SOCIAL CONSTRUCTIVISM AS A PHILOSOPHY OF MATHEMATICS

Paul ErnestUniversity of Exeter, UK

SOCIAL CONSTRUCTIVISM AS A PHILOSOPHY OF MATHEMATICS

THIS TALK IS BASED ON FOLLOWING PUBLICATIONS

• Ernest, Paul (1998) Social Constructivism as a Philosophy of Mathematics, Albany, New York: SUNY Press.

• Ernest, P. (1991) The Philosophy of Mathematics Education, London: Routledge.

(c) Paul Ernest 2015

MATHEMATICAL KNOWLEDGE NOT ABSOLUTELY VALID

• 1. BASIS Proof in mathematics assumes (a) truth, or (b) correctness, or (c) consistency of axioms and of logical rules. But truth of this basis cannot be established without vicious circle.

• 2. PROOF Mathematical proof absolutely correct only if unjustified assumptions made

• (a) Standards of absolute rigour attained. No grounds for assuming this exist.

• (b) Any proof rigorizable. But virtually all accepted mathematical proofs informal.

• (c) Check-ability of rigorous proofs for correctness is possible. Further formalising informal proofs will lengthen them and make checking even more uncertain.

• 3. SYSTEM Mathematical proof depends on the assumed safety of axiomatic system. This unjustified after Gödel

Failure of Absolutism

• Absolute rigour unattainable. So claim of absolute validity for mathematical knowledge unjustified.

• Increasing recognition by philosophers that this is so.

• Foundationist programmes of Formalism, Logicism, Intuitionism each failed `to establish maths knowledge as absolutely valid (cf. Russell, Gödel)

• Growing agreement that any such absolutist or foundationist enterprise must fail.

New Fallibilist Movement

• New 'maverick' tradition with Wittgenstein, Davis, Hersh, Kitcher, Tymoczko, Lakatos

1. Rejects narrow philosophy of mathematics focus on foundationist epistemology and Platonistic ontology

2. Rejects exclusion of history and practice of mathematics

3. Claims mathematical knowledge is fallible

What is fallibilism? 3 meanings

1. Fallibilism1: Humans make mistakes F1 is trivial. Clearly true.

2. Fallibilism2: (Some) mathematical knowledge is or may be false (not F1) It is enough to find one falsehood or contradiction and Gödel’s Theorem means we cannot eliminate this possibility But F2 assumes absolute truth judgements can be made.

Fallibilism3 denies this assumed absolutism

3. Fallibilism3: Maths is a relative, contingent, historical construct

Absolute judgements re truth, correctness can never be made. Criteria and definitions vary with time, context, never final form.This is a postmodern view of mathematics

Fallibilism3: Social Constructivism

• The concepts, definitions, rules of mathematics invented and evolved over millennia, including rules of truth and proof.

• Fallibilism3 rejects Absolutism which assumes the following:• Universalism: All knowing beings in all times & cultures would agree on

truth & mathematical knowledge.• This is false if ‘do’ is put for ‘would’. But how could we ever know if this were

true? (An article of faith)• Objectivism: Truth depends on objective reality, not views of persons/groups• This raises the problem of privileged access to ‘objective reality’, which begs

the question• Foundationalism: There is a unique permanent foundation for knowledge.• This foundation has not been identified historically. Foundationalist

philosophies of mathematics (Logicism, Formalism, Intuitionism) all failed.

Wittgenstein’s Philosophy of Maths

• Wittgenstein’s naturalistic and fallibilist social philosophy of maths based on key concepts of ‘language games’ - how we use language coordinated with our actions - inseparably part of ‘forms of life’ - our historico-cultural practices. Mathematics teaches you, not just the answer to a question, but a whole language-game with questions and answers.

The mathematician is not a discoverer: he is an inventor. (Wittgenstein)Proof serves to justify mathematical knowledge through persuasion, not by its inherent logical necessity. W’s philosophy fallibilist, because certainty grounded in accepted (but always revisable) rules of language games (Rorty).

