Number 1, July 1990 Editor: Paul Ernest

This is the first Newsletter of The Philosophy of Mathematics Education special interest group/network. The aims of this network are:

To explore current developments in the philosophy of mathematics such as Lakatos' fallibilism, and other humanistic views of mathematics

To explore philosophical perspectives of mathematics education, and to raise philosophy to the level of the other disciplines of mathematics education

To form an open international network of persons interested in the topic area conceived broadly, and to provide opportunities for the sharing and advancement of ideas and perspectives, at such venues as conferences. In particular, a Topic Group on the Philosophy of Mathematics Education has been proposed for ICME-7, and accepted at BCME (details overleaf).

To facilitate these aims, this newsletter is copyright free, and you are invited to pass copies to any interested person.

All or part may be reprinted if the source is acknowledged.

To ensure that multiple voices and perspectives are aired, the editorship will rotate among the organizing group.

The editor of The Philosophy of Mathematics Education Newsletter 2 will be Stephen Lerman. Please send any correspondence, items, comments, publication or conference announcements for issue 2 to him, at the following address.

Dr. Stephen Lerman, Department of Computing and Mathematics, South Bank Polytechnic, 103 Borough Road, London SE1 0AA, U.K.

This issue has been made possible by the generous support of the University of Exeter School of Education.


Contact Address:Dr. Paul Ernest

University of Exeter, School of Education, Exeter EX1 2LU, U.K.



Raffaella Borasi (USA), Stephen I. Brown (USA), Leone Burton (UK), Paul Cobb (USA), Jere Confrey (USA), Kathryn Crawford (Australia), Philip J. Davis (USA), Paul Ernest (UK), Reuben Hersh (USA), Christine Keitel (FRG), Stephen Lerman (UK), Marilyn Nickson (UK), David Pimm (UK), Sal Restivo (USA), Leo Rogers (UK), Anna Sfard (Israel), Ole Skovsmose (Denmark), John Volmink (USA).

This Topic Group has been proposed for the 7th International Congress of Mathematical Education, Quebec, August 16-23, 1992. Acceptance has not yet been notified. If accepted the likelihood is that we will have just 2 X 90 minute sessions.



Plenary presentations from invited speakers in the organising group on:

Recent Developments in the Philosophy of Mathematics

The Significance of Philosophies of Mathematics for Education


Parallel sessions of brief research reports and discussion from a large number of participants, on such themes as:

Recent Developments in the Philosophy of Mathematics

Social Constructivism: A New Paradigm for the Philosophy

of Mathematics

History and Mathematics as a Social Institution

The Impact of Fallibilist Philosophies of Mathematics on

Teaching and Classroom Practice

Personal Philosophies of Mathematics: Teacher and Student

Conceptions of Mathematics

Values, Mathematics and Feminism

The Theoretical Basis of Ethnomathematics, Multiculturalism and

Anti-Racism in Mathematics

Epistemological Agency and Problem Posing: The Student as

Mathematician, the Teacher as Researcher

The Philosophical Bases of Educational Research Paradigms

Radical Constructivism: Elaboration and Critical Review

The Implications of Semiotics, Post-Structuralism and

Post-Modernism for the Philosophy of Mathematics and

Mathematics Education

The Social Responsibility of Mathematics and Critical

Mathematical Literacy

Criticism and responses to these suggestions is solicited! Some relevant correspondence is reported on pages 7-8. A possibility being considered is the publication of contributions to the proposed Topic Group in book form.


There will be a Philosophy of Mathematics Education Discussion Group at the British Congress of Mathematics Education, Loughborough, July 13-16, 1991, led by Paul Ernest. The group will have 3 X 90 minute slots. We have been asked to organize the sessions in an imaginative and interactive way, and to involve and accommodate the interests practising teachers.


This has been made up of British residents, and currently consists of Leone Burton, Paul Ernest, Ruth Farwell, Christopher Knee, Steve Lerman, Marilyn Nickson, David Pimm, Leo Rogers. However it will be enlarged to include more persons interested in the topic, including serving school teachers.


Currently it is proposed that each of the three sessions will begin with one (or more) plenary talks on a theme followed by related practical activities and discussion by participants for up to 60 minutes. In global terms the three proposed themes concern the nature of mathematics, aspects of classroom interaction, and aspects of the mathematics curriculum.

