense patterns which emerge from the integrated use of language,mathematical symbolism and visual display (O'Halloran, 2000, 2004c).

SFL Discourse Analysis

In SFL discourse analysis, clauses are marked by slashes / / . . . / / and theelements within each clause are analysed according to the metafunctionallybased grammatical and discourse systems. Elements in the clause are ana-lysed several times, and functional labels are attached according to choicesmade from each system. Clauses are also classified as major or minor, andcomplete or ellipsed. In the case of spoken discourse, abandoned clausesmay also be tagged. Minor clauses are 'clauses with no mood or transitivitystructure, typically functioning as calls, greetings and exclamations'(Halliday, 1994: 63). Eggins (1994: 172) explains that these minor clausesare 'typically brief, but their brevity is not the result of ellipsis'. The classifi-cation of clause type is useful for examining interpersonal patterns ofdomination and deference.

Clauses are classified as to whether they contain rankshifted or embed-ded elements. Using Halliday's (1994: 63) notion of ranks (word, wordgroup/phrase, clause and clause complex), rankshifting is the processwhereby a clause or phrase functions at the lower rank of word or wordgroup. That is, embedded clauses (indicated by sets of square brackets[ [ . . . ] ]) and phrases (indicated by the square brackets [ . . . ] ) functionwithin the structure of a word or word group, thus shifting rank from clauseand phrase to the lower rank of word/word group.2 Halliday (ibid.: 242)explains that embedded clausal and phrasal elements may function as aPostmodifier in nominal groups; for example, //the job [[I want]] wasadvertised//; and adverbial groups; for example, //she reacted morestrongly [[than they expected]]//. Alternatively the rankshifted elementmay function as a Head of a nominal group; for example, //[[that so many

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(SFL) AND MATHEMATICAL LANGUAGE 67

staff are leaving]] is cause for concern//. As may be seen from theseexamples, rankshifting in language serves the important function of pack-ing information into the clause. The concept of rankshift is particularlysignificant in the grammar of mathematical symbolism in Chapter 4 wherethe lexicogrammatical strategies for encoding meaning in the symbolismare seen to be different from those found in mathematical and scientificlanguage.

In the following discussion, the major grammatical and discourse systemsare explained, and the nature of the selections found in mathematical andscientific language is discussed according to metafunction. The followinglinguistic extract from Stewart's (1999: 132) textbook Calculus is used toillustrate features of mathematical language:

From Equation 3 we recognize this limit as being the derivative off at x,, that is /' (x,).This gives a second interpretation of the derivative: The derivative/' (a) is the instant-aneous rate of change of y=f(x) with respect to xwhen x= a. The connection with thefirst interpretation is that if we sketch the curve y=f(x), then the instantaneous rate ofchange is the slope of the tangent to this curve at the point where x = a. This meansthat when the derivative is large (and therefore the curve is steep, as at the Point Pin Figure 4), the ^values change rapidly. When the derivative is small, the curve isrelatively flat and the ^values change slowly.

The multisemiotic text for this extract is reproduced in Plate 6.3(2) inChapter 6 where the meanings arising from intersemiosis between the lin-guistic, symbolic and visual choices in the text are analysed. It becomesapparent in the analysis of Stewart (1999: 132) that mathematical and sci-entific language must necessarily take into account the visual and symboliccomponents of the text.

3.2 Interpersonal Meaning in Mathematics

The analysis of interpersonal meaning is concerned with the nature of thesocial relations which are enacted through linguistic choices from the sys-tems listed in Table 3.1 (2). The description of the system of MOOD is givenin Halliday (1994: 71-105). In essence, the MOOD system (where choicesare made for Subject, Finite, Mood Adjuncts, Comment Adjuncts, Predica-tor, Complement and Circumstantial Adjuncts) is related to the SPEECHFUNCTION which is concerned with the giving/demanding information(statements and questions) and goods and services (commands and offers).The SFL analysis is concerned with choice: how do interactants negotiatethe exchange of information and goods and services, and what does thisreveal about their social relations?

A one-to-one relationship between the grammatical classes of MOOD(declarative, interrogative, imperative and exclamative) and the SPEECHFUNCTION (statement, question, command and offer) does not exist,so the co-text and context are taken into consideration in the analysis.However, the congruent or unmarked case is that statements are realized

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68 MATHEMATICAL DISCOURSE

through declarative Mood (SubjectAFinite, for example, 'this is . . . ' ) , ques-tions are realized through interrogative Mood (FiniteASubject, for example,'is this . . .?') and the WH-element (WH, for example, 'why . . .?'), com-mands through imperative Mood (Predicater, for example, 'solve . . .') andoffers through modulated interrogative Mood (modulated FiniteASubject,for example, 'would you . . .'). Incongruent selections result in inter-personal metaphors where there is variation (for whatever reason) in theenactment of the social relations.

The system networks for MOOD and SPEECH FUNCTION are displayedin Figure 3.2 (I),3 together with the Exchange Structure which consistsof sequences of moves. SPEECH FUNCTIONS include responses to thestatements, questions, commands and offers and Martin's (1992: 66-76)'dynamic moves' for spoken discourse, which are requests and responsesfor the tracking moves of Backchannel, Clarification, Check, Confirmationand Challenging moves. The SPEECH FUNCTIONS of Call and Greetinghave also been included, together with moves of Reacting in Figure3.2(1). These classifications frame the types of moves found in writtenmathematical texts.

As displayed in figure 3.2(1), the SPEECH FUNCTION and MOOD sys-tems relate to the Exchange Structure, which is Martin's (1992) discoursesystem of NEGOTIATION. The Exchange Structure is based on Halliday(1994) where the opposition is between information and goods and ser-vices moves. Following Berry (1981), moves are classified as primaryknower (Kl) and primary actor (Al) moves, and secondary knower (K2)and secondary actor (A2) moves. Kl is the speaker/writer who has theinformation which is being negotiated, and Al is the participant who per-forms the action. The Exchange Structure consists of obligatory moves Kland Al and three optional moves. These are delaying moves (dKl anddAl), secondary knower (K2) and secondary actor moves (A2). Berry's(1981) moves are developed from Halliday's (1994: 69) SPEECH FUNC-TION classifications of 'initiating and responding' to the 'giving anddemanding' of goods and services and information. While all the classifica-tions presented in Figure 3.2(1) do not typically occur in written mathemat-ical texts, it is useful to consider the selections afforded in dynamic spokencontexts to situate the type of discourse found in written mathematics.

In order to incorporate Ventola's (1987, 1988) 'move complexes', thecategories of initiation, request, response and closure moves are sup-plemented with 'K-Continuation' and 'A-Continuation', and 'K-x' and 'A-x'moves. These move-complexes are based on Halliday's (1994: 220)LOGICO-SEMANTIC RELATIONS of expansion (elaboration, extensionand enhancement) and projection (locution and idea) and interdepend-ency relations (see Section 3.4 on logical meaning). A 'continuation' moverealizes a continuation of the same SPEECH FUNCTION which was estab-lished in the previous move even though the clause selects independentlyfor MOOD. This move is realized through paratactic relations of inter-dependency between the two clauses. An 'x' move realizes a continuation of

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Figure 3.2(1) NEGOTIATION (Exchange Structure), SPEECH FUNCTION and MOOD

EXCHANGE

new

continuation

exchangenumber

Knowledge

Action

PropositionDynamic moveFollowupReactingCallGreeting

dK1K1K2K-lnitiationK-ContinuationK-xK-ResponseK-Request

ProposalDynamic moveFollowupReacting

dA1A1A2

A-lnitiationA-ContinuationA-xA-ResponseA-Request

SPEECHFUNCTION

SPEECHFUNCTION

statementexclamativeexpletivequestionacknowledgemecontradictionanswerdisclaimerclarificationchallengebackchannelcheckconfirmation

commandoffercompliancerefusalacceptancerejection

clarificationchallengebackchannelcheckconfirmation

MOOD

declarative

WH - interrogative

YN - interrogative

imperative

paralinguistic

none

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70 MATHEMATICAL DISCOURSE

the same SPEECH FUNCTION in hypotactically related finite and non-finite clauses. In this way, it is possible to map each clause as an element inthe unfolding Exchange Structure.

SFL analysis involves analysing choices in the text for mood, speech func-tion and negotiation (exchange structure). While this framework is used todiscuss the linguistic selections in Stewart (1999: 132), it is apparent thatdiscourse moves in written mathematics often involve shifts from one semi-otic resource to another. Stewart (1999: 132) makes reference to 'Equation3' and 'Point Pin Figure 4'. In a similar fashion, the command to solve aproblem is typically undertaken symbolically. However, the SFL grammat-ical and discourse frameworks provide the starting point for the develop-ment of grammars for the symbolism and visual display and the theoriza-tion of intersemiotic shifts and transitions in mathematical discourse. Inwhat follows, the nature of linguistic selections for interpersonal meaningin Stewart (1999: 132) are investigated.

The selections from the systems of MOOD, SPEECH FUNCTION andNEGOTIATION function to establish unequal relations between the writerand the reader of the mathematics text. In the following chapters, itbecomes apparent that the dominant position of the writer is reinforcedacross choices for mathematical symbolism and visual display. Similarly,unequal social relations are established between the teacher and the stu-dents in the context of the mathematics classroom. While mathematicalpedagogical discourse is dominated by the teacher (Veel, 1999), the natureof those social relations in classrooms differs on the basis of gender andsocial class (O'Halloran, 1996, 2004c). The nature of the linguisticselections which reinforce the position of dominance of the author of themathematics text is discussed below.

Given the monologic format of the written discourse of mathematics, thewriter assumes the speech roles. Foremost, the writer is the primary knower(Kl) who gives information in the form of statements through declarativeMood. These statements are typically complete, and so the writer providesdetailed information. In the case of mathematics textbooks such as Stewart(1999), the author also assumes the role of secondary knower (K2) whoasks the questions and, most typically, provides the answers (Kl). Thewriter takes the role of the one who commands (A2). In addition, as pri-mary actor (Al), the writer checks that commands have been completedcorrectly by providing the solutions to problems. Mathematics lends itselfto these types of social relations between the writer and the reader as math-ematics is a written discourse. The Exchange Structure typically involveslong sequences of moves as seen in Stewart (1999: 132). The extendedexchanges contribute to the steady interpersonal rhythm of mathematicaldiscourse, with its overarching aim of deriving results through longsequences of logical reasoning.

Stewart (1999: 132) attempts to vary the interpersonal nature of theextended exchanges through various strategies, which include the use of'we' as Subject. However, such selections as 'we' give rise to interpersonal

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(SFL) AND MATHEMATICAL LANGUAGE 71

metaphors where it is clear that the relations are being manipulated. Forexample, in the statement 'From Equation 3 we recognize this limit asbeing the derivative of/at xl that is,/' (xj) ' , the inclusive 'we' serves inter-personal rather than experiential meaning as the statement may be seen asa metaphorical variant for the command 'Recognize this limit as being thederivative of/at x1} that is,/' (xj) ' . Alternatively, the process 'recognize' inthe projecting clause may be seen to be metaphorical and unnecessary withrespect to the more direct statement 'this limit is the derivative of/at x1;that is/' (xj)'. Although the writer attempts to vary the social relations withthe reader through Subject choice and metaphorical expressions, thereader nonetheless remains the receiver of information, and the one whoseanswers and responses are checked against those provided by the author ofthe mathematics text.

The degrees of probability and obligation associated with the lin-guistic statements, questions, commands and answers in mathematicaldiscourse are similarly consistent. Halliday's (1994: 354—363) gradu-ations in probability and usuality (MODALIZATION) and inclination,obligation and potentiality (MODULATION) are associated with proposi-tions ('information') and proposals ('goods and services') respectively.The descriptive categories for MODALIZATION and MODULATIONand the value and orientation of the selection are displayed in Figure3.2(2).

C-MODULATION

VALUE

ORIENTATION

Figure 3.2(2) MODALIZATION and MODULATION

O-MODULATION

MODALITYprobability

usuality

obligation

potentiality

inclination

potentiality

-maximalhighmedianlow

objective

subjective

explicit

implicit

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72 MATHEMATICAL DISCOURSE

The statements are unmodalized in terms of graduations of probabilityand usuality in the extract from Stewart (1999: 132). For instance, theabsence of modality (realized through the Finite selections such as 'might','could', or 'should') functions to make the mathematics statements appearas correct and factual. The lack of modalization is accompanied by maximalobligation in the commands. POLARITY is simply positive ('is') and nega-tive ('is not'). This contributes to the steady interpersonal orientation of adiscourse which possesses an unqualified level of certainty. As noted inChapter 4, probability in mathematical discourse is typically expressedsymbolically through relational clauses. For example, probability may beexpressed through approximations such as x =» 0.5.

The typical absence of selections from Halliday's (1994: 82-83) system ofMOOD ADJUNCTS displayed in Figure 3.2(3) also functions to create anaura of factuality. For example, in Stewart (1999: 132) Mood Adjuncts indi-cating plays with probability (for example, 'possibly', 'perhaps' and 'cer-tainly') are not selected. Instead a certain presumption arises from theunmodalized statements and unmodulated commands, the nature of theprocesses which are selected (see Section 3.3) and the long implicationsequences in the Exchange Structure arising from selections for logicalmeaning (see Section 3.4). This is not to say that Mood Adjuncts are notselected in mathematical discourse. However, the nature of such adjunctsmay replicate the high level of presumption and obviousness found in thepedagogical discourse of mathematics (O'Halloran, 1996, 2004c).

The objective, rational and factual stance of mathematics is the productof the nature of the selections for interpersonal meaning as they combinewith a limited range of process types and participants (see Section 3.3) withan emphasis towards logical meaning (see Section 3.4). Descartes' removalof the human realm in mathematics and science is apparent in modernmathematics. The types of interpersonal choices from language, math-ematical symbolism and visual display function to simultaneously restrictand expand experiential and logical meaning in mathematics. In the con-text of the mathematics classroom, teachers introduce a variety of inter-personal strategies to maintain solidarity and group cohesiveness and torelieve the interpersonal stance of the subject matter (O'Halloran, 1996,2004c). However, if one considers high school mathematics texts, bookson mathematics and other generic forms of mathematical discourse, the

Figure 3.2(3) MOOD ADJUNCTS

MOODADJUNCT

degree

time

inclination

presumption

usuality

probability

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(SFL) AND MATHEMATICAL LANGUAGE 73

interpersonal stance of mathematical written discourse is largely consistent.This observation explains the inclusion of metaphorical forms of expres-sion and non-generic choices such as cartoons, drawings and photographswhich are inserted to disrupt the interpersonal orientation of mathematicstexts. Furthermore, the issue of lexical choice for interpersonal meaning inmathematics is addressed below.

Halliday (1961: 267) states that 'The grammarian's dream is ... to turnthe whole of linguistic form into grammar, hoping to show that lexis can bedenned as "most delicate grammar" '(see Hasan, 1996a). Two major areasof interest are lexis specific to the field of mathematics which is consideredunder experiential meaning (see Section 3.3), and lexical items which areinterpersonally marked. The relevant notion is one of 'core vocabulary'where certain lexical items are more central than others in describingexperiential or intersubjective reality. Carter describes tests for corenesswhich involve syntactic and semantic relations, and neutrality. From thisperspective, the coreness of lexical items is the extent to which they are'more tightly integrated than others into the language system; that is, theyoccupy places in a highly organized network of mostly structurally-semanticand syntactic interrelations' and are 'more discoursally neutral than others,that is, generally they function in pragmatic contexts of language use asunmarked and non-expressive' (Carter, 1998: 36).

Expressive linguistic selections orientated towards interpersonal mean-ing are included in Martin and Rose's (2003: 54) system network forAPPRAISAL, which is reproduced in Figure 3.2(4). APPRAISAL is con-cerned with evaluation: how the text functions to align the reader orspeaker with the various propositions or proposals which are put forth. Thisincludes lexical items and cases of amplification, special forms of addressand so forth. Further research will see the development of the system ofAPPRAISAL so that different strategies for positive and negative evaluation

APPRAISAL

ENGAGEMENT

ATTITUDE JUDGEMENT

APPRECIATION

GRADUATION

Figure 3.2(4) APPRAISAL Systems Reproduced from Martin and Rose(2003: 54)

FORCE

FOCUS

PROJECTION

MODALITY

CONCESSIONAFFECT

monogloss

heterogloss

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74 MATHEMATICAL DISCOURSE

may be uncovered. The three main appraisal systems given by Martin andRose (2003: 54-55) are attitude, amplification (or graduation) and source(or engagement):

Attitude comprises affect, judgement and appreciation: our three major regions offeeling. Amplification covers grading, including force and focus; force involves thechoice to raise or lower the intensity of gradable items, focus the option of sharpeningor softening an experiential boundary. Source covers resources that introduceadditional voices into a discourse, via projection, modalization, or concession; the keychoice is one voice (monogloss) or more than one voice (heterogloss).

The absence of lexical items orientated towards expressive or evaluativeinterpersonal meaning is apparent in the extract from Stewart (1999: 132).Lexical choice in mathematics is largely orientated towards experientialand logical meaning rather than interpersonal meaning. This does notmean, however, that the mathematics writer does not make evaluations.Appraisals of what is presented in mathematics exist in different genres indifferent forms. For example, the authors of research papers in mathemat-ics presumably cast a favourable impression on the results which are estab-lished. However, such judgements presumably appear as factual rather thanevaluative. For example, Mood Adjunct selections (for example, 'of course'and 'typically') which may combine with an explicit objective orientationtowards modality ('it is certain' and 'it appears that') mean that evaluationsare made through grammatical choices which do not necessarily includeinterpersonally expressive lexis (for example, 'that is excellent' or 'that isreally ridiculous'). The linguistic strategies for evaluation in mathematicsand scientific discourse require further research. In addition, perhapsAPPRAISAL may be understood as a meta-system arising from the layeringand juxtaposition of functional choices across experiential, interpersonal,textual and logical systems, rather than a discourse system in its own right.Further work is needed, however, to establish how such layers function toorientate the reader.

The apparent lack of the need to explicitly evaluate contributes to theview of mathematics as a rational discourse of truth. However, as discussedin Chapter 2, mathematics dispensed with many realms of human activity.As the semiotic which provides the meta-discourse for that which is per-formed symbolically and visually, language choices in mathematics arecircumscribed within certain semantic domains. The limited fields ofmeaning are considered in the discussion of experiential meaning inSection 3.3 and the packing of that information through grammaticalmetaphor in Section 3.6. However, the point is that the discourse ofmathematics appears as factual and objective truth because of the types ofinterpersonal choices which are made using language, and the preciseorganization of those choices in the mathematics text (see Section 3.5 fortextual meaning). This orientation is supported by the nature of experien-tial and logical choices (see Sections 3.3 and 3.4), and by the availableoptions in the system networks for the grammar of visual images and the

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(SFL) AND MATHEMATICAL LANGUAGE 75

symbolism. This point is explored further in the chapters concerned withthe symbolism, visual images and intersemiosis in mathematics.

3.3 Mathematics and the Language of Experience

Following Halliday (1994), experiential meaning at the rank of clause isrealized through the system of TRANSITIVITY displayed in Figure 3.3(1).The construction of experience takes the form of choices for process, parti-cipants and circumstance. Halliday (1994) includes the ergative interpret-ation of experience in the form of AGENCY. The associated functionalelements are the Medium, Agent, Beneficiary, Range and Circumstance.Halliday's (1994: 166) descriptive categories have been extended toinclude mathematical 'Operative' processes in mathematical discourse asdisplayed in Figure 3.3(1). This new process type initially appeared inmathematical symbolism in the form of mathematical processes such asaddition, subtraction, multiplication, division, powers and roots, and othermathematical operations. The meanings of these processes in math-ematical symbolism do not accord with existing processes categories. Thelinguistic versions of these process types have thus been categorized asOperative processes. The rationale and justification for the inclusion ofOperative processes in the mathematical symbolism is found in Chapter 4.

The stages through which mathematics became concerned with particu-lar realms of meaning to the exclusion of others are discussed in Chapter 2.Mathematics dispensed with the human realm, and became concernedwith dynamic relations which could be viewed visually and described sym-bolically. Relations took a visual form, and linguistic descriptions shifted tothe symbolic formulations. As language functions as the meta-discourse forthese descriptions and visual instantiations, the nature of experientialmeaning in mathematics simultaneously expanded to incorporate the newmeanings, and contracted to the limited semantic realms with which thevisualizations and symbolic descriptions were concerned. The impact onthe nature of mathematical language arising from the semantic expansionsmade possible through the symbolism and visual display may be seenin Stewart (1999: 132). This includes the relatively high incidence ofrelational processes and the metaphorical nature of the participants. Thesefeatures of experiential meaning in mathematical language are consideredbelow.

The major process type found in mathematical language appears to bethe relational process, which Halliday (1998: 193) explains is the favouredprocess type in science. It appears that as mathematical symbolism becameconcerned with the description of relations, the same shift occurred withinlanguage which was being used to describe and contexualize the visualiza-tions and symbolic descriptions. Halliday (1993a, 1993b, 1998), Hallidayand Matthiessen (1999) and Martin (1993a, 1993b) explain that theregrammaticization of experience which takes place through scientific lan-guage involves relational processes and entities in the form of grammatical

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76 MATHEMATICAL DISCOURSE

Mental

Verbal

Relational

Existential

Behavioural

Extent

Location

Existent

Manner

Cause

Accompaniment

Matter

Role

fcomitation

uaddition

Figure 3.3(1) The TRANSITIVITY System

ActorGoalRangeRecipientClient

SenserPhenomenon

SayerVerbiageReceiver

duration

distance

temporal

spatial

meansqualitycomparison

reasonpurposebehalf

CarrierAttributorBeneficiaryAttribute

Intensive

Circum-stantial

Possessive

Intensive

Circum-stantial

Possessive

Behaver

Behaviour

Operator

Participant

CIRCUMSTANCE

Identified

Identifier

TokenValue

Attributive

Identifying

PROCESS

PARTICIPANTS

Operative

Material

X1X2X3Xs

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(SFL) AND MATHEMATICAL LANGUAGE 77

metaphors, in particular, nominalizations. Such metaphorical participantspermit experiential meaning to be economically packaged within the nom-inal group structures which are aligned through relational clauses. Theimpact of this regrammaticization of experience is discussed in relation togrammatical metaphor in Section 3.6.

The relatively high incidence of relational processes and metaphoricalparticipants is found in mathematical language. For example, relationalprocesses realized through 'is' and grammatical metaphors (bold) appearin Stewart (1999: 132): '//The connection with the first interpretation is[ [that if we sketch the curve y =f(x), \ \ then the instantaneous rate of changeis the slope of the tangent to this curve at the point [[where x= a ]]]]'.The repacking of experiential content through relational processes andgrammatical metaphor is reconsidered in Chapter 6 where the notion ofgrammatical metaphor is linked to semiotic metaphor. The reasons for thecurrent forms of scientific and mathematical language include the impactof the functions which are fulfilled by mathematical symbolism and visualimages.

The basis of the discourse system of IDEATION for experiential meaningis lexis (Martin, 1992). The discourse units underlying the lexical itemsare lexical relations which are concerned with (i) taxonomic relations,(ii) nuclear relations and (iii) activity sequences. The discourse structuresrealizing lexical relations are called lexical strings which run through thetext. In the case of taxonomy, the two types of lexical relations are superor-dination involving subclassification and composition involving part/wholerelations. The types of taxonomic relations are summarized in Eggins(1994: 101-102). In mathematics, the taxonomies for mathematical termsare extended and precise; for example, triangles are defined according tothe size of the angles and sides. This serves to order mathematical reality inexact ways, leading to condensation in mathematical texts; for example, theterm 'isosceles triangle' incorporates a range of meanings. Mathematicaltaxonomies, however, are not explored here.

The second category of lexical relations involves nuclear relations.'Nuclear relations reflect the ways in which actions, people, place, thingsand qualities configure as activities in activity sequences' (Martin, 1992:309). These relations have previously been handled in SFL under colloca-tion. In the case of mathematics, nuclear relations stretch across linguistic,visual and the symbolic components of the mathematics text. Nuclearrelations are realized through configurations of Halliday's functionalcategories of process, participant and circumstance in the system of TRAN-SITIVITY, and the corresponding systems in mathematical symbolism andvisual display. The model of nuclearity adopted for language and themathematical symbolism follows Martin (1992: 319).

Centre Nucleus Margin PeripheryPROCESS = + MEDIUM + AGENT x CIRCUMSTANCERange: process + Range: entity + BENEFICIARY

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78 MATHEMATICAL DISCOURSE

The third category of lexical relations is expectancy and implication rela-tions between activities in activity sequences: 'These relations are based onthe way in which the nuclear configurations . . . are recurrently sequencedin a given field' (Martin, 1992: 321). The relations in mathematics arerealized through conjunctive relations, with implication relations typicallyinvolving conditional and consequential type relations. As suggested by theextract from Stewart (1999: 132), given the emphasis on logical meaningand the derivation of results, implication chains involving the semiotic con-struction of mathematical knowledge through language, symbolism andvisual images are extended and complex (O'Halloran, 1996, 2000).

Halliday (1978) explains that the need to conceptualize abstract relationsin mathematics using linguistic modes of expression causes grammaticalproblems. Apart from borrowing everyday linguistic terms, mathematicallanguage is technical and often involves complex taxonomies of terms innominalized forms. Halliday (1993b: 69-85) describes the difficulties inmathematical and scientific language which involve interlocking def-initions, technical taxonomies, special expressions, lexical density, syn-tactical ambiguity, grammatical metaphor and semantic discontinuity.However, these problems cannot be viewed in isolation. Rather the difficul-ties with mathematical language must be viewed in connection with symbolicand visual descriptions. Further to this, the texture of mathematical dis-course (linguistic, visual and symbolic) involves grammatical intricacy (likespoken discourse) and lexical density (like written discourse) which resultsin grammatical density (O'Halloran, 1996, 2004c). In other words, thelanguage of mathematics is best investigated in relation to functions andgrammar of mathematical symbolism and visual display to understand thefunctions of contemporary linguistic constructions in mathematics.

3.4 The Construction of Logical Meaning

Martin's (1992) discourse systems of CONJUNCTION and CONTINUITYare informed by Halliday's paradigm for clause complex relations in theform of INTERDEPENDENCY and LOGICO-SEMANTIC RELATIONS.Halliday's (1994: 221) description of clause complex relations is based onthe system of TAXIS which is common to word, group, phrase and clausecomplexes alike. Halliday (1994) distinguishes hypotaxis as a dependentmodifying relation and parataxis as an independent continuing relation. Asillustrated in Figure 3.4(1), clause complexes are classified as paratacticand hypotactic. In addition, cohesive or intersentence logical relations arebased on Halliday (1994: 220) and Martin (1992: 179).

Halliday's (1994: 219-220) system of LOGICO-SEMANTIC RELATIONSis also concerned with EXPANSION and PROJECTION. The categories ofEXPANSION describe the relations whereby a secondary clause expandsthe primary clause through Elaboration ('='), Extension ('+') andEnhancement ('x'). Secondary clauses realizing Elaboration ('that is' typerelations) function to restate, specify, comment on or exemplify the

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Figure 3.4(1) LOGICO-SEMANTIC RELATIONS and INTERDEPENDENCE

independent

dependent

paratactic

cohesive

.none

hypotactic

expansion

projection

implicit

explicit

internal

external

elaboration

extension

enhancement

locution

idea

elucidative

additative

comparative •

temporal

consequential

oppositionclarification

additativealternation

similaritycontrast

simultaneoussuccessive

purposeconditionconsequenceconcessionmanner

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80 MATHEMATICAL DISCOURSE

content of the primary clause. Secondary clauses realizing Extension('and') add new elements to the primary clause by giving exceptions oroffering alternatives. Secondary clauses realizing Enhancement ('so', 'yet','then') serve to qualify the primary clause with circumstantial features oftime, place, cause or condition. Projection describes the situation wherebythe secondary clause is projected through the primary clause as a Locutionor Idea. Locution is the realization of the secondary clause as wording (")while Idea realizes the secondary clause as an idea ('). As illustrated inFigure 3.4(1), the type of INTERDEPENDENCE (paratactic or hypotactic)is cross-referenced with the type of LOGICO-SEMANTIC RELATION(expansion or projection).

The discourse semantic systems of CONJUNCTION and CONTINUITYare modelled through covariate dependency structures called conjunctivereticula (Martin, 1992). The discourse system of CONTINUITY differs inthat items are realized in the Rheme as opposed to textual Theme. Follow-ing Martin (1992), these systems are organized by listing the clauses downthe page.4 Succeeding moves are shown to be dependent on precedingones by dependency arrows pointing upwards towards the presumed mes-sage. Typically conjunctive relations are anaphoric but in the case of theforward relations, an arrow is placed at both ends of the dependency line.Implicit conjunctions are shown where they could have been made explicitin the discourse.

The systems of CONJUNCTION and CONTINUITY may be used todescribe logical relations in mathematics (O'Halloran, 2000: 378) whichtypically involve discourse moves across linguistic, symbolic and visual partsof the text. The step-by-step development of logical reasoning is an import-ant function of symbolic mathematical discourse discussed in Chapter 4.The analysis of logical meaning in mathematics involves long and complexchains of reasoning which favour consequential-type relations (O'Halloran,1996, 1999b, 2000). Typically these chains of reasoning (at least in thesymbolic text) are primarily based on pre-established mathematical results.

The significance of logical meaning in mathematical linguistic text isevident in the analysis of the extract from Stewart (1999: 132) displayedin Figure 3.4(2). There are complex nested structures of logical rela-tions realized through structural conjunctions and conjunctive adjuncts,and there are also clause complex relations within rankshifted clauseconfigurations. The analysis also reveals that logical meaning is realizedmetaphorically in the form of processes. That is, logical meaning is realizedthrough the processes 'gives' and 'means' in the following clauses: 'Thisgives a second interpretation of the derivative', and 'This means [[thatwhen the derivative is large «(and therefore the curve is steep, as at thePoint P in Figure 4)» || the y-values change rapidly]]. Such processes areexamples of grammatical metaphor (see Section 3.6) where logical mean-ing is encoded through process type. However, as evidenced in the shortextract from Stewart (1999: 132), logical relations typically stretch acrosssymbolic, visual and linguistic components of the text.

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(SFL) AND MATHEMATICAL LANGUAGE 81

From Equation 3 we recognizethis limit as being the derivative of f at x 1that is, f'(x1)This gives a second interpretation of the derivative:(that is) The derivative f'(a) is the instantaneous rate of change of y = f(x) with respect to xwhen x = a.The connection with the first interpretation is [[that if we sketch the curve y = f(x), || then theinstantaneous rate of change is the slope of the tangent to this curve at the point [[where x= a]]]]This means [[that when the derivative is large «(and therefore the curve is steep, as at thePoint P in Figure 4)» ||the y-values change rapidly]].When the derivative is smallthe curve is relatively flatand the y-values change slowlyStewart[, 1999 #465: 132]

Figure 3.4(2) Logical Relations in Stewart (1999: 132)

3.5 The Textual Organization of Language

At the lexicogrammatical stratum, textual meaning is realized as GivenA

New through the system of THEME (Halliday, 1994) which is composed oftwo functional elements: the Theme and Rheme. Following Halliday (1994:38), ' [t]he Theme is the element which serves as the point of departure forthe message; it is that with which the clause is concerned. The remainder ofthe message, the part in which the Theme is developed, is called . . . theRheme'. The system network for THEME is given in Figure 3.5(1). TheTheme analysis, which is concerned with the organization of New infor-mation, permits the development of the text to be tracked at the rank ofclause and clause complex. In addition to Theme, Martin and Rose (2003:175-205) discuss thematic development in terms of phase and the wholetext. That is, hyperThemes function to organize information at the rankof paragraph, and macroThemes provide the focus for the text. Thus the

THEME

textual

interpersonal -

ideational

RHEME

Figure 3.5(1) The System of THEME

Conjunction - structural

Conjunctive Adjunct

Continative

Vocative

Modal Adjunct

Finite

topic

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82 MATHEMATICAL DISCOURSE

organization of the text is investigated as regular periodic waves of increas-ing amplitudes at the ranks of clause, clause complex, paragraph and text.

The THEME analysis for Stewart (1999: 132) (where Theme selectionsappear in bold) is given below:

//From Equation 3 we recognize////this limit as being derivative of/at x^////that is, /'(*!)////This gives a second interpretation of the derivative:////(that is) The derivative/' (a) is the instantaneous rate of change of

y—f(x) with respect to x////when x = a////The connection with the first interpretation is [ [that if we sketch the

curve y = f (x), 11 then the instantaneous rate of change is the slope ofthe tangent to this curve at the point [ [where x = a] ] ] ] //

//This means [ [that when the derivative is large « (and thereforethe curve is steep, as at the Point P in Figure 4)» 11 the y-values changerapidly]]//

//When the derivative is small////the curve is relatively flat////and the y-values change slowly//

The analysis demonstrates that the mathematical linguistic text is care-fully organized to carry forth the argument. Marked Themes are selected('From Equation 3', and 'When the derivative is small') to foregroundimportant experiential content. Information is not only packaged intonominal group structures through grammatical metaphor, but clausalrankshift also appears to be a significant method of organizing experientialmeaning. In addition, selections such as 'this' link the clause to previouslyestablished results. Martin (1992: 416) sees these types of selections as acase of textual grammatical metaphor (see Section 3.6). The linguistic textin Stewart (1999: 132) reflects grammatical intricacy as well as lexical dens-ity. More generally, these two types of complexity combine in mathematicaldiscourse to give grammatical density (O'Halloran, 2000, 2004c) asdiscussed in Chapter 7.

The discourse system of IDENTIFICATION is used to track participantswhere the basic opposition involves phoricity whereby information is recov-erable from the text or context. That is, a participant is either newly pre-sented ('addition'), or alternatively the identity of the presumed participanthas to be retrieved in some way from the text or context (Halliday, 1994:312-316). The means of retrieval are described by the types of phora (seethe system network in Martin, 1992: 126). This includes 'bridging refer-ence' where the referent has to be inferentially derived from the contextrather than by direct reference, and 'multiple reference' which results inambiguity. There is also 'generic' or 'specific' reference: 'Generic referenceis selected when the whole of some experiential class of participants is atstake rather than a specific manifestation of that class' (Martin, 1992: 103).

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(SFL) AND MATHEMATICAL LANGUAGE 83

These classifications are used to track the participants in mathematicaldiscourse in order to understand how reference functions in mathematics.

The linguistic text in Stewart (1999: 132) illustrates that tracking partici-pants in mathematics necessarily involves the linguistic, symbolic and visualcomponents of the text. In addition, reference chains in mathematics arecomplex as they split and cojoin as mathematical participants arerearranged for the solution to problems and the mathematical relationswhich are described are visualized (O'Halloran, 1996, 2000). The complex-ity of tracking participants may be seen in Stewart where 'the derivative' isvariously referred to as '/'(xj)' and '/'(a)' and participants are recon-figured in other ways, for example, 'at x^ and 'x= a\ Tracing participantreconfigurations across the three semiotic resources necessarily involvesknowledge of the grammars of language, mathematical symbolism andvisual display.

3.6 Grammatical Metaphor and Mathematical Language

Grammatical metaphor is an important concept for understanding thenature of scientific language (for example, Chen, 2001; Derewianka, 1995;Halliday, 1994; Halliday and Martin, 1993; Martin, 1992, 1997; Martin andVeel, 1998; O'Halloran, 2003b; Simon-Vandenbergen et al, 2003). This dis-cussion forms the basis in Chapter 6 for the extension of the concept ofgrammatical metaphor to semiotic metaphor. In this formulation, thenotion of grammatical metaphor is extended to take into account the typesof meaning expansions which take place intersemiotically in multisemiotictexts. The nature of the systems and lexicogrammatical strategies forencoding meaning in language, visual images and symbolism are theproduct of the interaction of the three resources.

Grammatical metaphor is a 'variation in the expression of a given mean-ing' which appears in a grammatical form although some lexical variationmay occur as well (Halliday, 1994: 342). The typical or unmarked form isreferred to as the congruent realization and the other forms which realizesome transference of meaning as the metaphorical form. The presence ofgrammatical metaphor necessitates more than one level of interpretation,the metaphorical (or the transferred meaning) and the congruent. Martin(1993a: 237) states: 'the fact that we have to read the clause on more thanone level is critical - the metaphor makes the clause mean what it does'. If,therefore, an expression can be unpacked grammatically to a congruentmeaning, it is a case of grammatical metaphor. Halliday's categorization ofthe types of grammatical metaphor (see Table 1.9 in Martin, 1997: 32) isgiven in Table 3.6(1).

The types of grammatical metaphor are organized metafunctionallyaccording to rank in Table 3.6(1). There exist logical, experiential andinterpersonal metaphors at ranks of clause complex, clause and wordgroup. Grammatical metaphor involves the shifts to 'entity', 'quality', 'pro-cess' and 'circumstance' from congruent realizations listed in Table 3.6(1).

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84 MATHEMATICAL DISCOURSE

Table 3.6(1) Halliday's Grammatical Metaphor (see Table 1.9 in Martin, 1997: 32)

RANK AND METAFUNCTION GRAMMATICAL METAPHOR

Clause complex:LOGICAL

relatorExamples:soifbecause

entity(nominal group)

cause/proofconditionreason

relatorExamples:thenso

Clause:LOGICAL (internal relations)EXPERIENTIAL and INTERPERSONALprocess

Examples:eventauxiliary- tense- phase- modality

process

Examples:event

transform

will/going totry tocan/could/may/will

poverty is increasing

quality(nominal group)

subsequent/followresulting

relator

Examples:thensoand

relator

Examples:wheniftherefore

process(clause)

followcausecomplement

circumstance(clause)

in times of/in . . . timesunder conditions of/under . . .conditionsdue to

entity(nominal group)transformation

prospectattemptpossibility, potential, tendency

quality(nominal group)increasing poverty

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(SFL) AND MATHEMATICAL LANGUAGE 85

Table 3.6(1) - contprocessauxiliary- tense- phase- modality

processExamples:divide

circumstanceExamples:withto

circumstance

Examples:mannerother

other

circumstance(clause)Examples:be aboutbe instead of

was/used tobegin tomust/will [always]may

[decided] hastily[argued] for a longtimecracked on thesurface

quality

previousinitialconstantpossible/permissable

circumstance

'h' [on V]

entity(nominal group)

accompanimentdestination

quality(nominal group)

hasty decisionlengthy [argument]

surface [cracks]

process(clause)

concernreplace

Word group:LOGICAL (internal relations),EXPERIENTIAL and INTERPERSONAL

quality

Examples:unstable

entityExamples:the government [decided]

the government couldn't decide

entity(nominal group)

instability

modifier [expansion](nominal group)

the government [decision][a/the decision] of/by thegovernmentthe government's [indecision][the indecision of thegovernment]governmental [indecision]

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86 MATHEMATICAL DISCOURSE

The new case where a process is realized by circumstance (as illustrated by'h on r' to mean 'h divided by r') which has appeared in mathematicalclassroom discourse (O'Halloran, 1996) is added to Halliday's categories inTable 3.6(1).