Lakatos’ Philosophy of Mathematics

• Lakatos contributed Logic of Mathematical Discovery (LMD) or Method of Proofs and Refutations as methodology of mathematics with 3 functions:

• Epistemology: To account for the genesis and justification of mathematical knowledge naturalistically as part of his Fallibilism3. (Present)

• Theory of historical development of mathematics -past• Methodology for practising mathematicians (future)• Lakatos' theory starts with a mathematical conjecture C –

an attempted proof of it - P and a background informal theory T

Lakatos’ Logic of Math’l DiscoverySTAGE CONTEXT PART OF CYCLE

Stage n1 Problem P, Informal Theory T

Conjecture C

n2 - thesis Informal proof I of C

n3 - antithesis Informal refutation R of C

Lakatos' LMD is a cyclic process with a dialectical form

Stage n has attempted proof I of C - generates a refutation R of C. Next stage n+1 has new modified conjecture C’ (in context of modified problem P’ and new informal theory T’)

(n+1)1 - synthesis

Problem P’, Informal Theory T’

Conjecture C’

Lakatos’ Fallibilism3

• The most radical aspect of Lakatos’ PM is his fallibilist epistemology.Why not honestly admit mathematical fallibility, and try to defend the dignity of fallible knowledge from cynical scepticism (Lakatos)

• Because:1. Any attempt to find a perfectly secure basis leads to infinite regress2. Mathematical knowledge cannot be given a final, fully rigorous form - “one so pay for each step which increased rigour in deduction by the introduction of a new and fallible translation.” (Lakatos)

• NB. Lakatos would not support Social Constructivism. He believed mathematics is fallible3 but wholly rational and not basically social

SC: Need to reconceptualise PM

• The philosophy of mathematics begins when we ask for a general account of mathematics, a synoptic vision of the discipline that reveals its essential features and explains just how it is that human beings are able to do mathematics. (Tymoczko)

• The philosophy of mathematics should account for more than just Mathematical knowledge and the objects of mathematics

NEW CRITERIA NEEDED FOR PM • An adequate philosophy of mathematics should account for all of:• Epistemology: Mathematical knowledge; its character, genesis and

justification, with special attention to the role of proof• Theories: Mathematical theories, both constructive and structural: their

character and development, and issues of appraisal and evaluation• Ontology: The objects of mathematics: their character, origins and

relationship with the language of mathematics, the issue of Platonism• Methodology and History: Mathematical practice: its character, and the

mathematical activities of mathematicians, in the present and past• Applications and Values: Applications of mathematics; its relationship

with science, technology, other areas of knowledge and values• Individual Knowledge and Learning: The learning of mathematics: its

character and role in the onward transmission of mathematical knowledge, and in the creativity of individual mathematicians

Social Basis Of Social Constructivism• (i) Based on Wittgenstein's notions of 'language game' and 'forms

of life'. Mathematical knowledge rests on socially situated linguistic practices, including shared rules, meanings and conventions, i.e. on both tacit and explicit knowledge and symbolic practices.

• (ii) Based on Lakatos' Logic of Mathematical Discovery for negotiation and acceptance of mathematical knowledge, concepts and proofs.

• (iii) Objectivity reinterpreted as social and intersubjective. Objective knowledge understood as social, cultural, public and collective knowledge (not personal, private or individual belief) following Bloor and Harding

• (iv) Adopts conversation as the basic underpinning representational form for its epistemology. Thus views mathematics as basically linguistic/semiotic, embedded in social world of human interaction.

CONVERSATION AND EPISTEMOLOGY

• Beyond metaphor of the 'great conversation' conversation taken literally taken as a basic epistemological form by many.

• our certainty about the Pythagorean Theorem ... Is a matter of conversation between persons, rather than an interaction with nonhuman reality. (Rorty)

Forms of Conversation in SC

INTRAPERSONAL CONVERSATION

ORIGINAL FORM

INTERPERSONAL CONVERSATION

CULTURAL CONVERSATION

Thought as constituted and formed by conversation

Language games situated in human forms of life

Extended version: creation and exchange of texts in permanent form

Thinking: internalised conversation with imagined other

Actual conversation: based on shared experiences, understandings, values, respect, etc.

Reading of any text: dialogical, with reader interrogating it and creating answers from it

All forms of conversation are social in manifestation (interpersonal and cultural) or origin (intrapersonal)

Conversational Nature of Mathematics

• Conversation – underlying form of ebb and flow, alternation of voices in assertion and counter assertion.