THEME 1: Recent Developments in the Philosophy of Mathematics.

Different non-absolutist philosophies, Uncertainty in Modern Mathematics, Change in Historical Conceptions, Personal Philosophies of Mathematics, Teacher and Student Conceptions of Mathematics.

THEME 2: The Implications of Fallibilism for Education.

Process-oriented work, Problem posing and solving, Discussion, Rejection of duality with regard to answers, Students' alternative conceptions and procedures.

THEME 3: Mathematics, Values and Equal Opportunities.

Anti-racist and multicultural Mathematics, Ethnomathematics, Anti-sexist and girl-friendly mathematics, Empowerment through Problem Posing and Political Mathematics.

This programme is tentative and critical responses and further suggestions are sought.

The purpose of the British Congress of Mathematics Education is twofold. First, to facilitate preparation for ICME-7 in 1992. Second, to provide free-standing experience for those unable to attend ICME-7. There will be about 300 places at the conference, and it will cost about œ120. Ten discussion groups have been arranged, including this one.

The BCME Conference Secretary is Ms Rita Nolder, School of Education, University of Technology, Loughborough, Leicestershire LE11 3TU, U.K.


It seems appropriate to provide a rationale for the topic of this special interest group in this newsletter.

The philosophy of mathematics is in the midst of what might be termed a 'Kuhnian revolution'. The Euclidean paradigm of mathematics as an objective, absolute, incorrigible and rigidly hierarchical body of knowledge is increasingly under question. One reason for this is that that the foundations of mathematics are not as secure as was supposed. Godel's first Incompleteness theorem has shown that axiomatics must fail to capture the truths of most interesting mathematical systems. Another reason is a growing dissatisfaction amongst mathematicians, philosophers and educators with the traditional narrow focus of the philosophy of mathematics, limited to the foundations of pure mathematical knowledge and the existence of mathematical objects.

A number of authors have proposed that the task of the philosophy of mathematics is to account for mathematics more fully, including the practices of mathematicians, its history and applications, the place of mathematics in human culture, including issues of values and education - in short - the human face of mathematics. Publications by Davis, Hersh, Kitcher, Lakatos, Putnam, Tymoczko and Wittgenstein, for example, are suggesting new fallibilist, quasi-empirical or social constructivist paradigms for the philosophy of mathematics. At the same time developments in the sociologies of science, knowledge and mathematics, and post-structuralist and post-modernist thought are looking towards social constructionist accounts of knowledge. These have important implications for mathematics and particularly for educational theory and practice.

A parallel with the philosophy of science is worth remarking. Science education has been drawing its inspiration from developments in the philosophy of science for a number of years, leading to such developments as research into teacher and curriculum philosophies of science, learner's alternative conceptions, the nature of scientific inquiry and in 1989 to the first international conference on the history and philosophy of science in science teaching (see page 6).

Philosophical considerations are also central to empirical research in mathematics education. Educational researchers are becoming increasingly aware of the epistemological foundations of their methodologies and inquiries, and referring to them explicitly. Multiple research paradigms are now in use, and proponents of the quantitative/positivistic, qualitative/ethnomethodlogical and critical theoretical research methodologies are discussing and comparing their philosophical bases (when not engaged in internecine strife!)

Finally, in the field of mathematics education itself there is an increasing awareness of the significance of epistemological and philosophical issues. Theories of learning are becoming more epistemologically orientated, such as constructivism. A growing number of areas of inquiry are drawing on the philosophy of mathematics and philosophical perspectives. These include problem solving and investigational pedagogy, curriculum theories, teacher education and development, teacher beliefs, applications of the Perry Theory, ethnomathematics, gender-fair and multicultural mathematics, and the sociology and the politics of mathematics education.

The Philosophy of Mathematics Education Network has been formed to explore some or all of these and related issues under the umbrella of an overall philosophical perspective on mathematics education. The psychology, history, and to a lesser extent, the sociology of mathematics education have a tradition behind them. It is proposed to take steps to add a philosophical perspective to mathematics education. Ultimately, the aim is provide a means to link philosophers of mathematics and mathematicians interested in education with mathematics educators with an interest in philosophy. All should gain from this cooperative aim.

In fact, whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests on a philosophy of mathematics.Rene Thom

Announcement of New Book Series


Falmer Press, London, Philadelphia, New York

Book proposals are sought by the series editor: Paul Ernest.

