This issue is due to the generous support of the editor's institution: Leo Rogers, Digby Stuart College, Roehampton Institute, Roehampton Lane, London, SW15 5PH, U.K. Phone: (0)81-876-8273 Fax: (0)81-392-2384

This Newsletter is the publication of the

PHILOSOPHY OF MATHEMATICS EDUCATION NETWORK

Organising Group

Raffaella Borasi (USA), Stephen I. Brown (USA), Leone Burton (UK), Paul Cobb (USA), Jere Confrey (USA), Kathryn Crawford (Australia), Ubiratan D'Ambrosio (Brazil), Philip J. Davis (USA), Paul Ernest (UK), Ernst von Glasersfeld (USA), David Henderson (USA), Reuben Hersh (USA), Christine Keitel (Germany), Stephen Lerman (UK), Marilyn Nickson (UK), David Pimm (UK), Sal Restivo (USA), Leo Rogers (UK), Anna Sfard (Israel), Ole Skovsmose (Denmark), Francesco Speranza (Italy), Les Steffe (USA), Hans-Georg Steiner (Germany), John Volmink (South Africa).

The aim of the newsletter is to encourage international awareness and cooperation between scholars researching philosophical aspects of mathematics education and mathematics, and to inform others interested in these areas, understood broadly.

As the POME mailing list has now grown beyond 300 it is no longer possible to supply this newsletter free of charge. A small charge must be made to cover printing and postage costs. Readers who would like to continue receiving it twice yearly are asked to send annual subscriptions as soon as possible, as follows, according to the currency of preference: (Cheques made out in the following names please).

UKœ 5 to Paul Ernest (address at foot of page, below)

US$ 10 to Stephen I. Brown, Graduate School of Education, SUNY at Buffalo,

Buffalo, NY 14260, USA.

Australian$ (equivalent to US$10) to Kathryn Crawford, Faculty of

Education, University of Sydney, NSW 2006, Australia.

Colleagues in Africa, Central & South America, Eastern Europe, Middle East, and the Far East who have any difficulties in obtaining these currencies are currently welcome to continue receiving the newsletter free of charge.

Any correspondence, news items, comments, reviews, publication or conference announcements for Newsletter 4 should be sent to its editor Marilyn Nickson, 10 Long Road, Cambridge, UK.

---------------------------------------------------------------

POME Group Chair, Dr. Paul Ernest

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

Tel. 0392-264857 Fax 0392-264736 E-mail: ernest.p@uk.ac.exeter

POME - AN INTERNATIONAL NETWORK

There has been a very positive response to this newsletter. Scholars from many countries continue to send in comments, details of their interests, copies of their books and papers, and the names and addresses of others. More of the same would be most welcome, especially items for publication. An increasing amount of correspondence has been received from from such regions as Africa, Central America, Eastern Europe, and the Far and Middle-East. If you know of scholars in the Southern or Eastern parts of the World who are interested in the concerns of this newsletter please send photocopies and encourage them to join the mailing list and to write in.

Membership of the POME network has spread quite widely. Currently the newsletter is mailed to the following countries.

AFRICA Nigeria 2, Mozambique 1, South Africa 6.

AUSTRALASIA Australia 23, New Zealand 2

CENTRAL & SOUTH AMERICA Brazil 2, Costa Rica 1, Mexico 3, West Indies 1.

EASTERN EUROPE & MIDDLE EAST Czechoslovakia 2, Hungary 3, Israel 6,

Poland 1, Russia 1, Yugoslavia 1.

FAR EAST India 2, Indonesia 1, Pakistan 1, Papua NG 1, Philippines 1,

Seychelles 1, Taiwan 1.

NORTH AMERICA Canada 15, USA 82.

WESTERN EUROPE Austria 2, Belgium 4, Denmark 3, Eire 1, Finland 1, France

6, Germany 17, Greece 1, Italy 22, Netherlands 4, Norway 2, Portugal

1, Spain 1, Switzerland 1, Sweden 3, United Kingdom 73.

Special thanks are due to Mrs Elaine Henderson, University of Exeter, for keeping the address list up to date and for compiling these figures.

PHILOSOPHY OF MATHEMATICS EDUCATION TOPIC GROUP AT ICME-7

This is Topic Group 15 at the 7th International Congress of Mathematical Education (ICME-7), Quebec, August 16-23, 1992 (ICME-7 Secretariat, Universite Laval, Quebec, QC, Canada, G1K 7P4). The chief organizer of the topic group is Paul Ernest. After some deliberation, it has been decided to have 2 X 90 minute plenary sessions, with 15 minute presentations by about 6 persons chosen from the Organising Group (see page 1) followed by open discussion. Currently the speakers are being finalised, and Newsletter 4 will carry full details. It is planned to edit a book containing the plenary presentations and chapters by other members of the POME Network. If you would like to contribute, please write in with an indication of your intended chapter (e.g. tentative title plus four sentences).