The majority of cases of grammatical metaphor involve the process ofnominalization whereby a grammatical class or structure realizing a pro-cess, circumstance, quality or conjunction is turned into another gram-matical class, that of a nominal group realizing a participant. FollowingHalliday (1993a, 1993b, 1998) nominalization is conceived as 'the pre-dominant semantic drift of grammatical metaphor in modern English'(Martin, 1992: 406), which has largely resulted from changes in the Englishlanguage to realize a scientific view of the world. That is, 'a new variety ofEnglish' was created 'for a new kind of knowledge' (Halliday, 1993b: 81),one in which the main concern was to establish causal relations. As Hallidayexplains, the most effective way to construct logical arguments is to estab-lish steps within a single clause, with the two parts 'what was established'and 'what follows from it' reified as two 'things' or participants realizedthrough nominal group structures. These two participants are then con-nected with a process in a single clause. The strategy of recursive modifica-tion of the nominal group is also employed in scientific discourse. Thesetwo devices are typical of contemporary written discourse, and as Hallidayand Martin (1993: 39) point out, nominalizations may serve importantideological functions because they are less negotiable than the congruentform: 'you can argue with a clause but you can't argue with a nominalgroup'. Cases of grammatical metaphor may be mapped through thesystem network as displayed in Figure 3.6(1).

In addition to experientially based types of grammatical metaphor, inter-personal metaphors occur in conjunction with the systems of MODALIZA-TION and MODULATION (see Martin et al, 1997: 70). Following Halliday(1994: 354—363) metaphors of modalization and modulation are realizedthrough the use of modal auxiliaries (modal Finites) with high, medianand low values of probability and usuality, and obligation, inclination andpotentiality respectively. MODALIZATION and MODULATION vary inorientation with respect to two criteria: first, objectivity and subjectivity; and

Table 3.6(1) - contNon-entity:LOGICAL (internal relations),EXPERIENTIAL and INTERPERSONAL

entity(nominal group)the phenomenon o f . . .

process(clause). . . occurs/ensues

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(SFL) AND MATHEMATICAL LANGUAGE 87

Figure 3.6(1) System Network for Grammatical Metaphor

second, implicitness and explicitness. A subjective explicit orientation isrealized through a projecting clause; for example, 'I know this is correct'.An explicit objective orientation is realized through encoding of the object-ivity; for example, 'it is certain this is correct'. Interpersonal metaphorsare also realized through incongruence between MOOD and SPEECHFUNCTION selections (Halliday, 1994: 363-367). As explained in Section3.2, the unmarked MOOD realizations of the SPEECH FUNCTIONS arestatement realized by declarative (SubjectAFinite), question by inter-rogative (FiniteASubject and WH), command by imperative (Predicator),and Offer by modalized interrogative (modalized FiniteASubject).Mathematical discourse includes Modal and Mood metaphors (see Section3.2).

Martin (1992: 416) introduces textual grammatical metaphor which isorientated towards organizing the text as ' "material" social reality'. Martingives four types of textual metaphor which contribute to this organizationof text: (i) 'meta-message relations' as found in Francis' (1985) anaphoricnouns (for example, 'reason', 'example', 'point' and 'factor'); (ii) 'textreference' which identifies facts rather than participants (for example,'this'); (iii) 'negotiating texture' which can, for example, exploit mono-logic text as dialogic (for example, 'let me begin by'); and (iv) internalconjunction which orchestrates text organization as opposed to field organ-ization (for example, 'as a final point'). As Martin (1992: 416) points out,rather than being orientated towards logical meaning, these types of text-ual metaphors may be orientated towards the interpersonal. For example,That point is just silly' (Martin, 1992: 417) is a textual metaphor of thetype 'meta-message relation' which is orientated to interpersonal meaningin the form of APPRAISAL. Derewianka (1995: 238) explains that the

experiential

logical

interpersonal

textual

circumstance

process

quality

entity

nodifier

process

entity

relator

circumstance

process

quality

entity

-zero

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88 MATHEMATICAL DISCOURSE

functioning of a nominal element to 'summarise or "distil" a figure orsequence of figures' does not necessarily mean that 'any instance of thistype is inherently metaphorical'. According to Derewianka (1995: 238),what needs to be taken into account is a change in the level of generaliza-tion and abstraction. Nonetheless, Martin (1992: 395) sees grammaticalmetaphor as an important strategy for creating texture: 'The resources forweaving chains and strings through different grammatical functions . . . areimportant ones: but they provide only a very partial picture of the way inwhich meanings are packaged for grammatical realisation. The real gate-keeper is grammatical metaphor.'

As evident in the discussion of experiential meaning in Stewart (1999:132) in Section 3.3, mathematical discourse involves grammatical meta-phor. The analysis of multisemiosis in mathematical and scientific textsenhances our understanding of the role and function of grammaticalmetaphor. Once the notion of semiotic metaphor is introduced in the formof metaphorical realizations which take place with intersemiotic shiftsacross semiotic resources, the semantic drift in language where grammat-ical metaphor developed intrasemiotically in language as a means ofre-packaging information becomes understandable in the context of thefunctions and roles which are fulfilled symbolically and visually. Thisimportant point necessitates further discussion of the nature and functionsof grammatical metaphor in Chapter 6.

As well as grammatical and semiotic metaphors, lexical metaphors(Halliday, 1994: 340-342) may also be examined in mathematics, althoughthis is not undertaken in this study. Lexical metaphors are metaphors in themore classical sense of the term where 'a particular lexeme is said to have a"literal" and a "transferred" meaning'(Derewianka, 1995: 109). In terms ofdistinguishing grammatical and lexical metaphors, both have 'a semanticcategory which can be realized congruently or metaphorically' but withgrammatical metaphor, 'what is varied is not the lexis but the grammar'(ibid.). Although this field of study is worthy of investigation, the majorconcern here is shifts in meaning which arise grammatically in mathematicsthrough the interactions between the semiotic resources of language, visualimages and mathematical symbolism.

3.7 Language, Context and Ideology

SFL views language as a social-semiotic, a system of meanings that construethe reality of a culture. This construction is described metafunctionally: theideational metafunction construes 'natural reality'; the interpersonalmetafunction construes 'intersubjective reality'; and the textual metafunc-tion construes 'semiotic reality' (Halliday, 1978; Matthiessen, 1991). Thisintrinsic functional organization of language is modelled as interactingwith the organization of social context in what Halliday (1985) terms aslanguage's extrinsic functionality. In other words, language is viewed asconstruing the social context with the net result being the reality of a

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(SFL) AND MATHEMATICAL LANGUAGE 89

culture. Conversely, the social context impinges upon language use. Martin(1992) describes the relationship between language and social context asone of mutual engendering where instances of language use, collectivelycalled texts, are social processes which are analysed as manifesting the cul-ture they in part largely construct. These SFL formulations are extendedto other semiotic resources (for example, Baldry, 2000b; Halliday, 1978;Kress and van Leeuwen, 1996; O'Halloran, 2004a; O'Toole, 1994; Ventolael al., forthcoming) which, in the context of this study, are language,mathematical symbolism and specialized forms of visual display.

The analysis of text becomes 'the analysis of semantic choice in context'(Martin, 1992: 404) where context is conceived as consisting of the contextof the situation and the context of culture. Context is viewed as a semioticsystem manifested in whole or part through language and other semioticresources. The levels of semiosis articulated by this process of realizationare referred to as communication planes. The difference between lan-guage, mathematical symbolism and visual display on the one hand, andcontext on the other, is that the former have their own means of organizingexpression (through typography/graphology and so forth) while contextdepends on other semiotic planes for realization (Ventola, 1987).

The context of a text consists of two communication planes: register atthe level of context of situation, and genre at the level of the context ofculture. Register is constituted by contextual variables of field, tenor andmode which work together to achieve a text's goal. Field is concerned withexperiential meaning (what is actually taking place), tenor with inter-personal meaning (the nature of the social relations) and mode with text-ual meanings (the role language is playing) (Halliday, 1978, 1985). Thethree register variables of field, tenor and mode can be viewed as workingtogether to achieve a text's goals, 'where goals are defined in terms of thesystems of social processes at the level of genre' (Martin, 1992: 502-503).Genre networks are formulated on the basis of similarities and differencesbetween text structures which define text types. A culture consists of par-ticular ways of meaning, which are described through genre, register andthe integration of different forms of semiosis. In this study, the focus isdirected towards the language and expression plane, rather than registerand genre. However, a brief discussion of the register of mathematicallanguage in terms of field, tenor and mode is included below.

The written mode of mathematics means that semiosis in the form oflanguage, visual images and the symbolism is constitutive of the mathemat-ics which is developed, rather than contextual meaning arising from theimmediate material setting. While mathematics does involve genres otherthan the written (such as the academic lecture, the conference paper andso forth), in essence modern mathematics developed as a written discourse.The textual organization of linguistic, symbolic and visual componentsand the compositional arrangement of those selections in mathematicsare sophisticated. Mathematical discourse is concerned with particularrealms of experiential content according to the field of mathematics (for

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90 MATHEMATICAL D I S C O U R S E

example, elementary mathematics, calculus, pure mathematics and appliedmathematics) and the genre. At this stage, the nature of the tenor relationswhich are established in mathematics is worthy of closer inspection as theserelations orientate the reader towards the mathematics which is presented:

'Tenor refers to the negotiation of social relationships among partici-pants' (Martin, 1992: 523). Tenor is the projection of interpersonal mean-ing realized through discourse semantics and lexicogrammatical systems inthe language stratum. Tenor is mediated along the three dimensions ofpower (which Martin refers to as 'status'), contact and affect (Martin, 1992;Poynton, 1984, 1985, 1990) as displayed in Figure 3.7(1). Status refers to'the relative position of the interlocutors in a culture's social hierarchy',contact is 'their degree of institutional involvement with each other' andaffect includes 'what Halliday (1978) refers to as the "degree of emotionalcharge" in the relationship between participants' (Martin, 1992: 525).

The principle of reciprocity of choice is significant in terms of the realiz-ation of status in spoken discourse. Patterns of dominance and deferencein which the status of the writer/speaker is reflected take place through thekinds of linguistic choices which are made. Equal status is realized throughselections of the same kinds of options for both interlocutors while unequalstatus is realized through non-reciprocal choices. As Martin (1992: 528)explains, there is 'a symbolic relationship between position in the socialhierarchy and various linguistic systems, especially interpersonal ones'. Thecontact, or degree of involvement, is equivalent to what Hasan (1985)describes as 'social distance', the frequency and range of interaction. Theprinciple of proliferation is used in which a high degree of contact meansa wider range of options are available, while a low degree of contactmeans a smaller range of options. The basic realization principle of affectis amplification in which speakers can vary the 'volume' from normalwriting/listening levels.

Martin (1992: 529-535) lists features of interaction patterns, discoursesemantics, lexis, grammar, and phonology which realize patterns of domin-ance and deference, involved and uninvolved contact, and dimensions ofaffect. The lexical and grammatical realizations of these tenor dimensions

TENOR

Figure 3.7(1) TENOR Dimensions adapted from Martin (1992: 526)

positive

negativemarked

amplification

AFFECT

STATUS

reciprocity

CONTACT

equal

unequal

involved

distantproliferation

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(SFL) AND MATHEMATICAL LANGUAGE 91

are represented in a three-dimensional space in Figure 3.7(2). While manyof these systems for spoken language do not operate in mathematics writ-ten texts (for example, swearing, slang and so forth), the three dimensionalspace is nonetheless useful as a means of framing the range of optionswhich are available in mathematics. Such a framework is also useful for theanalysis of pedagogical discourse in mathematics. The nature of the linguis-tic choices in written mathematics means that the discourse operates from anuncontested position of dominance. The linguistic choices are not recipro-cal, there is minimal affect, and the contact is involved but distant. Thenature of interpersonal relations is further discussed in relation to the sym-bolic and visual components of mathematics text in the following chapters.

Martin (1992: 507) explains that ideology may be seen as 'the system ofcoding orientations constituting a culture'. Incorporating Foucault's(1970, 1972, 1980a, 1984, 1991) formulations of knowledge, power anddiscourse, SFL analysis is concerned with how texts relate to each other, andhow one text relates to all texts that may have been. As texts are interpretedin a multidimensional intertextual semiotic space, this allows the selectionswhich have been made to be effectively placed alongside all other possi-bilities, thus revealing the ideological positioning of the choices that havebeen made. Ideology has genre, and hence register and language as itsexpression plane. The ideological orientation of mathematics is discussedin relation to the concepts of abstraction, contextual independence,reason, objectivity and truth in Chapter 7.

The adoption of a multisemiotic perspective of discourse facilitates a holis-tic understanding of text, context and culture. The inclusion of other formsof semiosis and the study of intrasemiotic and intersemiotic processesenhances the theoretical possibilities afforded by SFL. For example, the SFLframework presented in Table 3.1(1) is extended to incorporate other semi-otic resources in the 'Integrative Multisemiotic Model' (IMM) in Lim (2002,2004). Such a multisemiotic systemic functional model incorporates (i) thegrammatical systems for other semiotic resources, (ii) intrasemiosis withinthe semiotic resources, (iii) intersemiotic mechanisms for meaning acrosssemiotic resources, (iv) systems which operate on the Expression stratum (forexample, Colour and Font Style and Size), and (v) the materiality andmedium of the text (for example, print versus electronic medium).

A multisemiotic approach reveals differences between the functions andsystems of semiotic resources across different strata. For example, Lim's(2002: 37) division of metafunctionally based systems shows a separation ofmetafunctional boundaries with respect to the systems which operate at thegrammatical stratum for language. However, the 'system-metafunctionfidelity' (Lim, 2002, 2004), or the measure of dedication of a system toone particular metafunction, breaks down on the expression plane. Thesesystems (for example, systems such as Font, Colour and so forth) do nothave the clear metafunctional orientations which are found in grammaticalsystems. Choices from the system of Colour, for example, can functioninterpersonally to attract attention, textually for cohesive purposes, and

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Figure 3.7(2) Lexicogrammatical Aspects of the Realization of TENOR adapted from Martin (1992:529-535)

GRAMMARResidue ellipsispolarity matchedattitude concurcomment invitedvocationrespectfulperson 2ndtagging checkingagency: I/mediummodalization low

" modulation:inclination

LEXISeuphemizetempered swearingcovert

GRAMMARminor clausesMood ellipsisMood contractionvocation' range of names

nick-name

LEXISattitudinaltabooswearing

POWER- Defer

CONTACTIntimate/involved

LEXISspecializedtechnicalslanggeneral words

GRAMMARexclamativeattitudecommentminor expressiveintensificationrepetitionprosodic nm gpdiminutives;mental affectionmanner degree

AFFECTPositive

AFFECTNegative

GRAMMARexclamativeattitudecommentminor expressiveintensificationrepetitionprosodic nm gpdiminutives;mental affectionmanner degree

LEXISattitudinaltabooswearing

POWERDominate

CONTACTDistant/uninvolved

GRAMMARmajor clausesno ellipsisno contractionno vocationsingle namefull name

LEXIScorenon-technicalstandardspecific words

GRAMMARno ellipsispolarity assertedattitude manifested

~ comment presentedvocation familiarperson 1sttagging invitedagency: I/agentmodalization highmodulation:obligation

LEXISexplicit bodily functionsswearingovert

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(SFL) AND MATHEMATICAL LANGUAGE 93

experientially for representational meaning. In a similar fashion, semioticresources have different grammatical systems, and such differences in themeaning potential have implications for the functions which are fulfilled bythat resource, as seen in the discussion of the grammar of mathematicalsymbolism and visual images.

The formulation of SF frameworks for mathematical symbolism andvisual display and the analysis of intersemiotic processes reveal that the keyto the success of mathematical discourse is the ability to create a semanticcircuit across the linguistic, symbolic and visual components of the math-ematics text through the specialized grammars of each resource. Thesesemantic circuits give rise to metaphorical expressions in the form of semi-otic metaphors. The analysis of the mathematical texts in Chapter 7 leadsto a further discussion of the ideology and orientation of mathematics.Martin's (1992: 507) focus is situated within the dynamic view of ideologywhich is 'concerned with the redistribution of power - with semioticevolution'. These concerns provide the impetus for this study.

Notes

1 Martin's (1992: 14-21) arguments for stratification of the content planeinclude the following limitations of the lexicogrammar: semantic motifscannot be generalized because of diverse structural realizations; themultiple levels of semantic layers resulting from grammatical metaphorcannot be fully accounted for; generalizations across structural and non-structural textual relations such as those found in cohesion are notpossible; and the semantic stratum is more abstract and the systems arecomposed of larger units which differ in structure from those foundelsewhere. Martin (1992) and Martin and Rose (2003) are concernedwith overcoming these limitations and capturing semantic interdepend-encies in the whole text which are otherwise only partially accounted forby the lexicogrammar. The type of structures are open ended in so far asthe issue is not one of constituency, but rather interdependency.

2 Although word groups and phrases occupy the intermediate position onthe rank scale, Halliday (1994: 180) distinguishes between the two: 'APHRASE is different from a group in that, whereas a group is an expan-sion of a word, a phrase is a contraction of clause.' Halliday's (ibid.: 242)classification of rankshifting thus covers clausal and phrasal elements.

3 In SF system networks, the curly brackets mean 'select from each of thesystems' (that is, 'select from this and this'), while the square bracketsmean 'select only one of the options' (that is, 'select this or that').

4 Typically in conjunctive reticula, external relations are modelled on theright-hand side, and the internal relations are modelled down the leftwith external additive relations positioned in the centre. This allows theconjunctive relations to be separated into those which function in arhetorical (internal) sense compared to those which function in a moreexperiential (external) sense.

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4 The Grammar of Mathematical Symbolism

4.1 Mathematical Symbolism

The historical perspective covered in Chapter 2 reveals how mathematicalsymbolism developed as a tool for reasoning through the discovery thatcurves could be described algebraically and the increasingly important aimof rewriting the physical world in mathematicized form. Mathematicaldescriptions eventually replaced metaphysical, theological and mechanicalexplanations of the universe (see for example, Barry, 1996; Kline, 1972,1980; Wilder, 1981). Today, many fields of human endeavour are written inmathematicized or pseudo-scientific form. The scientific view of the worldis not confined to the physical universe; rather it underlies our day-to-dayconception of reality.

Mathematical discourse succeeds through the interwoven grammarsof language, mathematical symbolism and visual images, which meansthat shifts may be made seamlessly across these three resources. However,each semiotic resource has a particular contribution or function withinmathematical discourse. Language is often used to introduce, contextual-ize and describe the mathematics problem. The next step is typically thevisualization of the problem in graphical or diagrammatic form. Finallythe problem is solved using mathematical symbolism through a variety ofapproaches which include the recognition of patterns, the use of analogy,an examination of different cases, working backwards from a solution toarrive at the original data, establishing sub-goals for complex problems,indirect reasoning in the form of proof by contradiction, mathematicalinduction (if Sk is true, and Sk+1 is true whenever Sk is true, then Sn is true forall n) and mathematical deduction using previously established results(Stewart, 1999: 59-60). The generalized solutions and mathematicalmodels are used for predictive purposes. Before discussing intersemioticprocesses which take place across language, the symbolism and visualimages in Chapters 6-7, intrasemiosis within mathematical symbolism andvisual display is explored in Chapters 4—5 respectively. The unique func-tions of each resource are discussed through SF frameworks and an investi-gation of choices from the systems which are found in symbolic and visual

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 95

parts of the mathematical texts. In this way, the grammatical strategies forencoding meaning in each resource may be understood before proceedingto the complex problem of understanding how meaning is madeintersemiotically across the three resources.

The general nature of meanings afforded by language, mathematicalsymbolism and visual images is described by Lemke (1998b, 2003). That is,language is seen to be orientated towards making categorical-type distinc-tions (for example, Bateson, 1972; de Saussure, 1966; Messaris, 1994); thatis, typological-type meanings. Mathematical symbolism, on the other hand, isseen to make meanings by degree in the form of continuous descriptions ofpatterns of co-variation; that is, topological-type meanings. For example, onecan observe that the tiger population in an ecosystem is decreasing, andone may even comment: 'there are not many tigers around these days'. Thelinguistic statement makes a categorical type assessment of the situationregarding tigers: 'there are not' (Existential process with negative polarity)'many tigers' (Existent) 'around' (circumstance-Location of place) 'thesedays' (circumstance-Location of time). However, through 'predator-prey'type mathematical models, the relationship between the number of tigersand the number of men, for example, can be specified in order to study thepatterns of the interaction between the two species, and to predict the tigerpopulation at any one time, including when they may be expected tobecome extinct. The mathematical model expresses the relationshipbetween the number of men and the number of tigers as a continuousfunction over time. For example, if M represents the predator 'man', Trepresents the prey 'tiger', and t represents time, then such a model would

take the form (Stewart, 1999: 662). In

addition to describing patterns of variation over time, the symbolism hasthe potential to express the exact relations of parts to a whole. For example,a triangle with base b and height h is related to the area of the triangle A (A)

through the symbolic statement

As the nature of patterns of variation is not easily discernible from thesymbolic statements, the graphs and diagrams are used to give more intui-tive understanding of the relationships which are encoded symbolically(Lemke, 1998b). For example, the predator-prey mathematical model forthe relationship between the population of tigers and the number of mencan be displayed graphically to give a sense of the type of relationshipencoded in the mathematical model. Visual images supersede language interms of the ability to represent continuous spatial relations. However,mathematics visual patterns are often only partial descriptions over alimited domain, and they are limited in terms of their ability to be used forcalculations. This shortfall is becoming less marked with the developmentof the power of computers to display and manipulate visual patterns, atheme which is explored in Chapter 5. The computer is revolutionizing the

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96 MATHEMATICAL DISCOURSE

types of mathematics being developed due to the increased facility for per-forming numerical calculations and displaying the resulting visual patternsthrough computer graphics.

The functions and grammar of mathematical symbolism are examinedthrough the development of an SF framework. This framework is used toexplore the nature of interpersonal, experiential, logical and textual mean-ings afforded by symbolism, and the strategies through which these mean-ings are encoded. A similar exploration of intrasemiosis in visual imagestakes place in Chapter 5. From this point, it is possible to examine howlanguage, the symbolism and the visual images combine intersemioticallyto create meaning in mathematical discourse. The examination of thegrammars of mathematical symbolism and visual display on a separate basisis a somewhat artificial approach as historically these semiotic resourcesdeveloped together in mathematical discourse. The key to the success ofmathematics is that the three grammars function integratively. However,if the process of semiosis is 'frozen' in stages where meaning is madeprimarily within one resource rather than across the three resources, thecontribution of that one resource may be appreciated. This appears to bea necessary preliminary first step to understand how the three semioticresources function together. The functions and grammar of the math-ematical symbolism and visual images are therefore first investigatedindividually.

As will become evident, the grammatical strategies for encoding meaningin mathematical symbolism differ from those found in scientific language.This is not surprising as the symbolism was designed to fulfil different func-tions, and its grammar evolved accordingly. The nature of scientific lan-guage, with its propensity to pack experiential meaning into extendednominal group structures in the form of grammatical metaphors which areconfigured with relational processes (for example, Halliday and Martin,1993; Martin and Veel, 1998), is the resultant product of the impact of theuse of the symbolism and the visual display in mathematical and scientificdiscourse. From the discussion of intrasemiosis in mathematical symbolismand visual display, intersemiotic processes and their impact on scientificlanguage are investigated in Chapter 6.

4.2 Language-Based Approach to Mathematical Symbolism

The language-based approach to the SF framework for mathematical sym-bolism adopted in this study is justified by the fact that the symbolismdeveloped as a semiotic resource which evolved from language. The stagesof the development of algebra, for example, have been characterized asrhetorical where instructions were in the form of linguistic commands,syncopated where recurring linguistic elements for participants and pro-cesses were symbolized, and symbolic where mathematical symbolismdeveloped as a semiotic resource (see Chapter 2 and Joseph, 1991). Thesymbolism developed a functionality through new grammatical systems

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 97

which permitted semantic expansions beyond that capable with language,but at the same time it depended upon employing certain linguistic elem-ents and a range of grammatical strategies inherited from language. Fur-thermore, symbolic statements are typically embedded within linguistictext. Thus, despite the new functionality of mathematical symbolism, itnonetheless requires a surrounding linguistic co-text to contextualize thesymbolic descriptions and procedures that take place. The dependence onthe linguistic semiotic suggests that the symbolism did not develop a well-rounded functionality, which becomes evident in the discussion of thetypes of meaning which are possible using mathematical symbolism. Thelanguage-based approach permits the semantics of the mathematical sym-bolism to be understood and contextualized in relation to the types ofmeaning afforded by the linguistic semiotic.

The unique relations between language and mathematical symbolismexplain the nature of the mappings that may be made between the twosemiotic resources. For example, there exist acceptable wordings in naturallanguage for mathematical symbolic statements, although this is not anexact one-to-one correspondence. Mathematical statements are recover-able from linguistic statements, although in some cases this is problematicbecause the linguistic construals are metaphorical (see the discussion ofsemiotic metaphor). The unique relations between language and math-ematical symbolism serve to highlight an important difference betweenthese two resources and other semiotic resources such as art, sculpture andarchitecture where such accurate mappings do not exist. For instance,unlike a mathematical symbolic statement, a painting or a sculpture is notrecoverable from any combination of words.

After introducing the SF framework for mathematical symbolism, thetypes of semantic shift in the evolution of the symbolism are discussed interms of the expansion and contraction of experiential meaning, the nar-rowing of interpersonal meaning, the development of selected types oflogical meaning, and the refinement of textual meaning. These types ofsemantic shift mean that mathematical symbolism developed as a semioticresource with a grammar through which meaning is unambiguously encodedin ways which involve maximal economy and condensation. The economicalmeans of encoding meaning in the symbolism permit the easy rearrange-ment and manipulation of relations so that mathematical models can beconstructed and problems can be solved. This perspective is developed inthe following discussion of the grammar of mathematical symbolism. Asummary of the major points concerning the grammar of mathematicalsymbolism appears in Section 4.8

4.3 SF Framework for Mathematical Symbolism

The SF model for mathematical symbolism displayed in Table 4.3(1) isbased upon Halliday (1994) and Martin's (1992; Martin and Rose, 2003)systemic model for language. The communicative planes of ideology, genre

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and register are applicable to the multisemiotic mathematical texts con-sidered in Chapters 6-7. In the language plane, the content stratum formathematical symbolism consists of discourse semantics and grammarstrata with the ranks of statement (clause complex), clause, expression andcomponent. The model parallels the discourse stratum and lexicogram-matical ranks of clause complex, clause, word group/phrase and word forlanguage. The 'display plane' for mathematical symbolism corresponds tothe 'expression plane' for language in the model. The term 'display plane'is used rather than 'expression plane' because a new grammatical rankof 'expression' is introduced for mathematical symbolism in Table 4.3(1).The need for the inclusion of the rank of expression in the grammar ofmathematical symbolism will become apparent in Section 4.4.

The SF framework for a grammar for mathematical symbolism is pre-sented in Table 4.3(2). This framework provides a description of the majorsystems through which mathematical symbolism is organized as a semioticresource for experiential, logical, interpersonal and textual meaning forthe content and display planes. The discourse systems for mathematicalsymbolism parallel those found in language. However, as discourse movesoften span linguistic, symbolic and visual components of the text, Martin'sdiscourse systems are extended in Chapter 6 in the attempt to theorizeintersemiosis between the three resources. In the model presented in Table4.3(2), systems which operate at the level of the display plane are alsoincluded. It is recognized that options in the expression of the semioticchoices in the mathematics text (for example, Colour, Font Size and Style)function to create meaning (for example, Kress and van Leeuwen, 2002;Lim, 2004; O'Halloran, 2004a). Traditionally, the expression stratum in lan-guage has been under-theorized in SFL where the major concerns havebeen the language plane and the communicative planes of register, genreand ideology.

98 MATHEMATICAL DISCOURSE

Table 4.3(1) SF Model for Mathematical Symbolism

MATHEMATICAL SYMBOLISM

CONTENT Discourse SemanticsInter-statemental relations

GrammarStatements (or clause complex)Clause (// //)Expressions ([[ ]])(rankshifted participants of the clause which are the resultof mathematical operations)Components(the functional elements in expressions)

DISPLAY Graphology and Typography

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 99

Table 4.3(2) Grammar and Discourse Systems for Mathematical Symbolism

DISCOURSE SEMANTICS

EXPERIENTIAL LOGICAL INTERPERSONAL TEXTUAL

IDEATION• Activity Sequences

consisting ofOperative process andparticipantreconfigurations(progressive steps ofsimplification andsolution)

• Nuclear relations(participant andprocess)

• Collocation (symbolicrelations and stringsthrough taxonomies,definitions, axiomsand theorems)

CONJUNCTION andCONTINUITY (basedon EXPANSION)• Sequential

placement ofstatements (explicitlymarked when thelogical connection isnon-sequential)

• Extension of TAXISinto long implicationsequences

Structure: conjunctivereticula

NEGOTIATIONExchange Structureand SPEECHFUNCTION at themove rank• Consists of moves

and move-complexes

Structure: ExchangeStructure linkingmoves

IDENTIFICATION• Direct Repetition• Referential cohesion

(based on definition,operationalproperties withexplicit repetition ofreference)

• Positional notation(the sequentialdownward placementof statements andpositional placementfunctionalcomponents)

Structure: strings fortracing activitysequencereconfigurations

Structure: referencechains linkingparticipants

INTER-STATEMENTAL RELATIONS

EXPERIENTIAL LOGICAL INTERPERSONAL TEXTUAL

• Positional notation toindicatecontinuations ofActivity Sequences

• Repetition ofprocesses andparticipants in newconfigurations

EXPANSION• Conjunctions and

cohesiveconjunctions

• Implicit and explicitconjunctions(external symbolicand linguisticconjunctive devices;internal substitutionand operativeproperties)

• Apposition• Parenthesis• Labelling

SPEECH FUNCTION(statements andlimited forms ofcommand)• Intricacy of symbolic

representation• Abstractness (nature

of participants,processes)

• Discursive links(using verbal code ofmain text within themathematical array)

• Labelling

• Positional notation(the sequentialdownward placementof statements andpositional placementfunctionalcomponents)

• Dependent clauses(thematic or spatiallymarked)

• Ellipsis (marked byspatial position)

• Discursive links(using verbal code ofmain text within themathematical array)

• Labelling

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100 MATHEMATICAL DISCOURSE

Table 4.3(2) - cont

STATEMENTS // //

EXPERIENTIAL

TRANSITIVITY• Processes (Operative,

relational andexistential)

• Participants arerankshiftedconfigurations ofOperative processes

• Circumstantialfeatures (minorclauses, dependentclauses or fusedwithin participantstructure)

• Ellipsis of Operativeprocesses

• Rule of Order ofoperations

LOGICAL INTERPERSONAL

• Rhetorical 'temporal' MOOD (with oneconjunctive relations symbol for the Finiterealized through and Predicator)Rule of Order of • MODALITYoperations and use of (consistently high,brackets implicit objective

orientation)POLARITY(presence orabsence of a slashthrough the processsymbol)

• Intricacy (embeddedprocesses)

• Abstractness(participants andprocesses)

TEXTUAL

• THEME (unmarkedchoice is Subject ofthe clause withmarked caseindicates steps insimplification)

• Multiple Theme(textual elementspatially placed)

• Ellipsis (spatialpositioning)

• Dependent clauses(thematic orotherwise spatiallyseparated)

• Conventional spatialorganization

• Rule of Order anduse of brackets forunfolding ofOperative processes

EXPRESSIONS [[ ]]

EXPERIENTIAL LOGICAL INTERPERSONAL TEXTUAL

• Operative processes(condensation occursvia high level ofrankshift within andbetween participants)

• Degree of rankshiftindicated by [ [ ] ]

• Circumstantialelements (throughprocesses and fusedparticipant structures

• Ellipsis of Operativeprocesses

• Rule of Order ofoperations

• Rhetorical 'temporal'conjunctive relationsrealized throughconventionalizedRule of Order ofoperations and use ofbrackets

• Intricacy (degree andexplicitness ofembedding)

• Degree ofabstractness (natureof participants andprocesses)

• Rule of Order anduse of brackets forunfolding ofOperative processes

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 101

Table 4.3(2) - cont

COMPONENTS

EXPERIENTIAL LOGICAL INTERPERSONAL TEXTUAL

DISPLAY PLANE

EXPERIENTIAL LOGICAL INTERPERSONAL TEXTUAL

• Variations in the formof case, font, scriptsand size for specialsymbols,abbreviations, icons,punctuation,brackets, andcombinations ofsymbols

• Use of spatial andpositional notation

• Spatial organizationof symbolic text

• Style of production(hand written,computer generated)

• Contrasts in font,script and size

• Spatial arrangementof text at each rank

• Font style and format• Ellipsis of process

The SF framework in Table 4.3(2) provides insights into the ways inwhich the grammar of mathematical symbolism is organized to fulfil thefunctions of mathematics, and the ways in which the systems and lexico-grammatical strategies in the symbolism depart from those found in lan-guage. Further research is needed in the analysis of mathematical texts,however, in order to fully document the systems, which remain at a pre-liminary stage of theorization. The framework presented in Table 4.3(2) isbest viewed as a first step towards a comprehensive SFG for mathematicalsymbolism.

The metafunctional systems in the SF framework for mathematicalsymbolism are discussed with reference to the mathematical symbolic textdisplayed in Plate 4.3(1). This mathematics problem is concerned with

• Restricted range ofunits in the nominalgroup (the absence ofDEIXIS, attitudinaland experientialepithets)

• Qualifiers (form partof the nominal groupwithout the need forembedding asphrases)

• Classifiers• Conventionalized use

of specific symbols(numerals, Roman,Greek, Hebrewalphabet)

• Conventionalizedcombinatorypractices

• Degree ofabstractness

• Degree ofmodification

• Function ofconstituents (spatial,serial position andbrackets)

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Plate 4.3(1) Mathematical Symbolic Text (Stewart, 1999: 139)

Here we r a t i o n a l i / e the numerator

EXAMPLE 4 If f ( x ) = J x - 1 , f i n d t h e d e r i v a t i v e o f / . S t a t e t h e d o m a i n o f / ' .

SOLUTION

We see that f'(x) exists if x > 1, so the domain of/' is (1, o°). This is smaller thanthe domain of/, which is [1, °°).

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 103

finding the derivative/'(x) of a function/(x) by finding the limit off(x) asx is approached; that is, as h tends to zero. The derivative is the rate ofchange of the function, which may be interpreted geometrically as theslope of the curve at the point (x,f(x)). The visualization of the derivative inthe form of a graph for the geometrical interpretation of the derivative isexamined in Chapter 5.

4.4 Contraction and Expansion of Experiential Meaning

Experiential meaning in mathematical symbolism is largely concerned witha semantic field in the form of the description and manipulation of rela-tions. The semantic field of mathematics therefore includes a limitedexperiential domain compared to language. With the narrowing of thesemantic domain, an expansion of meaning took place in mathematics withrespect to the description of relations and patterns of variation. The ways inwhich mathematical symbolism achieves this simultaneous contraction andexpansion of the experiential meaning are discussed below.

One major innovation in mathematical symbolism is the evolution of anew process type, the Operative process, which takes the form of arithmeticoperations and other processes found in the different fields of mathemat-ics. Operative processes initially arose in early numerical systems, whichwere among the earliest forms of mathematical symbolism. Numericalnotation appeared in different cultures arising from practical needs suchas recording quantities and marking time intervals for social and economicactivities. The nature of the early numerical systems in cultures whichinclude the European, Egyptian, Mesopotamian, Indian, Arabian andChinese, and independent traditions such as the Mayan in South America,depended upon the functions which were required to be fulfilled and theavailability of material resources. Once established, numerical systems cir-cumscribed mathematical activities and new developments in much thesame way that grammatical systems in language function to structure realitythrough the nature of the linguistic choices which are available. As we haveseen, the adoption of the Hindu-Arabic numerical system, for example,had a major impact on the development of mathematics in Europe.

Symbolic processes in early numerical systems developed from Materialprocesses (Lemke, 1998b; O'Halloran, 1996) which were concerned withcounting, adding, multiplying, subtracting, dividing and measuring. How-ever, new mathematical Operative participants and processes began toappear with the development of numerical systems. For example, newparticipants in the form of very small and very large numbers, which couldnot materialize in concrete form, arose in the symbolism. Moreover, Opera-tive processes replaced the semantics of Material processes. That is, Opera-tive processes of adding, multiplying, subtracting and dividing symbolicnumbers initially paralleled existing Material processes of combining,increasing, decreasing and sharing physical objects. However, the complex-ity of Material processes undertaken by human participants in a physical

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104 MATHEMATICAL DISCOURSE

world had practical limitations and intuitive expectations which did notnecessarily extend to the semiotic Operative processes performed usingsymbolic notation. It became possible to perform complex combinations ofOperative processes which were not otherwise feasible or even conceivable,and to obtain results unlike those previously expected. An example of thistype of semantic extension occurs in the case of multiplication of fractionswhere the product is less than the numbers which are multiplied. Thiscontravenes the common-sense understanding where 'to multiply' means'to increase'. A similar situation arises with the division of fractions wherethe result is larger than the number which is being divided.