Mathematics education is established worldwide as a major area of study, with numerous dedicated journals and conferences serving national and international communities of scholars. Research in mathematics education is becoming more theoretically orientated. Vigorous new perspectives are pervading it from such disciplines as psychology, philosophy, sociology, anthropology, feminism, semiotics and literary criticism. This series will consist of studies in mathematics education based on disciplined perspectives, but aiming to link theory with practice. It is founded on the philosophy that theory is the practitioner's most powerful tool in understanding and changing practice. Whether the practice is mathematics teaching, teacher education, or educational research, the series Studies in Mathematics Education is intended to offer new perspectives to assist in clarifying and posing problems and stimulating debate. The series Studies in Mathematics Education will encourage the development and dissemination of theoretical perspectives in mathematics education as well as their critical scrutiny. It aims to have a major impact on the theoretical development of mathematics education as a field of study in the 1990s.


The Humanistic Mathematics Network shares many of the concerns of this network, being concerned to reconceptualize mathematics as a humanistic discipline, as well as to teach it humanistically. The May 1990 issue of the newsletter contained interesting and provocative papers including: Humanistic Aspects of Mathematics and their Importance (Philip J. Davis), Heuristic Thinking and Mathematics (J.F. Lucas), Mathematics and Ethics (Reuben Hersh), Teaching Global Issues through Mathematics (Richard H. Schwartz), A Social View of Mathematics & What has Mathematics to do with Values? (Stephen Lerman). For more information contact Alvin White, Harvey Mudd College, Claremont, California 91711, USA.

The International Study Group on Ethnomathematics is interested in culturally-embedded, non-academic mathematics, which provides the basis for a social view of mathematics. Contact Rick Scott (Newsletter editor), College of Education, University of New Mexico, Albuquerque, NM 87131, USA.

The International History, Philosophy, and Science Teaching Group have published their first newsletter, following their first international conference (Florida, November, 1989). Contact Michael R. Matthews, School of Education, University of NSW, Kensington, NSW, Australia.

Hans-Georg Steiner has proposed a Topic Group for ICME-7 called The Theory of Mathematics Education (TME), with the subtitle relations between Philosophy of Mathematics and Mathematics Education. Evidently there is a great deal of overlap with this network, and we must keep in touch and work in cooperation! Currently TME is having its fourth conference, in Mexico, preceding PME 14. Contact H-G Steiner, IDM, Universitat Bielefeld, Postfach 8640, 4800 Bielefeld 1, Federal Republic of Germany.

New Thinking on the Nature of Mathematics

This one day conference was organised by the Mathematics Applicable Group, Norwich, June 2, 1990. The papers included:

Philip J. Davis, on The Human Face of Mathematics

A number of parallels with literature were drawn. Mathematics has metaphor, ambiguity, aesthetics, unformalizable components, paradox, mystery and awe, catharthis, theological and philosophical implications, redemptive and destructive features, escapism, historical contexts, and mythologies.

The conference was chaired by Eric Blaire, and papers were also given by Christopher Ormell, Paul Ernest, Warwick Sawyer and others. Proceedings including short versions of the papers will be available from MAG, School of Education, University of East Anglia, Norwich NR4 7TJ, U.K.


Ivor Grattan-Guinness writes of the danger of separating the philosophy of mathematics from its history, and "falling into a non-historical sociological relativism, as an absurdity opposite but equal to the traditional certainties."

As Lakatos says "the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics has become empty." A sentiment that many of us doubtless share. However I still want to distinguish the properly empirical concerns of the history of mathematics from purely philosophical questions, whilst still claiming that the realms are inter-related. Indeed, in my view, Lakatos falls into the trap of failing to make this distinction in Proofs and Refutations (Appendix I). Here he offers the detailed method of proofs and refutations as a theory of historical invention, and also, by implication, as a philosophical theory of mathematical knowledge creation. But the details of the historical thesis could be refuted without undermining the more general philosophical view of knowledge creation by the process of conjectures, proofs and refutations.

Ivor Grattan-Guinness is editing the Encyclopaedia of the History and Philosophy of the Mathematical Sciences, Routledge, planned for 1992.