• Conversation is the source of feedback, in the form of acceptance, elaboration, reaction, criticism and correction essential for all human knowledge and learning

Different conversational roles originate in the interpersonal but occur in all forms:

• Proponent / friendly listener following line of thought or thought experiment sympathetically, for understanding

• Critic - argument is examined for weakness and flaws.

Knowledge lives in the World

• Conversation based in language games and forms of life (Wittgenstein).

• Conversation as epistemological basis re-grounds mathematical knowledge in physically-embodied, socially-situated acts of human knowing and communication.

• Rejects Cartesian dualism of mind versus body, knowledge versus the world.

• Acknowledges multiple valid voices and perspectives on knowledge, leading to ethical implications (cf. Habermas)

Mathematical Text is Conversational• Mathematics primarily symbolic activity - to create, record and

justify its knowledge (Rotman). • Viewed semiotically as comprising texts, maths addresses a reader.

In all cases the word is orientated towards an addressee (Volosinov)Maths texts, proofs use verb forms in indicative and imperative moods. • Indicative mood for statements, claims, assertions describing future

outcomes of thought experiments for reader to perform or accept• Imperatives are shared injunctions / orders issued by the to reader. Reader of mathematics is either • agent of mathematician's will, response is imagined / actual action• critic seeking to make a critical response. In all cases the mathematical text is conversational

Concepts Dialogical / Conversational

TOPIC DIALECTICAL CONCEPT

Analysis - definitions of the limit

Constructivist Logic

Interpretation of quantifiers: xy...

"You choose x, and I show how to construct y"Recursion theory Arithmetical Hierarchy - ... Set theory Diagonal argument: for any enumeration, omitted

elementSet theory Game-theoretic version of Axiom of Choice

Game Theory Alternation of moves by opponents

Number theory J. Conway's game theoretic foundations of number

Statistics Hypothesis testing (H0 versus HA)

Probability Analysis of wagers, betting games

Knowledge Acceptance Conversational

• Proof structure is a means to epistemological end of persuading mathematical community. A proof becomes a proof after the social act of ‘accepting it as a proof’. (Manin)

• Acceptance depends on largely tacit criteria and informed professional judgement.

• Likewise teacher's decision to accept mathematical answers from student depends on professional judgement. Based on criteria including rhetorical style, not just rigid rules of correctness.

Acceptance of mathematical knowledge depends on dual roles developed and internalised through conversation :

• Proposer of would be new knowledge, or sympathetic reader / listener• Critical reader / listener: reviewer, assessor, gatekeeper

Generalised Logic of Maths DiscoveryConversational mechanism for acceptance / modification of maths knowledge

SCIENTIFIC CONTEXT for Stage n Background scientific and epistemological context: problems, concepts, methods, informal theories, proof criteria and paradigms, and meta-mathematical views.

THESIS Stage n (i) Proposal of new/revised conjecture, proof, solution or theory. ANTITHESIS Stage n (ii) Dialectical and evaluative response to the proposal:• Critical Response

Counterexample, counter-argument, refutation, criticism of proposal• Acceptance Response

Acceptance of proposal. Suggested extension of proposal.SYNTHESIS Stage n (iii) Re-evaluation and modification of the proposal:• Local Restructuring Modified proposals: new conjecture, proof, problem-solution,

problems or theory. • Global Restructuring of Context: changed problems, concepts, methods, informal

theories. Changed proof paradigms, criteria, meta-mathematical views.OUTCOME Stage n+1 (i)

Accepted or rejected proposal, or revised scientific / epistemological context.

(Generalisation of Lakatos’ LMD to overcome criticisms.)

Context of Discovery and Justification• Lakatos' Logic of Mathematical Discovery (and GLMD) shows

that proof (and concept) criticism and improvement is central to the business of mathematics. (Manifested in conversation)

• This process is socially situated, and cannot be divorced from context. Both proof-structure and its social function of persuasion essential for warranting mathematical knowledge. tacit or craft knowledge is involved in these judgements.

• Critical scrutiny of proof by the mathematical community leads to either (a) criticism, requiring development and improvement (context of discovery), or(b) acceptance as a knowledge warrant (context of justification)

• The same Logic of Mathematical Discovery is at work in both cases. There can be no proofs in mathematics which are above critical scrutiny and this Logic, no matter how rigorous.