The limits of Operative processes within early numerical systems weredependent upon parameters such as the base of the system, the existence ofplace value, the inclusion of a symbol for zero, a means of separating frac-tional components and the intricacy and number of symbols. When calcu-lations became complex, material computational devices based on thenumber systems, such as counting boards, table reckoners and the abacus,were employed. With the development of symbolic algebra, attentionturned to generalized descriptions of relations using algebraic methods.The success of these descriptions meant that mathematical symbolismdeveloped as a semiotic resource with grammatical systems which wereunique to that resource. These systems developed in accordance with theaim of mathematics: the descriptions of patterns and the means to solveproblems relating to those descriptions. This largely involved capturingand rearranging generalized descriptions of relations between variablesthrough Operative processes. With the evolution of mathematical symbol-ism as a semiotic resource, arithmetic Operative processes were supplemen-ted with processes concerned with powers, roots, complex numbers, limitsand other processes found in different branches of mathematics, as seen inthe calculus example in Plate 4.3(1) where the limit as h —> 0 is derived for

Operative processes are typically performed by human agents onsymbolic semiotic participants in the form of numbers and later variableparticipants, as seen in Plate 4.3(1) where the reader is instructed 'Iff(x) = J x — l , find the derivative of/'. In the development of symbolicalgebra, the human agent was not included in the mathematical symbolicstatements which were more concerned with describing the nature of rela-tions based on established mathematical results, rather than encoding therhetorical commands which accompanied the solution to the problem. As aresult, the human agent tends to be located within the linguistic part ofthe text which is concerned with the commands (for example, ' [you] findthe derivative of/' in Plate 4.3(1)) and statements such as 'We recognizethis limit as being the derivative of /at xl} that is,/' (xj' (Stewart, 1999:132).This statement takes the form of the metaphorical projecting clause 'werecognize' for 'this limit is the derivative of/at xlt that is, /'(*i)' (seeSection 3.2). As we have seen, the human agent also disappeared in math-

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 105

ematical visual images as the concern with lines and curves grew during theseventeenth and eighteenth centuries.

At the same time, the notion of agency, where one participant impacts onanother, appears to have developed in the symbolism in rather a differentfashion from that found in language. For example, in a mathematical func-tion the value of the independent variable x 'impacts' on the value of thedependent variable y in so much as the value of y depends on the value of x.However, the grammatical strategies for encoding such relations take theform of interactions between multiple participants rather than directimpact of one participant on another participant. This idea is developedbelow through an examination of the way in which Operative processes andparticipants are configured in mathematical symbolic statements.

Operative processes appear to be grammatically different from the lin-guistic processes documented in Halliday's systems of TRANSITIVITY andthe related system of ERGATTVTTY which is concerned with agency. Theprocess types in language are Material, Mental, Behavioural, Verbal,Relational and Existential processes. Halliday (1994: 163) explains that inlanguage there is a key participant, the Medium, which is associated witheach process. In the clause 'Jack opened the door', the verb 'opened' is aMaterial process with 'Jack' as the Actor/Agent who acts on 'the door'which is the Goal/Medium. In this case the Medium is 'the door'. Without'the door', the action of opening could not have been performed by Jack,the Agent. Halliday (1994: 163) calls the key participant the Medium in theergative interpretation of the clause:

Every process has associated with it one participant that is the key figure in thatprocess: this is the one through which the process is actualized, and without whichthere would be no process at all. Let us call this element the MEDIUM, since it is theentity through the medium of which the process comes into existence.

Every process in language has an associated Medium, and only in somecases is there an Agent. For example, 'Jack talked' is a Verbal process withthe Sayer/Medium 'Jack'. In this example there is no Agent. In the case of'Jack and Jill walked up the hill', the Medium is 'Jack and Jill', realized as acomplex nominal word group and the Range is 'up the hill'. There is noAgent associated with the process of walking up the hill. In a clause such as'the best idea [ [that Jack and Jill had all day] ] was [ [to walk up the hill'] ],the Token/Identified (Medium) in the Relational process is 'The best idea[ [that Jack and Jill had all day] ]', as evidenced by the probe 'what is the bestidea [[that Jack and Jill had all day]]?' The Value/Identifier (Range) is' [ [to walk up the hill] ]'. In this case, the Medium is the rankshifted clause'The best idea [[that Jack and Jill had all day]]'. Within this rankshiftedclause Jack and Jill' function as the Medium for the process 'had'. Thetypical nuclear configuration of functional elements for experientialmeaning in language has the form:

Participant (Medium) + Process ± Participant (Agent) ± Range ±Circumstance/s

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106 MATHEMATICAL DISCOURSE

However, Operative processes in mathematical symbolic clauses do notappear to replicate the nature of the experiential meaning in language.While the notion of a Medium and an Agent exists at the rank of clause inmathematical relational processes (for example, realized through '=') inthe form of the 'Token (Medium or Agent) = Value (Range or Medium)',the corresponding mathematical participants do not take the correlateform of a word, word group/phrase or a rankshifted clause (with oneembedded Medium) as discussed in the case of 'Jack', 'Jack and Jill', and'the best idea [[that Jack and Jill had all day]]'. Rather, the Medium andother participants in relational symbolic statements are most typically con-figurations of Operative processes with multiple participants which appear toplay equally key roles. For example, the notion of 'a single key participant'in the configuration of Operative processes which constitute the value of

the derivative does not seem to apply. Rather,

there appear to be several key participants, x, h and 1, in the algebraicexpression for the derivative/' (x).

In a similar fashion, there appear to be multiple key participants whichare central to the Operative process configurations in the examples givenbelow.

Arithmetic Operations:

Exponents: (xyz) n—x"y" znFactoring Special Polynomials: x* — y2 = (x + y)(x— y)

Geometric Formulae:

Cosine Law: a? = tf + <? — 2bc cos A

It appears that the semantics of language for experiential meaninghave been extended with the inclusion of Operative processes in math-ematical symbolic clauses. This proposal is investigated through further

consideration of

The generalized algebraic law for the expansion of exponential expres-sions is given by the mathematical symbolic statement (xyz)" - x" yn zn. Thisstatement contains a Relational Identifying process '=' equating theToken/Identified (Medium) (xyz)" with Value/Identifier (Range) x"y" z" asdemonstrated through the probe 'what is the expanded form for (ryz)"?'(Identified). In this 'decoding' clause '(xyz)" represents x" y" z"' (Halliday,1994: 165-167) the Medium is (xyz)" with Range x" yn z". However, theseexpressions are configurations of the participants x, y and z which interactthrough the Operative process of multiplication. That is, the Token/Medium is (x x y x z)" and the Value/Range is x x x x x. . . x (n times) x y x y

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In the case of Operative processes, it appears that the idea of one Agentimpacting on one Medium, or one Medium with a Range, is not necessarilyappropriate. Rather Operative processes appear as complex rankshiftedconfigurations with participants Xt, X2, X3 . . . Xn. This becomes evident ifthe presence of Operative processes is indicated by the systemic functionalconvention of square brackets [[]]. In each case, complex rankshiftedconfigurations of Operative processes and participants arise as seen in the

jconfigurations of Operative processes with multiple key processes and par-ticipants, in this case division, addition and multiplication with participantsa,b,c,d,e and / It is interesting to note that the majority of mathematical

THE GRAMMAR OF MATHEMATICAL SYMBOLISM 107

x y . . . y (n times) x z X z x z . . . z ( n times). However, it is not perhapsfeasible to ask which is 'the key participant' in these configurations of x, yand z where each variable appears to play an equally significant role. Thiscase is different to 'Jack and Jill' which is a complex nominal group inlanguage, or the case of the rankshifted clause 'the best idea [ [that Jack andJill had all day] ]' where the Medium in the rankshifted clause is 'Jack andJill'. The x, y and z appear to interact through the Operative process ofmultiplication as equally key figures in the configurations which constitutethe Medium and Range of the relational process.

Similarly, the generalized algebraic form for the addition of two fractionsmay be seen to consist of the six multiple key participants a,b,c,d,e and fin

This is a statement which contains a Relational

Identifying process with '=' equating the Value/Identified (Medium)

with Token/Identifier (Agent) as demonstrated by

the probe 'what is the generalized form for (Identified). In this

'encoding' clause (Halliday, 1994: 165-167), the Medium with

Agent However, once again the Medium and the Agent are

clauses for encoding relations seem to appear in the passive form

is represented by rather than represents

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108 MATHEMATICAL DISCOURSE

In order to mark this difference in the grammar of mathematical symbol-ism, the rank below the clause is called 'the expression' (see Table 4.3(2))which corresponds to the equivalent rank of word groups/phrases in lan-guage. At the rank of the mathematical expression, the potential for agencyand the nature of the participants within the nuclear configurations of pro-cess/participants need further research in order to classify the participantfunctions according to the types of Operative processes. This remains animportant research goal. At this stage, the typical form of the configur-ations of expressions in ranking clauses in symbolic mathematics is seento be:

[ [Participant + Processj + Participant + Process2 + Participant. . .Processn]]

n (Medium) + Process ± Agent ± Range ± Circumstance/s

where the Agent, Range and Circumstance also have the potential to takethe form [ [Participant + Process! + Participant + Process2 + Participant. . .ProcessJ]". The potential for rankshifted configurations of Operative processesand participants is one of the key factors in the success of mathematicalsymbolism because this strategy preserves process/participant structureswhich may be reconfigured for the solution to problems. This is a signifi-cant point in understanding how the grammar of mathematical symbolismis functionally organized to fulfil the goals of mathematics: to order, tomodel situations, to present patterns, to solve problems and to makepredictions.

The degree of rankshift found in mathematical symbolism exceeds thatfound in language. This may be demonstrated through closer inspection of

The degree of rankshift is displayed in Table 4.4(1), where it may beseen that there are six levels of embedding of Operative processes in themathematical expression for the derivative. At each stage, the process/participant configuration is preserved so that the expression can berearranged and simplified. This is an important grammatical strategy in theevolution of mathematical symbolism as the semiotic which is used to solveproblems. This is a different grammatical strategy to that found in scientificlanguage, where the potential of the nominal group structures expanded inorder to encode experiential meaning. The two different grammaticalstrategies reflect the different functions of mathematical symbolism and lan-guage. This point is further developed in relation to grammatical metaphorin Chapter 6.

shifted Operative processes are marked by [[]], we see that the level ofembedding is complex.

the derivative in Plate 4.3(1). If the rank-

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The process types in mathematics and their attendant participantsare concerned with capturing and reformulating patterns of relations.The processes in mathematical symbolism largely consist of Operativeprocesses, Relational processes in the form of identifying and attributiveprocesses (intensive, circumstantial and possessive) and Existential pro-cesses. While the potential of mathematics as a semiotic resource expandedin the field of description of relations, the overall scope of experientialmeaning with which mathematics was concerned narrowed. That is, pro-cesses involving the human realm of feeling, behaving and talking werelargely excluded in the quest to describe patterns. Mathematical symbolicparticipants became numbers and variables which function as generalrepresentations rather than the specific entities or groups realized throughlexical choice in language. Circumstance became limited to those typeswhich had applicability in the description of relations. Circumstance in thesymbolism is realized through minor clauses, dependent clauses or fusedelements within participant group structures in symbolic mathematics, asdescribed below. A summary of the range of experiential meaning in termsof processes, participants and circumstance in symbolic mathematics isdisplayed in Table 4.4(2). Mathematics became concerned with an

THE GRAMMAR OF MATHEMATICAL SYMBOLISM 109

Table 4.4(1) Levels of Rankshift in Mathematical Symbolism

Rank Process Operative Process/Participant Configuration

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110 MATHEMATICAL DISCOURSE

Table 4.4(2) Restricted Range of Experiential Meaning in MathematicalSymbolism

PROCESS

PARTICIPANTSMedium

Relational:Identifying

Token Value/

Relational: OperativeAttributive

Carrier Participants

Existential

Existent

Agent

Range

Token/Identifier Attributor Operator

Value/Identifier Attribute ParticipantsX], X2 . . . Xp

CIRCUMSTANCEExtentDuration/Distance

Location:Temporal/Spatial

Contingency:

Accompaniment

Temporal and Spatial

Time and Place

Condition

Commitation/Addition

expanded realm of meaning within a restricted experiential field. Furtherstudy, however, is needed to fully document the process types, the asso-ciated participants, and the types of circumstance which are found insymbolic mathematics.

In the effort to efficiently encode meaning in mathematical symbolismin an economical and exact manner, special means have developed forrealizing the Operative process/participant configurations in mathemat-ical symbolic statements. These strategies include ellipsis of Operative pro-cesses, the Rule of Order for Operative processes, and the use of brackets tore-organize the order of operations. These strategies for efficiently encod-ing experiential meaning are discussed in relation to textual meaning andthe organization of the symbolic statements, and to logical meaning andthe temporal unfolding of the Operative processes. Another significantstrategy for efficiently encoding experiential meaning in mathematicalsymbolism is the use of positional and spatial notation. This strategyis discussed below in relation to systems for experiential meaning in thedisplay stratum.

Another strategy to efficiendy encode experiential meaning in the sym-bolism relates to circumstantial elements which appear within participantand process structures, in addition to being separate functional elements inthe form of prepositional phrases. For example, temporal Location 'at atime of four seconds' may be represented by the dependent relationalstatement 'if t = 4', and the circumstance of Extent 'after four seconds' may

Identifier X], X2 . . . Xp

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 111

be included in participant structure '5(4)'. Combinations for packingcircumstantial information in the symbolic clauses also occur, for example,'the displacement in the first four seconds is ten metres' may berepresented by 5(4)-5(0) = 10. Circumstantial prepositional phrases andadverbial groups also occur in the surrounding linguistic co-text withinwhich mathematical symbolism is embedded.

At the rank of discourse semantics in mathematical symbolic text, thesystem of IDEATION is largely concerned with Activity Sequences in theform of Operative process/participant re-configurations for the solution ofmathematics problems. Such Activity Sequences are clearly marked in astep-by-step fashion as seen in the case of the derivation of the derivative /'(x) in Plate 4.3(1). In this example, the steps are organized according tovertically placed lines where participants and process configurations arerepeated, substituted, re-organized and simplified according to mathemat-ical definitions (for example, the definition for the derivative of a func-tion), algebraic laws (for example, x2 — f = (x + y) (x — y) which is used torationalize the numerator), and other established results for algebraicoperations. The Activity Sequence consisting of the strings for Operativeprocesses and participants is clearly marked through the textual and spatialorganization of the solution to the problem (see textual meaning in Sec-tion 4.8). The Rule of Order determines the sequence in which Operativeprocesses are performed within statements and expressions (see logicalmeaning in Section 4.7). Within the Operative participant structures, thereis a restricted range of elements; for example, experiential epithets andchoices from the system of DIEXIS do not occur.

With the development of the grammar for mathematical symbolism, theconventionalized use of specialized symbols in mathematics took place. Atthe rank of component, symbols include letters from the Roman alphabet(with upper and lower case letters of varying sizes written with varying fontsand scripts), the Greek alphabet and a range of choices from other alpha-bets. Other elements include punctuation symbols, brackets, iconic repre-sentations, abbreviations and the invention of new symbols as displayed inTable 4.4(3). Cajori (1993) provides evidence to prove that the path tostandardized symbolic notation was not smooth. Cajori (1993: 338)explains that' [o]ften the choice of a particular symbol was due to a specialconfiguration of circumstances (large group of pupils, friendships, popu-larity of a certain book, translation of text) other than those of the intrinsicmerit of the symbol'. In reality, the choice of mathematical symbols oftendepended upon circumstantial, personal and political contingencies ratherthan merit. This is not so surprising as mathematics is a field of humanendeavour.

Choices from the systems of the font style, size and format on the displaystratum function to realize experiential meaning. For example, symbolic

statements such as appear in italicized form

to mark them as separate elements in the mathematics text. Elements in the

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112 MATHEMATICAL DISCOURSE

Table 4.4(3) Examples of Mathematical Symbols

Type of Symbol

Roman alphabet (upper case)

Roman alphabet (lower case)

Roman alphabet (lower case,bold)

Roman alphabet (upper case,bold)

Greek alphabet

Abbreviations

Punctuation

Brackets

Brackets

Iconic

New symbols

Example

G

z

u

R

etanA

/(*)

3 [(8 + 2) -2]

/<«)_L

oo

Meaning

group

complex number

vector

set of real numbers

angle

the tangent of A

the derivative f (x)

grouping of terms indicatingchanged order of operations

the value of the function at a

is perpendicular to

infinity

mathematical statements typically appear in standardized fonts accordingto functional status; for example, different fonts are assigned for functions,variables, text, vector matrices, the Greek alphabet and so forth, with vary-ing sizes according to their function as a subscript (for example, 7 points),sub-subscript (5 points), superscript (7 points), sub-superscript (5 points),symbol (18 points) and sub-symbol (12 points). The different choices from

systems in the display stratum may be seen in the expression

display of mathematical symbols is standardized through special softwareapplications for mathematical symbolic text. As with all system choices forexperiential meaning in mathematical symbolism, the objective is to pre-cisely encode the relations in a condensed format ready for re-organizationand solution to the mathematics problem.

Mathematical symbolism realizes experiential meaning through spatialand positional notation in a form that is not found in language. This is asignificant grammatical strategy for encoding meaning efficiently in math-ematics. Simple examples of the use of positional notation are x3, whichmeans x x x x x through the spatial position of the 3 as a superscript, andthe process of division which is realized through the spatial arrangement of

the case of matrices where each variable has a value depending upon its

The

The use of spatial positioning for experiential meaning is highlighted in

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 113

spatial position in the matrix as seen in the example

Examples of the use of positional notation for experiential meaning inH A B

relation to the following configuration G x,f(x) C where x is a variable,F E D

f(x) is a function and the spatial positions are marked as A, B, C through toH are given in Table 4.4(4). These examples show that the use of spatialnotation extends to Operative processes, participants and expressions ofclassification and qualification.

The contraction of options in the systems for experiential meaning, theuse of strategies such as the Rule of Order for Operative processes, ellipsisand the use of spatial notation and special symbols mean that maximalstructural condensation can take place in the mathematical symbolicstatements. This issue is further developed in relation to textual meaningand the organization of symbolic mathematics. The contraction of optionsfor experiential meaning extends to the other metafunctions, in par-ticular to the realm of interpersonal meaning. The objective and factual

Table 4.4(4) Positional Notation Examples

Position

A

B

B

B

B

C

C

D

E

G

Functional Unit

entity

Operative process

entity

classifier

entity

classifier/ Operativeprocess

entity

deictic

entity

Operative process

Example

X

xy

/(*)MT

/-'

-x

xl

Xo

X

.X

Meaning

the mean [of x]

x raised to the power [of y]

the derivative [off]

the transpose of matrix x

the inverse of function f

opposite of x or (-1) timesx

x factorial

a specific value of x

vector x

multiplied by x

Combination circumstanceOperative process the definite integral

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114 MATHEMATICAL DISCOURSE

appearance of mathematics results from a combination of the restrictedselections in the fields of experiential and interpersonal meaning in themathematical symbolism (which is also reflected in the linguistic parts ofthe text), the textual strategies of condensation through which meaning isefficiently encoded, and the emphasis directed towards logical meaning.

4.5 Contraction of Interpersonal Meaning

The range of interpersonal meanings in language narrowed as mathemat-ical symbolism became concerned with the description and reformulationof patterns of variation between generalized variables and numerical quanti-ties. The quest for exactness in the encoding of those relations in the simplestmanner possible for reconfiguration in the solution to problems meantthat all 'superficial' information had to be removed. In combination withthe restricted experiential meaning, the rigorous ordering of fewer com-ponents of interpersonal meaning in the mathematical symbolism allowedaccessible and intelligible conventions to be established. The resultantimpact on the nature of interpersonal meaning in mathematical symbolismis explored through a discussion of the symbolic text in Plate 4.3(1).

EXAMPLE 4 I f f i n d the derivative off. State the domaino f f .

SOLUTION

At the rank of clause, a range of SPEECH FUNCTIONS (statement, ques-tion, command, offer and so forth) with different values of MODALITY(high, medium, low) are possible using language through the changeableorder and selection of different functional units in the Mood structure asdescribed in Chapter 3. For example, 'darling, could you please think howwe might get out of this jam' is a metaphorical variation of the command'think how we can get out of this jam' realized through modulated inter-rogative Mood ('could you'). The Vocative 'darling' and the colloquial lexis'jam' add further interpersonal dimensions in the play of the social rela-tions being enacted. Mathematical symbolism, on the other hand, is con-cerned largely with descriptive statements and a more restricted sense ofcommands which are realized linguistically through lexical choices such as'let' or, in the case of Plate 4.3(1), 'find', 'state', and 'we see that' for thecommand 'see that'. As discussed below, mathematical symbolism does notinclude lexical choices which are orientated towards interpersonalmeaning.

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 115

The Mood structure in relational symbolic clauses such as/(#) = Jx— 1may be seen to correspond to that found in language; that is, Subject/(%) A

Finite/Predicator '='. In mathematical symbolism, the Finite element doesnot appear as a separate functional element (for example, 'did =', 'was =',or 'could =', 'might =') to realize plays with TENSE (past, present or future)and MODALITY (in the form of probability and usuality). Rather, theFinite element is fused with the process in selections such as '=' for 'equals'.Similarly MOOD ADJUNCTS which function to realize probability (forexample, 'possibly' and 'probably'), usuality (for example, 'sometimes' and'usually') and other semantic domains such as readiness, obligation, time,typicality, obviousness, intensity, degree are typically excluded in math-ematical symbolic statements. In mathematics, choices for MODALITY inthe form of probability may be realized through symbolic statements formeasures of probability; for example, levels of significance: p <0.5 (wherethe notion of uncertainty is quantified) and different forms of approxima-tions. Mathematics thus narrows the options of language which permitexpression of a wide range of probability through the Finite element, thePredicator and MOOD ADJUNCTS.

As a further condensatory strategy, positive and negative POLARITY ('is'or 'is not') is realized through one element for the process in mathematical

symbolic statements such as For example, '=' and

'e ' represent positive polarity, and negative polarity is typically indicated bya slash through that same element, for example, V and '£ '. The multiplestrategies of condensation, which function to reduce the number of func-tional elements in a mathematical symbolic clause, mean that a maximallevel of certainty is typically associated with mathematical statements and ahigh degree of obligation is associated with commands.

Lexical choices such as 'darling', and expressions of attitude such as 'thatis horrific' are not found in the mathematical symbolism. Using the notionof coreness of lexical items (see Chapter 3), mathematical symbolism onlyconsists of items which have a specific meaning in the register of mathemat-ics. Shades of meaning derived from the selection of symbolic elementswhich occupy a non-central position do not exist in mathematical symbolicstatements. Every element is precisely defined in relation to other func-tional elements where the emphasis is orientated towards experiential andlogical meaning. In addition, the symbolic elements appear in generalizedforms as variables, special symbols and Operative processes rather than theindividuality which is possible through lexical choice.

At the level of discourse semantics, the choices from the discourse systemof NEGOTIATION largely result in an Exchange Structure where informa-tion is provided and commands are issued so that Activity Sequences unfoldin a strictly regulated fashion. The mathematical symbolism typically con-sists of a series of statements where the Exchange Structure is constructedfrom the position of the primary knower (Kl moves and move-complexes)and the primary Actor (Al moves) (see Chapter 3). In the Exchange

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116 MATHEMATICAL DISCOURSE

Structure, there is a strong sense of direction which appears to be non-negotiable. This strong sense of direction is also realized compositionallythrough the spatial arrangement of the mathematical symbolic statementswhich literally 'run' down the page. In addition, choices from the discoursesystem of APPRAISAL for graduations of evaluation and attitude (affect,judgement, appreciation) (for example, 'I like that', 'that is good' and'thanks, I appreciate that') are typically absent in mathematical symbolism.The discourse appears as non-evaluative and value-free.

The nature of the interpersonal choices in mathematical symbolismresults in a discourse which appears as factual and objective, largely as aseries of symbolic statements which lack modalization, modulation andchoices for affective realms of meaning. The truth-value of the symbolicstatements appears as consistently high with an implicit objective orienta-tion; there appears to be no question about symbolic statements such as x =y. The expression of probability takes different forms in mathematics,including probability statements and unmodalized approximations. Inaddition, the restricted range of process types and the nature of the partici-pants as generalized variables and numerical quantities function to makethe text appear as abstract. The symbolic clauses and statements are real-ized in a tightly organized Exchange Structure. Indeed, mathematicsappears as the ultimate truth which is difficult to re-negotiate, question oreven interrupt.

The factual and objective stance of mathematical symbolism is alsocommunicated through the style of production and contrasts in choices forscripts and fonts from the systems in the display plane. The development ofcomputer software devoted to the expression of mathematical symbolic textfunctions to ensure that conventionalized styles of typesetting and symbolicrepresentation are standardized. The interpersonal meaning realizedthrough such professionally produced texts, which are extraordinarilyintricate and complex due to the high degree of rankshifted expressionsand use of specialized symbols, contributes to the image of the text asdominating in terms of the social relations which are established betweenthe writer/producer and reader. Without a knowledge of the grammar ofmathematical symbolism, the texts are impenetrable as may be appreciatedfrom Plate 4.5(1).

Mathematical symbolic discourse is positioned as authoritative. However,what needs to be made explicit is that this position is created by the nature ofthe choices that are available from the systems in the language and displayplanes for mathematical symbolism. Mathematical symbolic texts are con-cerned with a limited semantic field which appears as factual and objectivedue to the range of interpersonal and experiential meanings which areadmitted. The reason for this is that mathematics is designed to fulfil cer-tain functions which do not include the re-negotiation of social relationsor the expression of typically human processes of feeling and emotion.Mathematics is designed to capture, model and predict patterns in themost economical fashion, and this overriding aim has resulted in pre-

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Plate 4.5(1) Mathematical Symbolic Text (Wei and Winter, 2003: 159)

determined patterns of experiential, interpersonal, textual and logicalmeaning. Mathematics is typically assigned a high truth-value, most likelythrough its success as a tool for science and technology which plays a majorrole in determining economic, social, cultural and political relations.Mathematicians and scientists, however, are aware of the fallibility andimpoverished nature of mathematical descriptions and models. To the

THE GRAMMAR OF MATHEMATICAL SYMBOLISM 117

(5.8)

In the case ft —»• 0, we calculate at .v = Pf, i = 1 K:

Lei

Then we have

Writing in matrix form, we obtain

where

X is the identity matrix and

(5.7)

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118 MATHEMATICAL DISCOURSE

outsider who cannot engage with the discourse, the myth is harder to coun-teract. The concept of truth and the cultural value attached to mathemat-ical discourse is explored further in Chapter 7.

4.6 A Resource for Logical Reasoning

In his pursuit of sure knowledge, Descartes desired a tool that could beused to reason with, and that tool was designated to be mathematical sym-bolism. Mathematical symbolism has been specifically designed as a semi-otic resource to describe patterns which can be rearranged for the solutionto problems. The encoding experiential meaning in the form of Relational,Operative and Existential processes which preserve process/participantconfigurations through rankshift, and the narrow range of interpersonalmeanings result in a grammar which encodes meaning in the most efficientmanner possible. Logical reasoning flows smoothly down the page as seenin the solution to the problem in Plate 4.3(1). In what follows, that reason-ing is explored.

Mathematical symbolism is concerned with relations of EXPANSION(Halliday, 1994: 328-329) in the form of (i) elaboration, or a re-statementin the form of apposition or clarification; (ii) extension, or additative andvariation type relations; and (iii) enhancement, which is predominantlycausal-conditional and spatio-temporal type relations. Logical relations ofPROJECTION in the form of quotations and reporting of locutions andideas (ibid.: 220) are not typically found in symbolic mathematics. Thisrepresents a narrowing of the options for logical meaning admitted intomathematical symbolism.

The 'temporal' conjunctive relations which are concerned with the orderin which Operative processes unfold in statements and expressions arerealized through the Rule of Order for Operative processes. The con-ventionalized order is brackets, powers, multiplication/division andaddition/subtraction, and this means that Operative processes do notnecessarily unfold in a left-to-right fashion. The Rule of Order is a conden-satory strategy in that the explicit order with which the mathematical oper-ations are to be performed need not be explicitly marked in mathematicalsymbolic statements. The order may be changed, however, through the useof differing types of brackets which function to re-organize the standardsequence of operations. For example, the use of brackets in the numerator

of means that the order is changed so that the

Operative processes of addition in (x+ h— 1) and (x— 1) are completedfirst, rather than the left-to-right sequential processing of the Operativeprocesses of addition and subtraction.

At the rank of discourse semantics and inter-statemental relations, CON-JUNCTION and CONTINUITY relations are realized through explicit andimplicit structural conjunctions which link clauses, and non-structural con-junctive adjuncts which function cohesively across stretches of text (see

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 119

Section 3.4 where Halliday's systems for logical meaning are explained).Some structural conjunctions and conjunctive adjuncts are symbolized inmathematics (for example, .'. and while others occurin linguistic form (for example, 'and', 'or', 'for example', 'also' and soforth). Statements are explicitly marked in cases where the logical connec-tion is non-sequential. For example, statements labelled (1) and (2) may bereferred to later as 'from (1) and (2)'. The conceptions upon which logicalmeaning at the rank of discourse semantics and inter-statemental relationsare based are investigated below.

Figure 4.6(1) contains the analysis of logical meaning in terms of theconjunctive relations which occur in the solution of the problem displayedin Plate 4.3(1). Conjunctive relations in mathematical symbolism typicallyfunction in an internal-rhetorical sense rather than an external-experiential sense. Following Martin (1992), these types of relations aremodelled on the left-hand side as shown in Figure 4.6(1). Relations aremodelled through arrows which indicate the dependency structure. Movesare shown to be dependent on preceding ones by dependency arrowspointing upwards, and the arrows point outwards in the cases where theresult depends on mathematical definitions, axioms, laws, theorems andother results. In Figure 4.6(1), the implicit conjunctions are causal-conditional type relations of the form 'therefore'. On the right-hand side,the basis for the logical reasoning is given.

Mathematical symbolic discourse typically involves long implicationsequences as seen in Figure 4.6(1). However, it becomes evident from thisanalysis that mathematical deductive reasoning depends to a large extenton laws, axioms, theorems and established results which are not madeexplicit in the derivation of the solution to the problem. Tiles (1991) claimsthat the image of mathematics as dealing solely with deductive reasoningbased on syllogisms (that is, deductive inference consisting of two premisesand a conclusion which are categorical propositions) is misleading. As seenin Figure 4.6(1), the steps in the Activity Sequence in the solution of theproblem involve implicit procedures using established results. Azzouni(1994: 79) characterizes mathematics as 'a collection of algorithmic sys-tems, where any such system, in general, may have terms in it that co-referwith terms in other systems'. It appears that the success of mathematics isdependent on this co-referral of terms based on previously establishedresults which are formalized as definitions, axioms, theorems, laws and soforth.

The fallibility of mathematics as a formal system, however, has beendemonstrated through contributions by Frege, Hilbert and Russell andGodel. New technological advances are contributing to this process: 'Com-puters are in the process of dismantling the very image of reason whichgenerated them' (Tiles, 1991: 169). The loss of certainty in mathematics(Kline, 1980) has been accompanied by new mathematical approaches tophysical systems; for example, chaos and dynamical systems theory. Perhapsin the pursuit of knowledge (see also Chapter 2), the realms of meaning

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Figure 4.6(1) Logical Meaning (Conjunctive Relations) in Plate 4.3(1)

f(x) given inExample 4

Definition of derivative

Substitution for f(x)

Rationalize numeratorMultiplication PropertyofOne(MPOne)

Factorizationx2-y2=(x+y)(x-y)

Distributive Propertyof Multiplication overAddition (DPM/A)Simplification based onalgebraic laws

Definition limitSimplication based onalgebraic laws

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 121

which were excluded in mathematical symbolism resulted in an inadequateset of semiotic tools to describe physical phenomena. However, techno-logical innovations such as those accompanying the latest development incomputers permit patterns to be mapped computationally and visually innew ways. While mathematical symbolism remains the major semioticresource for the construction of logical meaning in mathematics, perhaps anew semiotic resource will evolve through the increased power of com-puters to represent patterns of relations. This may be in the form of thedevelopment of a new grammar of visual images, or the construction of anew resource yet to be imagined. As may now be appreciated, mathematicsis constrained by the nature of the semiotic resources it has at its disposal.

4.7 Specification of Textual Meaning

The textual organization of mathematical symbolism is sophisticated andhighly formalized in order to facilitate the economical encoding of rela-tions which permits immediate engagement with the experiential andlogical meaning of the text. The nature of textual meaning is evident inPlate 4.3(1) where, at the rank of discourse semantics, the statements areplaced sequentially down the page to permit easy tracking of the nuclearreconfiguration of process/participant structures. The overall spatialarrangement of mathematical symbolic text is generic so that key equa-tions, definitions and solutions are immediately accessible in texts.

In addition to spatial sequential organization of the symbolic text, themathematical statements and clauses are organized syntagmatically inpre-defined forms for Relational and Operative process/participantconfigurations. Examples of spatial and syntagmatic relations which aretypically found in mathematical symbolism are given in Table 4.7(1). Thetextual organization of the symbolism may be contrasted to experientialmeaning in language which is typically represented as clusters ofparticipant/process/circumstantial functions rather than definitive syn-tagms which have a specific order. That is, alternatives are possible in theconfiguration of participants, process and circumstance in language; forexample, T will sit quietly for ten minutes', 'For ten minutes I will sitquietly', T will quietly sit for ten minutes' and 'Quietly will I sit for tenminutes'. Such re-organization is not a typical feature of mathematicalsymbolic statements. The tendency towards standardized syntagmaticorganization of symbolic statements is evident in Table 4.7(1) and the

definition of the derivative in Plate 4.3(1). The

specificity of textual organization relates to the need to organize the ex-periential meaning of the mathematical symbolic statement in an exact andsimplified format so that problems may be solved. 'To simplify' in math-ematics means to arrange mathematical Operative processes and partici-pants in the most simple standardized format using algebraic rules andpre-established mathematical results.

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Spatial positioning in combination with syntagmatic organization playsan important role in the semantics of mathematical symbolism as illustratedin Table 4.7(1). At the clause level of the lexicogrammar, with exceptions(for example, 'if, 'and', and 'or'), conjunctions typically occur on theleft- hand side of a mathematical sentence, slightly separated spatially fromthe mathematical clause as seen in Plate 4.7(1). It should be noted, how-ever, that mathematical texts vary in terms of spatial layout and the extentto which the symbolic text is integrated with language. Many other possi-bilities occur including the use of language conjunctions and completeintegration of mathematical statements within the linguistic text as foundin 'If f(x) = J x — l , find the derivative of/' in Plate 4.3(1). Key symbolicstatements of special interest, however, are typically spatially separated fromthe body of the text.

The use of spatiality is one key element of mathematical symbolism whichdiffers from the line-wrapped syntagmatic arrangement of linguistic text.Such visual arrangements permit easy engagement with the text. This isnecessary as the reading path is not necessarily linear in mathematical andscientific texts which consist of language, visual images and symbolism. Theuse of spatial arrangement also permits ellipsis on a scale which is notfound in language. Lemke (1998b) explains that the table, for example,

122 MATHEMATICAL DISCOURSE

Table 4.7(1) Textual Organization: Syntagmatic and Spatial Relations

1 Equations and formulae

Circ Qualifier Conj Participants Process Participants2x + 4 > 7

{x; 3x > y}if AB 7i CD

inAABD ABD = 60°

2 Simplification of expressions and formulae

Process ParticipantParticipant7(2x-3) -5( l -x )14x - 21 - 5 + 5x

3 Mathematical statements may be combined

Participant Process Participant Process Participant-3 < x < 2

4 Combinations may occur with certain conjunctions

if x + 2y = 7 and 2x + 3y = 4

=

= =

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Next,

Next

so that

THE GRAMMAR OF MATHEMATICAL SYMBOLISM 123

Plate 4.7(1) Textual Organization of Mathematical Symbolism (Clerc,2003: 117)

For the last term,

On the other hand.

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124 MATHEMATICAL DISCOURSE

carries textual ellipsis to the extreme so that the result is a 'textualizedvisual display', meaning that the table uses spatial arrangement to compen-sate for the lack of grammatical constructions. Baldry (2000a) discussesmeaning compression and the integration of visual, verbal and symbolicresources in scientific tables. Mathematical symbolic statements functionin a similar manner, permitting a visual arrangement for condensatorypurposes so that the experiential and logical meaning is immediatelyaccessible.

The organization of mathematical symbolism as a message is thereforehighly conventionalized in terms of genre, inter-statemental relations, andthe statement. Within statements, the notion of the Theme as the point ofdeparture of a message and the Rheme as the part in which the Theme isdeveloped is fundamental to the textual organization of mathematicalsymbolic clauses. The left-hand side of an equation usually involves theconflation of Subject with Theme. If a Circumstance Adjuncts occurs first,such as 'in A ABD' in Table 4.7(1), this functional element is typicallyplaced further apart from the mathematical equation so that the Subject/Theme conflation is not disrupted. Given the concern with logical mean-ing, multiple Themes containing textual elements in the form of conjunc-tions and conjunctive adjuncts (for example, 'if, 'then', 'therefore') arecommon in mathematical symbolic text. Marked Themes in the form of adependent clause (for example, 'if x= 4') also frequently occur. The spatialarrangement of the mathematical symbolic clauses permits ellipsis; forexample, /' (x) need not be repeated in the derivation of the derivative inPlate 4.3(1). Within the symbolic statements, the Rule of Order for Opera-tive processes (see Section 4.6 on logical meaning) determines the tem-poral unfolding of processes/participant configurations, which does notnecessarily follow a left-to-right format. Mathematical symbolism possessesits own rules through which mathematical Operative processes unfold.