ICME-7 CONFERENCE SECOND ANNOUNCEMENT

This is now available from the above address, and suggests that much of interest will be on offer. Speakers include Benoit Mandelbrot, and David Wheeler tells me, Thomas Tymoczko. There is much else of interest to readers of this newsletter, including WG5 Theories of Learning Maths, WG21 Public Image of Mathematics, WG23 Methodologies for Research in Math Ed, TG2 Ethnomathematics, TG6 Proof, TG10 Constructivism, TG12 Basic Theoretical Positions in Math Ed, TG14 Cooperation between Theory and Practice, and of course TG16 Philosophy of Math Ed. Unfortunately, all WGs are in one time slot, as are TGs, so there are some tough choices to make.

PHILOSOPHY OF MATHEMATICS EDUCATION DISCUSSION GROUP AT BCME

This will take place at the British Congress of Mathematics Education, Loughborough, July 13-16, 1991. Although many people were involved in the initial planning of the group, the final planning was carried out by a subcommittee of four: Paul Ernest, Ricky Lucock, Sue Sanders and David Wells, who will chair and record the sessions. The published group description is as follows.

THE PHILOSOPHY OF MATHEMATICS EDUCATION

A growing number of us feel that how the nature of mathematics is perceived has a tremendous impact on its teaching and learning. So this group will discuss recent thinking on the nature of mathematics, and then explore what the educational outcomes of different views of mathematics might be. One area we feel to be especially important is the question of whether mathematics is neutral or value-laden. How we answer this has profound implications for how we treat the issues of race and gender in mathematics. The sessions will be divided, roughly equally, between short presentations and open discussion and debate, organised into the the following themes.

SESSION 1 Conceptions of the nature of mathematics

David Wells Games as a Metaphor for Mathematics

Chris Knee & Ruth Farwell A Sociological View of Mathematics

Leo Rogers The Rational Reconstruction of the History of Mathematics and

its Implications for the Learning of Mathematics

Ray Godfrey Fallibilism, Historicism and Elitism

SESSION 2 The impact of views of mathematics on its teaching

and learning

Eric Blaire Implications of the National Curriculum

Sandy Dawson Lakatos and Process aspects of Mathematics in the Classroom

Sue Sanders Teachers' Beliefs about Mathematics

Peter Greenland Teachers' Beliefs about Maths and their Classroom Practices

SESSION 3 Mathematics, values and equal opportunities.11

Europe Singh Ideology and Anti-Racist Mathematics

Paul Ernest Mathematics, Hierarchy and Inequality

Thesis 1 Generally speaking, all more or less elaborated conceptions, epistemologies, methodologies, philosophies of mathematics (in the large or in part) contain - often in an implicit way - ideas, orientations or germs for theories on the teaching and learning of mathematics.

Thesis 2 Concepts for the teaching and learning of mathematics - more specifically: goals and objectives (taxonomies), syllabi, textbooks, curricula, teaching methodologies, didactical principles, learning theories, mathematics education research designs (models, paradigms, theories, etc.), but likewise teachers' conceptions of mathematics and mathematics teaching as well as students' perceptions of mathematics - carry with them or even rest upon (often in an implicit way) particular philosophical and epistemological views of mathematics.

Hans-Georg Steiner

NEWS AND CORRESPONDENCE

Paul Cobb writes in to pass on "a reference that might be of interest to the anti-foundationalists in the crowd": Fish, Stanley (1989) Doing what comes naturally: Change, rhetoric, and the practice of theory in literary and legal studies, Durham [North Carolina] and London: Duke University press. (Just reissued in paperback by Oxford University Press). I notice that R. Kimball in his reactionary attack on 'Tenured Radicals' (in his book of that name) slams "the deconstructivist movement...and a couple of notorious academics, Paul de Man and Stanley Fish" (The Times Literary Supplement, 25 January 1991, p.5). He makes the book sound interesting...

Nerida Ellerton of the School of Education, Deakin University, Geelong, Victoria, Australia 3217 is Chair of the Psychology of Mathematics Education (PME - not POME) working group 'Research on the Psychology of Mathematics Teacher Development'. This will be holding its annual meeting at PME in Assisi. The working group focus for 1991 is: Mathematics teachers' beliefs, conceptions and attitudes - Models and conceptualisations of the domain - The relationship between conceptions/beliefs and the practice of teaching mathematics - Ways of operationalizing or assessing beliefs and conceptions.

