Mathematics Education and Philosophy

An International Perspective

Paul Ernest

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781135716257
• Edition: 1
• Topic: Education
• Subtopic: Education General

Currently there is a great deal of interest in philosophical issues in the teaching and learning of both mathematics and science education. In this book Ernest has collected together papers from the foremost researchers and practitioners in the philosophy of mathematics education and related areas, together with a selection of papers from the International Congress of Mathematics Education held in Quebec in 1992. Throughout, the outstanding feature of the collection is its multidisciplinary approach to the field of study. This book is the second in Paul Ernest's "Studies in Mathematics Education" series.

Part I
Reconceptualizing the Philosophy of Mathematics

But all that mathematics is still tricked out in, its absolute character and perfect accuracy, its generality and autonomy, in a word, its truth and eternity, all this (if I may be forgiven the expression) all this is pure superstition! (Mannoury, 1947, in Beth and Piaget, 1966, 58)

Today the philosophy of mathematics is a specialist academic field, with its own literature and community of scholars. It has become professionalized as a branch of philosophy, employing the styles of thought and reasoning acceptable to the philosophical community at large, and publishing in general philosophical journals. Its current concern is largely with issues of the warrants for mathematical knowledge and the status of mathematical objects.¹

However this ‘professionalized’ model of the philosophy of mathematics is a recent construction (Kitcher and Aspray, 1988). Earlier this century, in the works of Frege, Russell, Hilbert, Brouwer, Heyting, Weyl, Carnap, Curry, Gödel, and others, the field looked more like a branch of mathematics, and was identified with the foundations of mathematics. Even today the field is not static, and currently there is a growing interest relating individual knowing in mathematics to theories of cognition and empirical findings in psychology (Detlefson, Kitcher). Thus due to internal forces, a broadening of the scope of the philosophy of mathematics is taking place.

In addition to the ‘professionalized’ philosophy of mathematics, there is also a broader domain of questions, interests, literature, and non-specialist scholars, concerned with reflecting on the nature of mathematics. These include mathematicians, historians, anthropologists, sociologists, and educationists specializing in mathematics, and interested in issues concerning the nature of mathematics. It also includes historians of ideas, and interdisciplinary ‘European’ (e.g., French and German) philosophers, including Husserl, Enriques, Bachelard, Mannoury, Serres, who have always related the nature of mathematics to the conditions of human experience, or to culture and history.

There is a also a multidisciplinary ‘maverick’ tradition emerging in the philosophy of mathematics itself, which is broadening the focus to include historical and methodological issues in the philosophy of mathematics (Davis, Hersh, Lakatos, Kitcher, Wang, Tymoczko).

Overall, it can be concluded that academic philosophy of mathematics has not the unity that is sometimes assumed. It is an area of contestation, and its history reveals different perspectives and standpoints on the nature of the field itself.

The authors in this section share the view that mathematical knowledge is fallible and quasi-empirical, and that consequently in mathematics (as in science and other areas of human knowledge), the contexts of discovery and justification interpenetrate. Because of this feature, social and cultural issues cannot be denied legitimacy in the philosophy of mathematics and must instead be admitted as playing an essential and constitutive role in the nature of mathematical knowledge. Just as has happened in the modern philosophy of science, the philosophy of mathematics needs to be broadened and reconceptualized to include ‘external’ factors related to the historical, methodological and wider cultural and social aspects of mathematics.

One view of what the reconceptualized field should treat is the following:²

• epistemology (the nature, genesis and justification of mathematical knowledge, and proof);
• ontology (the nature and origins of mathematical objects and relations with language);
• mathematical theories (constructive and structural, their nature, development, and appraisal);
• the applications of mathematics (and relations with other areas of knowledge and values),
• mathematical practice and methodology (including the practices and methods of mathematicians in the past and present); and
• the learning of mathematics (and its role in knowledge-transmission and creativity).

Three of the researchers whose contributions to the reconceptualization of the philosophy of mathematics have been decisive are David Bloor (1976), Reuben Hersh (1979), and Thomas Tymoczko (1986), so their contributions to this section are especially welcome.