Within mathematical statements, two further textual strategies for eco-nomical encoding of meaning are the generic forms of ellipsis establishedin mathematical symbolism, and the use of spatial position to realizeOperative processes (see Section 4.4 on experiential meaning). Forexample, x for the Operative process of multiplication in expressions is notrequired in expressions such as xyz (for x x y x z) and (a + b) (c + d) for (a +b) x (c+ d). In addition, the use of spatial position in x3 realizes the Opera-tive process of multiplication for x x x x x. In a similar fashion, spatial

notation for division is used in expressions such as

strategy for economy of expression is the serial placement of the com-ponents of an expression which follows conventions which differ from thestructure of the nominal group in language. An example of this type ofextension occurs within the decimal place value system (for example,3.1415926) where each place or location in the sequence has a particularvalue; that is, the decimal system involves place value.

At the rank of discourse semantics, the reference chains linking partici-

for x ~ y. Another

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THE GRAMMAR OF MATHEMATICAL SYMBOLISM 125

pants for the system of IDENTIFICATION are complex as the nuclear con-figurations of Operative processes and participants are constantlyrearranged in the derivation of the solution to a problem. However, thespatial organization is such that these chains are typically organized downthe page in a manner where the configurations of Operative processes andparticipants are standardized. Mathematical symbolic participants areexplicitly repeated clause by clause to permit tracking, and the fate of parti-cipants is based on definitions, algebraic laws, theorems and so forth. Refer-ential cohesion, the tracking of participants, however, requires a knowledgeof the implicit basis upon which participants are rearranged, as discussed inrelation to logical meaning. That is, participants may be transformedthrough axiomatic definitions, derived results, operational properties andtheorems. In addition, there may be discursive links to situate the symbolicnotation in relation to the main body of the text. Statements are also oftenlabelled for easy reference.

Mathematical symbolic texts are textually organized in a way that permitsmaximum condensation in the instantiation of the texts. Strategies includethe spatial or visual arrangement of the text which permits easy tracking ofprocess/participant configurations, standardized sequential placement ofelements in the statements and clauses in terms of Theme and Rheme,standardized forms of ellipsis, the Rules of Order for Operative processesand the use of brackets to change that order, and place value at the rank ofcomponent. The use of space as a textual means of organizing and display-ing mathematical symbolism is an important strategy which, as we haveseen, extends to the experiential and logical meaning. Spatial arrangementand positional notation thus developed in the grammar of mathematicalsymbolism as an important means for fulfilling the functions of thisresource.

4.8 Discourse, Grammar and Display

Hand in hand with language and visual images, mathematical symbolismdeveloped as a semiotic resource which had clearly defined functions.These functions include the description of patterns of relations and the re-ordering of those relations to create models of the physical world, to solveproblems and to make predictions. The focus on experiential and logicalmeaning necessitated the development of a specific grammar in order tofulfil those functions. The discourse, grammatical and display strategiesthat consequently developed in mathematical symbolism were based on theneed for an economy of expression so that meaning could be encoded ina manner which permitted the reconfiguration of process/participantstructures. The grammar also needed to be efficient in terms of making useof mathematical definitions, axioms, laws and theorems and other estab-lished results. In addition, that grammar interacts with linguistic and visualsemiotic resources in the manner described in Chapters 6-7. The dis-course, grammatical and display strategies described in this chapter are

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summarized below in terms of two major aims: (1) to encode patterns ofrelations economically and exactly; and (2) to permit reconfiguration ofprocess/participant structures through the strategy of rankshift.

Condensatory Strategies for Exact Encoding of Meaning

Experiential MeaningMathematical symbolism became concerned with limited fields ofexperiential meaning in the form of Relational, Operative and Existentialprocesses, the associated participants and restricted forms of circumstance.The removal of experiential realms of meaning considered extraneous (forexample, Mental, Behavioural and Verbal processes, their attendant parti-cipants and a range of circumstances) aided the expansion of experientialmeaning in terms of capturing and rearranging descriptions of patterns ofrelations through the configurations of Operative process/participants.Activity sequences which document the solution to mathematical problemsthrough the reconfiguration of Operative processes are clearly denned.

The development of an economical and precise grammar for encodingexperiential meaning permitted the development of mathematical symbol-ism as a tool for logical reasoning as process/participant structures areeasily reconfigured in the steps for deriving the required mathematicalresult. The grammatical strategies which contribute to the economicalencoding of experiential meaning include the use of spatial positioningwhich results in an economy of expression that is impossible in language. Inessence, mathematical symbolism incorporated and built on resourcesfrom language (in the form of syntagmatic relations and sequential andserial positioning) and visual images (in the form of spatial arrangementand spatial notation). Within the syntagmatic structures in mathematicalstatements, symbolic processes do not unfold in a left-to-right fashionbecause of a variety of strategies, including the Rule of Order for Operativeprocesses and the use of brackets. The order of operations is brackets,powers, and multiplication/division and addition/subtraction. The tem-poral unfolding of mathematical operations need only be grammaticallymarked through brackets when the Rule of Order is altered.

The grammar of mathematical symbolism permits Operative processes tobe ellipsed. This aids condensation in terms of encoding relations in asimplified format. There is a restricted range of elements in the participantstructures which take the form of generalized variables, numerical quan-tities and specialized symbolic expressions for mathematical terms. Quali-fiers form part of participant group structures without the need forembedding as phrases. Circumstance is encoded within process and par-ticipant structures in the clause in addition to appearing as separate func-tional elements. The condensatory grammatical strategies in mathematicalsymbolism include the use of special standardized symbols and place value.At the display stratum, meaning is encoded through Font Style, Formatand Size; for example, there are specific formats for functions, variables,

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text, vector matrices, Greek alphabet and so forth. These grammaticalstrategies contribute to the efficient encoding of relations in mathematicalsymbolism.

Interpersonal MeaningInterpersonal meaning in mathematical symbolism is largely restricted tounmodalized statements and commands. Probability is encoded as prob-ability statements and approximations. Lexis is replaced with variable andnumerical participants and other symbolic terms. Functional elements suchas the Finite element and Mood Adjuncts are excluded. The ExchangeStructure consists of rigid sequences of statements and commands whichlack choices of judgement, affect and evaluation. The restricted range ofinterpersonal meanings in mathematical symbolism contributes to theobjective and factual appearance of the discourse. The style of the produc-tion of the mathematical symbolic text arising from software applicationsreinforces the dominating type relations established between the writer/producer and the reader of the text.

Logical MeaningMathematical symbolism developed as a tool for reasoning in the form ofelaboration (clarification and reinstatement), extension (addition andvariation) and enhancement (predominantly causal-conditional andspatio-temporal type relations). The extended implication sequences ofinternal/rhetorical-type relations are most typically based on implicitresults such as definitions, laws, axioms, theorems and other establishedresults. The ease at which the grammar of mathematical symbolism permitsthe reconfigurations of mathematical participants and processes for thedescriptions of relations means that logical reasoning flows in the symbolictext. The spatial arrangement of the text enables the conjunctive relationsto be apprehended at a glance. At the rank of the statement, the temporalunfolding of Operative processes is determined by the Rule of Order forOperative processes. This means that selections for conjunctive adjuncts fortemporal relations need not be made within mathematical statements.Brackets are used to change the conventionalized order of Operativeprocesses.

Textual MeaningMathematical symbolic notation developed specific techniques fororganizing experiential and logical meaning at the ranks of discourse,grammar and display strata. These include the use of spatiality in wayswhich are not found in language. Visual spatial layout includes the organ-ization of the entire mathematical text (genre), the sequential ordering ofstatements within that text (discourse semantics), the syntagmatic orderingof components within the clauses (grammatical) and the use of positionalnotation at the rank of expression and component (expressions and com-ponents). That is, mathematical statements are sequentially positioned

THE GRAMMAR OF MATHEMATICAL SYMBOLISM 127

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vertically and horizontally, while syntagmatic structures unfold horizon-tally in a manner which depends on the Rule of Order for operations andthe use of brackets. Textual organization also functions at the rank ofcomponent, for example, place value.

Rankshifting of Operative Process/Participant Configurations

Mathematical symbolism and scientific language developed different lexico-grammatical strategies for encoding experiential meaning. Scientificlanguage evolved to expand the meaning potential of nominal group struc-tures through grammatical metaphor, where the relations between theseabstract entities were encoded as relational processes (Halliday and Martin,1993; Martin and Veel, 1998). However, the grammatical strategy whichdeveloped in mathematical symbolism is the preservation of Operative pro-cess/participant configurations through rankshift. This strategy is essentialbecause it permits the re-organization of the patterns of relations for thesolution to problems. In other words, mathematical symbolism succeedsprecisely because it preserves the nuclear configurations of Operative pro-cess/participant structures in an exact and economical format which maybe rearranged for the solution to mathematical problems. Mathematicalsymbolism succeeds as a tool for logical reasoning because the grammarpreserves relations in a dynamic format which can be manipulated toobtain the required result. The mathematical grammatical strategy of rank-shift is the platform upon which the grammar is built.

4.9 Concluding Comments

This discussion provides a framework for a grammar of mathematical sym-bolism. As becomes apparent, there is much work to be completed in orderto construct a grammar which compares with the comprehensiveness ofMichael Halliday's SFG for language and Jim Martin's discourse systems forlanguage. For example, the grammar for Operative processes needs to befurther investigated to understand how the notions of the Medium andAgent apply to this process type. In addition, Martin's discourse systemsneed extending to incorporate shifts across semiotic resources (see Chap-ter 6). More generally, however, the discussion presented here demon-strates that although mathematical symbolism is impoverished with respectto the meanings afforded by language, it nonetheless is capable of meaningbeyond that which is achieved linguistically. An outline of the strategiesthrough which this meaning expansion occurs in mathematical symbol-ism has been given above. These strategies require further investigation,especially in relation to the new developments in mathematics which aretaking place through computer technology. The impact of this technologyis considered with respect to visual images in Chapter 5.

128 MATHEMATICAL DISCOURSE

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5 The Grammar of Mathematical Visual Images

5.1 The Role of Visualization in Mathematics

Visual images are such an important component of our resources for mean-ing (Kress, 2003; Staley, 2003) that they have been defaced and destroyedin scientific, religious and artistic contexts (see Latour and Weibel, 2002),sometimes for reasons which are not even immediately clear. Visual imagesin mathematics give an intuitive understanding of the reality constructedthrough the symbolism and language (for example, Galison, 2002; Lemke,1998b). Mathematical visual images mirror our perceptual understandingof the world and thus connect and extend common-sense experience to themathematical symbolic descriptions. However, there have been longstand-ing tensions among mathematicians over the place of the visual imageversus the symbolic in mathematics. Traditionally, the functions of the vis-ual image are seen to be important, but limited compared to those fulfilledby symbolism. In what follows, the tensions and differing perceptions of theroles of mathematical symbolism and visual images are explored. From thispoint, an SF framework for mathematical visual images is used to analysethe types of metafunctional choices found in abstract mathematical graphs.In the final section, the changing role of mathematical images arising fromthe use of computer graphics programs to display patterns generated fromdigitalized data (Colonna, 1994; Davis 1974, 2003) is considered.

The struggle concerning the role of the visual image versus the symbol-ism in mathematics has a long history (Davis, 1974; Galison, 2002; Shin,1994). Since the time of Descartes, traditionalists have seen mathematics asprimarily being the symbolic. From this perspective, the visual image is seenas a heuristic tool rather than a means of establishing a valid proof. A'proper' theorem takes the form of a statement derived from axioms bya sequence of logical steps, rather than a visual pattern which requiresneither verbal statement nor symbolic proof. The opposing perspective isthat visual displays are a valid form of establishing results, leading to whathas been conceptualized as 'a theorem of the perceived type' (Davis, 1974).Galison (2002: 323) sees the ongoing tension in mathematical circles as anoscillation between two poles: 'We must have images; we cannot have

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images.' In fields 'from geometry to quantum mechanics, from astrophysicsto microphysics, the richness of the image and the austerity of the numer-ical are always falling into each other' (Galison, 2002: 323). The strugglebetween the image and the symbolic as legitimate forms of semiotic practicein mathematics deserves special attention.

Shin (1994) gives two common reasons why mathematicians and scien-tists claim a limited functionality for visual images: the limitations of visualimages in presenting 'knowledge'; and the possible misuse of diagrams,such as making unwarranted assumptions in geometry. In the first instance,the meaning of mathematical visual images is seen to be less precise thanthe symbolism which allows for logical deduction from a set of establishedresults and formalized procedures for mathematical induction, proof bycontradiction, calculation of exact values and so forth. In the secondinstance, the trends displayed in abstract graphs, for example, are seen tobe incomplete descriptions which vary according to how they are displayed.There is also room for error in interpreting visual images, for example, inthe case of optical illusions. The symbolism is thus seen to be more power-ful (Lemke, 1998b) and less prone to error than visualization. Shin (1994),however, claims that the negative prejudice against the use of diagrams as ameans for establishing logical proof is not entirely warranted. Shin (1994)uses C. S. Peirce's system of logical diagrams, called 'existential graphs', todemonstrate that Venn diagrams can be used for valid proof, and thatdiagrams are not inherently misleading.

Mathematicians have in fact always drawn conclusions from visualimages, but the results have traditionally been expressed through symbolicmeans. Davis (1974: 115-116) gives a comprehensive list of reasons why thisform of practice developed in mathematics:

1 The tremendous impact of Descartes' Discours de la Methode (1637) by which geom-etry was reduced to algebra; also the subsequent turnabout wherein the medium(algebra) became the message (algebraic geometry).

2 The collapse, in the early 19th century, of the view, derived largely from limitedsense experience, that Euclidean geometry has a priori truth for the universe; thatit is the model for physical space.

3 The incompleteness of the logical structure of Euclidean geometry as discoveredin the 19th century and as corrected by Hilbert and others . . .

4 The limitations of two or three physical dimensions which form the naturalbackdrop for visual geometry.

5 The limitations of the visual ground field over which visual geometry is built asopposed to the great generality that is possible abstractly (finite geometries, com-plex geometries, etc.) when geometry has been algebraicized.

6 The limitations of the eye in its perception of mathematical 'truths' (e.g., theexistence of continuous everywhere nondifferentiable functions, optical illusions,suggestive but misleading special cases, etc.).

Davis (1974) explains that mathematics never really recovered from theimpact of the Cartesian project which involved the algebraicization ofgeometry and other fields of mathematics with geometric content. Math-

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ematicians such as Henri Poincare (1854-1912) sought to establish a legit-imate place for visual images, which he saw as being the source of intuitionand the means for keeping mathematics in contact with the real or 'con-crete world', rather than spiraling off into 'the abstract' through thesymbolism. 'As far as Poincare was concerned, without intuition the math-ematician was like a writer shackled forever in a cell with nothing butgrammar' (Galison, 2002: 302). However, Poincare came to realize that thecomplexity of the solar system defied visualization, and his efforts in effectsparked the study of chaos. But in the quest for a general rule, Poincare hadturned to visual images. Galison (2002: 302) summarizes the situation: 'Inthe sciences of the last century and a half, the pictorial and the logical havestood unstably perched, each forever suspended over the abyss of theother.'

From a semiotic perspective, it is possible to see why modern mathemat-icians prefer to use mathematical symbolism as the means for establishingresults. Descartes and later mathematicians favoured algebra as the meansfor describing curves, and the grammar of mathematical symbolismdeveloped accordingly so that these exact relations could be rearrangedand manipulated for the solutions to problems. This is possible because thegrammar of mathematical symbolism preserves nuclear configurations ofprocess/participant structures in an economical and exact fashion. Thesestructures are encoded through elaborate forms of rankshift to form mul-tiple levels of embedded process and participant configurations. Theseconfigurations are rearranged to establish the required mathematicalresults. The grammar of mathematical symbolism is thus based on a rangeof condensatory strategies which facilitate rankshift for the rearrangementof relations. As described in Chapter 4, the condensatory strategies includelimited forms of process types, the use of spatial and positional notation,the use of special symbols, the Rule of Order for operations and soforth. The result is a semiotic resource which can be used as a tool forreasoning. The dynamic semiotic resource, the one that can be rearrangedand manipulated for the proof of results, was designated to be mathemat-ical symbolism, and the grammar developed accordingly. From Descartesonwards, the emphasis shifted from the construction of curves to symbolicdescription and manipulation of relations, which were visualized usingmathematical graphs, diagrams and other forms of visual display.

Mathematical symbolism thus developed a grammar which permitted itto be used for establishing mathematical results, and so the dynamic aspectof the process of mathematicization was allocated to the symbolism. Thesame effort was not extended to the grammar of visual images, whichretained the important but seemingly less significant role of the display ofthe patterns. In addition, modern mathematics developed as a written dis-course which circulated in manuscript and print form. The grammars formathematical symbolism and visual images were certainly the product ofthe functions allocated to each resource, but they were also the productof the technology through which the texts could be produced. The advent

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132 MATHEMATICAL DISCOURSE

of the printing press was a key factor in the increased popularity of theHindu-Arabic numerals, their accompanying algorithms, and the eventualdevelopment of standardized algebraic procedures. However, the visualiza-tion of the patterns traditionally appeared in the form of static graphs anddiagrams in printed texts. These visual images could not be manipulated,and they were also time-consuming to produce. Mathematical visual imagesthus developed as a static representation of the dynamic procedures under-taken through the symbolism, the tool through which the formal reasoningtook place.

There was a loss of meaning incurred in mathematics through the use ofthe symbolism. In the process of developing a specific grammar for thefunctions to which it had been assigned, mathematical symbolism becameconcerned with a limited semantic domain which not only excluded realmsof meaning in language as explained in Chapter 4, but also excluded therealms of meaning afforded by visual semiotic. Mathematical symbolismmay be seen as more powerful, but the descriptions of continuity nonethe-less arise from categorical distinctions of variation made from a limitedrange of choices within the symbolism. For example, only certain categoriesexist for symbolic process types, participants and circumstance. Visualimages, on the other hand, are capable of representing graduations ofdifferent phenomena (Bateson, 1972; de Saussure, 1966; Lemke, 1998b;Messaris, 1994). 'Words [and at a more delicate level, mathematical symbol-ism] divide the world into black and white (and in some languages, gray),large and small, strong and weak, good and bad. Images, however, canrepresent shades of gray, ranges of size, and degrees of those external attrib-utes that viewers use in making inferences . . .' (Messaris, 1994: 121). Theloss of the input from the visual semiotic in the symbolized environment ofmathematics cannot be underestimated. Davis (1974: 119) explains:

The algebraicization of geometry must be regarded as a prosthetic device of greatpower which maps certain aspects of space into analytical symbols. The blind might be[en]abled to manipulate space through the instrumentality of these symbols, butsince one channel of sense experience is denied to the blind, one feels that a corres-ponding fraction of the mathematical world must be lost to them [and to us] . . . Theanalytic program [the symbolism], then, is a prosthetic device, acting as a surrogatefor the 'real thing' [the visual image].

This situation is rapidly changing, however, with the development of com-puter graphics which is revolutionizing the role of the visual image inmathematics and science, and, more generally, across most other fields ofhuman activity. Visual images may now be manipulated and synthesized intoday's computer environment. This revolution is considered in terms ofthe increased functionality of the visual image in Section 5.7. However, inorder to make sense of what can be achieved visually, a framework for an SFgrammar for mathematical visual images is first presented in Section 5.2.The framework, based on O'Toole's (1994, 1995) SF framework for paint-ings, is extended to account for systems of meaning in mathematical visual

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images. The systems for representational (experiential and logical), inter-personal and compositional meaning are discussed through an analysis ofan abstract graph in Sections 5.3-5.6. From this point, the changing natureof visualization in mathematics is considered from the point of view ofcomputer graphics.

5.2 SF Framework for Mathematical Visual Images

Mathematical visual images include abstract and statistical graphs, a rangeof genres of diagrams and computer-generated graphics. Abstract graphsdisplay the functional relationship between two or more variables in theform of lines, curves and three-dimensional figures. The points are plottedon a set of co-ordinate axes and include only those points which satisfy thegiven relation. Statistical graphs show the relationship between sets ofquantities in the form of bar graphs, column graphs, line graphs, histo-grams, pie charts, scatter diagrams and so forth. The term 'diagram' is usedhere in the broadest sense to include pictorial representation of entitiesand relations such as Venn diagrams, geometrical figures and other figuressuch as those found in graph theory and topology (Borowski and Borwein,1989). Computer-generated graphics include traditional forms of abstractand statistical graphs and diagrams, in addition to new forms of dynamicimages of graphs which unfold over time. The visual images generatedthrough computer graphics include fractal geometry, views of mathemat-ical models and methods, and other images in applied mathematics, suchas graphical representations of diffusion, turbulence and flow, for example(see Colonna, 1994). The evolving range of new genres of mathematicalvisual images requires investigation and documentation. However, in whatfollows the focus of attention is the discourse and grammatical systemswhich constitute the grammar of mathematical visual images. The SFframework is used for the analysis of printed mathematical abstract graphs.

The systemic model for visual images in mathematics is presented inTable 5.2(1). The content plane includes a discourse semantics stratum

Table 5.2(1) SF Model for Mathematical Visual Images

MATHEMATICAL VISUAL IMAGES

CONTENT Discourse SemanticsInter-Visual RelationsWork/Genre

DISPLAY

GrammarEpisodeFigureParts

Graphics

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134 MATHEMATICAL DISCOURSE

which is concerned with Inter-Visual Relations established across asequence of visual images such as those found in mathematics journal art-icles, books and websites (see Plate 5.2(1) and Plate 5.7(ld)). Discoursemoves across visual, linguistic and symbolic parts of the text are theorized inChapter 6. Following O'Toole's (1994: 24) ranks for paintings, the entirevisual image is the Work and the grammar is concerned with the Episode orthe configurations of process/participant and circumstance in the visualimage, the Figures which are the participants in the Episodes, and the Partsof the display. The Work is the diagram, graph or computer graphics whichresult from interactions of Episodes with Figures which are composedof Parts. The sub-division of the mathematical visual image into separateEpisodes is not always relevant as there may only be a single Episode andFigure; for example, a drawing of a line. As with language and mathemat-ical symbolism, the systems which operate on the display plane are alsoconsidered. Despite the classification of mathematical discourse into threedifferent semiotic resources as suggested by the three SFGs for language,mathematical symbolism and visual display presented in Chapters 3-5,mathematical visual images are typically multisemiotic texts which containlinguistic and symbolic elements in the form of Titles, Labels and Captions.The multisemiotic nature of mathematical visual images are investigatedaccording to metafunction in Sections 5.3-5.6

O'Toole (1994: 24) organizes the systems for paintings according to therepresentational, modal and compositional metafunctions which corres-pond to ideational, interpersonal and textual metafunctions in language.The same approach is adopted in Table 5.2(2), which displays the SFG formathematical visual images. A similar approach for mathematical statisticalgraphs and biological schematic drawings is adopted by Guo (2004a,2004b). In Table 5.2(2), the representational metafunction is sub-dividedinto the experiential and the logical. The metafunctionally organized sys-tems are displayed according to rank and strata: discourse semantics for theSequence of Graphs/Diagrams/Computer Graphics; grammar with ranksGraphs/Diagrams/Computer Graphics, Episode, Figure, Parts; and thedisplay plane which, following Lim (2002, 2004), is called the 'graphics'plane. The systems for the logical metafunction appear in the discourseand grammar strata at the rank of Graphs/Diagrams/Computer Graphicsbecause logical meaning is seen to primarily arise from sequences of math-ematical visual images and interactions of Episodes. The categories forlogical meaning are based on Halliday's (1994) system of EXPANSION,that is, elaboration, extension and enhancement. In particular, logicalmeaning in the form of spatio-temporal relations in visual images isdiscussed in Section 5.5.

The applicability of O'Toole's (1994: 24) framework for displayed artto visual images in mathematics makes sense given the relationshipsbetween mathematics and visual art and design from the times of antiquityand the Renaissance to the present. The engagement between math-ematics and art includes the mathematicization of the human figure in

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G R A M M A R OF MATHEMATICAL VISUAL IMAGES 135

DISCOURSE SEMANTICS

SEQUENCEOF GRAPHS/DIAGRAMS/COMPUTERGRAPHICS

CONCEPTUALDEVELOPMENT

Development ofmathematical contentthrough sequences ofEpisodes, Relations,Figures and Parts

ENGAGEMENT(Inter-Visual Relations)

Discourse moves throughvisual sequence (throughrepetition and change inEpisodes)

THEMATICDEVELOPMENT

Textual organization fortracking participants,processes and relations

LOGICAL RELATIONS

EXTENSION in the formof elaboration, extensionand enhancement(through multiple Time-Frames)

GRAMMAR

Units REPRESENTATIONAL/ INTERPERSONALEXPERIENTIAL

COMPOSITIONAL

GRAPHS/DIAGRAMS/COMPUTERGRAPHICS(Genre)

• Display of patterns ofrelations (as lines, curvesand three-dimensionalshapes)

• Process types:- Relational (graphs of

functions)— Transformational

x ^ f ^ y• Perceptual Reality (for

example, geometricaldisplays)

• Mathematical SymbolicReality (for example,Venn diagrams, datagraphs)

• Interplay of Episodes• Multiple Time-Frames

with TemporalUnfolding throughSpatiality

• Comparisons of patternsof variation

• Circumstance

• Metaphorical Narrative• Modality and the Degree

of Idealization,Abstraction,Quantification

• Accompanying text inthe form of Caption,Title and Labellingwhich are emphasized bySize, Positioning,Underlining and Font

• Colour• Line Width, Shading,

Line Solidarity, Slope,Arrows

• Rhythm• Curvature• Perspective• Framing• Style of Production• Nature of participants• Production process• Intricacy of display• Directionality

• Gestalt: Framing,Horizontals, Verticalsand Diagonals

• Positioning• Perspective (2D, 3D)• Use of Lines and Curves• Inter-connections

established throughsymbolism andlanguage throughLabels

• Cohesion (Parallelism,Contrast, Rhythm)

• Reference throughlanguage, symbolism)

DynamicTemporal/SpatialUnfolding (ComputerGraphics)

Table 5.2(2) Grammar and Discourse Systems for Mathematical Visual Images

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136 MATHEMATICAL DISCOURSE

Table 5.2(2) - cont

GRAMMAR

Units REPRESENTATIONAL/ INTERPERSONAL COMPOSITIONALEXPERIENTIAL

LOGICAL RELATIONS• Spatial relations (from

Interplay of Episodes)• Temporal relations

(Multiple Time-Frames)• Spatial/temporal

Relations• Types: Elaboration,

extension andenhancement

EPISODE • Interplay of Actions orRelations betweenFigures

• Portrayal of process (forexample, relation asCurves or Lines)

LOGICAL RELATIONS• Spatial relations (in

Episodes)• Spatial/temporal

Relations (MultipleTime-Frames)

• Types: Elaboration,extension andenhancement

FIGURE • Participants• Circumstantial features

PARTS • Title• Axes, Scale, Arrows• Labels• Lines, Curves, Shading,

Intersetion Points• Slope

• Prominence of Interplay(Size, Colour, Labelling,Framing, Prominence,Position and so forth)

• Display of process (Line,Curve)

Prominence of Figures(Size, Colour, Labelling,Framing, Prominence,Position and so forth)

StylizationConventionalization

• Labelling of Interplay(through symbolism,language)

• Portrayal of Process andParticipants (relativePositioning, Size ofFigure and salientfeatures as displayed byLines, Curves, Colour,Line Width, Shadings)

• Labelling of Figure(through symbolism,language)

• Stylistic Features (Size,Shape, Dynamics,Colour, and marking ofParts)

• Textual markedness(through Labelling,Colour, Size and soforth)

• •

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GRAMMAR OF MATHEMATICAL VISUAL IMAGES 137

Table 5.2(2) - cont

DISPLAY

GRAPHICS • Variations in line width,dotted lines and arrows

• Variations in font, scriptsand size

• Colour, Shading,Brightness and Hue

• Variations in line width,dotted lines and arrows

• Variations in font, scriptsand size

• Colour• Stylization and

production (computergenerated, drawn and soforth)

PerspectiveCohesiveness andContrast (throughColour, Font, Size andso forth)Spatial arrangement

terms of proportion, the development and use of perspective, 'de-representationalized artistic productions' involving geometric construc-tions and computer-generated and assisted art (for example, Danaher, 2001;Davis, 1994: 167; Emmer, 1993; Field, 1997). Given the interconnectednessbetween art and mathematics, overlaps between O'Toole's (1994) systemsand those proposed in the SF model for mathematical visual images inTable 5.2(2) occur. However, mathematical visual images developed spe-cific grammatical systems which permit the integration of the symbolismand language with the visual images. The nature of these systems is dis-cussed according to metafunction in Sections 5.3-5.6.

The SF systems documented in Table 5.2(2) are discussed in relation tothe analysis of the abstract mathematical graphs (a) and (b) in Plate 5.2(1).These graphs are the geometrical interpretations of the derivative of afunction/' (x) as the slope of a curve. The algebraic form of the derivativeis considered in Chapter 4. Abstract graphs are chosen for the analysisas these forms of graphs are central to mathematical descriptions ofpatterns of variation and, further to this, these graphs employ uniquestrategies to encode experiential meaning. The application of the frame-work for images generated by computer graphics (see Plates 5.7(la-d)) isalso considered.

The investigation of the choices in the grammatical systems for math-ematical symbolism in Chapter 4 reveals a contraction and simultaneousexpansion of experiential meaning, a narrow range of interpersonal mean-ing, a refinement of textual meaning and specific forms of logical meaning.In what follows, it will become apparent that these types of meanings arelargely replicated in conventionalized forms of mathematical diagramsand graphs. As mathematical symbolism dispensed with different formsof meaning in the quest to encode and rearrange patterns of relations,it appears that a similar trend occurred in mathematical visual representa-tions which were designed to function in collaboration with the symbolicdescriptions. The dominant metafunction in the visual display is repre-sentational (experiential and logical) meaning, which is aided by theelimination of what is considered to be superfluous contextual informa-tion. Compositional styles of the visual images are conventionalized, and

• •

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"Si Interpretation of the Derivative as the Slope of a Tangent

In Section 2.6 we defined the tangent line to the curve y —f(x) at the point P(a,f(a)) tobe the line that passes through P and has slope m given by Equation 1. Since, by Defini-tion 2, this is the same as the derivative /'(«), we can now say the following.

The tangent line to y — f(x) at (o,/(a)) is the line through (a,/(a)) whose slope isequal to /'(a), the derivative of / at a.

Thus, the geometric interpretation of a derivative [as defined by either (2) or (3)] is asshown in Figure 1.

FIGURE 1Geometric interpretation

of the derivative

Plate 5.2(1) Interpretation of the Derivative as the Slope of a Tangent(Stewart, 1999: 130)

interpersonal meaning is contracted and direct. Once the viewer's atten-tion is focused on the relevant parts of the mathematics visual image, therepresentational meaning is the major function of the mathematical visualdisplay. The experiential meaning is concerned with the Episodes andrelevant Parts of the Figures, and logical meaning is largely concerned withspatio-temporal relations.

Computer graphics are contributing to the development of new sys-tems of meaning; for example, Colour, Shading, Brightness and Hue. Thesystems for representational, interpersonal and compositional meaningsthrough computer graphics are included in the grammar and discoursesystems in Table 5.2(2). In what follows, the metafunctionally based dis-course and grammatical systems for visual images in mathematics arediscussed through the analysis of Plate 5.2(1). In the final section, thecomputer graphics displayed in Plate 5.7(lc-d) are discussed. However,the framework is not complete, and further research is needed to docu-ment the systems through which meaning in mathematical images ismade. Table 5.2(1) and Guo's (2004a, 2004b) SFGs for mathematicalstatistical graphs and biological schematic drawings are first steps in thisdirection.

If we use the point-slope form of the equation of a line, we can write an equation of thetangent line to the curve y = f(x) at the point («,/(«)):

y- / (a)=/ ' ( f l )U-f l)

= slope of tangent at P= slope of curve at P

= slope of tangent at P= slope of curve at P

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5.3 Interpersonally Orientating the Viewer

The choices from the systems for interpersonal function in a visual imagedetermine how the viewer engages with the Work. O'Toole (1994: 7), forexample, carefully traces how the viewer becomes involved with Botticelli'sPrimavera through choices of Rhythm, Gaze, Frame, Light and Perspective:'The painting has a gentle, undulating rhythm which is in harmony withthe graceful gestures and stance of the figures, with the flow of drapedclothing, with the placing of fruit, flowers and foliage and with the easilyblending colours.' In the painting, Venus engages the viewer 'directly withher eyes' and 'even signals a greeting or benediction' with 'the gesture ofher right hand, the tilt of her head, and the poise of her body' (1994: 8).O'Toole discusses how the Compositional forces at work, such as the use ofconcentric Frames and Colour, also contribute to the viewer's engagementwith Venus. Further to this, the position of other Figures demarcates thecentrality of Venus. As O'Toole (1994: 11) comments, if the painting isviewed Episode by Episode, there is an interplay of modalities with the'Rhythm changing from episode to episode as our eye moves from right toleft across the canvas'. The result is an exquisite unfolding of Botticelli'smasterpiece.

Similarly, the viewer's gaze is also directed to certain aspects of math-ematical visual images. The result is not a gentle rhythmic engagementthrough the subtle use of Rhythm, Gaze, Frame, Light and Perspective,however. Rather, the choices for interpersonal meaning function directly sothat the viewer immediately engages with the significant aspects of therepresentational meaning of the graph or diagram. Botticelli's carefulchoices for individualization of the Figures through Stylization may be con-trasted to the uniformity of the mathematical visual images through Con-ventionalization. This feature is functional in mathematical discourse as itenables experienced viewers to immediately apprehend representationalmeaning of a display and it also lowers the likelihood of misinterpretation.The interpersonal meaning of traditional mathematical visual images isbest examined through an analysis of Plate 5.2(1) using the SF frameworkprovided in Table 5.2(2).

The geometrical interpretation of the derivative of the curve y = f(x)as the slope of tangent line at the point P(a,f(a)} is displayed in Plate5.2(1). There are two graphs for the two equivalent algebraic forms of the

derivative: (a) and In

what follows, Graph (a) is analysed for interpersonal meaning. The inter-personal meaning arising from the Sequence of Graphs for (a) and (b) isalso examined.

Interpersonally, at the rank of Work the viewer is presented with a Meta-phorical Narrative where the Figure of the Line intersects the Figure of theCurve in Graph (a). The Episode or 'plot' attracts the viewer's attentionthrough the Prominence of the Individual Figures and the Prominence of

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the Interplay; that is, the straight line is 'marked' through the Colour red(not displayed here) and attention is also drawn through the choices forSlope, Line Width and Line Solidarity. The viewer's attention is also drawnto the Curve through the dynamic aspect of the Curvature, particularly the'flourish' at the tail of the curve positioned around the origin of the axeswhich is marked with the Label '0'. Closer inspection reveals that the curveis Labelled y—f(x). This Label attracts attention through the curved arrowwhich is used to point directly to the Curve. The viewer is also drawn to thecritical aspect of the Metaphorical Narrative, the point of intersectionbetween the line and the curve which is marked by a dot and Labelled pointP in italicized font. The Episode involving the Line and Curve appears asdynamic because the Interplay is framed by the set of horizontal and verti-cal axes, respectively Labelled x and y. After the initial impact, the Rhythmachieved through the Slope of the Line and the Curvature of the Curvecreates a reading path which initially tends from left to right.

A series of 'minor' Episodes are marked by the dotted vertical Linespositioned above the values a and a + h on the x axis in Graph (a). Thesedotted lines intersect the Curve, and a triangle is formed by the horizontalLine Labelled h and the hypotenuse extending from point P. The minorEpisodes lack the Prominence of the Interplay of the major Episode as theColour of the lines is yellow (not displayed here) and the Line Solidarity isdotted rather than solid. In addition, the hypotenuse of the triangle ismarked by the Colour blue (not displayed here), which is not prominentbecause of its Position and close proximity to the Curve. These minorEpisodes function as rankshifted instances of Circumstance in Graph (a).

The x and y axes provide the visual context for the major and minorEpisodes, and attention is drawn to certain parts of the graph through theTitle, Labelling and Colour of the Figures and Parts. Apart from this, thecontext is provided by the symbolic and linguistic Caption, and the sur-rounding linguistic text. Captions and Titles of Graphs and Diagrams aretypically placed in a prominent position and usually are marked throughSize and Font. These Labels connect the Episodes, Figures and Parts to thesymbolic and linguistic descriptions appearing in the Captions for Graphs(a) and (b).

The Style of Production indicates that the graph has been professionallyproduced through a specialized software application. At the rank of Part,Stylization and Conventionalization are standardized through the means ofproduction. The graph appears to be contextually independent as back-ground information and 'noise' are erased. The graph appears as an abstrac-tion which consists of generalized participants involved in a metaphoricallyabstract interaction. The Modality and the degree of Idealization, Abstrac-tion and Quantification are high, leading to a maximal truth-value. In otherwords, the Graph appears to have a high degree of certainty as to the cor-rectness of the representational meaning it encodes. The Modality value isthe cultural value assigned to those choices made from the available systemsfor interpersonal meaning in mathematical graphs.

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At the rank of discourse semantics, the Sequence of Graphs (a) and (b)in Plate 5.2(1) involves repetition. The graphs are identical with the excep-tion of the Labels, which change from a and a + h to a and x, and h and/( a +K) —f(a) to x— aandf(x) -/(«)• In this way, a minor Episode, which appearsas Circumstance in Graph (a), is given prominence through Inter-VisualRelations while the nature of interpersonal meaning is reinforced with theexact repetition of the major Episode. The reason for marking the promin-ence of the Circumstance in Graphs (a) and (b) is explained in Section 5.3,which is concerned with experiential meaning.

