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• W329863269 abstract "early version of it appeared some twenty years ago; author died young, before Cambridge edition was completed. It a small essay, narrow in scope and wide in range of its implications. The second a whole collection of essays in and about mathematics by two mathematicians who are both well informed and reflective about history and philosophy of their discipline. The book rich in detail, and for illustration it samples well from kingdoms, phyla and classes of what all, somehow, mathematics. The third book by a philosopher who wants to bring discussion of nature of mathematical knowledge into a more fruitful relation to its own history and to that of sciences generally, when studied with serious concern for understanding of conceptual and methodological change. Its argument in a style which you will recognize if you have followed same philosophical paths, not easy for outsider. I bring these three books together not for detailed review but for reference along way. The way I wish to go to open up, for discussion, some questions about relevance of philosophy of mathematics to mathematics teaching, and on early mathematics learning in particular. I shall not break fresh ground in philosophy, though I'll try to crack a clod or two. Philip Kitcher wants to take a philosophical position which has for a long time been most scorned, empiricism of John Stuart Mill. Convinced as he was that all knowledge comes by way of inductive generalization from sense experience, Mill was then also committed to belief that subject matter of mathematics simply world of nature: different from physics or biology not in kind, but only in degree of generality. The only alternative for such an empiricist that of Formalism, to which Kitcher gives little attention. Davis and Hersh give it more, particularly in its logical positivist version. Their criticism neat and effective, and could be a text for several sermons. For any strict formalist mathematical statements are empty, their truth value only that of tautology. Mathematics only a language for describing phenomena of physics, for example. Formalism drops out everything except formalized end-product, and therefore also drops out working mathematician and all great world of mathematical conjecture, argument and discovery. At opposite pole metaphysics called Platonism. The term commonly used without much attention to writings of Plato, who never became his own disciple and thus cannot properly be called a Platonist. But we are stuck with isms; Plato did formulate this one, and his dialogues hover around it, in quite unforgettable ways. But be careful. He was a dramatist of ideas, and sometimes dramatic line not soberly to answer questions raised by his cast of characters; it can be, rather, a kind of insobriety which only deepens questions and blocks easy answers. Courage, says old soldier, is standing your ground. Before dialogue over courage has been successively redefined as the knowledge of hope and fear. That seems a puzzle, but dialogue leads to it. It a mathematician's kind of answer. It seems far from initial conjectures, but if you define courage that way it tightens all arguments. I mention this aspect of Plato because it puts Platonism in a more interesting light than any pat definition can manage. I think we have to recognize a strong family relation, for example, between this Platonic dialectic and that which dominates dialogue of Lakatos. If you study his Proofs and refutations and then look carefully at some of dialogues like Gorgias or Hippias Minor or parts of The Republic parallel clear. Despite all differences in subject matter there an implicit structure, implied by ways our ideas engage each other, in ethics or mathematics. By naive formulations, examples and proofs, counterexamples and refutations, reformulations et seq., we can hope to see these ideas as part of a larger and more coherent structure, transformed beyond our initial grasp of them, still familiar but, for new context embedding them, strangely so. In one case virtue of courage gets embedded in a context of modes of knowing, in other elements of polyhedra reappear as structures within a vector algebra. In both cases there residual doubt as to whether translation has not left something out. Davis and Hersh discuss Lakatos' analysis of method of discovery and provide us with their own casehistory illustrations. They set this account in context of George Polya's rich and many-sided work on heuristics, where it should be." @default.
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• W329863269 date "1985-01-01" @default.
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• W329863269 title "The Edge of Platonism." @default.
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