Prof. J. Fiala writes from the new Department of the Philosophy of Mathematics at Charles University, Prague, founded in 1990. They begin apparently from zero, but in fact have an unofficial tradition of the past 20 years of seminars in private apartments organised by Ivan M. Havel (brother of president Vaclav Havel) who participates in the department. The head of department is Prof. P. Vopenka (Czech minister of education). Prof. Fiala is very sympathetic with the subject matter and stance of this newsletter because of his concerns to "change the leading paradigm of mathematics education and about the humanisation of mathematics." The department is short of literature and any papers, pre-prints, newsletters and other information would be greatly appreciated if sent to: Prof. J. Fiala, Department of Philosophy of Mathematics, Faculty of Mathematics and Physics, Charles University, 118 00 Prague 1, Malostranske nam. 25, Czechoslovakia.

Chris Weeks writes that "There is a growing group of researchers in the social sciences who are engaged in 'New Paradigm Research'. Their philosophical position is post-positivist but they reject what are effectively moderations or adaptations of a positivist philosophy to special constituencies e.g. feminism, Marxism, liberation theology,...all of which they suggest still carry the faults of positivism, namely a fixed view about ontology and epistemology. What the 'New Paradigm' people propose is a form of 'constructivism'.

At the heart of the unease that New Paradigm research workers have is that research which does not recognise the humanness (and therefore self-directedness) of 'subjects' is to ignore an essential." Central to this tradition is the work of E. Guba and Y. Lincoln such as "Can there be a human science? Constructivism as an Alternative" Person Centered Review 5(2) 1990, 130-154.

The most neglected existence theorem in mathematics is the existence of peopleK.C. HammerLes Steffe, who has joined the organising group of POME, is well known for his research in constructivist approaches to mathematics education. Amongst other things he has been busy putting together a number of edited books in and around this field. Transforming Early Childhood Mathematics Education: International Perspectives (coedited with Terry Wood; Erlbaum, 1990) is the proceedings of the early childhood working group at ICME-6. It includes 45 papers by contributors including H. Sinclair, P. Cobb, E. von Glasersfeld, and F. Marton. Epistemological Foundations of Mathematical Experience is in the process of being sent to Springer-Verlag for publication. It is the proceedings of a conference Les held at the University of Georgia in 1988, and contains 12 chapters. Finally, Les is currently editing a book on Constructivism in Education in which it is planed to contrast social constructionism, social constructivism, radical constructivism, Socio-Cultural Theory, information processing constructivism, and cybernetic systems. This is very encouraging, because there has been a paucity of theoretical accounts of constructivism in mathematics education (excluding the seminal work of Ernst von Glasersfeld).

Jean Paul Van Bendgem, Vrije Universiteit Brussel, Sectie Wijsbegeerte, Pleinlaam 2, B - 1050 Brussels, Belgium, edits Philosophica and has brought out attention to issue 43 (1) 1989 (197-213); Recent Issues in the Philosophy of Mathematics II. His paper, "Foundations of Mathematics or Mathematical Practice: is One forced to Choose?" In this paper he distinguishes between two types of philosophers of mathematics: Type I who as questions on foundations, and who are Formalists, Logicists, Intuitionists, Constructivists, and the like, and Type II who wants answers to questions like, How is mathematics done? What is a real mathematical proof? etc.

He points out that there is not theory, nor even a bare framework for a description of mathematical practice; and in order to study the problems of education, psychology, sociology and cultural aspects of mathematics, we need some kind of theory of what the practice of mathematics is about. He engages in an interesting discussion on the philosophical views of a series of typical fictitious characters, from the "God-like mathematician" through to the "Real social mathematician in society at large".

The discussion aims to address the question: If X is any type of mathematician, then what does it mean for X to know A, where A is a mathematical statement? "Knowing A" entails a description of how we come to know A, and hence he claims we become involved in epistemic logic. Also, he suggests "that it is still possible to maintain the existence of a unique mathematical universe while holding the view that the way mathematics is done is best described using Real Social Mathematician type of model". The aim of the discussion is not necessarily to bring together the extreme types, but to develop a common rhetoric so that the differing views can be considered, and false dichotomies are not drawn.

Jean Paul will be happy to send colleagues copies of other papers and reports.

Paul Wolfson, Department of Mathematics and Computer Science, West Chester University, West Chester P.A. 19383 is offering a summer study tour in Greece, July 6-27. There may still be places, write for details to Paul.