In statistical graphs and mathematical diagrams, the viewer's attention issimilarly drawn to salient parts of the visual display through Captions,Labelling, Size and features such as Colour, Slope, Line Width, Shadingand Perspective. The viewer's gaze is also drawn to certain points of thediagram through Perspective. The degree of Abstraction and the level ofIntricacy of the figure interacts modally with the Style of Production. In lineand bar graphs, the viewer is engaged directly through certain featuressuch as Shading, Patterns and Colours. Parts may be labelled and the Slopemay be exaggerated. The 'art' of presenting information lies in makingsalient particular Parts of the Graph so that the viewer's gaze is directed tothose parts of the display. Misleading graphs are those which presentinformation in such a fashion that an unwarranted Prominence is attachedto particular dimensions of the display. Prominence is achieved throughspecific choices for interpersonal, experiential and/or compositionmeaning. The visual display of information in fields such as advertising,newspaper discourse, politics and so forth is worthy of further investigation.

From the analysis of choice from systems for interpersonal meaning inPlate 5.2(1), three features of mathematical visual images become evident.First, the viewer is explicitly directed to the relevant parts of the visualdisplay through the nature of the interpersonal choices at each rank. Inter-personal meaning is not a delicate balance of a variety of unobtrusive strat-egies working in harmony as for example, in O'Toole's (1994) descriptionsof the Primavera. The important Episodes, Figures and Parts are explicitlymarked through choices for Labels, Position, Colour, Framing, Line Width,Line Solidarity and so forth.

Second, the Modality or the truth-value of the visual image is high, butthis is not because mathematical visual images faithfully depict material'reality'. While mathematical visual images relate to our perceptual under-standing of reality, there is not the same degree of correspondence asportrayed in photographs, for example. Rather the high level of certaintyattached to mathematical and scientific representations has been culturallyassigned: 'visual modality rests on culturally and historically determinedstandards of what is real and what is not, and not on the objective cor-respondence of the visual image to a reality defined independently of it'(Kress and van Leeuwen, 1996: 168). The perfection and exactness of thevisual displays, the lack of contextual information, the Style of Productionand the metaphorical and abstract nature of the Episodes, Figures and

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their Parts mean that the certainty of the mathematical and scientific visualimages is difficult to counter. The interpersonal meaning of the mathemat-ical visual images in fact replicates that found in mathematical symbolism,which similarly selects for a high value of Modality. The appearance ofcertainty in mathematical discourse arises from the nature of the selectionsfor interpersonal and experiential meaning (the narrow range of partici-pant, process and circumstance selections) and the truth-value which hasbeen culturally accorded to these selections. These patterns of meaning arereinforced across the linguistic, symbolic and visual parts of mathematicaldiscourse.

Third, the nature of interpersonal engagement with mathematical visualimages is sharp and direct, and the social relations which are consequentlyestablished between the writer/producer and the viewer are unequal. Thewriter directs the reader to certain parts of the graph so that the experien-tial and logical meanings become the dominant metafunctions once theshort, sharp interpersonal impact subsides. The discourse creates a highlevel of certainty with respect to the representational meaning subsequentlyrealized. Moving from the interpersonal orientation of mathematical visualimages, the nature of the experiential meaning is explored with respect toGraphs (a) and (b) in Plate 5.2(1).

5.4 Visual Construction of Experiential Meaning

Visual images are typically conceived as dealing with the 'concrete realworld' rather than the 'abstract world' of the symbolism (for example,Galison, 2002). The reason for this view is that graphs replicate our per-ceptual experience of the world. Mathematical visual images are, however,concerned with particular forms of experiential meaning. For example, theabstract graphs of functions and transformations such as those shown inPlate 5.2(1) display patterns of relations as lines, curves and three-dimensional objects. Mathematical visual images are also concerned withperceptual reality in the form of geometrical displays, and mathematicalsymbolic reality in the form of statistical data graphs, Venn diagrams andother forms of diagrams which have no counterpart in the real world.While the form of the display, the curves, lines and three-dimensionalshapes are intuitively accessible, it becomes evident in what follows that anunderstanding of the grammar of mathematical visual images is necessaryfor a reading of the experiential meaning realized by images. This isbecause the grammar of mathematical visual images is designed to functionwith the grammars of mathematical symbolism and language.

Attention is directed to salient features of the mathematical visual imagesand the Titles, Labels and Captions. Once the immediate interpersonalimpact subsides, representational meaning takes over. The viewer isengaged with the experiential and logical meanings of the graph or dia-gram which are pre-empted by the selections for interpersonal meaning.The conventionality of the visual display allows experienced viewers to

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access the representational meaning of graphs and diagrams at a glance. Inwhat follows, the experiential meaning of Graphs (a) and (b) in Plate5.2(1) is analysed.

At the rank of Work, Graphs (a) and (b) each consist of an Episodeinvolving two Figures, the Curve y = f(x) and the Line which depicts theSlope of the Curve at point P. The Episode is an Interplay of Relationsbetween the two Figures. The Line appears in red (in the original version)which compensates for the lack of Labelling. The point of intersectionarising from the interaction of the two Figures is marked with P. This Part ofthe Figures is prominent through the marking of the point with a dot andthe italicized Label P. In Graph (a), the minor Episodes in the form ofCircumstance are emphasized with the Labelling of points a and a + h onthe x-axis and the distances h and/(a+ h) -/(«) which are marked by curlybrackets. The hypotenuse of the right triangle is drawn to form a trianglewhere the lengths of the other two sides are marked as the distancebetween two points.

The graph is partly accessible as it replicates our perceptual understand-ing of spatial relations; for example, the Line, Curve, and the lengths ofthe sides of the triangle as the distances between two points in the righttriangle formed by the line segments. However, the experiential mean-ing of abstract Graphs (a) and (b) in Plate 5.2(1) requires a knowledge ofthe grammar of mathematical visual display and the grammar of math-ematical symbolism and the grammar of language. The Captions, Titles andthe surrounding co-text, for example, are linguistic and symbolic. Tounderstand what the graph is depicting, the viewer needs to know thedefinition of the derivative which is given in the symbolic form: (a)

forms are re-instated linguistically in Plate 5.2(1) as the 'slope of tangent atP' and 'slope of curve at P\ Figure 1 Caption reads 'Geometric interpret-ation of the derivative'. The Labels on the Graphs (a) and (b) are symbolic.Mathematical visual images are multisemiotic texts.

The viewer must also understand the grammar of mathematical abstractgraphs. In this case, the Curve in Graph (a) represents the relationobtained by the mapping/: x—*f(x) where y =f(x). The Curve is the set ofordered pairs (x,f(x)) such that/(x) is the value of the function for x. Eachvalue of x corresponds to a value of f(x) which is y. Strictly speaking, the xand y axes, the Curve, Line and the Axes in Plate 5.2(1) should also haveArrows to indicate ongoing continuity. The function f(x) in Plate 5.2(1)remains in generalized form, rather than a specific case (for example,/(x)= x3 + 1 or y = x" + 1). The general values /(a) and /(a + h) are indicated bythe dotted vertical line segments which are drawn from a and a + h tointersect the curve y=f(x). In Graph (b), a + h has been replaced by thegeneral value x, and this has repercussions for the lengths of the sides of thetriangles which are conceptualized as the distance between two points.

The relations captured symbolically appear in the form of a Curve and a

a n d ( b ) T h e s y m b o l i c

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Line in Graph (a). While there is an element of dynamism associated withthe Curve and the Line, these visual displays may nonetheless appear asfixed entities, the Curve and the Line which intersect at point P. The sym-bolic description y = f(x) is dynamic; it represents the mapping/: x —> y,where f(x) is realized through configurations of the rankshifted Operative

processes and participants, The complex of the

relations captured by the symbolism is displayed in the form of an entity,the Curve. Despite appearances, the curve y = f(x) is dynamic as a set ofrelations which unfold temporally and spatially, and this dynamism isFramed by the axes x and y. Moreover, Graphs (a) and (b) in Plate 5.2(1)are frozen representations of the dynamic processes h —> 0 and x —> a. Thatis, the Graphs depict discrete instances of time for particular values of h andx. Further to this, the tangent line at point P represents the slope of thecurve at the point when these limits h —> 0 and x —> a are reached. Inessence, there are Multiple Time-Frames juxtaposed in Graphs (a) and (b)in Plate 5.2(1). These time frames are:

1 y—f(x) unfolds as a set of mappings/: x—*y.2 The processes h —> 0 and x —> a unfold.3 The limit is reached at point P.4 The Line is the tangent line at point P.

Variations in temporality, realized symbolically as h —> 0 and x —> a in

, are given a spatial interpretation in Graphs (a)

and (b). Temporal relations are visualized at instances of time in terms ofdecreasing distances. Continuity is made discrete in the form of spatialrelations at different points of time. Once the limit at point P is reached,the tangent line may be drawn. The Interplay of Episodes and Circum-stance result in Multiple Time-Frames with the Temporal Unfolding beingrealized through Spatiality. The repercussions in terms of logical meaningare discussed in Section 5.5.

The interpretation of experiential meaning in abstract graphs relies on aknowledge of the grammar of mathematical visual display, symbolic nota-tion and language, and the intersemiotic relations between the threeresources. In the case of Graphs (a) and (b), this includes the notion of agraph as a set of points, algebraic functional notation, the notion of Carte-sian co-ordinates whereby each point of the curve represents an orderedpair (x, f(x)), the notion of a function as a mapping from x —» f(x), thegraph as the set of these mappings, the geometrical interpretation off(x)values, the algebraic definition of the derivative as a limit, and the notionof the tangent as a line which intersects the curve at one point only. Inaddition, the reader must recognize that the graphical display is a partialrepresentation of the pattern of covariation described by f(x), and thegraph shows discrete instances of time as the limits h —> 0 and x —> a are

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approached. The Inter-Visual Relations between Graphs (a) and (b) repeatthe Interplay of Episodes with the exception that Circumstance changesfrom a and a + h to a and x. In this way, Circumstance is given prominence.The reason for Graph (b) becomes apparent in the linguistic text: Graph(b) establishes the equation of the tangent using the 'point-slope form ofthe equation of a line' (Stewart, 1999: 130).

Although visual display in mathematics is more intuitive than thesymbolic descriptions, the experiential meaning encoded within the visualdisplay is complex. Plate 5.2(1) involves Multiple Time-Frames where thetemporal unfolding of the processes is given a spatial interpretationat different instances of time. To access this experiential meaning, thegrammar of the graph and the relations in the symbolic and linguistic textmust be understood. The experiential meaning also depends on previouslyestablished mathematical results which have been derived symbolically andvisually. The Inter-Visual Relations are significant in accessing theexperiential meaning of Plate 5.2(1), which includes the ConceptualDevelopment of the derivative of a function. The meaning of visual imagesreplicates that found in mathematical symbolism. That is, reasoningdepends on previously established results which are implicit. In whatfollows, the logical meaning of mathematical visual images is discussed inrelation to Graphs (a) and (b) in Plate 5.2(1).

5.5 Reasoning through Mathematical Visual Images

Halliday (1994: 328-329) includes a linguistic category for spatial relationsin terms of 'extent' and 'place'; for example, 'here', 'there', 'as far as' and'wherever'. This category, corresponding to the semantics of spatiality, alsofunctions as a rhetorical organizing device rather than a direct reference toactual space or place; for example, 'as far as I can see, that is not possible'.Martin's (1992: 179) classification for logical relations in language, how-ever, does not include a category for spatial relations. The types of relationsare: Additive (addition, alternation), Comparative (similarity, contrast),Temporal (simultaneous, successive) and Consequential (purpose, condi-tion, consequence, concession, manner). It becomes apparent that logicalmeaning based on the semantics of spatiality is only minimally developed inlanguage. While the grammar is richer in terms of temporal relations in theform of structural conjunctions and cohesive conjunctive adjuncts (forexample, 'and', 'next', 'now', 'then' and 'simultaneously'), language didnot develop the same potentiality for spatial logical relations, presumablybecause visual images are functional in this respect. Our perceptual appar-atus permits 'logical deductions' based on spatiality to be performedthrough visual means rather than depending upon formalized linguisticand symbolic selections. The potential for logical meaning through visualimages is discussed below.

Graphs (a) and (b) in Plate 5.2(1) represent an instance of time in anunfolding dynamic process as the limits h —> 0 and x —> a are approached.

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This is only ascertainable from the linguistic Caption which explains

that the derivative is given by (a) and (b)

The symbolism captures the process of the derivative

as the result of a limiting process. The visual image, however, gives a spatialinterpretation of that limiting relation at an instance of time. Graphs (a)and (b) also depict the final result of that limiting process: the tangent atpoint P. Therefore, the temporal relations have a spatial dimension inGraphs (a) and (b).

Consider the temporal frameworks represented in Graph (a): y =f(x) as amapping/: x —> y, the limit as h —» 0, the reaching of that limit at point P,and the slope of the curve at the point P which is represented by thetangent line. The logical inference is that as h —> 0, a + h moves towards thevalue a over time with the result that the spatial distance h approaches zero.Thus the distance between the two points, displayed as lengths of sides ofthe triangle, approaches zero over time. The graph thus contains a visual-spatial interpretation of relations over time. Furthermore, once the limit isreached at point P, the tangent line as the slope of the curve at that pointappears. The visual display encodes a multidimensional Time-Frame. Inother words, the graph is used to reason with. Graph (a) encodes logicalmeaning in terms of temporal relations which are expressed visually asspatial relations. The nature of this relationship permits logical meaning inGraph (a) in the form of elaborative type relations ('in other words'becomes 'in other pictures'); additive type relations ('this and this'); andcausal relations (cause in terms of reason 'so'). The logical relationacross Graphs (a) and (b) takes the form of causal relations 'therefore' toestablish the equation of the tangent line in point-slope form.

The dynamic unfolding of the process, as afforded by computer graphics,makes the ability to reason perceptually even more encompassing. Thedynamic displays of the digital medium encode temporal relations throughan unfolding spatial form, and also permit the manipulation of visual pat-terns in ways previously unimaginable. The potential for logical reasoningusing computer graphics is explored in Section 5.7. In the next section,compositional meaning in terms of the organization of experiential, logicaland interpersonal meaning in mathematical visual images is considered.

5.6 Compositional Meaning and Conventionalized Styles of Organization

The textual organization of the mathematical visual images is conventional-ized in order to permit the viewer to engage immediately with the experien-tial and logical meaning which is encoded in precise and exact form.O'Toole (1994: 22) explains: 'decisions [in paintings] about the arrange-ment of forms within the pictorial space, about line and rhythm and colourrelationships, have been made by the artist in order to convey more effect-

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ively and more memorably the represented subject and to make for a moredynamic modal relation with the viewer'. With mathematical visual images,the interpersonal meaning is direct, and this engagement is aided by thecompositional arrangement of the Work, Episodes, Figures and Parts. Aswith mathematical symbolism, the narrow range of interpersonal meaningsis accompanied by unambiguously encoded experiential meanings incondensed format in mathematical visual images. In what follows, thecompositional meaning of Graphs (a) and (b) in Plate 5.2(1) is discussed.

At the rank of Work, Gestalt is the term used for the complex relationsbetween a whole visual image and its parts: ' "Gestalt theory" claims that wealways have an overall perception of forms and objects and that when wefocus on their parts we perceive them in relation to the whole' (O'Toole,1994: 23). The Work is the Graphs (a) and (b), where the Episode isFramed by the x and y axes. These axes contribute to the stability ofthe visual display. The Curve and Lines appear as dynamic elements in thevisual display because they are diagonally positioned, thus attracting theviewer's attention. Diagonal elements are similarly formed by perspective ingeometrical representations, and diagonal alignments visually trace trendsin the data and in bar and line graphs.

Graphs (a) and (b) are organized through Labels which form Cohesivelinks to the main body of the text. The Curve and the Line are perfectlyPositioned against the backdrop of the x and y axes, and Balance isachieved through these spatial positions and the curvature of the Curve. Atthe rank of Episode, the Interplay is explicitly marked through the Labelfor point P. At the rank of Figure and Parts, Labels are attached to themajor participants and circumstance as discussed above. There is max-imum Cohesion arising from the explicit ordering of each Part of Graphs(a) and (b).

Inter-Visual Relations are established through the spatial position andorganization of Graphs (a) and (b). As may be seen in Plate 5.2(1), the twographs are placed next to each other, and they involve direct repetition ofthe Metaphorical Narrative. Inter-Visual Relations later establish the Linein Graphs (a) and (b) (which represents the derivative as the tangent of thecurve at point P) as the instantaneous rate of change (Stewart, 1999: 132).The Conventionalization of visual images in mathematics is significantin establishing such Inter-Visual Relations. The Slope of the Line is con-ventionalized to correspond to the semiotics of physical/psychologicalperception. The steeper the Slope of the Line, the greater the rate ofchange, and thus the gradient of the tangent. A small change in x producesa large positive or negative change in y. That is, the derivative/' (x) as Slopeof the tangent Line as the instantaneous rate of change at point P iscompositionally organized to correspond to perceptual reality.

At the ranks of Figure and Part, there is an exact relationship betweencompositional and experiential meaning of the Figures in abstract graphs.In Graphs (a) and (b), the experiential meaning of each point of the Curvecorresponds exactly to its spatial location with respect to the x and y axes.

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Compositionally, Spatial Position has been connected to experientialmeaning in terms of sets of co-ordinates; that is, the placement of a point isexactly determined by the x co-ordinate value and the y co-ordinate value.The resulting Curve or Line is the set of all such 'points' in the mappings/: x —> y. This means that Positioning, Size and Shape of the Figure aredetermined by the Scale selection on the axes.

Choices of compositional meaning for the organization of the math-ematical visual images function alongside selections for interpersonalmeaning to direct the viewer to the significant parts of the visual text. Interms of the display stratum, compositional meaning is aided by Perspec-tive, Colour and Spatial Arrangement. Experiential meaning is efficientlyencoded, and the patterns are immediately ascertained. Often the visualimage is consulted before the viewer attends to the symbolic and linguisticparts of the mathematics text.

The analysis of interpersonal, experiential, logical and compositionalmeaning in mathematical visual images is incomplete as only abstractgraphs are considered. Mathematical visual images also include numerousother genres in the forms of statistical graphs, diagrams and other forms ofvisual display in the form of computer-generated dynamic images. In whatfollows, the general nature of images realized through computer graphicsis discussed, and the implications for a changing role of visualization inmathematics through the medium of computer technology are considered.

5.7 Computer Graphics and the New Image of Mathematics

The influence of computers is such that they have given rise to 'a newworld view which regards the physical world not as a set of geometricalharmonies, nor as a machine, but as a computational process' (Davies, 1990:23—24). One outstanding feature of the shift to computation is the ability ofcomputers to generate, represent and manipulate the numerical results asdynamic visual patterns which unfold over time. Davis (1974: 115-116)predicted the resurgence of the visual image in mathematics through thedevelopment of computer graphics, and this is indeed proving to be thecase. The visual image plays an increasingly important role in differentbranches of mathematics, as evidenced in the modelling of non-lineardynamical systems. The impact of increased computational ability is dis-cussed in relation to the revolution which is taking place through computergraphics.

Following Foley et al. (1990), computer graphics is denned as ' "the pic-torial synthesis of real or imaginary objects from their computer basedmodels" and covers working areas like rendering, scientific visualization,animation, graphics in documents, or interactive user interfaces' (GroB,1994: 2). The visualization process consists of 'the transformation ofnumerical data from experiments or simulation [via mathematical models]into visual information' (Grave and Le Lous, 1994: 12). The other relevantprocess is 'image processing', which deals with 'the management, coding,

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and manipulation of images, the analysis of scenes, and the reconstructionof 3D objects from their 2D projectional presentations' (GroB, 1994: 2).Software programs encode the data displayed visually on the computerscreen.

The increasing sophistication of computer graphics may be appreciatedthrough the images from the 1970s to the present which are displayedin Plates 5.7 (la-d). Computer graphics have developed from picturesconsisting of (i) predominately lines and (ii) patterns arising from arraysof dots to (iii) coloured kaleidoscope images of complex patterns and(iv) displays that replicate the detailed intricacy of photographs. Forexample, the coloured fractal patterns in Plate 5.7 (Ic) (reproduced inblack and white here) represent an entirely new 'image' of Newton'smethod for finding the solution to equations, and the coloured imagesand the textured surfaces of the three-dimensional graphs in Plate 5.7(Id) (reproduced in black and white) illustrate the detail now possible incomputer-generated images.

The increased applications of computer graphics in applied mathematicsand science for the interpretation of complex data sets relate to humancapabilities of seeing visual patterns: 'Because visual analysis techniques areparticularly well suited to the human cognitive capabilities, more emphasishas been placed on visual analysis tools for understanding computer simu-lations of complex phenomena' (Watson and Walatka, 1994: 7). Humanscannot process the information at the same rate if presented with thesymbolic output generated by supercomputer simulations or high-poweredscientific instruments. As Colonna (1994: 184) explains:

Vision is the most highly developed of our human senses for reception, isolation andunderstanding of information about our environment. Vision provides a global recep-tion of coloured shapes against a changing, moving, and noise-filled background. Theidea of using the eye as the main tool in the analysis of numerical results is thereforequite natural.

Computer graphics are increasingly being used for a range of functions; forexample, the visualization of large data sets (for example, in applied math-ematics, geology, meteorology and engineering), the creation of three-dimensional objects, the construction of view-dependent visualizations,multidimensional images of motion, and visualizations and reconstructionsin medicine and biology (for example, see Moorhead et al., 2002). Math-ematical ideas such as interpolation, approximation with polynomials, frac-tals and so forth have also rapidly moved into highly developed computersystems for military and industrial applications. Computer graphics areleading to new insights of complex mathematical problems (for example,topological problems). However, the symbolic still presides over the visualin higher dimensional mathematical theories. In addition, computergraphics have resulted in the growth of computer art and animated film,especially in the field of three-dimensional images (Danaher, 2001; Davis,1974): 'Over the last 20 years, all phases of film production have been

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Plate 5.7(la) Stills from a computer-made movie: wrapping a rectangle toform a torus (Courtesy T. Banchoff and C. M. Strauss) (Davis, 1974: 126)

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Plate 5.7(lb) MATLAB graphics, circa 1985 (courtesy of Philip J. Davis)

changed by computer technology' especially postproduction which has'been transformed by computer-generated imagery ("CGI")' (Bordell andThompson, 2001:26).

Davis (1974) suggested that 'visual theorems', where the traditional formof proof is not required, will become acceptable in mathematics: 'A figure,together with its rule of generation, is automatically and without furtherado a definition, theorem and proof of "the perceived type" ' (ibid.: 122).Davis (2003) sees that this trend has yet to develop to its full potential.Whether what is suggested visually requires symbolic proof 'seems todepend on the particular mathematical culture within which the pictureshave been derived' (ibid.). Colonna (1994) suggests that the notion ofexperiment rather than proof is more relevant in relation to computergraphics. Images generated through software programs are seen as tools for'virtual experimentation' in a computerized environment. The traditionalapproach based on 'real experimentation' involves finding analytical orsymbolic solutions to the model equations constructed from experimentaldata in order to explain the behaviour of physical systems. However, simplechaotic systems, for example, are difficult to capture analytically. The com-puterized numerical approach entails generating and visualizing numerical

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Plate 5.7(lc) The Complex Boundaries Of Newton's Method. (Gleick,1987: insert)

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Plate 5.7(ld) Graphical and Diagrammatic Display of Patterns (Bergeat al, 2003: 194)

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solutions, rather than finding an analytical solution to the model equa-tions: 'Virtual experimentation, therefore, assumes that the underlyinganalytical models of the simulation are correct. . . and explores the rangeof behaviours produced by that model' (ibid.: 183-184). The researchercan change the parameters or request different views of the numericalresults to explore the behaviour of the system: 'The researcher, the numer-ical calculations, and the displayed images form a feedback loop throughwhich the complex behaviour of the simulated system is explored' (ibid.:184).

Colonna (1994) sees 'picture synthesis' as a scientific tool where col-oured global patterns, rather than traditional point-by-point descriptions,may be generated: 'Complex forms will become distinct and the scientistmay be able to ascertain a hidden order in the numerical results. As othershave pointed out: Scientific visualization is the art of making the unseen visible1

(ibid.: 184). This statement echoes Newton's efforts at making the invisible'visible' through mathematical descriptions. The semiotic in question isnow the visual rather than the symbolic code. Through juxtaposition,connection, transformation and various other forms of manipulation, thevarious components 'come together to create a useful whole, the scientist,numerical calculation, and picture synthesis all work together to form ascientific instrument. . .' (ibid.: 184-185).

The new functions of computer graphics arises from the expandedmeaning potential afforded by these forms of visual image and the easewith which patterns can be generated, rearranged and combined. Whilestatic forms of graphs and diagrams in handwritten and print format havebeen time-consuming to produce, computer-generated visual images nowencode dynamic representations with minimal effort. New systems of mean-ing such as Colour Saturation, Hue, Shading and Brightness are playing animportant role in computer-generated images (Danaher, 2001; Levkowitz,1997). The visual image has thus evolved into a dynamic display that can beeasily manipulated in the same way that symbolic mathematics developed tobe the semiotic that could be rearranged in print format. However, visual-izations of the continuous patterns of relations have the advantage that theyencode spatial and temporal dimensions. In addition, the patterns are gen-erated with minimal effort. The development of mathematics is tied to theavailable technology, which has historically been limited to the pen, paper,the printing press and three-dimensional mathematical models. Computertechnology extends the meaning potential of mathematics in the digitalizedmedium.

New scientific methods arising from computer graphics may extendbeyond those suggested by Colonna (1994: 191). Traditionally data is firstfiltered through the lens of numerical quantification (through, forexample, the experiment). Following this, the data is distilled into general-ized mathematical models which are solved and visualized. This led to ade-contextualization and reduction of the complexity of the phenomenonunder study: 'The strength and novelty of seventeenth century science,

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both theoretical and experimental, was in its capacity to take things out ofcontext and analyze their relations in ideal isolation' (Funkenstein, 1986:75). However, the complexity of data can more easily be maintained in acomputerized 'virtual' world, and reasoning can take place visually throughthe dimensions of space and time, rather being based on the ordering ofspace and time through language and symbolism. The potential for logicalmeaning in digitalized images is further considered below.

The significant aspect of the digital computerization process is that visualimages may be reconfigured in forms which unfold dynamically as asequence of steps over time rather than a static display where instances oftime are displayed discretely. Mathematical symbolism developed grammat-ical strategies to ensure that logical sequences could unfold progressivelywith ease. Now visual images in the form of computer graphics possess anadded potential for sequential processing. Spatial and temporal dynamicsare afforded through digital encoding of information in a computerizedenvironment. A graph can be seen as an unfolding of relations throughtime and space on a computer screen. In addition, mathematical visualdisplays may be synthesized, rearranged and thus used to establish math-ematical results in much the same way that symbolism evolved to do. Thedifference is that computer graphics combine spatiality and temporality.The mathematician is no longer working as someone deprived of the use ofsight. The dependence on the grammar of mathematical symbolism (whichevolved from the grammatical systems of language) is supplemented withvisual forms of semiosis, which may develop further grammatical systemsfor encoding meaning.

As visual images increasingly take their place alongside mathematicalsymbolism, this semiotic resource may be seen to offer more than anintuitive understanding of the phenomena and a means for experimenta-tion and synthesis. Computer graphics may evolve formal systems for rea-soning. The intuition arises naturally as the visual image relates to ourperceptual understanding of the world. While this contribution isextremely productive for insights into the nature of the mathematics prob-lem, language and mathematical symbolism have formalized 'intuition'through the development of systems for logical meaning. The linguisticand symbolic function to impose order, and one major contribution tothat order, the part that is significant in mathematics, is logical reasoning.Language and mathematical symbolism possess systems for logical mean-ing in the form of elaboration-type relations (re-statement in the formof apposition or clarification), extension-type relations (additative- andvariation-type relations) and enhancement-type relations (predominantlycausal-conditional- and spatio-temporal-type relations). In order to makelogical connections in what is seen to be a 'valid proof in mathematics,reasoning is established in a step-by-step fashion through choices from theavailable linguistic and symbolic systems (for example, 'if this', 'thenthis'). A sequence is seen to be necessary for logical reasoning. Indeed, amathematical proof is considered to be 'a sequence of steps or statements

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that establish the truth of a proposition' (Wilkes, 1986: 1225). Mathemat-ical proof is thus equated to what can be achieved using language andmathematical symbolism. Our perceptual apparatus lets us perceive pat-terns, but temporal- and cause-effect-type relations have been formalizedusing language and mathematical symbolism. With the advent of com-puter graphics, however, such step-by-step reasoning may be possible usingvisual images which have the potential to unfold sequentially over time. Itremains to be seen if computer graphics will lead to the development offormalized systems for logical meaning in the grammar of the visualsemiotic.

The development of formal systems for logical meaning in visual imagesmay not be so far-fetched. Galison (2002: 321-322), for instance, points outthat the divide between the analytic and the visual is not so marked in acomputerized environment:

[i] t may be that the most significant development in the laboratory of the last fiftyyears has been the fusion of pictures and numbers into the production of the manipu-lable image. Controllable digitized images were built by computers from statistics andformed into pictorial renditions of non-visible worlds . . . After tracking the endlessdrive back and forth between images and data, it becomes clear that the powerfuldrive towards images and the equally forceful pressure towards analysis nevercompletely stabilized scientific practice. Quite the contrary, neither the 'pictorial-representative' nor the 'analytical-logical' exist as fixed positions. Instead . . . we seethat the image itself is constantly in the process of fragmenting and re-configuring . . .now, ever more intensively, the routinization of analog-to-digital and digital-to-analogconversions have made the flickering exchanges routine: image to non-image toimage . . . every-day science propels this incessant oscillation: 'Images scatter intodata, data gather into images.'

The continuous oscillation between visual and symbolic forms of display fordigitalized data described by Galison may contribute to the development ofgrammatical systems for logical meaning in visual images. If so, the visualsemiotic will possess an important advantage over the traditional semioticresources of language and mathematical symbolism: computer graphicscan display continuous spatial-temporal patterns of variation. The image isno longer the traditional frozen snapshot such as those displayed in thevisualization of the derivative as the tangent to the slope of line in Plate5.2(1). Traditional static visual images (such as those found in mathematicsbooks) are limited compared to the potential of the dynamic and inter-active complexity of computer graphics. The impact of this change remainsto be seen, but one may predict with some degree of confidence thatthe functions of graphical visual images will increase, as evidenced by therapid development of software programs and increased use of computervisualization across many disciplines and fields of study. Mathematicalvisual images are becoming increasingly functional alongside symbolicdescriptions of patterns of relations.

'Just as the microscope showed us the "infinitely small" and the telescopeshowed us the "infinitely large", so the computer will enable us to regard

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our world in a new, richer way' (Colonna, 1994: 191). The 'newer richer'way suggested by Colonna arises in computerized environments throughthe capacity to model, experiment, manipulate and synthesize four dimen-sions of experience: three-dimensional spatial world and one-dimensionaltemporal world to create a dynamic rendition of real and imagined worlds.This recreation takes place on the two-dimensional computer screen. Inaddition, virtual reality has the potential to incorporate other dimensionsfor semiotic input, such as touch, smell and taste. A variety of softwareapplications which can recreate perceived reality in space and time arealready available as documented by Danaher (2001) in the recent bookDigital 3d Design. The modelling process in a computerized environmentpermits noise, complexity and context (which are excluded in traditionalapproaches) to be incorporated for a more comprehensive understandingof 'reality'. Indeed 'reality' undergoes a transformation from 'real', thatwhich can be perceived, to 'abstract', that which can be imagined. Both realand imagined realities can now be modelled and experienced. Computer-ized environments have the potential to move beyond that perceivable andconceivable by the human senses. The implications of this semiotic re-ordering of the world remain to be seen. This will depend on the functionswhich are assigned to computerized medium, and the purposes for whichthe reconstructions of reality are employed. Typically, advances made forthe purposes of military concerns feed back into useful applications forother fields of human endeavour.

Chapters 6 and 7 are concerned with intersemiosis, the process wherenew meanings arise integratively through transitions from one semioticresource to another. The functionality of mathematical discourse doesnot only stem from accessing the three individual meaning potentials oflanguage, mathematical symbolism and visual images. Rather, the func-tionality of mathematics also arises from intersemiosis between the threeresources and the metaphorical construals which take place intersemiotic-ally. In what follows, the nature of intersemiotic mechanisms and systems,and the resultant meaning expansions across language, visual images andmathematical symbolism are investigated.

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6.1 The Semantic Circuit in Mathematics

Language, mathematical symbolism and visual images in the form ofgraphs, diagrams and, more recently, computer graphics have togethercreated a discourse which has transformed the face of the world, if technol-ogy is considered to be the direct result of mathematics and science. Math-ematical discourse is effective because the systems of meaning for language,symbolism and visual images are integrated in such a way that the behaviourof physical systems may be described. Traditional mathematical descrip-tions break down, however, when the behaviour of the physical systemsbecome non-linear and chaotic. In such cases, the entire system must beunderstood rather than the constituent parts. In practice, this means thatthe variety of constraints and conditions must be taken into account inorder to effectively model and predict the behaviour of the system. Thesecalculations and descriptions take place in digitalized form through the useof computers.

The functions of language, the symbolism and the visual image may besummarized as follows. Patterns of relations are encoded and rearrangedsymbolically for the solution to the problem. The symbolism has limitedfunctionality, however, so that language functions as the meta-discourse tocontextualize the problem, to explain the activity sequence which is under-taken for the solution to the mathematics problem, and to discuss theimplications of the results which are established. Visual images in the formof abstract and statistical graphs, geometrical diagrams, and other types ofdiagrams and forms of visual display, such as those generated throughcomputer graphics, show the relations in a spatio-temporal format whichinvolve multi-dimensional time-frames. As discussed in Chapter 5, the trad-itional role assigned to the mathematical visual image is changing with theincreasing power of computers to generate and manipulate complexdynamic visual patterns.

The metafunctionally based SFGs for language, mathematical symbolismand visual display in Chapters 3-5 provide the basis for the discussion ofintrasemiosis, or meaning within the systems which constitute the grammar

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of each resource. However, the three separate grammars are not sufficientfor the analysis of mathematical discourse because intersemiosis, the mean-ing arising across semiotic choices, must also be considered. The math-ematics text, and elements within that text (mathematical graphs and thesymbolic derivations to problems, for instance), are multisemiotic. Theanalysis of mathematical discourse therefore needs to take into account themeaning arising from intersemiosis at discourse, grammar and displaystrata both within and across elements in the text. Following Kok (2004),elements in mathematical texts which function as discernible units throughsystematic choices from the grammars of language, visual images andmathematical symbolism are called 'Items'. These include graphs, dia-grams, tables, stretches of linguistic text and the symbolic equations, as wellas photographs, maps and other forms of drawings. The Items are notcalled 'genres' because the use of this term is reserved for the communica-tion plane which is concerned with the goals and cultural context of theentire mathematics text. The SF framework adopted for the analysisof mathematics texts is displayed in Table 6.1 (1). The expansions in mean-ing arising from intersemiosis in mathematics are investigated at the dis-course, grammar and display strata. The register, genre and ideology ofmathematical discourse are considered in Chapter 7.

Royce (1998a, 1998b, 1999, 2002) refers to intersemiosis as 'intersemioticcomplementarity' where Visual and verbal modes semantically comple-ment each other to produce a single textual phenomenon' (Royce, 1998b:26). As Royce and also Lemke (1998b) explain, the product is 'synergistic'or 'multiplicative' in that the result is greater than the sum of the parts.Language, symbolism and visual images function together in mathematicaldiscourse to create a semantic circuit which permits semantic expansionsbeyond that conceivable through the individual contributions. Theresultant meaning potential of mathematics therefore stretches beyondthat possible through the sum of the three resources. Following this view,the success of mathematics as a discourse stems from the fact that it drawsupon the meaning potentials of language, visual images and the symbolismin very specific ways. That is, the discourse, grammar and display systems foreach resource have evolved to function as interlocking system networksrather than isolated phenomena. The ways in which the grammars oflanguage, mathematical symbolism and visual display are organized tofacilitate intersemiosis are explored in this chapter.

The three semiotic resources fulfil different functions as the mathemat-ics text unfolds. Therefore semiotic transitions, or movements between thesemiotic resources, occur according to the required functions at the differ-ent stages of the text. ledema (2003: 30) refers to the transition process as'resemioticization' or the translations from one semiotic resource intoothers as social processes unfold. Transitions are a feature of everydaydynamic discourse, which result in phases and sub-phases where there is achange in the semantic input from one or more of the semiotic resources(Baldry, 2004, in press; Baldry and Thibault, 2001, in press a, in press b;

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Thibault, 2000). For example, television advertisements combine thedynamic visual image, gesture, music, sound and spoken and written lan-guage into interwoven phases which favour metafunctional input from onesemiotic resource over another. Baldry and Thibault (Baldry and Thibault,in press b; Thibault, 2000) investigate how different types of transitions giverise to phasal shift in video texts. However, transitions are not always clearcut as phases and sub-phases merge and blend in what amounts to anorchestral organization of semiosis.

Table 6.1(1) SF Model for Mathematical Discourse

IDEOLOGY

GENRE

REGISTER

CONTENT(Items) INTERSEMIOSIS

LANGUAGE MATHEMATICALSYMBOLISM

MATHEMATICALVISUAL IMAGES

OTHERSITEMS

Discourse Semantics

Text Inter-statementalrelations

Inter-VisualRelationsWork/Genre

Grammar

ClausecomplexClauseWordGroup/PhraseWord

StatementsClauseExpressionsComponents

EpisodeFigureParts

DISPLAY INTERSEMIOSIS

Graphology and Typography Materiality

For example:PhotographsMapsDrawingsThree-DimensionalModelsEquipmentand so forth

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Modern mathematics developed as a written discourse, and so theintersemiotic shifts or transitions are deliberate and 'calculated' to fulfil therequired functions of the text at different stages. The increasing use ofcomputers and computerized mathematics, however, is changing the staticnature of semiosis which takes place in written mathematics. There is anincreasing drive towards computation where the results of the numericalcalculations are displayed visually through sophisticated software programs.Intersemiosis in mathematics may be viewed as a musical score, but one thatis increasingly played in a computerized environment. The shift to theelectronic medium is changing the nature of the theory and practice ofmathematics, especially in the field of applied mathematics. However, thediscussion presented here is concerned with intersemiosis in written math-ematical texts. The analysis of the written format permits us to appreciatehow language, visual images and the symbolism developed as integratedsystems to create modern mathematical discourse. Despite advances incomputerized mathematics, the three resources remain the primary toolsfor meaning in mathematics.