Anna Sfard "On the Dual Nature of Mathematical Conceptions: Theoretical Reflections on Processes and Objects as Different Sides of the Same Coin". This paper explores two sorts of complementarity. There is that between the abstract notions of mathematics which can be conceived of structurally - as objects - or operationally - as processes. There is also the complementarity between the historical (or historico-epistemological) and the psychological development of concepts. A theoretical model for concept formation suggests the following cyclic pattern: operations/processes on objects --> interiorization --> condensation --> reification --> new object.

Philip J. Davis points out the problems to be faced by any attempt to build a social constructivist philosophy of mathematics. (Paraphrasing) if mathematics is a social construction, then the problem is to account for how having set up matters in a particular way, then other things seem to be beyond our control. They seem to follow inevitably (perhaps logically or computationally). My analogy for this is chess. Having played so far, checkmate in four moves may be a certainty. But only because we accept the constraints of the rules and socially agreed meanings attached to the symbols and procedures involved

Heinrich Bauersfeld suggests (1) if a social constructivist philosophy of mathematics is to involve the attempt to combine social interactionism (H. Blumer) and radical constructivism (E. von Glasersfeld), then there is the incompatibility between the psychological and the social spheres to be accommodated. He also asks (2) Why a critical examination (sounds like the Spanish inquisition) of constructivism only? Is there nothing else to criticize or examine, perhaps "post-modernist" positions or materialist Activity theories (Rubenstein et al.) or ...?

I certainly agree, and will argue elsewhere against the post-modernist claim that chaos theory has forced us to reconceptualize mathematics. In my view, mathematics has been defusing uncertainty by incorporating it since its beginnings. What is the history of mathematics but the story of coming to terms with uncertainty? Consider Pythagoras and the irrationality of root 2, Zeno's paradoxes, unsolvability of the Delian problems, zero, negative numbers, probability, calculus and infinitesimals, quaternions, non-Euclidean geometry, transcendentals, statistics, sets and transfinite numbers, Peano curves, logical paradoxes, Godel, Tarski and Church's theorems, independence of the continuum hypothesis, catastrophe theory, chaos theory. (Actually, the post-modernist philosopher J.F. Lyotard acknowledges this colonialization of uncertainty.)

With regard to issue 1 - this is a real problem. But I must admit to attempting this reconciliation myself in a book (see below).

Gila Hanna Rigorous Proof in Mathematics Education, OISE Press, Toronto, 1983. This must be one of the first books in mathematics education to explore the philosophy of mathematics to any depth. It argues that placing rigorous proof at the heart of the mathematics curriculum is to misunderstand and to misrepresent developments in modern mathematics.

Gila Hanna and Ian Winchester have edited Creativity, Thought and Mathematical Proof, Special issue of Interchange, Volume 21, Number 1, 1990, OISE, Ontario. This contains papers by David Wheeler, Michael Otte, the editors and six others.

David Henderson (Cornell) questions the title The Philosophy of Mathematics Education, and suggests Philosophy in Mathematics Education or The Relationship Between Philosophy and Mathematics Education.

In choosing a title Philosophical Research in Mathematics Education (PRIME), was also considered, but in the end Steve Lerman's suggestion (Philosophy of Mathematics Education), was adopted, by analogy with Psychology of Mathematics Education.

The issue, then, is not, What is the best way to teach? but, What is mathematics really all about?..Controversies about...teaching cannot be resolved without confronting problems about the nature of mathematics.Reuben Hersh


Brian Rotman, Toward a Semiotics of Mathematics, Semiotica, Volume 72, Nos 1/2, pages 1-35.

This paper embarks on an ambitious project, addressing the question: what is the nature of mathematical language? The author/reader of mathematics is characterized by a tripartite model, which distinguishes the full Person, the more limited Mathematician, both timeless and locationless, and his Agent, an unreflective projection able to carry out infinite and other processes. Their role in mathematical proof is described as follows:

"Persuasion and the dialectic of thinking/scribbling which embodies it is a tripartite activity: the Person constructs a narrative, the leading principle of an argument, in the metaCode; this argument or proof takes the form of a thought experiment in the code; in following the proof the Mathematician imagines his Agent to perform certain actions and observes the results: on the basis of these results, and in the light of the narrative, the person is persuaded that the assertion being proved - which is a prediction about the Mathematician's sign activities - is to be believed."(page 29)

Three philosophies of mathematics are characterized on the basis of the semiotic analysis of mathematics. Formalism considers signifiers without the signified; intuitionism considers the signified without signifiers; platonism includes both signifiers and signified, but the latter are part of an ideal reality. The account digs deeper into philosophical issues than this precis suggests, offering genuinely novel and profound insights into mathematics and its philosophy.