David K. Johnston, Academic Foundations Department, Rutgers University, Newark, New Jersey 07102, USA, offers a challenge for constructivism:

"I would like to take issue with the radical constructivist claim that the social component of our knowledge is more that adequately canvassed by that theory's definition of others as "viable constructions" (POME Newsletter, November 1990; see also my remarks in this regard in that issue). Our prereflective beliefs about what constitutes a person contains at the very least, a notion of the logical (not causal) autonomy of those individuals who collectively constitute "society" or "the social". Most importantly, these beliefs accord with our settled convictions. In morals, if not in epistemology and metaphysics, a collection of independent intentional agents is a firm requirement: I remain a solipsistic knower, not a social one, as I attempt to locate within my own "experiential field" everything necessary for a social constructivist theory of knowing. True, our every epistemic grasp of those others has its source there, in our minds, yet it is something altogether different to claim that those individuals - in contrast to my view, grasp or understanding of them - are the workable creations of my imagination ("viable constructions"). A minimum requirement of being one of those "others" is that he or she is not at all reducible to the cognitive distinctions I, or anyone else, happens to make. I suggest that social constructivism rightly rests on this realistic foundation: We fallibly identify the independent nature of others in the course of our pragmatic intercourse with the rest of nature. Taking our cue from evolutionary theory, we say that knowers are natural creatures caught in the middle of things in a way that far outstrips any particular knower's "experiential field". This pragmatic realist conclusion i as commonsensical as it is consistent with contemporary science. We suppose that there exists an intransitive domain of space-time invariant properties (to use Roy Bhaskar's term), in addition to the less permanent constructions von Glasersfeld (and every other thinker, for the matter!) so aptly entertains, that serves as the source and limit upon the latter. Indeed, the attempt to find a social basis for radical constructivism is to give up the radical part, and admit its implicit reliance all along on this intransitive, natural domain of which humans and their cognitive products are a small part indeed."

Research over the past several years on teachers' beliefs gives strong testimony that teachers' conceptions make a difference in how mathematics is taught.Tom Cooney

In particular, the observed consistency between the teachers' professed conceptions of mathematics and the way they typically presented the content strongly suggests that the teachers' views, beliefs and preferences about mathematics do influence their instructional practice.

Alba Thompson

Anna Sfard's paper "On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin", referred to on p.7 of Newsletter No.1, is now published in Educational Studies in Mathematics 22 (1) 1991 (1-36).

THE CRITICALMATHEMATICS EDUCATORS GROUP NEWSLETTER

Contact person: Marilyn Frankenstein, College of Public and Community Service, University of Massachusetts, Boston MA 02125.

This contains a summary of the Conference on Critical Mathematics Education held at Cornell University, 13-14 October, 1990, which focused on three areas:

1. Epistemology and Philosophy of Critical Mathematics Education: A discussion of the origins and ownership of mathematical knowledge and its context in social change.

2. Mathematics in its Cultural Context: The effects of culture, language and ideology on the mathematics people develop.

3. Political, Economic and Social Issues: The knowledge - power relationship; emancipation and empowerment.

It also includes an account of the formation and the aims of the group, reports from the working groups at the CME conference (described in POME newsletter 2), and a definition of the term in the title, as follows:

criticalmathematics educator/n first used in the early 1980s as various mathematics educators began a dialogue about how to connect their concerns in mathematics education with their critique of society and their commitments to social, political, economic and cultural empowerment. By 1990, small groups of criticalmathematics educators were collaborating on projects, including writing and holding conferences. By the end of 1990, the definition was formally constructed and in 2000, after ten years of dialogue and debate, "criticalmathematics educator" was written in its present form as two words, and was synonymous with (the archaic) "mathematics educator" (see the related term "radicalteacher").

If you would like to join the mailing list or express sympathy with the aims of the group contact Marilyn Frankenstein, College of Public and Community Service, University of Massachusetts, Boston, MA 02125, USA.

Randall Collins, Department of Sociology, University of California, Riverside, CA 92521-0149 offers colleagues copies of a paper on "Math Blocks" which suggests teaching mathematics through the history of theoretical ideas and through problem-solving heuristics. Written about twelve years ago, he is modest about its current level of sophistication, but it still contains some useful ideas.

International History, Philosophy, and Science Teaching Group.

Michael Matthews, School of Education, University of New South Wales, Kensington, NSW 2033 Australia.

There are now two issues of the newsletter current since we last wrote; vol. 1 No.2 Oct. 1990, and vol. 1 No.3 March 1991.

Second International Conference on History, Philosophy and Science Teaching will take place in Canada on 11-15 May 1992. Write for details to Prof. Skip Hills, Faculty of Education, Queens University, Kingston, Ontario, Canada K7L 3N6. Fax: 613-545-6584.

Michael writes that Kluwer have shown interest in starting a Journal on HP & S Education; it might also include aspects of POME. Let Michael know what you think.

