When does diagrams support reasoning?

Diagrams can be accompanied by labels and sentences

When is a diagram precise and accurate?

Handmade drawings can be rather rough, but still intelligible

PROGRAMMECentro de Filosofia das Ciências da

Universidade de LisboaJanuary 14-16, 2008

• Workshop 1 (Jan. 14):Diagrams in logic and mathematics: epistemological

questions

• Workshop 2 (Jan. 15):Visual displays of information and the practice of

science: pragmatic approaches

• Public Conference (Jan. 16)Diagrammatic reasoning: Empirical research and

theoretical questions

Valeria GiardinoUniversità di Roma ‘La Sapienza’

Institut Jean Nicod

valeria.giardino@gmail.comhttp://valeria.giardino.googlepages.com/

Diagrams in Logic and Mathematics

Epistemological Questions

Valeria GiardinoUniversità di Roma ‘La Sapienza’

Institut Jean Nicod

valeria.giardino@gmail.comhttp://valeria.giardino.googlepages.com/

Centro de Filosofia das Ciências da Universidade de LisboaJanuary 14, 2008Workshop1

A very simple (and ‘visualized’) example:

The Pythagorean Theorem

(a+b)2 = c2 + 2ab

Outline

• Introduction: ‘Logocentric’ views of logic and mathematics

• I. First alternative: Heterogenous Logic

• II. Second alternative: A form of Platonism

• III. Third alternative: Discovery without justification as a conceptual possibility

• Questions for the discussion

Introduction The ‘dogma’ of logocentricity

Review of the second edition of Whitehead and Russell’s Principia

Mathematica

“Throughout the four volumes of Principia the authors are concerned with one main problem – namely, the proof of the thesis that pure mathematics is an extension of formal logic, and is nothing else.”

Logocentricity/1Sheffer (1926)

Dialectical and psychological speculation regarding the foundation of mathematics

Actual demonstration that mathematics either does, or does not, involve concepts or processes than are extra-formal

With the epistemology of mathematics – the question of the «objectivity» or «subjectivity» of mathematical «reality» - and with the psychology of the act of mathematicizing, Whitehead and Russell have no concern.

The logical status of mathematics – the question of the derivability or the non-derivability of pure mathematics from formal logic alone.

Just as the proof of certain theories in metaphysics is made difficult, if not hopeless, because of the «egocentric» predicament, so the attempt to formulate the foundations of logic is rendered arduous by a corresponding «logocentric» predicament. In order to give an account of logic, we must presuppose and employ logic.

Three basic norms

(a)They are assuming the validity of logic: their aim is not to validate logic, but only to make explicit, at least in part, that which we have assumed to be valid.

(b)They must keep the study of formal structure – the question of notation – entirely distinct from the investigation of the «loci» or interpretations of a structure-complex.

(c)Finally, if they are to surmount the difficulties created by the «logocentric» predicament, they must discriminate sharply between the problems of «Symbolic Analysis» - i.e. notation and interpretation - and the study of the conditions that make the notational and the interpretational phases significant and valid.

[The diagram] is only an heuristic to prompt certain trains of inference; [...] it is dispensable as a proof-theoretic device; indeed, [...] it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable way.”

A shift from a problem about foundations to a ‘dogma’ to an issue which concerns what a proof is

...what about the epistemology of mathematics?

Logocentricity/2Tennant (1984)

I. First alternative:Heterogeneous Logic

Barwise and Etchemendy’s project(1996)

“[...] visual forms of representation can be important, not just as heuristic and pedagogical tools, but as legitimate elements of mathematical proofs. As logicians, we recognize that this is a heretical claim, running counter to centuries of logical and mathematical tradition”

Against the dogma: why?Shin (2004)’s reconstruction

1. Students' performances: understanding semantic concepts can be of help for

carrying out formal proofs in a deductive system

2. Reasoning is a heterogeneous activitypeople use multiple representations of

information while reasoning, and those representations are often in non-sentential forms such as diagrams

3. Unity of teaching and reasearchthe practical power of multi-modal

reasoning together with modern logic and its formalization and rigour

expanding the territory of logic by freeing it from a mode of representation only

“All of us engage in and make use of valid reasoning, and in the process of reasoning human beings obtain information through many different kinds of media, including diagrams, maps, smells, sounds, as well as written or spoken statements”

Shin (2004)

First problem

Second problem“to do with visual/spatial reasoning

something analogous to what Frege and his followers did for the formal/linguistic one” towards a new standardized system

Axioms to account for diagrams and for our way of referring to them in reasoning?