One criticism is that conventionalism and social constructivism are not taken seriously as philosophies of mathematics, although the inescapable conclusion of the paper is that as a sign system mathematics is a human construction. But the notion that there might be a socially defined notion of objectivity (by social agreement) in between the traditional notions of subjectivity and objectivity is never quite reached, let alone elaborated as a philosophy of mathematics.

Overall, this paper represents a novel philosophical approach to mathematics. By applying the methods of semiotics to mathematics, the outcomes are surprisingly rich and deep. In fact the author has been thinking along such lines for almost 15 years, which partly accounts for the novelty and depth of the perspective he is able to construct. In my opinion, this paper represents any early contribution to what could well be a major area of growth in philosophizing about mathematics, namely the application to it of the methods and insights of semiotics, literary criticism and post-structuralism.


Paul Ernest, THE PHILOSOPHY OF MATHEMATICS EDUCATION, Falmer Press (Due November 1990). Studies in Mathematics Education 1

Although many agree that all teaching rests on a theory of knowledge, there has been no wide-ranging in-depth exploration of the implications of the philosophy of mathematics for education. That is the aim of this book. Building on the work of Lakatos and Wittgenstein and others it challenges the notion that mathematical knowledge is certain, absolute and neutral, and offers instead an account of mathematics as a fallible social construction.

At the heart of the proposed social constructivist philosophy of mathematics is a theory of the relationship between objective and subjective knowledge. Following Bloor, objectivity is interpreted as socially accepted, which is where social construction comes in. Subjective knowledge is understood to be the personal creation of each individual, acquired through social negotiations with others. It represents both the personal recreation of existing knowledge of mathematics, as well as the source of new contributions to the field, subject to the criticism and reformulation of the mathematics community.

Pivotal in relating the philosophy of mathematics to educational issues is a theory of how epistemology is embodied in individual belief systems. The Perry Theory plays this role in the book, and five intellectual and ethical perspectives are distinguished: one Dualistic, one Multiplistic and three Relativistic. The outcome is a well-grounded model of five educational ideologies, each with its own epistemology, values, aims, history and social group of adherents. An analysis of the impact of these groups on the British National Curriculum results in a powerful critique, revealing the questionable assumptions, values and interests upon which it rests.

Fallibilist and social constructivist accounts of mathematics have profound implications for social and educational issues, including gender, race and multiculture; for pedagogy, including investigations and problem solving; and challenges hierarchical views of mathematics, learning and ability. The book finishes on an optimistic note, arguing that an appropriate pedagogy allows the achievement of the radical aims of educating confident problem posers and solvers who are able to critically evaluate the social uses of mathematics.

Contents of the Book



The Philosophy of Mathematics

The Nature of Mathematical Knowledge

The Fallacy of Absolutism

The Fallibilist Critique of Absolutism

The Fallibilist View


The Scope of the Philosophy of Mathematics

A Further Examination of Philosophical Schools



Social Constructivism

Objective and Subjective Knowledge

Social Constructivism: Objective Knowledge

A Critical Examination of the Proposals


The Genesis of Subjective Knowledge

Relating Objective and Subjective Knowledge of Mathematics

Criticism of Social Constructivism


Philosophical Parallels

Sociological Perspectives of Mathematics

Psychological Parallels

Conclusion: A Global Theory of Mathematics



Epistemological and Ethical Positions

Aims in Education: An Overview


Overview of the Ideologies and Groups

The Industrial Trainers

The Technological Pragmatists


The Old Humanists

The Progressive Educators


The Public Educators

A Critical Review of the Model of Ideologies


The Aims of Official Reports on Mathematics Education

The National Curriculum in Mathematics


Hierarchy in Mathematics

Hierarchy in Learning Mathematics

The Hierarchy of Mathematical Ability

Social Hierarchy

Interrelating Mathematical, Ability and Social Hierarchies


Mathematics and Values

Anti-racist and Multicultural Mathematics Education

Gender and Mathematics Education


Mathematics Results from Human Problem Posing and Solving

Problems and Investigations in Education

The Power of Problem Posing Pedagogy

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