'Postmodernism and postmodern mathematics in schools' by Wacek M. Zawadowski (Department of Mathematics Education, Mathematical Institute, University of Warsaw, Palace of Culture room 706, 00-901 Warsaw, Poland)

It is suggested that postmodernism which has swept across art, literature and architecture is also making inroads into science, mathematics and school mathematics. Modernism is characterised by its designs based on one 'big idea' or scheme, which, sweeping away tradition as obsolete, is built up from 'clear and simple ideas' and 'explicitly stated postulates'. Modernistic mathematics curricula included Mathematique Moderne by Papy and Krygowska's 'Geometry', where the emphasis was on explicit visual or symbolic representation, respectively, but not both.

"The postmodern world is less self-assured, it is poly-centric, pluralistic, more connected to tradition."(p.2) In mathematics postmodernism is linked to the works of Lakatos (Proofs and Refutations, Cambridge, 1976), Davis and Hersh (The Mathematical Experience, Birkhauser, 1981) and Tymoczko (ed.) (New Directions in the Philosophy of Mathematics, Birkhauser, 1986). "Mathematics is changing its style and its role in our culture. The stress on axiomatics is less pronounced. The use of pictures and the vision, often neglected in the modernistic approach in favour of symbol manipulation, is again gaining ground."(p.3) Postmodernist curricula emphasise the interplay between 'concept images' (often produced by computer) and 'mental objects', out of which 'concept definitions' develop. Like postmodernist architecture it is 'double coded' to represent and make space for both the meanings of the designer and the meanings of the user - architect and dweller, or mathematician and learner (This is not in the paper, but was communicated verbally by the author). The paper includes examples from analysis, stochastics, number theory, Logo, Pythagoras, and links many of these with explorative computer based work.

This is a powerful vision, which links developments in our own narrow fields with the much grander currents of thought sweeping across our whole culture. Room is being made for human beings amidst the monolithic towers of modernism. There is a clear link with structuralism, whose rigid determining structures, be they architectural or mathematical, with their denial of the human presence, are being challenged and eroded.

However, such an account of postmodernism can be interpreted as merely softening and humanising mathematics superficially, as in the progressive education tradition. Surely the key feature of fallibilism, epitomised by the three references given above, is the reconceptualization of the epistemological basis of mathematics. Proof, according to this view, no longer guarantees certainty. Certainty is an unattainable ideal.

The paper does not address these epistemological issues. But the vision of postmodern mathematics communicated although downplaying the basis of mathematical knowledge is still fully consistent with fallibilism. Probably the references implicitly signify this epistemology.

Zawadowski has implemented his ideas in a post-modern school text Matematyka III (Wydawnicta Szkolne i Pedagogicczne, Warsaw, 1990; co-author Andrzej Walat). This is rich in graphics, diagrams, intuitive explanations, metaphors, computer investigations, iterative and recursive procedures, fractals, and so on. It is a small, unglossy paperback with a tentative, work-in-progress feel, consistent with its underlying philosophy.(PE)

'Postmodern curriculum: the mathematical basis; or, who was Kurt Godel anyway?' by Chris Bigum, in Noss, R. et al. (Eds) (1990) Political Dimensions of Mathematics Education: Action and Critique (Proceedings of the First International Conference), MSC Dept, Institute of Education, University of London, 25-32.

Chris Bigum argues that the 'basics' in the school curriculum, comprising literacy, numeracy and possibly computeracy, represent a return or at least a re-emphasis on certainty and predictability. Numeracy is framed in a mood of conservative nostalgia, and is accompanied by rituals of certainty, the rehearsing of signifiers (algorithms and chanting) not signs. The school discourse of certainty provided by the practice of numeracy is derived from the certainty, predictability and control that characterises deterministic Newtonian science and the modernist discourse of rationality. The naturalisation of numeracy as just another school subject conceals its metafunction - expressed in its cross-curricular role - as providing certain foundations for the curriculum in accord with scientific modernism. However the mathematics of numeracy is both limited and limiting. The irrational Other of school mathematics, Chaos, contra numeracy, should also be included. The school curriculum should be expanded to include more non-linear mathematics. Numeracy represents epistemological certainty, and chaos challenges this. In addition to chaos, pictures and imagery, the use of the computer and recursion are all important and should be included in a postmodern reconceptualization of the mathematics curriculum.

This is a brief summary of some of the key ideas in this conceptually rich paper. It contains a number of important theses:

1. Mathematics provides the foundations for the modern rational worldview, and in particular, for its epistemological certainty.