Conversation in Social Construction of Maths

Conversational Roles

In research mathematics, individuals use personal knowledge to • Construct mathematical knowledge claims (possibly jointly), • Participate in the dialogical process of criticism and

warranting of others' mathematical knowledge claims. In mathematics education individuals use personal knowledge

to direct and control mathematics learning conversation to • Present mathematical knowledge representations to learners

directly or indirectly (i.e. teaching), and • Warrant and critique others' maths knowledge claims /

performances (i.e. assessment of learning).Ultimately, individuals emerge with their personal knowledge warranted (certified), and potentially able to participate in these conversations as teachers or mathematicians

Expanded View of Maths Knowledge

Expanded View of personal Maths Knowledge (Kuhn, Kitcher , Ernest)

Mainly explicit components• Accepted propositions & statements• Accepted reasonings & proofs• Problems and questions

Mainly tacit components• Language and symbolism• Meta-mathematical views: proof & definition standards, scope &

structure of mathematics• Methods, procedures, techniques, strategies• Aesthetics and values

Implications for Maths & Education• Central role of conversation and language - both its use and

extended representational forms - in maths and education• Mathematical knowledge includes a tacit or craft concrete dimension:

knowledge of instances and exemplars of problems, situations, calculations, arguments, proofs, applications, etc.

• Maths problem solving depends on concrete knowledge of instances of past problem solutions (Schoenfeld).

• Mathematical craft-knowledge of concrete particulars and instances is vital in mathematics and learning mathematics- contrary to widely held view that over-emphasises abstract and general knowledge and neglects concrete and specific knowledge.

• This has significant implications for both epistemology and mathematics education

• tacit or craft knowledge plays a central role in the mathematics and teaching conversations, especially in the warranting of would-be-knowledge and assessment of learning

Situated Views of Maths & Learning

• tacit and craft knowledge learned in the practices of a culture. Embedded in shared social 'forms of life' (Wittgenstein)

• Need not be seen as exclusively sited in an individual mind, as assumed by cognitivism and constructivism

• Some developments in philosophy, psychology and mathematics education assume this, e.g. 'situated cognition' of Lave and Wenger, Vygotsky, Saxe, Walkerdine, Harré, Shotter, Gergen, Mead etc.

• Tacit knowledge may be elicited in its context(s) of origination as automatic component of engagement with situation.

Vygotskian Space For Mind/Knowledge

Conversation in Vygotskian SpaceInterpersonal Conversation

Appropriation Surface Learning

Intrapersonal Conversation (& Interpersonal Conversation)

Transformation Understanding – new personal knowledge

Interpersonal Conversation Publication New utterance – knowledge representation

Cultural Conversation Conventionalisation Warranted knowledge

Justification of Maths KnowledgeTo be knowledge it must have a warrant or justification.• tacit or craft knowledge validated by public performance and

demonstration. (To have knowledge is to have the power to give a successful performance. Ayer).

• Parallel between justification of knowledge, and assessment of learning. Identical for tacit or craft mathematical knowledge.

In assessment of learning:• Explicit recall of propositions is low-rated• The application of tacit or craft knowledge is rated higher, especially the

production of warrants for claims.• Critical evaluation of knowledge productions very high-rated (Necessary

skill for teachers and mathematicians - in role of critic / assessor. The role of critic is an important internalised conversational role )

The Rhetoric of Mathematics

• 'rhetoric' is not mere ornament or manipulation or trickery. It is persuasive discourse. In matters from mathematical proof to literary criticism, scholars write rhetorically. (Nelson et al.)

• Even in the most austere case, namely mathematics, a rhetorical function is served by the presentation of the proof. (Kitcher)

• The content and style of proofs and texts are judged with reference to experience of mathematical tradition (i.e. tacit and craft knowledge), rather than explicit criteria.

• Rhetorical styles vary for different mathematical communities, throughout research (and school) mathematics.