Transitions are seen to take two forms in written mathematics. Macro-transitions occur at the rank of discourse, where Items which consist ofpredominantly one semiotic resource give way to Items consisting ofanother semiotic resource; for example, language to visual images (graph)to symbolism (symbolic solution to the problem) back to language. Macro-transitions are conceptualized as discourse moves across Items in the math-ematics text. Theoretically speaking, the reading path of the text resultsfrom such discourse moves. In practice, readers scan multisemiotic math-ematical texts to ascertain the important information according to theirown requirements. On the other hand, micro-transitions in mathematicstexts occur at the rank of grammar where functional elements of oneresource are contained within Items which primarily consist of anotherresource; for example, symbolic elements appear in the graphs, and lin-guistic elements appear in the symbolic statements. More generally, macro-transitions involve discourse moves to access the meaning potential of asemiotic resource, while micro-transitions take place constantly because ofthe integrated grammars for language, mathematical symbolism and thevisual image where it is possible to embed functional elements from onesemiotic resource within a different semiotic resource. That is, micro-transitions are the result of the interlocking nature of the system networksfor the three resources. As seen in Sections 6.3 and 6.4, micro-transitions aidmacro-transitions or discourse moves to another Item in the mathematicstext.

The nature of the macro- and micro-transitions for intersemiosis inmathematical discourse depends upon the metafunctional requirements atdifferent stages in the text and the available choices in the discourse,grammatical and display systems. Regardless of the organization of themathematics text and the explicit choices marking the transitions whichshould take place, often the reader will examine the diagram or graph and

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the mathematical equations independently; for example, before readingthe linguistic text. The reading path is conceptualized as a recursive scan-ning process rather than the sequential path determined by the com-positional organization of the mathematics text. The use of Caption, Titlesand Labels aids this scanning process through the spatial separation andforegrounding of significant Items, such as the symbolic descriptions andthe visualization of relations. The resulting recursive path depends on theneeds and experience of the reader. As mathematical texts typically unfoldin a non-sequential fashion, macro-transitions therefore are not alwaysoperational in practice.

Intersemiotic transitions in mathematics offer the opportunity formetafunctional choices in system networks across the three semioticresources to combine in new ways across different strata. For example,choices from system networks for experiential meaning in language andsymbolism may combine with choices for interpersonal meanings in visualimages. For instance, 'the derivative/' (a)' is drawn as two red lines in theoriginal version of Figure 4 reproduced in Plate 6.3(2). The colour red inthe visual image functions interpersonally to give salience to the experien-tial meaning of the derivative as a geometrical entity. One dimension of the'multiplication of meaning' which takes place in mathematics arises fromthe combinations of system choices from different semiotic resourcesacross different strata. The metafunctionally based systems for the threeresources permit a range of different combinations. The number of pos-sible intersemiotic relations or combinations increases with the number ofsemiotic resources which are involved. Mathematics thus succeeds throughutilizing the meaning potential of language, visual images and the symbol-ism and the meanings which arise through intersemiosis. Significantly,intersemiotic shifts in mathematical discourse also permit metaphoricalexpansions of meaning beyond those which can occur within any onesemiotic resource. The metaphorical nature of meaning expansion isconsidered in relation to the concept of semiotic metaphor in Section 6.5.

These key ideas concerning intersemiosis in mathematics are developedthrough the notions of macro-transitions across Items and micro-transitions within those Items. However, the strategies or mechanismsthrough which intersemiosis takes place need to be extended beyond thenotion of transition. Thus, in addition to Semiotic Transition, intersemioticmechanisms are conceptualized as Semiotic Cohesion, Semiotic Mixing,Semiotic Adoption and Juxtaposition in Section 6.2. In addition, semioticmetaphor accounts for the metaphorical meaning expansions which occurduring transitions. This list of intersemiotic mechanisms is not exhaustive;on the contrary, further research is needed to understand how meaningexpansion takes place intersemiotically. In what follows, previousapproaches to the study of intersemiosis are discussed in order to con-textualize what are conceived to be four main issues pertaining tointersemiosis in mathematics, namely: (i) the mechanisms for intersemi-osis; (ii) the metafunctionally based systems at the discourse, grammar and

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display strata through which intersemiosis takes place; (iii) the semantics ofintersemiosis; and (iv) the metaphorical expansions which take place in theform of semiotic metaphors.

6.2 Intersemiosis: Mechanisms, Systems and Semantics

The underlying principle for the integration of semiotic resources inThibault's (2000) theoretical framework for the transcription and analysisof television advertisements is metafunctional salience where meaningsarising from choices from different resources function to contextualizeeach other. Thibault (2000: 362) explains that it is 'on the basis of co-contextualizing relations that meaning is created'. In order to analyse adynamic multimodal text such as the television advertisement, Thibaultsegments the text in phases and sub-phases: 'A discoursal phase, followingGregory (1995, 2002), is a set of co-patterned semiotic selections that areco-deployed in a consistent way over a given stretch of text' (Thibault, 2000:325-326). Thibault (2000: 325-326) deals with the complexity of multipleforms of dynamic semiosis through a transcription table where phases andmetafunctional salience are marked in what is a seminal effort at handlingthe complexity of the integration of multiple semiotic resources. Thisapproach is further developed into multimodal concordancing of patternsarising from different types of transitions (Baldry, 2004, in press; Baldryand Thibault, 2001, in press b).

Lim (2004) conceptualizes the expanded meaning arising fromintersemiosis as the 'Space of Integration' (Sol) in the Integrative Multisemi-otic Model (IMM). The Sol is designed to capture the meanings which arisethrough the interaction between language and visual images. Royce(1998a, 1998b, 1999, 2002) formulates this space as the 'intersemioticcomplementarity' between language and visual images: 'They [the visualimages and language] work together to produce a coherent multimodaltext for the viewers and readers, a text characterised by intersemiotic comple-mentarity (Royce, 2002: 193). Royce (1998b) identifies a number ofintersemiotic semantic mechanisms through which image and languageorchestrate the meaning of a text in the analytical metafunctionally basedframework reproduced in Table 6.2(1).

Royce (1998b, 2002) adopts Halliday (1994) and Halliday and Hasan's(1976, 1985) categories of lexical cohesion to account for ideational mean-ing arising in a multimodal text. Royce's categories include intersemioticrepetition, synonymy, antonymy, meronymy, hyponymy and collocationacross visual and verbal codes. For interpersonal meaning, Royce (1998b) isconcerned with the relations established between the reader/viewer andthe text through MOOD and MODALITY which function to reinforceaddress and attitudinal congruence or dissonance. In relation to textual orcompositional meaning, the layout and composition through informationvalue, salience, framing, inter-visual similarity and reading paths areconsidered. Royce attempts the difficult task of mapping intersemiotic

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Table 6.2(1) Analytical Framework for Visual-Verbal IntersemioticComplementarity, Royce (1998b: 29)

META-FUNCTION Visual Meanings IntersemioticComplementarity

Verbal Meanings

IDEATIONAL

INTERPERSONAL

TEXTUAL-COMPOSITION

Variations occuraccording to the CodingOrientation. In theNaturalistic Coding wecan look at:• Identification of

representedparticipants

• Activity portrayed• Circumstances of the

mean,accompaniment andsetting

• Attributes ofrepresentedparticipants

Variations occuraccording to the CodingOrientation. In theNaturalistic Coding wecan look at:• Address to the viewer• Level of Involvement of

viewer• Power relations between

the viewer and therepresentedparticipants

• Social Distance betweenviewer andrepresentedparticipants

• Modality — believabilityor acceptability of theportrayal

Variations in visualmeanings occuraccording to choicesmade in:• Information Value -

intra-visual placement• Visual Salience• Framing of Visual

elements

Various lexico-semanticways of relating theexperiential and logicalcontent or subjectmatter represented orprojected in both visualand verbal modesthrough theintersemiotic senserelations of:• Repetition• Synonymy• Antonymy• Meronymy• Hyponymy• Collocation

Various ways ofintersemioticallyrelating the reader/viewer and the textthrough MOOD andMODALITY throughintersemiotic semanticrelations of:• Reinforcement of address• Attitudinal Congruence• Attitudinal Dissonance

Lexical items whichrelate to the visualmeanings. These lexicalitems arise according to:• Identification

(participants)• Activity (processes)• Circumstances• Attributes

Elements of the clauseas exchange whichrelate to visualmeanings. These ariseaccording to:• The MOOD element

in the clause realizingspeech function

• The MODALITYfeatures of the clause

• Attitude - use ofattitudinal adjectives

Various ways of mappingthe modes to realize acoherent layout orcomposition by:• Information Valuation

• Salience on the page• Degree of framing of

elements on the page• Inter-visual synomymy• Reading paths

The body copy (verbalelement) as anorthographic wholerealized by varioustypographicalconventions:• General Typesetting• Copyfitting• Other Typesetting

Techniques• Also: Theme/Rheme,

Given/New Structures

expansions across visual and verbal elements by proposing mechanisms formeaning expansion that largely depend upon linguistic conceptions suchas those proposed for cohesion, and interpersonal dimensions of MOOD

on the page

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INTERSEMIOSIS 165

and MODALITY. Royce also considers composition as the differentmappings which result in the coherent layout of the whole text. Royce'sanalytical framework provides a point of departure for the descriptionof intersemiotic mechanisms which extend beyond those proposed forlanguage and visual images.

Following Thibault (2000: 325-326) and Royce (1998b), Lim (2004) seescontextualizing relations as a significant aspect of the meaning arising fromintersemiosis. Two types of contextualizing relations are proposed: 'In caseswhere the meaning of one modality seems to "reflect" the meaning of theother through some type of convergence, the two resources share co-contextualizing relations. On the other hand, in cases where the meaning ofone modality seems to be at odds with or unrelated to the other, theirsemantic relationship is one that creates divergence or dissonance. In thelatter case, the resources share re-contextualizing relations' (Lim, 2004: 239).From this perspective, the nature and the degree of the co-contextualizingor re-contextualizing relations are significant.

Cheong (1999, 2004) provides a meta-language for the semanticsof intersemiosis in terms of the degree of contextualization, and theimplications of those contextualizing relations. Cheong conceptualizesthe ideational meaning arising from the intersemiosis between visual andlinguistic selections in print advertisements as the Bi-directional Investmentof Meaning, which is measured through a scale known as ContextualizationPropensity (CP). CP 'refers to the degree/extent to which the linguisticitems contextualize the meaning of the visual images' (Cheong, 1999: 44).Cheong shows the CP has a direct influence on the Interpretative Space(IS), which results in the Semantic Effervescence (SE) of the text. Forexample, an advertisement with a high CP leads to a low IS resulting in alow SE.

Further research is needed to theorize the range of intersemiotic mech-anisms through which semantic expansions take place, and the implica-tions of those semantic reconstruals. In what follows, the nature ofintersemiotic mechanisms and the metafunctionally based systems throughwhich intersemiosis takes place at the discourse, grammar and display strataare considered. The metafunctionally based systems for intersemiosis aredeveloped through text analysis in Sections 6.3-6.4.

The four major issues in relation to intersemiosis in mathematics aredisplayed in Figure 6.2(1). First, the means or mechanisms through whichintersemiosis as a phenomenon takes place require investigation. A rangeof intersemiotic mechanisms is given below. The mechanisms reside withinand across the systems for language, mathematical symbolism and visualimages. The options within the system networks for the three resourcesfunction intrasemiotically as closed systems in theory only. In practice, thesystems for language and other semiotic resources have the potential tofunction intersemiotically. Semiotic resources have evolved to be used inconjunction with other semiotic resources, and thus considering them inisolation gives only a partial picture of their functionality. The second issue

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Figure 6.2(1) Aspects of Intersemiosis: Mechanisms, Systems, Semantics and Semiotic Metaphor

INTRASEMIOSIS(within one semiotic resource)

Meaning through choices fromsystems forming intra-connected

network options for each semioticresource

DISCOURSE, GRAMMARAND DISPLAY SYSTEMS

The functions of language,mathematical symbolism and

visual images include thepotential for intrasemiosis and

intersemiosis

INTERSEMIOTICMECHANISMS

Meaning through choices fromsystems functioning asinterlocking networks:

1 Semiotic Cohesion2 Semiotic Mixing3 Semiotic Adoption4 Juxtaposition5 Semiotic Transition

SYSTEMS AND SEMANTICS OF INTERSEMIOSIS

Textual/Compositional meaningInterpersonal meaningExperiential meaning

Logical Meaning

Co-contextualization (parallelism)Re-contextualization (divergence or dissonance)

Bi-directional Investment of Meaning. Contextualization Propensity (CP), Interpretative Space-(IS)Semantic Effervescence (SE) (Cheong, 1999, 2004)

SEMIOTIC METAPHOR

Metaphorical shifts across semiotic resources where functional status of elements is not preserved and new elements areintroduced across discourse, grammar and display

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INTERSEMIOSIS 167

therefore involves the description of the metafunctionally based inter-semiotic systems according to the discourse and grammar strata andthe display plane. A tentative description of these systems is proposed inTables 6.2(2a-d).) The systems are discussed in relation to the analysisof two mathematics texts in Sections 6.3 and 6.4. The third issue is thesemantics of intersemiosis which may be conceptualized in terms ofco-contextualizing and re-contextualizing relations. The semantics ofintersemiosis is considered in Sections 6.3 and 6.4. The fourth aspect is themetaphorical construals which result from shifts between semiotic codes.This is investigated in relation to semiotic metaphor in Section 6.5.

Metafunction Discourse Grammar Display

Table 6.2(2a) Systems for Intersemiosis: TEXTUAL MEANING

INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISMAND VISUAL DISPLAY

Textual INTERSEMIOTICIDENTIFICATIONCohesive devices forIntersemioticReference includingelements whichoperate acrossresources throughDirect Reference andIntersemioticRepetition (forexample, x) andsemantic reference(for example,Variable' and x)

INTERSEMIOTICMIXINGUse of selections ofdifferent semioticselections (forexample, A ABC)

DISCURSIVE LINKSacross text

CAPTIONSUse Captions whichuse multiple semioticresources

INTERSEMIOTICSUBSTITUTIONSubstitution of oneterm for another (forexample, x for AB +for addition)

INTERSEMIOTICADOPTIONUse of functionalelement acrosssemiotic resources(for example, x)

DEIXISUse of deictics inlanguage (forexample, 'this'curve) compensatedby generalizedparticipants insymbolism and visualdisplay

LABELSUse of Labels whichuse multiple semioticresources

JUXTAPOSITION(Textual andCompositionalArrangement)Use of spatialposition and layoutto juxtapose andseparate selectionsand items from eachsemiotic resource

FRAMING toorganize text

FONTUse font style, sizeand colour forcohesive purposes

COLOURUse of colour forcohesion across text

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168 MATHEMATICAL DISCOURSE

Metafunction Discourse

Experiential INTERSEMIOTICIDEATIONActivity Sequencesand relations whichstretch acrosssemiotic resourcesthrough directrepetition (forexample, let ABterminate at C) andintersemioticequivalence,synonymy,antonymy,hyponymy,meronymy andcollocation (Royce,1998b:29).Taxonomies whichstretch acrossresources (forexample, types oftriangles)

CAPTIONSUse Captions whichuse multiplesemiotic resources

Grammar

TRANSITIVITYRELATIONSThe use of relation processesto set up identifying relationsacross semiotic resources forexample, let AB = xTransitivity selections whichoverlap, for example, A and Bhave meaning in thegrammar of visual images,and the grammar oflanguage

LEXICALIZATION,SYMBOLIZATION andVISUALIZATIONMaintenance in process,participant and circumstanceand agency configurationsthrough:(i) Lexicalization of symbolicand visual functionalelements (for example,'flirt^r^ fi=,' fr~*v ' 7»' ^»-«*-l ( '\(ii) Symbolization of lexicaland visual functionalelements (for example, 'h'for 'line' and ' — ')(iii) Visualization of lexicaland symbolic functionalelements (for example, ' — 'for distance and 'h')

Display

JUXTAPOSITION(for ExperientialRelations)Use of space andposition to createlexical, symbolicand visual relations

FONTUse font style, sizeand colour forexperientialmeaning

COLOURUse of colour forexperientialmeaning

SEMIOTIC METAPHORShifts in functional statusand introduction of newprocess, participants andcircumstantial elements (forexample, introduction oftriangle visually whichbecomes symbolized andlexicalized), shifts in agency

LABELSUse of Labels which usemultiple semiotic resources

Table 6.2(2b) Systems for Intersemiosis: EXPERIENTIAL MEANING

INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISM ANDVISUAL DISPLAY

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INTERSEMIOSIS 169

Table 6.2(2c) Systems for Intersemiosis: INTERPERSONAL MEANING

Metafunction Discourse Grammar Display

Interpersonal INTERSEMIOTICNEGOTIATIONThe unfolding ofdiscourse movesacross semioticresources usingSPEECHFUNCTIONS,arrows, (toaccommodate lack ofgaze and so forth)

LABELS andCAPTIONS

SPEECHFUNCTION, MOODSpeech functions(includingcommands to viewparts of the text)

MODALITYConsistency ofModality acrossvisual, verbal andsymbolism

POLARITY

STYLE OFPRODUCTIONConsistency in styleof production

SALIENCE asdirecting discoursemoves across text

PROMINENCE asdirecting attention toverbal, visual andsymbolicromnonents

Use of Labels whichuse multiple semioticresources

INTERSEMIOTICAPPRAISALAppraisal acrosssemiotic resources

FONTUse font style, sizeand colour forinterpersonalmeaning

COLOURUse of colour forinterpersonalmeaning

The mechanisms of intersemiosis are categorized as:

1 Semiotic Cohesion: System choices function to make the text cohereacross different semiotic resources.

2 Semiotic Mixing: Items consist of system choices from different semi-otic resources.

3 Semiotic Adoption: System choices from one semiotic resource areincorporated as a system choice in another semiotic system.

4 Juxtaposition: Items and components within those Items are com-positionally arranged to facilitate intersemiosis.

5 Semiotic Transition: System choices result in discourse moves in theform of macro-transitions which shift the discourse to another Itemconsisting primarily of another semiotic resource, or alternativelymacro-transitions within Items occur.

INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISM ANDVISUAL DISPLAY

Displays of Polarityacross resources

SEMIOTICMETAPHORShifts in functionalstatus of expressionof modality acrosssemiotic resources

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170 MATHEMATICAL DISCOURSE

Metafunction Discourse Grammar Display

Logical IMPLICATIONSEQUENCESCohesive andstructural devicesacross semioticresources (forexample, linguisticand symbolicstruturalConjunctions,ConjunctiveAdjuncts, andcohesive ties, andarrows pointing toother semioticresources)

LOGICO-SEMANTICSandINTERDEPENDENCYCohesive andconjunctive devicesacross semioticresources

INTERPLAY OFSPATIALlTYandTEMPORALITYthrough visual, textualand symbolictransformations

SEMIOTIC

SPATIALPOSITIONAlignment of Itemsin the text insequence

FONTUse font style, sizeand colour forlogical meaning

COLOURUse of colour todirect the sequencefor the constructionof logical relations

METAPHORShifts in functionalstatus of logicalrelations acrosssemiotic resources

Intersemiotic mechanisms generally involve a two-way directional invest-ment of meaning as displayed in Figure 6.2(1). Semiotic transitions involveshifts in the discourse from one semiotic resource to another, and henceare indicated by one-directional arrows. However, the arrow pointing in theopposite direction shows that contextualization is a two-way process, des-pite the one-directional shift in the discourse. Intersemiotic transitions areparticularly significant in mathematics where semiotic resources are seento alternate between being primary and ancillary at different stages ofthe text. This alternation is explained by the functional requirements atdifferent stages in the generic structure of the mathematics text.

The intersemiotic mechanisms are realized through choices from thesystems in Tables 6.2(2a-d), which are categorized according to the dis-course and grammar strata and the display plane. The systems are organizedmetafunctionally, and thus form another semantic layer to the analysis ofmathematics texts. Up to this point, the analysis of mathematical discoursehas been based on the three SF frameworks proposed for language,mathematical symbolism and visual images in Chapters 3-5 respectively.The intersemiotic systems listed in Tables 6.2(2a-d), however, provide the

Table 6.2(2d) Systems for Intersemiosis: LOGICAL MEANING

INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISM ANDVISUAL DISPLAY

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facility whereby choices from a metafunctionally based system in one semi-otic integrate with choices from other systems in other semiotic resources.This includes the provision for integration of system choices across dis-course and grammar strata and the display plane. The framework providesa semantic layer whereby the 'texture' of the text may be seen as theinteraction between semiotic resource x metafunction x strata x system x choicethrough the mechanisms of interserniosis.

The systemic framework for interserniosis in Tables 6.2(2a-d) is notexhaustive. Further research is needed to examine how semiotic resourcesare organized to function interserniotically, and how that functionalityrelates to system options in the grammars for each resource. However, theimportant points are that semiotic resources are functional with respect tothe semantics of other semiotic resources, and that the systems throughwhich interserniosis takes place require further theorization. The frame-work in Tables 6.2(2a-d) is discussed with reference to Newton (1736: 46)and Stewart (1999: 132) displayed in Plate 6.3(1) and Plate 6.3(2) in Sec-tions 6.3 and 6.4. In the analysis of interserniosis, it becomes evident thatthe integration of semiotic resources is formalized in mathematics andscience in a fashion which is not typically found in other forms of discourse.Following this discussion, the notion of semantic expansion throughsemiotic metaphor is illustrated through text analysis in Section 6.5.

6.3 Analysing Intersemiosis in Mathematical Texts

Intersemiotic mechanisms through system choices at the ranks of dis-course, grammar and display are examined in the extract from Newton's(1736) writings The Method of Fluxions and Infinite Series displayed in Plate6.3(1). In Newton's text, '/' stands for the letter V in the linguistic text.'Fluxion' is Newton's term (now obsolete) for the derivate as the rate ofchange of a function with respect to x, where the geometrical inter-pretation of the derivative is the tangent to a curve at a point. Newton's

dynotation for the derivative x is used today along with other forms such as —

dxf'(x), and Dxf(x). Newton is concerned with drawing the tangent to thecurve in Plate 6.3(1). The Items in Newton's text are the linguistic text, thesymbolic text in (Point 3) (which is embedded in the surrounding linguistictext), and the diagram. In Stewart's (1999: 132) description of the deriva-tive as the instantaneous rate of change, the Items consist of Figure 4, thelinguistic text which is concerned with the derivative, and Example 4.

While the linguistic, symbolic and visual Items are spatially separated inthe modern mathematics texts such as Plate 6.3(2), Newton's (1736) workis typeset in the style of running text in Plate 6.3(1). Newton's text may becompared to the contemporary rendition of the derivative as the slope of atangent line displayed in Plate 5.2(1). Newton's work has been chosen foranalysis, however, because the linguistic choices are largely congruent andthus intersemiotic shifts between linguistic, visual and symbolic parts are

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Plate 6.3(1) Newton's (1736: 46) Procedure for Drawing Tangents

easier to track, unlike modern mathematical and scientific texts such asStewart (1999: 132) which are highly metaphorical (see Section 6.5). New-ton's writings in the eighteenth century illustrate how mathematics evolvedto integrate three semiotic resources, and the examination of contempor-ary discourse reveals the results of that integration. In what follows,intersemiosis in Plates 6.3(1) and 6.3(2) arising from systems displayed inTables 6.2(2a-d) is discussed according to metafunction.

172 MATHEMATICAL DISCOURSE

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The connection with the first interpretation is that if we sketch the curve y = fix), thenthe instantaneous rate of change is the slope of the tangent to this curve at the point where.v = a. This means that when the derivative is large (and therefore the curve is steep, as atthe point P in Figure 4). the y-values change rapidly. When the derivative is small, thecurve is relatively flat and the y-values change slowly.

In particular, i t 's — f ( t ) is the position function of a particle that moves along a straightline, then /'{«) is the rate of change of the displacement ,v with respect to the time t. Inother words, fin) is I/if ir/«<7/y of the particle at rime t a (see Section 2,6). The speedof the particle is the absolute value of the velocity, that is. i / '(a) \,

Thus, the velocity after 2 s is

Plate 6.3(2) The Derivative as the Instantaneous Rate of Change (Stewart,1999: 132)

Textual Meaning

At first glance, the primary resource for intersemiosis in mathematicsappears to be textual; that is, the organization of the message for the en-abling of interpersonal, logical and experiential meaning across semioticresources. The importance of textual organization explains the traditionalemphasis on compositional layout in graphical design where items arealigned, framed and juxtaposed in relation to other items to create certaineffects in advertising, newspapers and magazines. From the systemic per-spective, however, there are textual systems other than those on the displaystratum which aid intersemiotic tracking of participants across linguistic,visual and symbolic items as displayed in Table 6.2(2a). These systemsinclude INTERSEMIOTIC IDENTIFICATION where cohesion is achievedthrough Intersemiotic Reference. For example, in Plate 6.3(1) Intersemi-otic Reference occurs through the selections 'tangents' and the actual line(TD) which is drawn to intersect the curve in the diagram. Similarly, 'BD' is'a right line' which appears in the diagram. Often there is Direct Referenceacross the three semiotic resources; for example, x appears in the linguistic,

From Equation 3 we recogni/e this l imi t as being the derivative of / at . \ i . that is. f'(x\).This gives a second interpretation of the derivative:

FIGURE 4

The y-values arc changing rapidlyal P and slowly at Q.

E X A M P L E « The position of a particle is given by the equation of motioni ~ /(?) — 1/i I + /), where t is measured in seconds and ,v in meters. Find the velocityand the speed after 2 seconds.

SOLUTION The derivative of / when / = 2 is

The derivative /"(«) is the instantaneous rate of change of v -•/(.*) with respect tov when A = a.

m/s, and the speed is m/s.

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174 MATHEMATICAL DISCOURSE

visual and symbolic parts of the text in Plate 6.3(2). In addition,INTERSEMIOTIC MIXING aids the tracking of participants. For example,the graph in Plate 6.3(2) consists of visual, symbolic and linguistic selec-tions which refer to participants in the linguistic and symbolic parts ofthe text.

At the rank of grammar, textual resources include INTERSEMIOTICSUBSTITUTION (for example, x for AB in Plate 6.3(1)) andINTERSEMIOTIC ADOPTION (x appears as an element in each resourcein Plate 6.3(2)). DEIXIS in language aids textual organization across visualand symbolic modes; for example, 'the curve' in 'the curve y=f(x)' (whichcould have been labelled in Figure 4 in Plate 6.3(2)). As mathematicalsymbolism evolved from language, the grammars interlock in ways so thatselections are almost interchangeable as seen in the example 'the curvey = f ( x ) \ INTERSEMIOTIC MIXING, LABELS and CAPTIONS ensure thatthe semantic realm of the linguistic and symbolic is included in diagramsand graphs.

In terms of the display stratum, JUXTAPOSITION in terms of Textualand Compositional Arrangement aids intersemiosis. For example, theItems consisting of the graph, the linguistic text, and the symbolic solutionto the problem are spatially organized in Plate 6.3(2) so that the reader mayeasily access the experiential and logical meaning of each Item. Further tothis, COLOUR also aids textual organization; for example, in the originalversion of Plate 6.3(2), the definition of the tangent which is Framed (inred) coheres with the (red) line corresponding to the tangent lines inFigure 4.

The choices which function intersemiotically for textual meaning resultin a discourse where reference chains split and recombine in complexways across semiotic resources (O'Halloran, 2000). The resulting textureis dense as linguistic, symbolic and visual participants are reconfiguredand combined, especially within the symbolic parts of the text. Similarly,grammatical choices function to ensure that experiential and logicalmeanings arising from textual organization of the message are preciselyencoded in ways that permit the reader to access these meanings acrosssemiotic resources in the most immediate manner possible. While discour-sal, grammatical and display choices for textual meaning function toensure a coherent text, the problems of tracking participants across semi-otic resources in mathematics are addressed in relation to pedagogy inSection 7.3.

Experiential Meaning

Experiential intersemiosis is formulated in terms of INTERSEMIOTICIDEATION at the discourse stratum. INTERSEMIOTIC IDEATION con-tributes to the construction of Activity Sequences which stretch across thetext. For example, the reader is instructed in Plate 6.3(1) to 'first let BD bea right Line, or Ordinate, in a given Angle to another right Line AB, as a

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Base'. These participants correspondingly appear in the diagram. Fur-thermore, 'Calling AB = x and BD = / means that the symbolism is broughtinto play in the Activity Sequence. The sequences are realized throughinterpersonal commands and statements, and TRANSITIVITY RELA-TIONS which function intersemiotically to establish identifying relationsacross process, participant and circumstantial elements through LEXI-CALIZATION, SYMBOLIZATION, VISUALIZATION and the use ofLABELS. In this process, Intersemiotic Repetition (where, for example, xiscontinually repeated in symbolic, visual and linguistic parts of the text)is important. In addition, semiotic metaphor is conceptualized as animportant intersemiotic strategy for metaphorical meaning expansions inmathematics (see Section 6.5).

Royce (1998b, 2002) uses the notion of cohesion across visual and verbalsemiotic resource to conceptualize ideational intersemiosis (see Table6.2(1)). Although associated with textual meaning in SFL, the concept ofcohesion is nonetheless useful for tracing lexical relations in mathematicsat the discourse stratum. For example, taxonomies are constructed inmathematics so that cohesive relations are realized through classification (xis a type of y relationship in the form of hyponymy; for example, the tan-gent is classified as the line which intersects the curve in one point only)and composition (whole/part relations in the form of meronymy; forexample, a point is part of a line). The second main type of cohesive devicefor lexical relations is expectancy relations which are similarly direct giventhe limited semantic field with which mathematics is concerned. Forexample, Plates 6.3(1) and 6.3(2) are concerned with the geometricalinterpretation of the derivative, and so particular linguistic, visual andsymbolic choices are expected.

At the display stratum, JUXTAPOSITION for Experiential Relations (forexample, the relative position of the graphs and diagrams, and theaccompanying Labels and Captions) and COLOUR (for example, the useof red lines for the derivative in Figure 4 in Plate 6.3(2)) aid the construc-tion of intersemiotic experiential meaning. The use of space and positionto create lexical, symbolic and visual relations is an important resource forexperiential meaning in mathematics. In addition, FONT size, style andCOLOUR function experientially to cohesively link, for example, symbolicvariables, important parts of the linguistic text and significant Episodes,Figures and Parts in the visual display.

Interpersonal Meaning

INTERSEMIOTIC NEGOTIATION for interpersonal meaning at the dis-course stratum stretches across the three resources; for example, the readeris given a linguistic command 'first let BD be a right Line, or Ordinate, in agiven Angle to another right Line AB, as a Base'. Compliance with thiscommand takes place visually in the diagram in Plate 6.3(1). Also informa-tion is given through statements (for example, 'Calling AB — x and BD = y'

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in Plate 6.3(1)) which function intersemiotically to link the visual image tothe symbolism. LABELS and CAPTIONS attract attention in order to aidINTERSEMIOTIC NEGOTIATION across resources. Further to this,FONT, COLOUR and the resulting visual SALIENCE and PROMINENCEfunction to assist discourse moves across and within the Items in the text.Interpersonally there is typically a consistency in terms of high MODALITYvalues across the discourse and grammatical selections (in terms of prob-ability and obligation values) and the STYLE OF PRODUCTION at thedisplay plane also suggests a high truth-value. Interpersonal relations ofauthority with a high truth-value are reinforced intra and intersemioticallyin mathematics.

Logical Meaning

IMPLICATION SEQUENCES function intersemiotically through linguisticand symbolic Conjunctive Adjuncts and structural Conjunctions in Plates6.3(1) and 6.3(2) in combination with visual devices such as arrows (seePlate 5.2(1)). The interplays with spatiality and temporality in math-ematical visual images are aided by linguistic selections such as thoseappearing under the CAPTION 'Figure 4' in Plate 6.3(2): 'The ^values arechanging rapidly at Pand slowly at Q', so that intersemiosis permits a logicalexpansion beyond that possible with language and symbolic elementswhere typically space and time are separated into different elements. TheINTERPLAY OF SPATIALITY AND TEMPORALITY in Figure 4 is glossedlinguistically in terms of the manner of change, 'rapidly' and 'slowly'.Intersemiosis means that spatial and temporal dimensions are integrated inthe visual image. In addition, the logic of spatial visual perception is trans-lated into a text-based tool in the form of mathematical symbolism whichpermits reasoning to progress beyond that possible with language. Thegrammar of the visual image interlocks with the grammars of the symbolismand language so that these types of reasoning can take place. At the displaystratum, SPATIAL POSITIONING and the alignment and sequence ofItems also function to aid logical reasoning. For example, sequential stepsrealized through language and symbolism are placed alongside and under-neath one another in Example 4 in Plate 6.3(2). Once again, FONT andCOLOUR aid logical sequential processing in terms of organizing thesymbolic and linguistic steps.

Metafunctionally Based Co-Contextualization and Re-Contextualization

Textual, experiential, interpersonal and logical meanings are co-contextualized and re-contextualized through the intersemiotic mechan-isms which are described above. However, the types of contextualizationappear to be metafunctionally based. Interpersonal meaning in the form ofdominating social relations with high modality values is co-contextualizedacross linguistic, visual and symbolic selections. The writer directs the

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Activity Sequences with commands and statements which possess highdegrees of obligation and certainty which are reinforced through thenature of the de-contextualized visual representations and the modalityfound in the symbolic text. The Style of Production of the text and theabstract nature of the linguistic, visual and mathematical participants,processes and circumstances function to further co-contextualize theselections for interpersonal meaning. Experiential meaning, however, isre-contextualized as the Activity Sequences stretch across the differentsemiotic resources. This significant aspect of semantic expansion inmathematics is further developed in Sections 6.4 and 6.5. Similarly,logical meaning involves re-contextualization of spatial and temporalrelations through intersemiosis across linguistic, visual and symbolic com-ponents. Aided by visual perception, the tool for reasoning to derive themathematical results is ultimately the symbolism, although languagefunctions to contextualize the results which are established.

Textual and compositional meaning as the enabling function assumes anew realm of importance when viewed through the lens of intersemiosis.The organization of the message is critical in mathematics so that themeaning potential of each resource may be accessed and integrated atdifferent stages in the unfolding of the text. New systems for textualmeaning are called into play as participants, processes and circum-stances are semantically realigned across the three semiotic resources.The identification and tracking of those functional elements and theirsubsequent semantic reconfiguration is important in mathematics. In whatfollows, the integration of language, mathematical symbolism and visualdisplay is further examined in terms of the transitions which take place inorder to investigate the semantics of re-contexualization in mathematicaldiscourse. Newton's text in Plate 6.3(1) is examined for this purpose.

6.4 Intersemiotic Re-Contexualization in Newton's Writings

Newton's problem in Plate 6.3(1) is titled 'PROB. IV. To draw Tangents toCurves, First Manner', and the directions to achieve this objective take theform of a series of linguistic commands and statements. There are constantmacro-transitions back and forth between the linguistic text and the dia-gram as the problem is visualized. Micro-transitions occur in that the func-tional elements within the linguistic text, for example, 'Line AB' and'Curve ED', correspond to functional elements in the geometrical visualimage, the points A, B, E and D. Once visualization is achieved, there aremacro-shifts between the diagram and the symbolism to establish the rela-tionship between entities; for example, Newton writes 'So that it is TB : BD:: DC (or B£) : cd'. Finally the functional elements within the geometricalimage are given a symbolic algebraic form: 'Calling AB = x and BD = y\Newton thus describes the relations using mathematical symbolism: 'let

their Relation be Xs - ax2 + axy — yz = 0' and 'Therefore

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Newton concludes: Therefore the Point D being given, and thence DB andAB, or y and x, the length BT will be given, by which the Tangent TD isdetermined'.

On the surface it appears that the intersemiotic transitions are largely amatter of giving functional elements in one semiotic resource an identityin another semiotic resource so that the meaning potential of thatsecond resource can be accessed. For example, 'Calling AB = x and BD = y'means that functional elements in the visual image now have correspond-ing elements in the symbolism so that the problem can be solved algebra-ically. This certainly is the case as each semiotic resource has a uniquemeaning potential which permits particular functions to be fulfilled.Newton's problem in Plate 6.3(1) is introduced, contextualized andfinalized using language. The visual image permits a perceptual under-standing of the problem through re-contextualizing spatial and temporalrelations at discrete instances of time. The relations of parts to the wholemay also be ascertained through the visual image. The symbolism encodesthe exact nature of the relations over time in a continuous and completeform. Newton's problem of drawing the tangent to a curve is resolvedalgebraically as a set of relations involving Operative processes and parti-cipants. The grammar of mathematical symbolism is such that these rela-tions are easily reconfigured to solve the problem. The semantic circuitin the form of macro- and micro-transitions is seamlessly woven so thatshifts back and forth are a constant feature of mathematical discourse.The result is a highly coherent text. The functional grammars of math-ematical symbolism and visual display have evolved to ensure this is thecase.

However, the semantic circuit involves more than using relational clausesto give functional elements in one semiotic resource an identity in another.While this permits macro-transitions across resources and ensures repre-sentation of particular elements through repetition, (for example, x mayappear in the linguistic, visual and symbolic parts of the text), the expan-sion of meaning also involves metaphorical exchange whereby functionalelements do not always retain their original status when they are re-represented in a second semiotic resource. The x, for example, functions asa participant (in the form of a generalized noun) in all three semioticresources. However, close textual analysis reveals that functional roles arenot always maintained across different semiotic resources, and that, furtherto this, new entities are created through intersemiotic transitions. Forexample, Newton (1736: 46) writes:

Let this Ordinate [BD] move through an indefinitely small Space to the place bd, sothat it may be increased by the Moment cd, while AB is increased by the Moment Bi, towhich Deis equal and parallel.