2. Certainty in school mathematics serves to reproduce this perspective.

3. Numeracy signifies more than just a set of utilitarian skills.

However, there are aspects of the paper and the way it frames its versions of these claims that I wish to question. First of all, surely it is mathematics as a whole, when understood in absolutist terms, and not just numeracy which plays the role ascribed to it in theses 1 & 2? The introduction of new content into school mathematics such as chaos does not challenge the underpinning of certainty by mathematics. Was not the lesson of the Modern Mathematics movement of the 1960s that introducing new content such as set theory does not fundamentally change the nature of school mathematics (by raising the level of conceptual understanding, in that case)? Is it not the whole of the school mathematics curriculum which when presented in an absolutist way helps to install the modern rational, deterministic outlook?

Secondly, although numeracy signifies more than merely skills, in my view it is its minimal nature and the associated pedagogy of work, practice and effort which provides much of its significance. Basic skills have been associated with 'social training in obedience' and the reproduction of a class society. Is therefore not part of the significance of numeracy the exercise of power and social reproduction, as much as the inculcation of an epistemology?

These are personal responses to an all too brief paper. At the very least, what Chris Bigum succeeds in doing is offering provocative insights and stimulating an important debate which links the aims of schooling with epistemology. Such discussions are all too rare!(PE)

Post-modernism is a style, a trend, a mode of thought that has become widespread - fashionable perhaps - in art, architecture, cultural studies and criticism, social theory and recently in mathematics and mathematics education, as the above papers illustrate. Habermas (The Philosophical Discourse of Modernity, Polity/Blackwell, 1990) distinguishes the philosophical and cultural discourses of modernity and post-modernity. The latter seems to derive from the nihilism of Nietzsche, and is epitomized in the work of Jean Baudrillard. Beginning as a critic of Marx, Baudrillard developed a position in which there is no reality but the seductiveness of appearances. In Brian Rotman's words "Once seduced we are in the post-modern world of pure floating images - hypereality. First the image reflected reality, then it masked reality, then it masked the absence of reality, and now, in its final phase, the image bears no relation to any reality but has become its own simulacrum."(Guardian, 21 October 1988: 27).

At first blush, this seems far-fetched, and certainly distant from mathematics. But is it? In the universe of sets generated from the empty set by the axioms of Zermelo-Fraenkel Set Theory we have an infinite universe of froth - endless bubbles built up from an empty bubble. The symbols create their own empty and substanceless structure of forms - labyrinthine and byzantine - which becomes its own substance. Is this so different from the world of simulacra?

Jean-Francois Lyotard is often credited with the first use of the term 'postmodernism' with reference to philosophical discourse in his 'The Postmodern Condition: A Report on Knowledge' (Manchester University Press, 1984; French original published in 1979). Lyotard considers all of human knowledge to consist of narratives, whether it is in the traditional narrative forms, such as literature, or in the scientific disciplines. Each disciplined narrative has its own legitimation criteria, which are internal, and which develop to overcome or engulf contradictions. Lyotard describes how one discipline, mathematics, overcame the crises in the foundations of axiomatics brought about by Godel's Theorem, in this way by incorporating meta-mathematics into its enlarged research paradigm. He also notes that continuous differentiable functions are losing their preeminence as paradigms of knowledge and prediction, as mathematics incorporates undecidability, incompleteness, Catastrophe theory and chaos. Thus, from his perspective, a static system of logic and rationality does not underpin mathematics, or any discipline. Rather they rest on narratives and language games, which shift with the organic changes of culture. Lyotard claims that the traditional objective criteria of knowledge and truth within the disciplines are but internal myths, which attempt to deny the social basis of all knowing. The postmodern perspective, like a number of other intellectual traditions, affirms that all human knowledge is interconnected through a shared cultural substratum, and is a social construction.

Certainly the history of mathematics bears out his reading. Mathematics has been defusing uncertainty by colonialising it since its beginnings. This includes the incommensurability of lengths, Zeno's paradoxes, the Delian problems, negative numbers, probability, infinitesimals and infinity, transcendentals, statistics, Cantor's sets, Peano curves and logical paradoxes, for example, as well as the topics he mentions. Comparably, there is relativity and Uncertainty in physics. But all these topics are widely accepted as technical conceptual advances, and not as challenges to the underlying paradigm of rational control and scientific certainty. Chaos is only the latest branch of mathematics to be tamed and engulfed, and not the beginning of a wholly new game. I see Godel's Theorem, now 60 years old, as far more significant. For it reveals the structural flaw in the foundations of mathematics, on which all of its claimed certainty rests.Paul Ernest

BRIEF REVIEWS

Richard F. Kitchener, (ed) 1988. The World view of Contemporary Physics: Does it Need a New Metaphysics? SUNY Press, Albany, NY.