Criteria For Rhetoric of Maths TextsSome key criteria for acceptance of a mathematical claim text:• Use restricted technical language in conformity with accepted usage,

using the standard accepted mathematical notation• Avoid deixis: pronouns (except we) or terms re context of utterance• Use spare, clipped expressions with minimal grammatical conventions

observed, no superfluous prose used, formal symbols predominating.• Be succinct and preferably short (length lessens acceptance likelihood)• Use accepted models of style for ‘motivating’ prose, linking prose,

definition, exposition, proof, etc., at appropriate points in the texts• Use standard methods of computation, transformation or proof• Justify proof steps in accordance with accepted rhetorical practice• Refer to standard theorems, results, definitions, texts, mathematicians.• Relate any new problems, methods or results to recognised ones• Express text to minimize its novelty, re-arrangement of known

elements.• Provide a persuasive logical narrative establishing any claims.

Rhetoric of School Maths TextsAccepted rhetorical style of school mathematical text typically:• Uses a restricted technical language and standard notation• Uses spare, minimal overall forms of expression.• Uses certain forms of spatial organisation of symbols, figures and

text on the page (‘linear’ with side-illustrations)• Avoids deixis (pronouns or spatio-temporal locators).• Employs standard methods of computation, transformation or

proof.

When used in classroom by learner this style is strictly regulated (according to local, tacit norms)

When applied to published text books the style subject to different constraints (commercial, content as well as professional)

Rhetoric of the Classroom

Students work on symbolic tasks, writing text-sequences, learning through oral/written teacher-pupil 'dialogue':

• 1. content: symbols, concepts, definitions, procedures, etc.• 2. rhetorical style for school mathematics in written,

symbolic, iconic and oral modes of representation.• Rhetorical style incorporates public justifications: evidence for

teacher of the desired processes and concepts. formal mathematical texts with no trace of authorial subject,

• Project / investigational / problem solving work involves shift in rhetorical style to text including judgements and thought processes of the mathematical subject (learner), and writer may use pronouns (‘I’).

Varieties of Knowledge

Comparing School & Research Maths

• The overall scheme of social constructivism integrates the formation and warranting of both subjective and objective knowledge of mathematics.

• These concern • 1. The individual, in the context of education

(learning and assessment), and • 2. Shared or social knowledge, in the context of

research maths (creation and warranting of mathematical knowledge)

• There are many similarities and differences

Differences between (traditional) school and research mathematics

SCHOOL MATHEMATICS

RESEARCH MATHEMATICS

Learning of existing knowledge

Creation of new knowledge

Studied by all children Done by a tiny group of adults

A preparatory learning activity

Based on the preparatory activity

Interactions face-to-face Interactions at a distanceMatters only to the learner

Becomes part of public knowledge

Similarities Between (Traditional) School and Research Mathematics

SCHOOL MATHEMATICS RESEARCH MATHEMATICSBased on school texts Based on mathematics textsAnswers to school problems Answers to disciplinary

problemsSubmitted to teacher Submitted to editorProduced and validated by conversational pupil-teacher interaction

Produced and validated by conversational mathematician-editor interaction

Acceptance depends on judgement of teacher

Acceptance depends on judgement of editor/referees

Judgement based on shared tacit criteria of correctness and values of school mathematics culture

Judgement based on shared tacit criteria of correctness and values of research mathematics culture

Contribution of Social ConstructivismIT PROPOSES THAT:• 1. Mathematical knowledge is necessary, stable and

autonomous, but that • 2. This co-exists with its contingent, fallibilist3, and

historically shifting character.As Vico said, re geometry: the only things we can know

completely are those we have made (because there is nothing more to know beyond what we have constructed).

Social Constructivism links both the learning of mathematics and research in mathematics in an overall scheme in which knowledge travels either embodied in a person or in a text, and the processes of formation and warranting in the two contexts are parallel

Social Constructivism provides an explanation of how• Mathematics & logic seem irrefutably certain, yet are

contingent, historical creations• The objects of mathematics are cultural fictions emerging

from the use of mathematical language and symbolism, yet seem so solid

• Mathematics is so unreasonably effective in providing the conceptual foundations of our scientific theories about the world

The central explanatory concept is emergence: the evolutionary history of culture and the individual, and the shaping role of conversation

For more complete arguments see• Ernest, Paul (1998) Social Constructivism as a Philosophy of

Mathematics, Albany, New York: SUNY Press.