The command for Actor/Agent ('you') to make 'this Ordinate [BD]'(Goal/Medium) 'move' (Process: material) 'through an indefinitely small

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Space' (Range) 'to the place bd (Location: Place) // so that 'it' [BD](Goal/Medium) 'may be increased' (Process: Material) 'by the Moment cd(Range). First, the linguistic Material process 'move' BD (the goal) resultsin the introduction of new entities, the line segments foand B6, in the visualimage. Second, the linguistic circumstance of location in 'to place bdbecomes a major participant bd in the visual image. That is, the intersemi-otic transition to the visual images from the linguistic involves the creationof new entities (foand B&), and a shift in the functional status of an existingentity (Circumstance 'to the place bd becomes a participant bd). Lastly,through the meaning potential of visual images, these entities (be, B£ andbd) enter into spatial relations with one another. In other words, Newton'scommand to 'Move X' introduces visually new participants Y and Z whichare configured in the new semantic field of spatial relations with W. In asimilar fashion, 'Let Dd be produced till it meets with AB in T' involves theintroduction of the visual participant T from the linguistic circumstance 'inT'. Thus intersemiotic transitions in the semantic circuit do not onlyinvolve accessing the meaning potential of new semiotic resources, but alsoinvolve introducing new functional elements and changing the functionalstatus of existing elements, both of which may be reconfigured in anew semantic realm. The semiotic interchange involves semantic re-contextualization and metaphor. The significant phenomenon ofintersemiotic metaphor is further explored through the notion of semioticmetaphor.

6.5 Semiotic Metaphor and Metaphorical Expansions of Meaning

Semiotic metaphor is the phenomenon where an intersemiotic transitiongives rise to a metaphorical expansion of meaning (O'Halloran, 1996,1999a, 1999b, 2000, 2003a, 2003b, in press) as demonstrated in the preced-ing discussion of Newton's text. In terms of experiential meaning, forexample, the status of the functional element as a process, participant orcircumstance undergoes a transformation through the shift or transition toanother resource. Alternatively, the shift to another semiotic resourceinvolves the introduction of a new process, participant or circumstancewhich did not previously exist. The phenomenon is best illustrated throughan example from a secondary school mathematics lesson in trigonometry(O'Halloran 1996, 1999b). This lesson contains several instances of thetypes of semantic expansions which are made possible through semioticmetaphor. In this example, the metaphorical nature of the semantic expan-sion is traceable because the linguistic text includes the oral discourse inthe classroom. Being spoken discourse, the classroom discussion is congru-ent rather than metaphorical (see Sections 3.6 and 6.6 for a discussion ofgrammatical metaphor in written mathematics and science). Cases of semi-otic metaphor are therefore evident in the classroom discussion and boardtexts largely because meaning has not been pre-packaged metaphoricallythrough grammatical metaphor.

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The teacher introduces and describes the trigonometric problem usinglanguage, and the diagrams in Figure 6.5(1) are produced in stages on theblackboard as a result of the classroom discussion led by the teacher. Theproblem concerns finding a generalized expression for the height of a cliffh and the width of a river r using two angle measurements. From the dia-gram, mathematical symbolism is used to solve the problem using the tan-gent ratio, which captures the relationship between the size of an angleand the length of the opposite and adjacent sides in the triangles. Extractsfrom the lesson which involve shifts between language, visual image andmathematical symbolism are given below.

STEP ONE: Teacher's Linguistic Introduction to the Problem// a man is actually at this point here// he is climbing a cliff// and /ahh doesn't know// how high up he is// and he looks down of course// and looks at the river// and doesn't know// how wide the river is

// so with this information, he has a ten-metre rope and a device thatmeasures angles, we are asking the question

// how can the man determine, firstly, the height of the cliff at point Aand, secondly, the width of the river

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INTERSEMIOSIS

STEP TWO: Visualization of the Problem

(a) HOW HIGH and HOW WIDE ?

181

HOW HIGH and HOW WIDE ?

Figure 6.5(1) Diagrams for Trigonometric Problem

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STEP THREE: Symbolic Solution to the Problem

In right A CBR

(1)

In right &ABR.

(2)

There are multiple cases of semiotic metaphor which occur in the processof solving the trigonometric problem through the macro-transitionsbetween language, visual image and mathematical symbolism. Thesesemiotic metaphors are described below.

1 The process realized by the verb 'look' in 'and he looks down of course'becomes an entity in the form of a line segment AR in the visual dia-gram on the blackboard (see Figure 6.5(la)). This new entity is laterintroduced in the verbal discourse as 'the line of sight'.

2 The Material process of 'measuring' becomes the participants a and 6with arrows marking the direction through which the angles are meas-ured and the dashed line representing the horizontal 'line of sight' (seeFigure 6.5 (lb)).

3 The circumstance of 'how high' and 'how wide' realizing Extent (spa-tial distance) is transformed into participants h and later h — 10 whichare marked with arrows indicating the relative distances in the diagram(see Figure 6.5 (lb)).

4 New entities in the form of the two triangles, AABR and ACBR, areintroduced visually in the mathematical diagram. These entities did notexist prior to the visual semiotic representation of the trigonometricproblem. The Figures of the triangles emerge as configurations of theparticipants h, rand 10, and a and 0. This means that these participantsmay be now viewed as a connected whole rather than as isolatedentities.

5 The newly constructed Figures of the triangles permits the relations

of sides and angles to be expressed symbolically as and

Thus the new entities, tan a and tan 9 are introduced. From

this point, the problem may be solved algebraically.

The metaphorical shift from the linguistic processes and circumstance tothe visual participants leads to the introduction of new participants in theform of two triangles in the diagram. From this point, the diagram becomesthe point of departure for the expression of the mathematical relation-ship which exists between the sides and angles of a right triangle in the

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form of the tangent ratio. This relationship forms the basis for the symbolicsolution of the problem.

From this example, it becomes clear that mathematical discoursedepends on the types of metaphorical shifts which take place with transi-tions between the three semiotic resources. From the semantic expansionsmade possible through semiotic metaphor, participants like 'the angle ofelevation' and 'angles of depression' become choices in language. Theimportant relationship between grammatical metaphor and semioticmetaphor is discussed in Section 6.6. Grammatical metaphor seems to haveevolved as a response to the expansions of meaning which occurredthrough the use of other semiotic resources in mathematics and science,including those metaphorical expansions arising from semiotic metaphor.In the case of the trigonometric problem considered above, new visualentities are construed in the shift from language to visual image. The sys-tems for language must necessarily expand to encompass these new visualentities, in this case in the form of the linguistic choices 'the angle ofelevation' and 'the angle of depression'. However, such linguistic construc-tions have traditionally been theorized in SFL as grammatical metaphorsrather than the product of meaning expansions through intersemiosis.

There appear to be graduations in the types of semiotic metaphoraccording to the nature and the extent of the semantic expansion whichtakes place. The two opposite poles are conceived as parallel and divergentsemiotic metaphors (O'Halloran, 1999a). A parallel semiotic metaphorhas 'anexpanded semantic field but also one which is situated within the old'(ibid.: 348). Although there could be redundant meanings because of over-laps, 'new layers of meaning are [essentially] simultaneously added to theoriginal representation' (ibid.: 348). Examples include the shift from pro-cess and circumstance to participant in Newton's diagram. The reconstrualof functional elements in a divergent semiotic metaphor, however, means that'the functional element is reconstrued into a new semantic field' (ibid.:348). The possibility for meaning expansions in divergent reconstruals isextensive as the functional element is relocated in a new semantic fieldwhich is not typically inter-textually related to the first. An example is theintroduction of the tangent ratio in the mathematical symbolism whichconfigures the relationship between an angle and the sides of the triangle.However, the meanings arising from divergent semiotic metaphors becomenaturalized over time, as has occurred with trigonometric ratios such assine, cosine and tangent in mathematics.

There are semantic redundancies involved in parallel semiotic meta-phors, and thus these types of metaphorical shift serve a reinforcing andco-contextualizing function. On the other hand, divergent semiotic meta-phors have the potential to involve conflicting meanings. These examplesof 'ideological disjunction' are a possible result 'of the complex, oftenintricate, relations of inter-functional solidarity among the various semioticresource systems that are co-deployed' (Thibault, 2000: 321). Divergentsemiotic metaphors emerge to create re-contextualizing relations. In

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essence, semiotic metaphors are one further intersemiotic mechanism forthe realization of co-contextualizing and re-contextualizing relations. Theother mechanisms include Semiotic Cohesion, Semiotic Mixing, SemioticAdoption and Juxtaposition developed in Section 6.2.

The identification of the phenomenon of semiotic metaphor is import-ant because it helps to explain how intersemiosis contributes to the expan-sion of meaning, including co-contextualization and re-contextualization.There remains much research to be completed in this field, which includesinvestigating semiotic metaphors for interpersonal, logical and textualmeaning. From this perspective, using multiple semiotic resources is notjust a matter of accessing different meaning potentials, rather theintersemiotic mechanisms prove to be an important tool for the semanticexpansion. These types of semantic expansions feed back into the grammarof each resource so that not only do the transitions become seamless, butalso new grammatical strategies evolve in each semiotic resource as a resultof their co-functionality with other resources. As explained below, the evo-lution of grammatical metaphor in language may be seen as a response tothe semantic expansions which occurred through semiotic metaphor andthe use of mathematical symbolism and visual display in the construction ofa scientific view of the world. The significance of semiotic metaphor isdiscussed in relation to mathematics pedagogy in Section 7.3.

6.6 Reconceptualizing Grammatical Metaphor

Halliday (1993a, 1993b, 1998), Halliday and Matthiessen (1999) and Mar-tin (1993a, 1993b) describe the regrammaticization of experience whichtakes place through scientific language. There is a decided 'semantic drift'towards metaphorically reconstruing experience in terms of entities whichenter into relations with other entities. The nominalization of process,quality, relator and circumstance takes place through the lexicogrammati-cal strategy of grammatical metaphor, which is described in Section 3.6.Halliday (1998: 209-210) and Halliday and Matthiessen (1999: 246-248)categorize the different types of grammatical metaphor and demonstratethat the general semantic shift is towards regrammaticizing of experiencein the form of entities. Halliday (1998: 211) captures this shift in thefollowing form:

relator —» circumstance —> process —» quality —> entity

Halliday and Matthiessen (1999: 263-264) distinguish two general motifs inthe semantic shifts taking place through grammatical metaphor: '(i) Theprimary motif is clearly the drift towards "thing", (ii) The secondary motif. . . the move from "thing" into what might be interpreted as a manifest-ation of "quality" (qualifying, possessive or classifying expansions of the"thing")'. This secondary shift involves the expansion of the potential ofthe nominal group to encode experiential meaning in condensed format.Halliday (1993a, 1998) cites several examples from Newton's (1704) Opticks

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to demonstrate that the shift towards nominalization noted in modern sci-entific writing had begun to emerge in this work. Halliday (1998: 201-202)explains that although reasoning is carried forth congruently by conjunc-tions (for example, 'if, 'so', 'as' and 'and'), grammatical metaphor in theform of nominalization also carries forth Newton's argument.

Grammatical metaphors may occur with relational processes to repack-age experiential content, which Halliday (1998: 193) terms 'the favouritegrammatical pattern ("syndrome" of grammatical features) in modern sci-entific English'. In this process, the semantic sequence of two clauses(forming a clause complex) is reconstrued as a single relational clausewhere the two original clauses are reconfigured into nominal groupstructures. The logical relations in the original clause complex aretypically reconstrued as different forms of relational processes. Hallidayand Matthiessen (1999: 239) summarize congruent and metaphoricalrealizations in Table 6.6(1).

Halliday (1998: 202) uses an example from David Layzer (1990:61) toillustrate the metaphorical repackaging of experience:

If electrons weren't absolutely indistinguishable, two hydrogen atoms would form amuch more weakly bound molecule than they actually do. The absolute indistinguish-ability of the electrons in the two atoms gives rise to an 'extra' attractive force betweenthem. (David Layzer, Cosmogenesis, 1990: 61)

In this example, the clause complex 'If electrons weren't absolutely indis-tinguishable, [then] two hydrogen atoms would form a much more weaklybound molecule than they actually do' is replaced by the clause 'The abso-lute indistinguishability of the electrons in the two atoms gives rise to an"extra" attractive force between them.' The logical relations 'if and 'then'are replaced by the causative relational process 'gives rise'. In addition, theclause Tf electrons weren't absolutely indistinguishable' is replaced by thenominalization 'the absolute indistinguishability of the electrons in the twoatoms'. Close analysis of scientific writing reveals that this type of grammat-ical repackaging is typical of contemporary scientific writing (Guo, 2004a,2004b; Halliday and Martin, 1993; Halliday and Matthiessen, 1999; Martinand Veel, 1998).

Halliday (1998: 202) explains that the impact of grammatical repack-aging is twofold. First, there is an increased potential for categorization and

Table 6.6(1) Congruent and Metaphorical Realizations of Semantic Units: Hallidayand Matthiessen (1999: 239)

Semantic Unit congruently metaphorically

clausenominal groupverbal group

clause complexclauseconjunction (relating clauses in acomplex

sequencefigurelogical relation

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taxonomic organization. Second, effective reasoning and logical progres-sion are enhanced as the repackaging of a clause into a nominal group hasa discursive textual function of carrying forth the momentum of the argu-ment. Halliday (1998) sees the development of scientific language in termsof'technicalizing' and 'rationalizing'. Halliday and Matthiessen (1999: 239,270-271) explain that the significance of grammatical metaphor is relatedto the new textual organization of the clause where participants may beforegrounded (and backgrounded) as packaged information. Halliday andMatthiessen (1999) also explain the gain is an increased potential toencode in condensed format experiential meaning in nominal group struc-tures, which is essential for the construction of technical knowledge andtaxonomies (Martin, 1993a, 1993b, 1993c). However, there is a subsequentloss of experiential meaning with the collapse of the clause into nominalgroup structures and the blurring of categories of experience whererelators, circumstance and process become construed as entities.

In effect, the loss of experiential meaning includes the loss of arguabilitywhich accompanies a congruent configuration of process/participant in aclause. For example, in addition to questioning the effect on Z, the reader/listener can disagree that X transforms Y based on the following congruentclause complex sequence:

X transforms Y, so Z will be affected

However, the metaphorical construal does not admit such doubt concern-ing the relations between X and Y, and, further to this, the effect on Z seemsmuch more certain with the absence of the logical conjunction:

the transformation of X to Y affects Z

The packaging of experiential meaning through the use of grammaticalmetaphor and relational processes takes place in mathematical discourse.For example, the analysis of the linguistic text in Stewart (1999: 132) inPlate 6.3(2) reveals that the majority of clauses contain relational identify-ing processes with nominalized participants involving extended nominalgroup structures; for example, 'the derivative of/at x^, 'a second interpret-ation of the derivative', 'the instantaneous rate of change', 'the connectionwith the first interpretation', 'the instantaneous rate of change', 'the slopeof the tangent to this curve' and so forth. The metaphorical nature ofmathematical writing is evident in the following examples from Plate 6.3(2)where relational processes (in bold) configure entities in the form of nom-inalizations. In addition, logical meaning is re-packaged through processselections (for example, 'as being' and 'gives').

// this limit as being the derivative of /at xt//// this gives a second interpretation of the derivative//// The derivative/' (a) is the instantaneous rate of change ofy=f(x)//// the connection with the first interpretation is [[that if we sketch the

curve y=f(x) \\ then the instantaneous rate of change is the slope ofthe tangent to this curve at the point 1 1 where x = a ] ] //

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Halliday (1998) and Halliday and Matthiessen (1999) thus explain howrelational clauses and grammatical metaphors which expand the potentialof the nominal group function in scientific language. However, if the sym-bolic text in Example 4 in Plate 6.3(2) is analysed, we may see that theopposite trend has occurred in the mathematical symbolism where pro-cess/participant configurations are preserved through the grammaticalstrategy of rankshift, as developed in Chapter 4.

However, if the symbolic statement

it would read something like:

the derivative of fat t = 2 is [[the limit as h approaches zero of thedifference between the value of the function at 2+h and 2 [[divided byh]]]].

The linguistic version of the symbolic statement involves a relational pro-cess for '='. However, the rankshifted configuration of symbolic processesand participants on the right-hand side of the equation undergoes a recon-strual through the grammatical metaphor: 'the difference between thevalue of the function at 2+h and 2 divided by h'. In other words, in thetransition to language, the meaning of the symbolic statement is not retained becauseprocess/participant configurations become entities in the form of nominalized parti-cipants. This is particularly significant in the context of the mathematicsclassroom where the symbolism is regularly verbalized. The metaphoricalnature of such construals, however, is rarely if ever noted. The significanceof semiotic metaphor for pedagogy is discussed in Chapter 7.

It appears that scientific language evolved to construe a 'stable' view ofthe universe through relational processes with nominalized participantswhich carried forth the impetus of the argument because the function of the'dynamic' description of the relations was allocated to the mathematicalsymbolism, which accordingly developed a grammar enabling it to fulfilthis role. The functions of scientific language should be contextualized inrelation to the context of its development. However, given the relativeinaccessibility of mathematical symbolism to a general audience, scientificlanguage has assumed functions which are not understood in relation to

is verbalized,

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the limits of its applicability. For as Halliday and Matthiessen (1999: 271-272) explain, the linguistic reconfiguration of reality through scientificdiscourse has repercussions:

Metaphors are dangerous, however; they have too much power, and grammaticalmetaphor is no different in this respect. Because it leaves relations within a figurealmost inexplicit, this demands that they should in some sense already be in place[beforehand] . . . It is this other potential that grammatical metaphor has, for makingmeaning that is obscure, arcane and exclusive, that makes it ideal as a mode ofdiscourse for establishing and maintaining status, prestige and hierarchy, and toestablish the paternalistic authority of a technocratic elite.

The mathematical and scientific view of the world is used in contexts whichare not always appropriate. The metaphorical nature of the construction ofmathematical and scientific discourse and the accompanying uses of thescientific view of the world are considered in Chapter 7. From the preced-ing discussion, it becomes evident that more research is needed in the studyof the functions of language and other semiotic resources as an integratedphenomenon. Only from this perspective can semiosis be fully understood.

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7 Mathematical Constructions of Reality

7.1 Multisemiotic Analysis of a Contemporary Mathematics Problem

Mathematical discourse consists of a range of genres such as the researchpaper, the mathematics lecture, the mathematics book and textbook whichtypically contain a range of sub-genres. For example, the mathematics text-book consists of chapters and subsections which contain definitions, the-orems, explanations of theory, demonstration examples, practice examplesand solutions. Thus the mathematics textbook is a multisemiotic genreconsisting of sub-genres, which in turn consist of Items. In what follows,Example 2.24, a mathematics problem from Burgmeier et al. (1990: 76-77),an introductory university textbook on calculus, is analysed in order toinvestigate intersemiosis in contemporary mathematics discourse. Example2.24 is reproduced in Plate 7.1(1). The theoretical frameworks for theanalysis include SF models for language (Tables 3.1(1) and 3.1(2)), math-ematical symbolism (Tables 4.3(1) and 4.3(2)) and visual images (Tables5.2(1) and 5.2(2)), and the systemic framework for intersemiosis across thethree resources (Tables 6.1(1) and 6.2 (2a-d).

In Example 2.24, readers are asked questions concerning the 'receptivityof students', which is conceptualized as a function of the time elapsed in alecture. Example 2.24 contains a linguistic description of the problem andthe equation G(x) — - 0.1 x2 + 2.6x + 43 which describes receptivity, 'theability of the students to grasp a difficult concept', as a function of timewhere x is the number of minutes which have elapsed in a lecture (Burg-meier et al, 1990: 77). The description of the mathematics problemincludes a black and white photograph of a group of students in a universitylecture. In Example 2.24, the questions concern (a) the times at whichreceptivity is increasing and decreasing, (b) the situation with regard tostudent interest after ten minutes, (c) the time in the lecture where the mostdifficult concept should be placed, and finally (d) whether it would bepossible to teach the students a certain concept which 'requires a receptivityof 55' given 'the intelligence level of the students in this group' (ibid.: 77).

The solutions to the questions (a)-(d) are given in the form of explana-tory linguistic statements, mathematical symbolic statements and Figure

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2.5 Increasing and Decreasing Functions

Example 2.24 By studying the learning behavior of a group of studentsa psychologist determines that receptivity, the ability of the students tograsp a difficult concept, is dependent on the number of minutes of theteacher's presentation that have elapsed before the concept is intro-duced. At the beginning of a lecture a student's interest is stimulated,but as time passes, attention becomes diffused. Analysis of this group'sresults indicate that the ability of a student to grasp a difficult concept isgiven by the function

where the value G(x) is a measure of receptivity after x minutes ofpresentation.(a) Determine the values of x for which student receptivity is increasingand decreasing.(b) Is student interest being stimulated after 10 minutes or is attentive-ness falling off?(c) Where in the presentation should the most difficult concept beplaced?(d) For the intelligence level of the students in this group, a certainconcept requires a receptivity of 55. Is it possible to teach the studentsthis concept?

Solution (a) Student interest is being stimulated when G(x) is increas-ing, and attentiveness is falling off when G(x) is decreasing. Todetermine where G(x) is increasing and where it is decreasing, we usethe derivative of G(x) that is, G'(x) = -0.2* +2.6. We know that G(*)is increasing where G'(x)>0, so we solve the inequality

which is equivalent to

(inequality sign is reversed)

Thus, student receptivity G(x) is increasing for x < 13 and decreasingfor x > 13. (See Figure 2.24.)(b) Since G(x) is increasing at x = 10, student interest is still beingstimulated 10 minutes into the presentation.(c) Receptivity is increasing during the first 13 minutes and decreasingafter the first 13 minutes, maximum receptivity occurs 13 minutes intothe presentation. Thus, the most difficult concept should be discussed 13minutes after the presentation begins.(d) Since G(13) = 59.9 and the concept we want to present requires areceptivity of 55, it is possible to teach this concept to these students.

Plate 7.1(1) Mathematics Example 2.24 (Burgmeier et al, 1990: 76-77)

2.24, the graph of the function G(x) where different points of time havebeen marked. The solution is obtained by considering the derivative G (x)where receptivity increases for G (x) > 0 (i.e. an increase in x correspondsto an increase in G(x)) and decreases for G (x) < 0 (i.e. an increase in xcorresponds to decrease in G(x)). In what follows, the linguistic, visual andsymbolic choices in Example 2.24 are analysed with respect to the meaningswhich arise (i) intrasemiotically within the linguistic, visual and symbolicItems and (ii) intersemiotically within and across the Items. The register,genre and ideology arising from such choices are discussed.

Figure 2.24Graph of student receptivity func-tion G(x) in Example 2.24

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Compositional Organization of Example 2.24

Example 2.24 is a demonstration example, which is a common sub-genre inmathematical textbooks. These examples serve to illustrate standard solu-tions to typical mathematics problems. Such examples contain obligatoryItems in the form of the problem and the solution. In mathematics prob-lems involving the derivative (such as Example 2.24), a graph or sketch istypically included, although this is not always the case. The photograph isoptional. The obligatory Items in Example 2.24, the question and the solu-tion, are spatially separated from the optional Items, the photograph andgraph. The compositional arrangement is marked by the justification of thelinguistic/symbolic text in the column to the left of the two visual images.Within this text, the important symbolic statements are spatially separatedthrough the use of line spacing and centre alignment. Each of the symbolicstatements is marked through italicized font. Compositionally, the problemis organized so that the question and the solution (including the symbolicequations), the photograph and the graph are distinct Items. Example 2.24is textually organized to accommodate a reading path for these four Items.Prominence is given to the question, the solution and the graph throughTypography, where 'Example 2.24', the Questions '(a)-(d)', the 'Solution'for (a)-(d) and the Caption for Figure 2.24 appear in bold. At the rank ofdiscourse semantics, intersemiotic patterns of meaning take place acrossthe Items.

The Photograph

At the rank of Item, the photograph is visually prominent because of itsspatial location on the right-hand top corner of the page and the contrastwhich is provided by the density of black and white photograph against thewhite background of the printed text. The photograph is a picture of uni-versity students in a lecture theatre. Culturally, this is evident to the readerof the mathematics book (presumably a university student or lecturer)given the nature of the seating arrangements, the age of the students, thestyle of clothing and the activities which are depicted, such as listening andtaking notes. The Gaze of the students in the photograph is directeddownwards to the left, most likely towards the stage where the presentationis taking place.

The students in the photograph are two male black students and fourwomen. At least three of these women belong to ethnic minority groups.The two black male students physically lean sideward from each other asthey look at and listen to the lecture presentation. The contrast provided bytheir light-coloured clothing and the dynamism provided by the angleformed by their arms and the arm of the woman sitting in the row aboveconverge to an empty space that lacks a centrally framed Figure. However,given their Position and Size, the Figures in the photograph which emergeas prominent are the two male students whose body posture forms a

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dynamic 'V shape. At the rank of Part of the Figure, the facial expressionsof these two students suggest a somewhat sceptical reaction to the perform-ance which is taking place. The female students listen with varying degreesof concern, three looking to the left, and one looking down as thoughwriting. The majority of the students in the photograph rest their chinsupon their hands in what appears to be a studied reaction to the lecture.

The Gaze and action of the students in the photograph do not beckonthe reader of the mathematics textbook. Rather the reader's attention isdrawn to the picture due to the position and the density of the black andwhite photograph. However, the students' Gaze and accompanying bodyposture in the photograph guide the reader's attention downwards to theleft towards the block of text which consists of the question and solution inExample 2.24. There is a directional link which stretches from the photo-graph towards linguistic/symbolic parts of the text through the vectorprovided by the Gaze and posture of the student in the photograph.INTERSEMIOTIC NEGOTIATION thus includes a discourse move,however fleeting or recursive, from the photograph to the mathematicsproblem through Gaze and body posture of students in the photograph.The implications of this discourse move are considered below.

Intersemiosis: The Photograph and the Linguistic/Symbolic Text

The linguistic text introduces the mathematics problem in the followingmanner:

Example 2.24By studying the learning behaviour of a group of students a psychologist determinesthat receptivity, the ability of the students to grasp a difficult concept, is dependent onthe number of minutes of the teacher's presentation that have elapsed before theconcept is introduced. At the beginning of a lecture a student's interest is stimulated,but as time passes attention becomes diffused. Analysis of this group's results indicatethat the ability of a student to grasp a difficult concept is given by the function

where the value G(x) is a measure of receptivity after x minutes of presentation.

INTERSEMIOTIC IDENTIFICATION at the rank of discourse involvesidentifying and tracking participants across the Items in Example 2.24. Inthe case of the photograph and the linguistic text, INTERSEMIOTICIDEATION and TRANSITIVITY RELATIONS function to establish whatappears to be a VISUALIZATION of the mathematics problem: 'By study-ing the learning behaviour of a group of students' (photograph of a groupof students is provided) 'a psychologist' (a professional whose identityremains anonymous) 'determines that receptivity, the ability of the stu-dents to grasp a difficult concept' (photograph of students concentratingon the lecture), 'is dependent on the number of minutes of the teacher'spresentation' (a second professional whose identity remains anonymous)'that have elapsed before the concept is introduced.' 'At the beginning of a

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lecture a student's interest is stimulated' (the students in the photographappear to be attentive), 'but as time passes attention becomes diffused' (thephotograph is taken at some stage during the lecture, given the body pos-ture and facial expression of the students). 'Analysis of this group's results'(the performance of 'this group' as measured by the psychologist) 'indi-cates that the ability of a student to grasp a difficult concept is given by thefunction G(x) = -O.lx2 + 2.6x + 43 where the value G(x) is a measure ofreceptivity after x minutes of presentation' (the notion of 'this group' islinked to the ability of an individual student). From the description of theproblem, it becomes apparent that groups of students perform differently,and so the symbolic description of receptivity as a function of time elapsedin a lecture in Example 2.24 is a description of the results of one particulargroup. And the photograph does contain one group of students; that is,students from ethnic minority backgrounds. The results of the group arere-contextualized in relation to individual performance, 'the ability of astudent to grasp a difficult concept', in Example 2.24. The added dimen-sionality to Example 2.24 which arises intersemiotically from the choices inthe photograph and the linguistic text is further discussed below.

In what amounts to a VISUALIZATION of the mathematics question,only some of the major participants introduced in the linguistic text arerepresented in the photograph. The professionals, the 'psychologist' whopronounces the relationship between time and student receptivity and the'teacher', appear in the linguistic text only. On the other hand, the stu-dents are pictured as ethnic minority students. In this way, the linguisticconstruction 'receptivity', which is reconstrued as 'the ability of the stu-dents to grasp a difficult concept', becomes associated with one particulargroup of students. The view is reinforced that the ability of these minoritystudents is related to how long they can pay attention because 'as timepasses, attention becomes diffused'. Thus INTERSEMIOTIC IDEATIONrelations are established as to which group of students have problemsgrasping difficult questions given their attention span. Failure 'to graspconcepts' is linked to personal ability, attentiveness and, through thephotograph, ethnicity and race. The problem of educational achievementbecomes tied to internal psychological, mental and biological criteria suchas the attention span, ability and race rather than the complex practices ofeducational institutions and society as a whole. A whole range of issues suchas social, cultural and economic factors are thus distilled into what appearsto be an extraordinarily simplistic and misleading construction of thefactors underlying educational achievement.

Further to this, the professional psychologist is seen to be able to deter-mine the exact nature of the learning process, and the result is a concernwith presenting lecture material at the right time. The organization of thelecture is a textual concern where ideational meaning is reduced to thenotion of 'a difficult concept'. Features of the lecture and its presentation(for example, the actual content of the lecture, the language and visualimages used, the interpersonal stance of the lecturer, the educational

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institution, and other social and cultural contextual factors) are excludedin the formulation of 'student receptivity'. Finally, the notion of 'receptiv-ity' is converted into 'the intelligence level of the students in this group' inquestion (d). The relations between educational achievement and geneticmakeup are reinforced along racial grounds. The implications of theseconstructions are further considered through an analysis of the linguisticand symbolic text in Example 2.24.

Intrasemiosis in Language: The Use of Grammatical Metaphor

The linguistic text, which functions to introduce and contextualize theproblem in relation to the photograph and the symbolic equations, ismetaphorical as indicated by the numerous instances of grammaticalmetaphor, which appear in bold below:

By studying the learning behaviour of a group of students a psychologist determinesthat receptivity, the ability of the students to grasp a difficult concept, is dependent onthe number of minutes of the teacher's presentation that have elapsed before theconcept is introduced. At the beginning of a lecture a student's interest is stimulated,but as time passes attention becomes diffused. Analysis of this group's results indicatethat the ability of a student to grasp a difficult concept is given by the function

G(x) =-0.1x2 + 2.6x+43

where the value G(x) is a measure of receptivity after x minutes of presentation.

After the explanation of the problem, questions (a)-(d) also containgrammatical metaphors which include: (a) the time when student receptiv-ity is increasing or decreasing, (b) whether at different times interest isbeing stimulated or attentiveness is falling off, (c) where the most difficultconcept should be placed in the presentation, and (d) whether or not aconcept requiring a receptivity of 55 could be taught given the intelligencelevel of the students in this group.

In terms of experiential meaning, grammatical metaphor functions toconstruct a situation where processes and attributes are reconstrued asentities. In this shift, the real participants (the lecturer and students asMedium and/or Agent) and circumstance completely disappear or they areburied within the nominal group structures of the metaphorical entities.For example, 'learning behaviour' is an entity (rather than the process ofthe students 'behaving' in some fashion) which can be measured by thetrained expert, the psychologist. 'Receptivity' is an entity (rather than theprocess of the students 'receiving' material from the lecturer in some fash-ion) which is denned to be another entity 'the ability of the students'(rather than an attribute that the students demonstrate in some manner)to grasp the entity 'a difficult concept' (from the process of the students'conceiving' something from the lecturer in some way). The mathematicsproblem in effect reduces the complex and dynamic practices of universityeducation into metaphorical entities in the form of nominalized partici-pants which are defined and aligned using relational processes. Following

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this formulation, these metaphorical entities are exactly described usingmathematical symbolism in Example 2.24.

The use of grammatical metaphor extends to interpersonal meaning inthe form of modality metaphors. The series of statements are either unmo-dalized in terms of probability or usuality, or they are metaphorical in thatthe modality orientation is explicitly objective; for example, Ts it possibleto teach the students this concept?' The interpersonal orientation of thetext exudes a confidence and objectivity which is difficult to counteract,especially as the majority of clauses contain relational processes whereidentifying relations are established between the metaphorical entities. Inaddition, the textual organization of the clauses reveals marked choicessuch as foregrounding particular elements such as 'By studying the learn-ing behaviour of a group of students', 'At the beginning of the lecture', and'For the intelligence level of the students in this group'. Furthermore,logical meaning is metaphorically re-packaged as a process rather thanbeing instantiated as a conjunctive relation; for example, 'Analysis of thisgroup's results indicate that the ability of a student to grasp a difficultconcept is given by the function'. In summary, the linguistic text is highlymetaphorical, and, further to this, the shift to mathematical symbolicdescription involves a reconstrual which only includes one dimension ofthe context of the problem - the time elapsed in the lecture.

Intersemiosis: Linguistic Construction of the Problem and the Symbolic Equations

The metaphorical entity 'receptivity' is symbolized as G(x). Through therelational process '=', G(x) is given a definitive description in the form ofthe mathematical equation G(x) = -O.I;*2 + 2.6x+ 43 where the independ-ent variable is x, the number of minutes that have elapsed after the presen-tation has started. The mathematical symbolism thus provides a dynamicdescription of the metaphorical entity 'receptivity' in exact terms wherethe only variable is the time elapsed in the presentation. In the questions(a)-(d) which follow, 'receptivity' is related to another metaphoricalentity, 'the intelligence level of these students' in question (d). As seenabove, there are particular students in question as portrayed by thephotograph.

The nature of the experiential meaning in the linguistic construction ofthe problem and the intersemiotic shift to the symbolic description arerather incredible. First, the use of grammatical metaphor reduces the com-plex dynamic process of a university lecture into a series of metaphoricalentities which are related to other metaphorical entities through relationalprocesses. This is achieved with a high degree of certainty. The photographfunctions to contextualize these relations along racial lines. Second, thesymbolism permits the dynamic reconfiguration of that situation in termsof one variable only, the time which has elapsed in a lecture. Other vari-ables concerned with teaching and learning are excluded in the symbolicformulation. In this process, the issue of the viability of measuring and

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describing such an entity as 'receptivity' in precise symbolic form is side-stepped. Grammatical metaphor and other linguistic choices construemetaphorical entities which are re-inscribed symbolically as a dynamicprocess which unfolds over time, but only within a semantic field whichinvolves x, the time elapsed in the lecture.

In the shift from language to the symbolism, the phenomenon of semi-otic metaphor is invisible because of the layers of grammatical metaphor.For example, congruent linguistic constructions such as 'the lecturer useslanguage and other forms of semiosis to teach the students', or 'the studentreceives the lecture material from the lecturer in multiple semiotic forms'where the lecturer/student is the Medium or Agent are reduced to 'recep-tivity', 'the ability to grasp a difficult concept'. The mental/material/behavioural dynamic process of teaching/learning shifts seamlessly fromthe metaphorical construct 'receptivity' to the symbolic participant G(x).In other words, semiotic metaphor occurs in that the process 'receive'(linguistic) is construed as an entity G(x) (symbolic). Once constituted as asymbolic entity, G(x) becomes a linguistic entity; that is, the 'measure ofreceptivity after x minutes of presentation'. The development of grammat-ical metaphor in language is related to the types of metaphorical expan-sions which take place through intersemiosis. Such grammatical andintersemiotic metaphorical constructions now appear as commonsenseknowledge in contemporary discourse. In Example 2.24, the symbolic G(x)is reformulated in terms of a dynamic construal of mathematical partici-pants and Operative processes in relation to the variable of time. In theshift to Figure 2.24, another form of semiotic metaphor occurs in that anew entity in the form of a curve is introduced visually. Thus through theuse of language, mathematical symbolism and a graph, the complexdynamic process of teaching/learning is reduced to metaphorical entitiesand relations. This construal is hedged in rhetoric which implicitly relateseducational performance to the innate characteristics of the studentssuch as ability and intelligence which are literally 'viewed' along the lines ofethnicity and race.

Intrasemiosis: The Mathematical Symbolism

To answer the questions (a)-(d) posed in Example 2.24, mathematicalsymbolism is called into play. The derivative G' (x) is calculated andequated to zero to find the times for which receptivity is increasing anddecreasing, and the optimal time for the presentation of the most diffi-cult concept. To determine when receptivity is increasing, the value forG' (x) > 0 is found as follows:

-0.2x+2.6>0-0.2x > -2.6

0.2% < 2.6x < 1 3

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The solution to the problem unfolds seamlessly in the form of an IMPLI-CATION SEQUENCE. The grammar of mathematical symbolism has beenspecifically designed exactly for this purpose (see Chapter 4). That is, thesymbolism preserves the rankshifted process/participant configurations sothat the equation can be reconfigured and simplified in order to find thesolution to the problem. The algebraic inequality is rearranged so that thevalues for which the derivative is greater than zero are found. The solutionto Example 2.24 depends upon previously established results which includethe definition of the derivative, various algebraic laws, the definition of anegative number, and the Multiplicative Properly of Negative One (which issignalled in Example 2.24 by the statement 'inequality sign is reversed').Thus the following answers are provided in Example 2.24: (a) studentreceptivity is increasing for x < 13 and decreasing for x > 13, (b) studentinterest is found to be still increasing when x = 10, (c) the optimal time forthe discussion of the most difficult concept is x= 13 (although the fact thatstudent receptivity must immediately start to decrease after this point oftime is not considered) and (d) substituting x= 13 in the original equationG(x) = -0.1x2 + 2.6x+43gives 'the maximum value of receptivity' to be 59.9,which is higher than the 55 required for the concept to be taught to 'thisparticular group of students'. So apparently it is possible to teach theconcept to these students, given 'their intelligence level'.