This is a collection of essays by philosophers and physicists from a conference in 1986. Considering the attention we now pay to new approaches to the epistemology of mathematics education, it is interesting to find Kitchener defining "metaphysics" as a "set of principles - the reality - that underlies the outward manifestations of matter. Thus the central question being addressed is, "Does contemporary physics imply a need for a new view of reality?"

The outcome of this debate in the book is the view that there is no sharp distinction between physics and metaphysics, and that to understand physics it is also essential to understand its metaphysical implications.

Some similar remarks might be made about mathematics, particularly in the context of mathematics education, where, in coming to terms with "understanding" we are continually struggling with the problem of the "underlying reality".

With contemporary physics, the metaphysical struggle has been between relativity and quantum theory, and many recent writers have attempted to make a connection between these theories via special relativity. Kitchener and others believe that the conflict will be resolved by a holistic approach; i.e. that the "whole", be it the entire universe, or any particular domain of it is not the sum of independent and separable parts, but a totality with infinite varieties of manifestations. Two of the contributors are Fritjof Capra and Ilya Progogine, well known for their attempts to portray a holistic view.

Zheng Yuxin From the Logic of Mathematical Discovery to the Methodology of Scientific Research Programmes British Journal for the Philosophy of Science 41 (4) 1990 (377-399). It is interesting to compare this paper with the ideas expressed above: another conflict is being discussed here, that of Euclideanism and quasi-empiricism. The writer reviews Lakatos' philosophy of mathematics, and contrasts it with his philosophy of science, claiming that they start from different ends, but meet in the middle! His philosophy of mathematics is an attempt to extend Popper's fallibilism into mathematics while his philosophy of science uses the logic of mathematical discovery to establish scientific methodology. The link is the claim that both science and mathematics are based in a quasi-empirical context. It is interesting to note that Worrall, the editor of the book version of Lakatos' Proofs and Refutations was never wholly on Lakatos' side, and so the work used by most of us for reference to Lakatos' ideas was, (as was common knowledge at the time), edited to soften some of his more radical views. However, this is an interesting paper, if not entirely accurate in its representation of Lakatos.

The activity of human mathematicians, as it appears in history, is only a fumbling realisation of the wonderful dialectic of mathematical ideas. But any mathematician, if he has talent, spark, genius, communicates with, feels the sweep of, and obeys this dialectic of ideas.Imre Lakatos

Ian Hacking The Taming of Chance, Cambridge 1990.

Published in the series "Ideas in Context", this is classified by the British Library as philosophy, but could equally well be a commentary on the social history of the use of numbers to count things, people, economic power, states of health, wealth, sickness and poverty from the eighteenth century onwards. It shows how different approaches to the classification and importance of the things counted determine our ways of dealing with them, and how the reification of numbers leads to the view that statistical patterns become explanations in themselves. Our contemporary view of "normality" in all its forms, for example, is based on the eighteenth century view of rational man. The book concerns itself with the European scene, and the different motivations for the use of information for social measurement and control.

George Gheverghese Joseph The Crest of the Peacock: Non-European Roots of Mathematics I.B. Tauris 1991. Just published; the author has brought together many sources on Prehistoric, Chinese, Indian and Arab mathematics not easily available. New interpretations of Egyptian and Babylonian works are considered, and the thesis of the non-European roots of much mathematics is well presented. (This book was in the hands of Penguin Books for 2 years, but they baulked at publishing it because it treats Islamic Mathematics, and they are terrified of any such issue after the Rushdie affair. In fact, the book indicates the crucial and hitherto neglected part played by Islam, Africa and other non Euro-Hellenic civilizations in the development of mathematics.)

'Proof, social Process and Thomas Tymoczko' by David Wells, Studies of Meaning Language & Change (SMLC) 21 (1988), 13-26.

This is a review essay on the well-known collection edited by Thomas Tymoczko New Directions in the Philosophy of mathematics (Birkhauser, 1986). Wells regards the collection as promoting a frankly sociological view of mathematics, resulting in a quasi-empirical account of mathematics, but one which is flawed because it fails to accommodate the psychological component of mathematics (1). Wells applauds the attempt to locate the practising mathematician in a socio-historical context, but adds that the psychology of individual mathematicians is also extremely important. In particular, he takes Goodman and others to task for claiming that mathematics is a public activity, arguing that unpublished or private proofs are also mathematics (2). He cites Gauss' and Euler's unpublished proofs as counterexamples. Wells claims that to look to social processes or social approval as warranting proofs is mistaken. If a correct chain of reasoning is produced by a mathematician it is a proof, irrespective of if or when it is published (3). He also criticises Tymoczko for describing mathematics as a rational human activity, on the grounds that it also contains irrational elements (4). Finally. a major criticism levelled at quasi-empiricism is that it seems to omit any account of beauty or aesthetics in mathematics (5).