“[…] the work carried out by Barwise and Etchemendy on visual arguments in logic and mathematics is motivated in great part by the proof-theoretic foundational tradition”

Mancosu (2005)

II. Second alternative: A form of Platonism

Brown (1999)

“Some ‘pictures’ are not really pictures, but rather are windows to Plato’s heaven”

It is possible to have a realist view of mathematics and its objects, without being at the same time a realist view of pictures

“As telescope help the unaided eye, so some diagrams are instruments (rather than representations) which help the unaided mind’s eye”

Problems1. ontological statements are in general much more

problematic than the difficulties they are intended to find solution to

the infinite debate about the ontological nature of mathematical objects shows that it is very problematic to settle ontological questions once and for all

Balaguer (1998)

2. even if we accept that ontology can represent some methodological advantages, this is not always the case

depending on the problem in question, it could be more useful to think about the object of research differentely

III. Third alternative: Discovery (by visualizing) without justification

First...Something on discovery and

mathematics

Hadamard (1949)

Mathematical discovery – or invention – involves first a preparation and an illumination which are mostly subconscious; then, this illumination is verified and concluded by means of conscious thinking, which is nevertheless necessary in teaching mathematics.

Polya (1945)How do we solve it?- Understanding the plan- Devising the plan- Carrying out the plan - Looking back

Exposition “which progresses relentlessly from the data to the unknown and from the hypothesis to the conclusion”:

perfect for checking the argument in detail BUT far from being perfect for making understandable the main line of the argument!

“A detail pictured in our imagination may be forgotten; but the detail traced on paper remains, and, when we come back to it, it reminds us of our previous remarks, it saves us some of the trouble we have in recollecting our previous consideration.”

Rota (1997)“the laws of logic are not sculpted in stone,

eternal and immutable. A realistic look at the development of mathematics shows that the reasons for a theorem are found only after digging deep and focusing upon the possibility of a theorem. The discovery of such hidden reasons is the work of the mathematician. Once such reasons are found, the choice of particular formal statements which express them is secondary. Different but exchangeable formal versions of the same reason can and will be given depending on the circumstances”

“this pretence of ‘identifying’ mathematics with a style of exposition is having a corrosive effect on the way mathematics is viewed by scientists in other disciplines”

Giaquinto’s approach

Epistemological (evaluative) questions to ask

• Can a visual way of acquiring a mathematical belief justify our believing in it?

• Can a visually way of acquiring mathematical belief be knowledge in the absence of indipendent non-visual grounds?

• What level of confindence would be rational?

Conditions for mathematical discovery:

- To believe something independentelycoming to see it by one’s own lights,

rather than by reading it or by being told

- To believe something in a epistemically acceptable way

a way of acquiring a belief is epistemically acceptable if it is reliable as a way of getting belief and it involves no violation of epistemic rationality in the circumstances

- Absence of irrationality in getting a belief does not entail that the believer has a justification

- Justification is not a further conditions of discovery

Back to an epistemology of individual discovery

Discovery without justification is at least a conceptual possibility; in mathematics, this possibility is quite often realized

Proofs often post-date discovery

Questions for the discussion

I. Is an epistemology of mathematics supposed to discuss the role of visualization in mathematics?

II. Is it necessary to consider visualization in the proofs or as an “underlying mental operation”?

III. What is the difference between these two approaches?

IV. Mental operations or external supports?

ReferencesBarwise - Etchemendy (1996), ‘Visual Information and Valid Reasoning’,

in Allwein - Barwise (eds.) (1996), Logical Reasoning with Diagrams, 3-25.

Brown (1999), Philosophy of Mathematics: an introduction to the world of proofs and pictures.

Giaquinto (2007), Visual Thinking in Mathematics.

Polya (1945), How to solve it.

Rota (1997), ‘The Phenomenology of Mathematical Proof’, in Kamamori (ed.), Proof and Progress in Mathematics, 183-196.

Mancosu (2005), ‘Visualization in Logic and Mathematics’, in Mancosu et al. (eds.) (2005), Visualization, Explanation and Reasoning Styles in Mathematics, 13-30.

Sheffer (1926), ‘Review of A. N. Whitehead and B. Russell, Principia Mathematica’, in Isis (8), 226-231.

Shin (2004), ‘Heterogenous Reasoning and its Logic’, in The Bullettin of Symbolic Logic (10) 86-106.

Tennant (1984), The withering away of formal semantics.