Intersemiosis: The Mathematical Symbolic Solution and the Graph

The mathematical symbolic solutions to questions (a)-(d) are directlylinked to the graph of the function, which serves to provide a VISUALIZA-TION of the results which are established symbolically. For example,INTERSEMIOTIC IDENTIFICATION is realized through direct referencein the caption for Figure 2.24 which reads 'Graph of receptivity functionG(x) in Example 2.24'. Intersemiotic Reference takes place through theintersemiotic mechanism of Intersemiotic Mixing where the graph isLabelled G(x) = —O.lx2 + 2.6x + 43. A curved line pointing to the graph ofthe function accompanies the Labelling in Figure 2.24. The axes areLabelled linguistically 'receptivity' and 'time (in minutes)' and symbolicallyx and G. In addition, the scales are calibrated numerically. The use ofJuxtaposition and compositional arrangement of the labels in Figure2.24 means that there is no ambiguity as to the identification of the visualparticipants and circumstance in the graph.

The mathematical symbolic relationship consists of rankshiftedconfigurations of mathematical participants and processes indicated by[[ ]]:

G(x) = [[[[-0.1[[*x x]]]] + [[2.6*]] +43]]

This rankshifted configuration becomes a metaphorical visual entity inthe form of a curve in Figure 2.24. The dynamic aspect of time is there-fore related to spatiality in the graph. Significant values for (x, G(x)) for

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questions (b) and (d) are Labelled and marked by points on the graph as(10, 59) and (13, 59.9). Interpersonal attention is drawn to the criticalturning point of the graph at (13, 59.9) through the dotted vertical linewhich extends from the x axis to the curve. There is no visual represen-tation of the derivative as the slope of the tangent line at the points whichare marked on the curve. The definition of the derivative is assumed giventhe preceding theory presented in Sections 2.1-2.5 in Burgmeier et al.(1990:45-76).

The levels of assumption in this mathematics problem, in relation to itsconception (for example, the method through which the psychologistdetermines that 'receptivity' for a group of students is given by G(x) ——0.1^ + 2.6x+ 43) and the solution, are not addressed in Burgmeier et al.(1990). In Example 2.24, the variable 'receptivity' is a measurable entitywhich is dependent on time. The terms 'receptivity' and 'intelligence' arereduced to a function of the time elapsed in the teacher's presentation.The problem is solved using mathematical symbolism. The main functionof the problem is, however, to demonstrate how the derivative of a functionmay be used to solve mathematics problems involving increasing anddecreasing functions, and this is perhaps the key reason for the formulationof the problem. Nonetheless, it is worthwhile to examine the types ofsemiotic choices in Example 2.24 in order to investigate the underlyingideological assumptions of the problem.

Example 2.24: Register, Genre and Ideology

There are consistent patterns of registerial configurations across the lin-guistic, symbolic components of the Example 2.24. In terms of experientialmeaning, the Bi-directional Investment of Meaning through intersemiosiswith respect to the Contextualization Propensity (CP) (Cheong, 1999, 2004)is high. The linguistic items, visual images and mathematical symbolismdirectly function to contextualize each other so there is a low InterpretativeSpace (IS) which results in a direct Semantic Effervescence (SE). The for-mulation and solution to mathematics problems become straightforwardstandard procedures. Example 2.24, however, involves the application ofmathematical concepts and theory to an educational context. The assump-tions behind this application of mathematical theory and concepts are notdiscussed in Burgmeier et al. (1990).

In terms of interpersonal meaning, the tenor, or the relation between theauthor and the reader, is unequal. Following generic conventions, choicesfrom the systems for language, symbolism and visual images are such thatthe mathematics text appears objective and factual. Statements are made,directions are issued and the problem is solved. The abstract nature of theprocesses and participants and the style of production of the text contrib-ute to the dominating tenor. Textually, the mathematics problem is tightlyorganized at the ranks of Item, discourse, grammar and display. There isno ambiguity arising from the conventionalized modes of presentation.

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Logically, the solution to the problem unfolds smoothly according to themeaning potential made available through the grammar of mathematicalsymbolism. The reasoned logic realized through the symbolism is mirroredin the surrounding linguistic text and the graphical display of the results.

As seen in Example 2.24, the use of language, mathematical symbolismand visual representation in mathematics discourse has developed andexpanded into new realms of human experience since the days of Newtonand Descartes. In Example 2.24, order is imposed on educational practicessuch that identifiable entities such as 'learning behaviour', 'receptivity','the ability of students', 'teacher's presentation', 'difficult concepts' and'intelligence levels' become commonsense constructs. Firmly entrenchedin mathematical and scientific discourse, the use of grammatical and semi-otic metaphor functions to order experience such that metaphorical con-structs appear as real entities formulated in exact symbolic terms byaccredited experts in the field. Once described, exact values may be calcu-lated as they change over time. The nature of these exact relations may beviewed graphically. Within this discourse, certain groups are constructed assubjects in such a way as to provide causes and reasons for central inequal-ities in the educational system. The flux of experience is reduced to amatter of variables, in the case of Example 2.24, one variable in the form ofthe time which has elapsed in a university lecture. Explanations can beprovided to legitimize and rationalize educational practices which continueto support privilege in society. Following Foucault (1991), this is done in asociety that categorizes, measures, evaluates and normalizes. The tools forthese descriptions are semiotic.

Foucault (1980b, 1984) speaks of scientific discourse as the discourse oftruth in contemporary society. Thibault (1997: 108) elaborates: 'This para-digm has provided the basis for approaching the problems of causality notonly in physics, but in all other domains of enquiry in both the natural andsocial sciences.' The mechanisms through which semiotic mediation inmathematics contributes to this exercise are a central focus of concern inthis study. Further interpretation of the complexity and the simplicity ofthe social-semiotic reality of mathematical and scientific discourse isneeded, especially in the realm of mathematics and science education.In what follows, the implications of an SF approach to mathematics as amultisemiotic discourse for mathematics education are summarized.

7.2 Educational Implications of a Multisemiotic Approach to Mathematics

An SF perspective of mathematics as a multisemiotic discourse has manyimplications for mathematics education. In what follows, these are dis-cussed in relation to language, mathematical symbolism, visual display,intersemiosis across the three resources, and semiotic and grammaticalmetaphor. In addition to providing a theoretical framework through whichmathematics may be viewed in mathematics education, the SF approachalso permits the analysis of discourse in mathematics classrooms. The

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nature of pedagogical discourse in mathematics classes is discussed inSection 7.3. The analysis reveals the complexity of classroom discourse andthe difficulties which inadvertently arise from using English as the meta-language to teach mathematics. Finally, concluding comments concerningthe implications of the semiotic makeup of mathematics and science aremade in Section 7.4.

The SF Approach to Language

The SF approach to language offers a comprehensive model through whichstudents may come to understand that language is a tool used to createorder, and that meaning is a matter of choice. The metafunctionalapproach shows that content is only one aspect of the order which isimposed on the world. Equally important are the social relations which areenacted, the logical reasoning which takes place and the ways in which themessage is organized and delivered. In addition, although meaning may bea matter of choice, students can appreciate that there are culturally specificways in which language is used in different contexts. The understanding ofthose ways, and the interests served by such language selections, opens theway for a critical engagement with texts. An understanding that reality isenacted contextually through particular configurations of linguisticchoices is critical for interpreting discourse in diverse contexts which spandigital media, printed media and material lived-in-day-to-day reality. In con-temporary times where information is increasingly a commodity, the abilityto critically read, interpret and write is becoming a necessary resource forsurvival. This is particularly important in the context of education wheregroups of students are marginalized (Bernstein, 1971, 1973, 1977, 1990,2000).

The nature of linguistic selections in mathematics and science may becontextualized with respect to other possible choices using the SFLapproach. Mathematical and scientific language involve particular types oflinguistic choices which organize reality in particular ways. Mathematicsand science are registers where particular configurations of experiential,logical, interpersonal and textual meanings are found. The choices relateto the functions of language in mathematics; that is, to contextualize themathematical problem and to draw implications from the results. However,the nature of scientific writing and reading (Halliday and Martin, 1993;Lemke, 1990; Martin and Veel, 1998) need to be situated in relation to thefunctions of mathematical symbolism and visual images. For example,the semantic drift towards the constructions of metaphorical entitiesand the expansion of nominal group structures in language needs to becontexualized in relation to the functions and lexicogrammatical strategiesfound in mathematical symbolism and visual images. Mathematical andscientific language developed in particular ways because the dynamicconstrual of reality was delegated to mathematical symbolism where therelations could be displayed visually.

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In terms of experiential meaning, mathematical language is largely con-cerned with relational processes where identifying relations configuremetaphorical identities. Logical meaning is concerned with elaborating,extending and enhancing relations for the description of problems andthe prediction of outcomes. The interpersonal stance of mathematicallanguage includes a high modality or truth value where statements andcommands have maximal probability and obligation. The abstract andnominalized forms of the participants contribute to the authorative stanceof mathematics texts, which are organized in generically specific ways. Thetypography and compositional choices in mathematics texts include thespatial organization of items according to significance; that is, importantlinguistic, symbolic and visual components are spatially marked so that thereader may easily access information through a recursive scanning-typereading path. Finally, to reiterate an important point, mathematical andscientific language cannot be viewed in isolation. The nature of the selec-tions and the lexicogrammatical strategies for encoding meaning needs tobe seen in relation to the mathematical symbolism and visual display. SFLoffers a theoretical approach through which this is possible.

The SF Approach to Mathematical Symbolism

The SF approach to mathematical symbolism demonstrates how this semi-otic resource developed new grammatical strategies for encoding meaning,and the reasons for this development. A historical perspective reveals howrealms of meaning were set aside in order to develop a semiotic resourceconcerned with the descriptions of relations in a de-contextualizedenvironment. The algebraicization of geometry meant that spatial relations(the visual image) could be connected to temporal and logical relations(symbolic descriptions of relations over time) in an exact manner. A seman-tic circuit was created in that linguistic and mathematical descriptions weretied to visual images. Mathematical symbolism developed as a tool thatcould be used for reasoning, and the grammar thus developed specialtechniques where process/participant configurations were preserved forreconfiguration for the solution to problems. These techniques includethe development of new grammatical systems, the simultaneous con-traction and expansion of process types to include Operative processes, andthe use of multiple levels of rankshift so that the symbolic processes andparticipants can be rearranged in the solution to mathematics problems.

The grammar of mathematical symbolism functions differently from thegrammar of language, and this needs to be made explicit in an educationalcontext. Language functions to construe metaphorical entities through theexpansion of the potential of the nominal group. These entities are relatedto other entities so that clause complex logical relations are reconfiguredas single clauses, and clauses are reconfigured as nominal groups. Inthis process, logical deduction using language is realized metaphoricallythrough the selection of causative relational processes. However, these

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grammmatical strategies took place in language because the responsibilty forthe dynamic description of relations and logical deductions was allocatedto the symbolism. Consequently mathematical symbolism developed newgrammatical systems (such as the use of spatial notation, special symbolsand the Rule of Order for operations) so that meaning was economicallyand precisely encoded in rankshifted configurations which could be easilyrearranged for the solution to mathematics problems.

Halliday (1985: 87) compares the 'density of substance' or the 'lexicaldensity' of written text to the 'intricacy of movement' or the 'grammaticalintricacy' of spoken language. However, the multiple rankshift or embed-ding of nuclear configurations of Operative processes in mathematicalsymbolic text gives what can be described as grammatical density. Eventhough the participants and processes are typically generalized variablesand Operative processes, the symbolic mathematics is experientially densebecause of the precision and economy with which meaning is encoded.The preservation of process/participant configurations involves an un-precedented flexibility through a range of grammatical systems whichpermit those configurations to be rearranged as the symbolic text unfolds.Mathematical symbolism combines the flow of spoken discourse with thedensity of written language. The difference is that density in mathematicalsymbolism involves specialized systems for economy of expression andmultiple levels of rankshift, while written language packs meaning intoextended word group structures.

The price for the semantic expansion where mathematical symbolismprovides exact descriptions of relations which are reconfigured to solveproblems is a limitation of the semantic realm with which mathematics isconcerned. Mathematical symbolism is largely concerned with relationaland Operative processes with limited forms of circumstance. Logical mean-ing is aided by textual organization so that solutions of mathematics prob-lems are organized in very specific ways which utilize spatial positioning.Interpersonal meaning is largely restricted to maximal values of modalityand modulation where expressions of probablity and uncertainty areencoded through relational processes such as approximations and prob-ability statements. Mathematical symbolism is functional, but only within acertain semantic realm compared to language. This reduction of meaningin the evolution of mathematical symbolism permitted semantic expan-sions so that the exact description of the relations can be displayed visually.

In an educational context, students are typically presented with modernmathematics in a pre-packaged form where the functions and grammar ofmathematical symbolism are not discussed. The discourse is presented insuch a way that many students do not understand what mathematics is, orhow mathematical symbolism developed historically as a semiotic resourcein order to fulfil particular functions. Given that the grammar of math-ematical symbolism is not taught from a linguistic perspective, the culmula-tive effect is that many students fail mathematics because they simply do notunderstand how (or why) mathematical symbolism functions as a resource

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for meaning. Students cannot write in the formats required by educationalinstitutions if they cannot make use of the options provided by the gram-mar of language and, in a similar manner, they cannot solve mathematicalproblems without recourse to the grammar of mathematical symbolism,especially as the resource employs new strategies for encoding meaning.The functions and grammar of mathematical symbolism need to beaddressed in the teaching and learning of mathematics.

The SF Approach to Visual Images

The SF approach to visual images in mathematics includes a description ofthe functions and grammar of visual representation and how these relate tothe algebraic and linguistic descriptions. Mathematical visual imagesdeveloped to link spatiality with temporality so the visual relations could beexactly encoded using the symbolism. A semantic circuit with language,mathematical symbolism and visual images thus exists. While visual imagesare accessible as they correspond to perceptual reality, the grammar ofvisual images, however, needs to be understood because there are specificsystems which permit links to be made to the symbolism.

A historical perspective demonstrates how mathematics evolved as a dis-course, and one means for understanding this development is the changeswhich occurred in the visual image. A de-contextualization of the visualdisplay took place as human actors and material circumstances wereremoved as concern focused on the display of relations in the form of lines,curves and three-dimensional objects. Experientially, the participants andprocesses in visual images are the relations encoded symbolically, which arevariously shown as intersecting lines, planes and other forms of visual rep-resentation. Visual images are typically multisemiotic as accompanyinglabels function to identify the relations with respect to symbolic descrip-tions. These types of selections in mathematical images mean that con-temporary mathematical graphs and diagrams appear abstract with a highmodality value. However, the development of computer technology permitsdigital data to be transformed into new forms of visual images. The increas-ing sophistication of computer graphics are leading to more complexforms of mathematical visual images. This includes the use of systems suchas colour and three-dimensional displays that replicate and extend ourperception of the world in what has become virtual reality. Modern math-ematics developed as a written and printed discourse. However, becauseof advances in visualization and computation, mathematics and scienceare entering into a new era where new ways of construing reality arebecoming possible. The relations between semiosis and technology couldbe incorporated into mathematics education.

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Intersemiosis across Language, Mathematical Symbolism and Visual Images

The SFL approach to mathematics as a multisemiotic discourse is signifi-cant because it provides a theoretical framework to explain how language,mathematical symbolism and visual images function intersemiotically.Apart from concerning itself with a limited semantic domain, mathematicaldiscourse is successful because:

1 The meaning potentials of language, symbolism and visual images areaccessed.

2 The discourse, grammatical and display systems of each resourcefunction integratively.

3 Meaning expansions occur when the discourse shifts from one semioticresource to another.

The functions and resulting grammars for language, symbolism and thevisual images may be conceptualized as three integrated systems whichpermit intersemiotic transitions to take place. Intersemiotic transitionsconsist of macro-shifts from one resource to another across Items, andmicro-shifts where choices from one semiotic resource are integratedwithin another semiotic resource. These transitions give rise to semanticexpansions in mathematics. In addition to transitions, mechansisms forintersemiosis include Semiotic Cohesion, Semiotic Mixing, Semiotic Adop-tion and Juxtaposition. The examination of intersemiosis and intersemioticmechanisms provides a conceptual framework which adds a new semanticlayer to analysis beyond that provided by the SF frameworks for eachresource.

The systems for intersemiotic transitions and other intersemioticmechanisms are metafunctionally based, and a new meta-language is usefulfor examining the semantic overlays which consequently arise. These over-lays are formulated as co-contextualizing relations where there is a con-vergence of meaning, and re-contextualizing relations where divergenceoccurs. The extent and degree of these contextualizing relations arisethrough the Bi-Directional Investment of Meaning where the Contextual-izing Propensity (CP) and Semantic Effervescence (SE) result in an Inter-pretative Space (IS) (Cheong, 1999, 2004). Traditionally, mathematicsfunctions to co-contextualize and re-contextualize meanings inter-semiotically in such a way that the Interpretative Space (IS) is limited. Thepotential for re-contextualization is changing with the advent of computertechnology. This is not confined to the display of mathematical visualimages where computer graphics display numerical rather than analyticalsolutions in rapidly changing dynamic formats. Computer technology isalso revolutionizing mathematics in terms of approach (non-linearity ver-sus linearity), method (computation versus analytic solution) and mediumand materiality (computerized environment rather than print).

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Semiotic and Grammatical Metaphor

Semiotic metaphor is an important means for semantic expansions inmathematical discourse. Semiotic metaphor occurs when a choice fromone semiotic resource undergoes a change of functional status in the tran-sition to another semiotic resource. For example, a linguistic process ('tomeasure') becomes a symbolic or visual entity or circumstance (an xin thesymbolism, or a distance in a diagram). In turn, these metaphorical seman-tic expansions are absorbed into the systems for each resource and thuscontribute to the dynamic evolution of semiosis. The types of co-contextualizing and re-contextualizing relations which take place withsemiotic metaphor extend beyond those possible within one semioticresource. Co-contextualization in the form of parallel semiotic metaphors,and re-contextualization in the case of divergent semiotic metaphors,occur. Alternatively or concurrently, the transition to another semioticresource may also permit new functional elements to be introduced.

These metaphorical shifts which take place across language, symbolismand visual images are not always noticed because the high incidence ofgrammatical metaphors in mathematical and scientific language create asemantic layer over the semiotic metaphors occurring through intersemi-osis. Grammatical metaphor in language is most likely a response to thedynamic function fulfilled by mathematical symbolism and the types ofsemantic expansions which take place intersemiotically. For example, ifthere is a shift from a linguistic process to a visual entity then the linguisticreconstrual of that visual entity results in a linguistic entity. However, thatlinguistic entity has traditionally been viewed as a case of grammaticalmetaphor. In other words, if one is only examining language, then the shiftis perceived to be language -> language rather than language -» visualimage/mathematical symbolism —> language. Thus grammatical metaphormay be viewed as the product of semiotic metaphor.

Scientific and mathematical constructions appear as commonsenseknowledge despite their metaphorical nature because these types of con-structions have become 'the way' of establishing order. The scientific viewof the world is the means through which truth is established, regardless ofthe context or the field of human endeavour. Semiotic metaphor is furtherdiscussed in relation to the nature of pedagogic discourse in mathematicsclassrooms. In this context where the oral discourse tends to be congruent,semiotic metaphors are traceable.

7.3 Pedagogical Discourse in Mathematics Classrooms

The theorization of mathematical discourse in this study concerns printedmathematical texts and the evolution of new forms of mathematics madepossible through computer technology. In the context of the classroom,however, mathematics is taught using a variety of modes which include theblack/white board, printed material including mathematical textbooks,

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practice sets of examples and test papers, computer software, graphicscalculators, and mathematical models and equipment. The analysis ofmathematical pedagogical discourse must necessarily take into account themultisemiotic nature of mathematics, and the shifts between the written/spoken modes and the shifts between language, symbolism and visualdisplay. From this perspective a new multimodal approach to discourseanalysis emerges (O'Halloran, 1996, 2000), one that integrates the multi-semiotic written and spoken modes of mathematics with the use of othersystems of meaning such as gesture, body movement, proxemics and soforth. In what follows, the implications of the SF perspective to pedagogicaldiscourse in mathematics classrooms are discussed.

The constant shifts between the written and spoken modes in mathemat-ics classrooms mean that the preceding theory of the functions and gram-mars for language, mathematical symbolism and visual display is relevant interms of both intrasemiosis and intersemioisis. While mathematics evolvedas a written discourse, the interaction between the teacher and the studentin the context of the mathematics classroom involves spoken language. Themeta-language for teaching and learning mathematics in this study isEnglish, which includes the verbalization of the symbolic and visualdescriptions. While gestures such as pointing and hand movements, facialexpression, and body movement are significant in classroom interactions,the teacher and students inevitably resort to language at each stage ofthe lesson. Although teacher-talk dominates mathematical classrooms(O'Halloran, 1996; Veel, 1999), SF discourse analysis reveals that thepatterns of interaction in classrooms are different (O'Halloran, 1996,2004c). The nature of the discourse in three mathematics classrooms dif-ferentiated on the basis of gender and socio-economic status are summar-ized after the more general features of mathematical pedagogic discourseare outlined below.

The texture of mathematical pedagogical discourse is dense as thespoken mode provides the meta-language for the action which takes placein the temporal material unfolding of the lesson which includes the writtenmathematics. For example, the INTERSEMIOTIC IDENTIFICATION ofparticipants extends across language, symbolism and visual images in thewritten and spoken modes. As a result, major reference chains continuallysplit and conjoin as the solution to the problem is derived. The grammar ofmathematical symbolism contributes to the dense texture of classroomdiscourse. The constant reconfiguration of the symbolic rankshiftedOperative processes and participants and the nature of taxonomic relationsincrease the difficulty of tracking participants. In terms of logical meaning,there are long chains of logical reasoning which typically involve a highincidence of consequential-type relations. In many cases these relations arebased on previously established mathematical results which are often leftimplicit. Interpersonally, the authorative interpersonal stance of writtenmathematics may be replicated in the oral discourse, although analysisdemonstrates that the nature of relations established between the teacher

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and the students varies across classrooms and schools (O'Halloran, 1996,2004c).

The use of English as the meta-language for the symbolic and visual textsinvolves metaphorical-type constructions that typically remain unnoticed(O'Halloran, 1996, 2000). This may be one important factor accountingfor learning difficulties in mathematics. The verbalization of symbolicmathematics, for instance, often results in semiotic metaphor. For example,the teacher may refer to G(x) = -O.I;*2 + 2.6x+ 43 in Example 2.24 in thefollowing manner: 'the function G(x) is negative zero point one x to thepower of two plus two point six xplus forty-three'. The symbolic statementmeans G(*) = [[[[-0.1 [[xx *]]]] +[[2.6*]]+43]] where [[ ]] indicaterankshifted configurations of Operative processes and participants. How-ever, the Operative processes become participants in the verbalization ofthe symbolic statement, for example, 'the function G(x) is [[negative zeropoint one x to the power of two plus two point six x plus forty-three] ]'. Inother words, the linguistic entity in the form of the nominal group 'nega-tive zero point one x to the power of two' and 'two point six x' replaces theprocess/participant configurations -0.1 X [[xx x]] and 2.6 x x Similarly,'the sum of the square root of x squared and five x to the power of 4' isrealized linguistically as an entity because it takes the form of a nominalgroup. However, there are multiple configurations of process/participantsin the symbolic form. Jx2 + 5x* as indicated by [[^/[[x2]]]] + [[5x[[xx x xxx x]]]].

This raises an important issue. Although verbal descriptions of the sym-bolic statements and visual images constantly take place in the mathematicsclassroom, the metaphorical nature of those linguistic constructions is notusually discussed. Although the verbalizations permit exact translationsback to the mathematical symbolic statement, the impact of the meta-phorical form of those constructions needs to be considered. Otherwise,students are presented with linguistic entities which are, in fact, complexconfigurations of mathematical processes with their associated participants.The consequent shifts in meaning that take place in oral pedagogical dis-course need to be addressed so that students understand the metaphoricalnature of those linguistic constructions in relation to the symbolic andvisual forms.

The nature of pedagogical discourse in mathematics varies greatlyacross institutions and teachers (O'Halloran 1996, 2004c). The analysis ofdiscourse in an elite private school for male students reveals that themetafunctionally based choices in the pedagogical discourse and black-board texts mirror those typically found in mathematical discourse. Thesestudents subsequently performed exceptionally well in university entranceexaminations in mathematics and other subjects. Interpersonal dimensionsof the interaction between the male teacher and female students in a simi-larly placed private school for girls were found to be orientated towardsdeferentiality compared to the patterns of dominance enacted in the lessonwith male students. The female students, nonetheless, performed well in

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university entrance exams, although not to the same level as the male stu-dents. The most marked difference in classroom discourse was found in thelesson in the low socio-economic government school, where the peda-gogical discourse was different in all respects from that found in the othertwo classrooms. The analysis indicated that the major concern of this lessonwas the maintenance of interpersonal relations rather than the experientialcontent of the lesson. As far as the mathematical content was concerned,the register selections in the classroom discourse were colloquial ratherthan technical, and, more generally, the oral discourse and non-genericellipsed format of the blackboard texts tended to be context dependent.Coherence in the lesson depended to a large extent on the immediatephysical context of the classroom and accompanying physical gestures ofthe teacher and the students. In this classroom, '[t]he orientation towardsparticularistic/local/context-dependent meanings . . . does not accordwith the universal, remote and seemingly context-independent meaningsof mathematics' (O'Halloran, 2004c). The students who attended thisschool did not perform well in university entrance examinations. Theseresults indicate that while we may speak of classroom discourse in math-ematics classrooms, the reality is that the discourse differs across institu-tions. The practices and processes of teaching/learning mathematics are asdiverse as the outcomes.

This discussion has indicated some areas in which the study of semioticsof mathematics is useful for mathematics education. In particular, the SFapproach offers a comprehensive social-semiotic theory of languagethrough which mathematical symbolism and visual display may be viewed.Using this approach, the complexity of classroom interactions, includingthe oral discourse, the physical context and the non-verbal behaviour of theteacher and the students, may be analysed. The SF approach includes thedevelopment of SFL theory to account for the context of culture (educa-tion), the context of situation (school type and class), the school curric-ulum in the form of macro-genres (Christie, 1997), the lesson genre (thetype of lesson), the micro-genres and activity sequences which constitutethe lesson, and the use of language, visual images, language and othersemiotic resources in those activity sequences (O'Halloran, 1996). In add-ition, the use of computers and software applications and the internet forteaching and learning mathematics may also be investigated. There aremany applications of the SFL approach to mathematics education whichrequire further research and development.

7.4 The Nature and Use of Mathematical Constructions

Mathematical and scientific discourses are semiotic constructions whichhave moved beyond the printed page into new realms of meaning. Thesesemiotic constructions have proved enormously powerful for the reshapingof the physical world and developing technology. The uses of mathematicsand science have also extended into other realms such as the social sciences

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where the problematic nature of the metaphorical constructions whichconsequently arise may be demonstrated through close textual analysis asundertaken in Section 7.1. Further to this, a commonsense reading ofreality today is often based on a mathematical and scientific view of theworld. The nature of that semiotic construction has been the subject of thisstudy. In conclusion, a few final comments are made below.

From the SF perspective developed here, critical concepts in mathemat-ics and science such as abstraction, contextual independence, reason,objectivity and truth are viewed as particular types of semiotic choices madefrom the available systems from the grammars of mathematical symbolism,visual display and language. Abstraction is the re-contextualization whichtook place historically where 'superfluous' information was put aside in thepursuit of knowledge which entailed the description of relations in a formwhich could be visualized and rearranged for the solution to problems.Similarly, objectivity is the organization of particular experiential andlogical realms of meaning which are accompanied by a contracted inter-personal stance. From this view, objectivity becomes a 'valued' culturalproduct which is enacted semiotically. Reason becomes the rearrangementof relations which can be undertaken with the available semiotic tools. Andthe nature of truth is reduced to the nature of the semiotic constructionsfound in the mathematical and scientific views of the world.

The scientific method initially involved describing the physical worldwhere new entities were introduced to explain physical phenomena. Mod-ern mathematical and scientific constructions are effective in that theysuccessfully model physical systems up to the point where they become non-linear. From here, non-linear dynamical systems theory introduces newcomputerized techniques to describe and predict the behaviour of chaoticsystems. However, mathematical and scientific descriptions have extendedinto the realms of the economic, the political, the social, the educationaland the private. The re-contextualization which took place through Des-cartes, Newton and countless other mathematicians to produce modernmathematics has undergone further re-contextualization in what couldonly be described as the mathematicization of the human condition.Meanwhile, science and technology advance rapidly, especially in the fieldsof computer technology, the life sciences and the military. We need tocome to terms with the functions and limitations of mathematical and sci-entific descriptions which increasingly appear to be harnessed to thedemands of capitalism, before the political rhetoric surrounding suchadvances betrays us all.

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Index

abacus, use of 25abstract theory 36activity sequences 111,115,119,126,

176-7Adorno, T.W. 18advertising 8, 160, 163agency, concept of 105-8algebra 23-4, 27-8, 33, 45-6, 57, 96,

104-7, 130-2, 201algorithms 25anaphoric nouns 87applied mathematics 33, 36Aquinas, Thomas 34Archimedes 31Aristotle 34Azzouni,J. 119

Baldry,A.P. 19, 124Beevor, A. 1-7Berlin, fall of (1945) 3-7Bernstein, B. 9Berry, M. 68Bordell, D. 149-51Botticelli, Sandro 139Bourdieu, P. 9brackets, use of 118, 125-8BurgmeierJ.W. 189-90, 198Bush, George W. 2

Cajori,F. 23,27,111Carter, R. 73certainty in mathematics, loss 119Chandler, D. 6chaotic systems 17, 58, 119, 151, 209Cheong.YY. 8, 165Chomsky, N. 62

Christie, F. 60Chuquet, Nicolas 33classroom discourse 205-8Cobley, P. 6Coca Cola (company) 8code-switching 16cohesion, concept of 175Colonna,J.-F. 149-57 passimcommunication planes 89compositional meaning 146-8computational devices 25, 104computer graphics

applications of 149, 154—6definition of 148

computer technology 15, 23, 58-9,95-6, 119-21, 128, 132, 138,146-58, 161, 203-5, 208-9

condensatory strategies 126—8conjunctive relations 119-20,127conjunctive reticula 80context of text 89contextualization propensity (CP) 165,

198, 204contextualizing relations 165—7coreness of lexical items 73,115cosmology 55culture 7-8, 89

Danaher, S. 157Davies, P.C.W. 58, 148Davis, PJ. 19, 34, 46, 129-30,132, 148,

151deductive reasoning 119Derewianka, B. 87-8Descartes, Rene 17,22-3,33,38-48,

53-9, 72, 118, 129-31, 199, 209

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224 INDEX

discourse systems 60-1, 64-8, 98-101,128

dynamical systems theory 17, 58, 119

education, mathematical 16-17, 72,199-208

Eisenstein, E.L. 33ellipsis 122-6encoding of meaning 97, 108, 110, 114,

118, 124, 155,202-3condensatory strategies for 126-8

Euclid 27-31, 34, 46, 49, 130exchange structures 68-72,115-16,127existential graphs 130experiential meaning 71,126-8,174-5,

179, 186,195,198contraction and expansion of 103-14encoding of 148visual construction of 142-5

experimentation 58virtual 151, 154

expression 108

Febvre, L. 31female students 207-8'field' variable 89FoleyJ. 148Fontana, Niccolo see TartagliaFoucault, M. 91, 199Francis, G. 87Funkenstein, A. 154—5

Galileo 35-7, 58Galison, P. 129-31, 142, 156genre 20,89-91geometry 45-8,130, 132, 201Gestalt theory 147Goldhagen, D J. 1grammatical density 202grammatical metaphor 83-8, 96, 128,

183-8,194-6,199, 205grammatical repackaging 185-6Grave, M. 148Gregory, M. 163GroB, M. 149Guo, L. 134,138

habitus 9Halliday, M.A.K. 6-7,13, 20, 60-90, 97,

105, 119, 128, 145,163,184-8, 202

Hamming, R.W. 24Hasan, R. 20, 63, 90, 163Hersh,R. 46historical development of mathematical

discourse 22-4Hjelmslev, L. 61Hooper, A. 58Horkheimer, M. 18

ideational meaning 62, 88ideology 91, 93ledema, R. 11,20, 159image processing 148-9induction, mathematical 94integration of semiotic resources 171integrative multisemiotic model 91,163internet resources 208interpersonal meaning 62, 65, 67, 71-4,

88-91, 97,113-14,127,139,175-6,198, 202, 206-8

contraction of 114—18interpretative space (IS) 165, 198, 204intersemiosis 11-12, 16, 65-6, 93, 96,

98,157,160-3,189-98, 204-5and expansion of meaning 184in mathematical texts 171-7mechanisms of 169-71semantics of 163-9systemic framework for 171

intrasemiosis 16, 65-6, 88, 91, 94, 96,158-9

Iraq war (2003) 18'items' in mathematical texts 159-61,

189-91

Koestler,A. 24,55Kok,K.C.A. 159Kress, G. 20Kuhn, T.S. 24

langue 62Layzer, David 185LeLous,Y. 148learning difficulties 207Leibniz, G. 33,45,57Lemke, J.L. 13, 95, 122-4, 159lexical density of text 202lexical metaphor 88lexicogrammar 63-7, 81, 92, 184, 200-1Lim, F.V. 91, 134, 163, 165

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INDEX 225

logical meaning 78-81,127logical reasoning 128, 155-6

McDonald's (company) 8macro-transitions 161-2, 178Martin, H.-J. 31Martin, J.R. 60-8, 73-93 passim, 97-8,

119, 128, 145, 184mathematical spirit 19mathematical books 24-33, 189-91mathematics, status of 57-8matrices 112-13Matthiessen, C.M.I.M. 75, 184-8medium, definition of 20Messaris, P. 132metafunctionality of language 7,13,

62-7,88,91,101,158,176,200metaphorical terms 14-17, 57, 73-7,

188; see also grammatical metaphor;semiotic metaphor

metaphorical exchange 178micro-transitions 161-2, 178mode 20,89models, mathematical 94—7multimodality 19-21,206multisemiosis 10, 13-21, 59-60, 88, 91,

97-8educational implications of 199-205see also integrative multi-semiotic

model

natural language 97Nazi regime 2, 6, 10Newton, Isaac 17,22-3,31-3,40-58,

149, 152-4, 171-2, 177-9, 183-5,199, 209

Nike (company) 8non-linear systems 58, 209notation, mathematical 23, 112-13, 127Noth, W. 6-7nuclear relations 77numeration systems 24-7, 103-4

objectivity 209O'Halloran, K.L. (author) 208operative processes 103-13, 121-8, 178,

201-2, 206-7; see also rule of orderO'Toole, M. 5, 13-14, 132-41, 146-7

Pacioli, Luca 33

Page, T. 2parole 62Peirce, C.S. 130photographs, use of 191-3picture synthesis 154Plato 34Poincare, Henri 130positional notation 112-13Poynton, C. 62printing technology 27-33, 131-2probability 72

rankshifting 66-7, 82, 105-9, 118,128,131,187,197,201-2,206-7

rationalism 18realization, concept of 62, 65reciprocity of choice 90-1re-contextualization 177-9, 184,

204-5, 209'register' of mathematical language

89-91Reisch, G. 34Renaissance mathematics 33-8resemioticization 11, 159Rose, D. 65, 73-4, 81Rotman, B. 13Royce, T. 11,159,163-5,175rule of order for operative processes

118,124-8

Saussure, F. de 62scientific language 14, 60, 96, 108,

128, 186-7, 200-1, 205scientific view of the world 6,10,17,

22, 24, 94, 184, 188, 205, 209secondary clauses 80semantic drift 200semantic effervescence (SE) 165, 198,

204semantic shift 184semiotic capital 9semiotic metaphor 16, 88, 93, 179-84,

196, 199, 205, 207September llth 2001 attacks 2Shea,W.R. 44,46,54Shin, S. 130Smith, David Eugene 24social distance 90social-semiotic theory 6-10, 17

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226 INDEX

space of integration (Sol) 163spatial positioning of mathematical

symbols 122,201-2spoken discourse, status of 90statistical graphs 141-2, 148, 158Stewart, J. 67-83 passim, 88,186Sweet Stayer, M. 57Swetz,FJ. 24,27,33syllogisms 119symbolism, mathematical 6-7,10-17,

38, 44-8, 53-7, 60, 75, 78, 80, 89,93-104,112-25,155-6

as distinct from visual images 129-32framework for 97-103grammar of 15, 22-3, 57-9, 65-7, 96,

108, 111, 126,131-2,142,178,187,199-206

and intersemiosis 196-7language-based approach to 96-7operative processes in 106as a semiotic resource 125SF approach to 201-3textual organization of 122-5as a tool for logical reasoning 128

syntagmatic organization 121-2, 126-8systemic functional (SF) approach 93-

8,132-9,159,170,199, 204-9and language 11-15, 60-5, 200-1and mathematical symbolism 201-3and visual images 133-8, 203

systemic functional grammar (SFG)13-15, 18-20,128,134, 138, 158

systemic functional linguistics (SFL) 8,60-1, 66-7, 70, 88-91, 98,101,175,183,200-1,208

Tartaglia, Niccolo 34-42taste 14taxonomic relations 77

tenor 89-92textual meaning 121-8textual organization of language

81-3Thibault, P. 19, 62,163,165,183,

199Thomas-Stanford, C. 27-30Thompson, K. 149-51Tiles, M. 119topological-type meaning 95transitions in mathematics 159-62Treviso Arithmetic 22-7trigonometry 179-83'true knowledge' (Descartes) 54—5truth-values 117-18,201,209typological-type meaning 95

van Leeuwen, T. 20Venn diagrams 130,133,142Ventola, E. 68virtual experimentation 151,154virtual reality 203visual forms of semiosis 155-6visual images

grammar of 15reasoning through use of 145-6in SF framework 133-8, 203

visual theorems 151visualization 129-33, 148-9, 203

Walatka, P.P. 149war

attitudes to 1-3scientific impact of 18

Watson, V. 149Weitz,E.D. 1Whitrow, GJ. 33Wilder, R.L. 23-4Wilkes,G.A. 155-6