Notes

1. Ironically the only dedicated journal Philosophia Mathematica has in its editorial policy explicitly adopted a broader view of the field than the narrow professional focus described here.

2. Adapted from Ernest (in press).

References

BETH, E.W. and PIAGET, J. (1966) Mathematical Epistemology and Psychology, Dordrecht, Reidel.

BLOOR, D. (1976) Knowledge and Social Imagery, London, Routledge and Kegan Paul (Second revised edition, University of Chicago Press, 1991).

ERNEST, P. (In press) Social Constructivism as a Philosophy of Mathematics, Albany, NY, SUNY Press.

HERSH, R. (1979) ‘Some proposals for reviving the philosophy of mathematics’, Advances in Mathematics, 31, pp. 31–50.

KITCHER, P. and ASPRAY, W. (1988) ‘An opinionated introduction’, in ASPRAY, W. and KITCHER, P. (Eds) History and Philosophy of Modern Mathematics, Minneapolis, University of Minnesota Press, 1988, pp. 3–57.

TYMOCZKO, T. (Ed) (1986) New Directions in the Philosophy of Mathematics, Boston, Birkhauser.

Chapter 1
Fresh Breezes in the Philosophy of Mathematics

Reuben Hersh

Since the time of Pythagoras, philosophy of mathematics has tried to account for mathematical existence and the nature of mathematical objects.

The mystery is that numbers, circles, analytic functions, n-dimensional manifolds, all seem to be different from everything else we think about. They’re neither physical nor mental. Not mental, because the Pythagorean theorem or any other well-established mathematical fact is independent of what you or I think. Whether we know it and believe it or don’t know it and don’t believe it, the Pythagorean theorem is still true. Yet it’s not physical either. Plato and Aristotle explained clearly that the triangles and circles of the geometer are not the same as the carpenter’s triangles or circles, but something ‘ideal’.

Spiritual, empirical, psychological, formalist, and logicist explanations have been offered. None give a credible account of what we do when we do mathematics. Presently some dozen authors are working to construct a social-historical-cultural answer.

An Israeli mathematics education researcher, Anna Sfard, recently published an interesting insight. In learning a new mathematical concept, children first learn it as an algorithm—a procedure, a method. Later, the algorithm or procedure is transformed into an entity or object. She calls this transformation ‘reification’. It’s difficult to achieve, often requiring help from the teacher. This story is close to the theory of a Russian psychologist, Lev Vygotsky, who died in the 1930s and has recently become well-known.

For example, subtraction is an algorithm. It isn’t hard. It reifies into the negative numbers—very hard. Again, it isn’t hard to connect two points by a straight line. It is hard to conceptualize the straight line as an entity in itself, apart from operations. Everyone understands the operation of collecting distinct individuals into a set. The set as an entity in itself, apart from any act of collecting, is subtle.

Is every entity in mathematics just a frozen algorithm? Which are? Which aren’t? What about high-level novelties like submartingales, dual functors, kittygories, generalized functions? What is the relationship, the interaction between doing and being, between algorithm and entity? This is an example of a question in the philosophy of mathematics that’s based on mathematical practice, on seeing mathematics as a human activity. It’s an answer to people who say, ‘lf philosophy of mathematics isn’t foundations of mathematics, then what is it?’

Foundations Lost

In books on philosophy of mathematics (S. Körner, or P. Benacerraf and H. Putnam) one reads of the leading problem, ‘foundations’. How to establish mathematical knowledge as certain, indubitable, free of any possible doubt? Three historically important solutions to that problem were offered—logicism (Platonism), formalism, intuitionism. All three proved unsuccessful. For logicism and formalism, as far as I know, no major new idea has been suggested in over half a century. Intuitionism, and its offspring constructivism, did strive to carry out the programme enunciated by Brouwer and streamlined by Bishop. But their goal of replacing classical mathematics is more remote now than sixty or seventy years ago.