These are just some of the claims that are made. I wish to respond to some of them from my own perspective. Point 1 fails to take account of the distinctions between philosophy, sociology and psychology. Quasi-empiricism is a philosophy of mathematics which treats mathematics as a social phenomenon. It is not sociology. Nor should it include the psychology of mathematicians. A social philosophy of mathematics attempts to account for mathematical knowledge and practice in terms of the intellectual activity and products of persons, but without presupposing any empirical data or theories from sociology or psychology.

Point 4 that mathematics contains a non-rational or irrational dimension is well made. Mathematical production or creation cannot be reduced to the rational, for there is no inevitable or step-by-step procedure underlying mathematical conjecture and creativity. The heuristics of Lakatos, and indeed of his source Polya, is descriptive, not prescriptive. Likewise beauty plays an essential role in mathematics (point 5) but it is not clear to me what account must be taken of it by the philosophy of mathematics. It must figure in an overall theory of mathematics because of its centrality in the values of mathematicians.

Points 2 and 3 depend on what one's views of proof and mathematics are. By my lights mathematics and proof are essentially public activities, because (i) mathematics - both as a practice and as a body of knowledge - is based on a socially constructed symbol system, and (ii) community assent for proof is essential. According to my conception a proof is only a proof if it is a persuasive argument which satisfies accepted public criteria. None of this is to deny that there is a private/psychological basis for mathematics in individual persons. But this alone is not enough without the public/social dimension. The view I am proposing is that prior to community acceptance a putative proof is only a potential proof. It is the acceptance of the mathematical community, not something inherent in its argument, which confers status of 'proof' on it.

David Wells offers a provocative and stimulating if at times opinionated and irritating essay review of an important collection. His 'publish and be damned' approach is to be welcomed, and elsewhere he has turned his pen on a number of shibboleths beyond quasi-empiricism such as investigations and problem solving (SMLC 18). However readers should be warned of his idiosyncratic vocabulary. SMLC is written and published by David Wells. Issues are available (œ1 each) from Rain Publications, 6 Carmarthen Road, Westbury on Trym, Bristol BS9 4DU, England.(PE)

RESPONSE BY DAVID WELLS

The breadth and depth of disagreement on the most fundamental issues between individuals who on the face of it might be expected to see eye to eye on the bases, if not more, of their common subject of interest is a continual source of fascination to me.

Paul Ernest's remarks, which I very much enjoyed reading, are a case in point. They are clearly based, like Tymoczko's book, on assumptions which I do not share. In particular, I reject any sharp distinction between philosophy, psychology and sociology.

More particularly, I reject the very possibility of 'A social philosophy of mathematics (which) attempts to account for mathematical knowledge and practice in terms of the activity and products of persons, but without presupposing any empirical data or theories from sociology or psychology'. All three subjects are concerned with problems of knowledge and existence, among others. How do we know mathematics? What could it mean to say that I understand a theorem? In what sense does mathematics exist?

Psychologists will tend to emphasise the individual as individual, the sociologist the individual interacting with others, The most effective psychologists, I will add, are aware of these social dimensions and the best sociologists take psychological features into account.

Where does this leave philosophers, who have and use data about 'the activities and products of persons', that is, data which is both psychological and sociological?

At the risk of being provocative - no, make that irritating! - I will suggest that, firstly, philosophy is most effectively interpreted as science, whether philosophers like this or not; secondly, even the finest philosophers tend to be only moderately good scientists, because they so often decline to dip their fingers into the pot of empirical experiment and rely instead on what they can think of in the quiet of their studies.

Therefore I interpret a social philosophy of mathematics as Paul Ernest describes it, as inevitably flawed by its deliberate avoidance of vast amounts of data about 'the activity and products of persons', and its reliance on very selective use of data.

Tymoczko's enterprise, and the failure of Philip Kitcher, for example, to consider aesthetic factors in mathematics are flawed for just such reasons. (See 'Beauty, mathematics, and Philip Kitcher' SMLC 21, 28-44, 1988.) They are not selecting out data clearly marked 'philosophical' rather than 'psychological' or 'sociological', (data does not come in packages with such clear labels). They are picking out aspects of mathematical persons and activity which suit their theories and ignoring the rest - which is simply unscientific. The result is weak and flawed theories.

I have not yet made the role of miscalculating clear. The role of the proposition: 'I must have miscalculated'. It is really the key to an understanding of the 'foundations' of mathematics.

Wittgenstein

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