Logicism. the doctrine of Gottlob Frege and Bertrand Russell, said ‘Build on logic!’ In fact, they believed that mathematics is actually a part of logic. This would be a good thing, they thought, because they thought logic was the most indubitable, rock-bottom part of human thought. But in building mathematics out of logic they needed set theoretic axioms such as the axiom of infinity (There exists an infinite set’) which aren’t logic in the original sense. Russell’s famous paradox exposed unsuspected treachery in set theory. So as an explanation of the nature of mathematics, logicisn died. Some investigators still study the relation between mathematics and logic.

Next came intuitionism, the doctrine of Henri Poincaré, Emile Borel, Henri Lebesgue and above all, L.E.J. Brouwer. The intuitionists said, ‘Build on counting—the natural numbers! Only what’s obtained constructively from the natural numbers must be allowed in mathematics’. They didn’t think it necessary or possible to define ‘constructive’. They did say it didn’t include indirect proof—proof by contradiction. For the intuitionists, mathematics that depends on indirect proof is not valid.

Errett Bishop produced a streamlined intuitionism called ‘constructivism’, closer to mathematical practice, not coloured with mysticism as is Brouwer’s intuitionism. Groups of Bishop’s followers can still be found here and there. But their dream of constructivizing mathematics, of converting the mathematics profession, is dead. The overwhelming majority of mathematicians rejected intuitionism and constructivism years ago.

Formalism is credited to David Hilbert. It’s been said that his leap into foundationalism was a reaction to the flirtation of his favourite pupil, Hermann Weyl, with Brouwer’s intuitionism. Hilbert was an implacable foe of Brouwer and intuitionism. He said that depriving the mathematician of proof by contradiction was like tying a boxer’s hands behind his back.

Hilbert intended to prove that mathematics was consistent—not tautologically true as Russell had thought, nor objectively true when stripped of unacceptable reasoning, as Brouwer thought. To do this, he had a brilliant, original idea: work with formulas, not content. Think of axiom and theorem as strings of meaningless symbols. Think of the transformation from axioms to theorem—the proof—as a permutation of symbols. Permutations of symbols are elementary mathematical objects, studied in ‘combinatorics’ or ‘combinatorial analysis’. To prove mathematics is consistent, Hilbert had to prove that the permutations allowed in mathematical proof, starting with the axioms, could never yield a falsehood: 0=1, for example.

A few years later, Kurt Gödel proved that such a thing could never be proved by the proof methods Hilbert allowed. This incompleteness theorem of Gödel’s is commonly cited as the death blow to Hilbert’s programme, and to formalism as a philosophy of mathematics.

The surviving remnants of foundationalism have been named ‘neo-Fregeanism’ by Philip Kitcher. This doctrine today dominates the philosophy of mathematics as done by academic philosophers. It says: ‘Philosophical thinking about mathematics need only concern itself with sets, and with set theory’s twin sister, logic.’ The goal of foundationalism admittedly is hopeless and forgotten, but the assumption persists that ‘Foundations of mathematics is the philosophy of mathematics.’ But the great majority of researchers, users, teachers, and historians of mathematics aren’t primarily concerned with set theory. Consequently, philosophers of mathematics ignore mathematics and mathematicians, and mathematicians find nothing of interest in philosophy of mathematics.

A deplorable situation! The principal problem in philosophy of mathematics left unsolved by all schools, which for over half a century are in paralysis! Mathematicians and philosophers of mathematics virtually ignorant of each other’s existence!

A Harvard philosopher Hilary Putnam once published a foundationalist paper titled ‘Mathematics Without Foundations’. Is philosophy of mathematics pointless and unnecessary? Or is it time for a new start?

Phil/m and Phil/sci

A weird fact about modern US philosophy is that philosophy of science and philosophy of mathematics are almost disconnected from each other. Authors in philosophy of science rarely refer to philosophy of mathematics, and vice versa. Even an author who writes on both subjects, in any one art...

Table of contents

• Cover
• Half Title
• Full Title
• Copyright
• Contents
• List of Figures and Tables
• Preface by Series Editor
• Introduction
• Part 1: Reconceptualizing the Philosophy of Mathematics
• Part 2: Post-modernist and Post-structuralist Approaches
• Part 3: The Human Face of Mathematics
• Part 4: The Social Context of Mathematics and Education
• Notes on Contributors
• Index