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Mathematical Knowledge and Pattern Cognition

Published online by Cambridge University Press:  01 January 2020

Michael D. Resnik
Michael D. Resnik*
Affiliation:
University of North Carolina, Chapel Hill
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Extract

This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. (One might do so by developing a theory of meaning and truth, which together with observations from the sociology of mathematics would imply that mathematical knowledge exists. Mathematicians do seem to make knowledge claims, so all one needs is a theory which shows that here at least appearances are real.) I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to me than any philosophical view of mathematics I know of — including my own. So I am going to take mathematics as my starting point.

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Type
Research Article
Information
Canadian Journal of Philosophy , Volume 5 , Issue 1 , September 1975, pp. 25 - 39
Copyright
Copyright © The Authors 1975

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References

1 Cf. Frege, G.. Foundations of Arithmetic. Oxford, 1950Google Scholar. K. Gödel. “Russell's Mathematical Logic” and “What is Cantor's Continuum Hypotheses?”, both reprinted in Benecerraf and Putnam, Philosophy of Mathematics. Englewood Cliffs. 1964.Google Scholar

2 Heyting, A.. Intuitionism. Amsterdam, 1966. pp. 1–12.Google Scholar

3 Cf. my “The Frege-Hilbert Controversy,” Philosophy and Phenomenological Research XXXIV, 1974, and Putnam, H. “The Thesis that Mathematics is Logic,” in Bertrand Russell: Philosopher of the Century. Edited by Schoenman, R.. London, 1967.Google Scholar

4 Hilbert, and Bernays, . Grundlagen der Mathematik I, II. Berlin, 1934 and 1939Google Scholar. Mendelson, E.. Introduction to Mathematical Logic. Princeton, 1964.Google Scholar

5 Bishop, E.. Foundations of Constructive Analysis. New York, 1967.Google Scholar

6 Heyting, op. cit., p. 6.Google Scholar

7 Cf. Poincaré, . “Mathematical Creation,” reprinted in The World of Mathematics, IV. Edited by Newman, J.R.. New York, 1956.Google Scholar

8 Cf. Benacerraf, P. “Mathematical Truth,” The Journal of Philosophy, LXX. 1973.Google Scholar

9 Cf. Benacerraf, P.. “What numbers could not be”, The Philosophical Review, LXXIV. 1965.Google Scholar

Parsons, C.. “Frege's theory of number” in Philosophy in America. Edited by Black, M.. Ithaca, 1965.Google Scholar

10 Cf. Castonguay, C.. Meaning and Existence in Mathematics. New York, 1972CrossRefGoogle Scholar. This book contains an extensive critique of Platonism.

11 Putnam, op. cit.

12 Cf. Benacerraf, “Mathematical Truth.“

13 This theorem usually takes the form: if x is the same size as a subset of y and y is the same size as a subset of x then x and y are of the same size. I am referring to the proofs which do not use the axiom of choice.

14 We still credit Newton and Leibnitz with the discovery of the calculus, and Cantor with the theory of transfinite numbers, although their original presentations were seriously flawed.

15 Views similar to mine have been discussed by C. Parsons, P. Benacerraf, Piaget and O. Chateaubriand.

16 I use the term object in scare quotes since it is no longer clear whether numbers, sets, etc. should count as objects. I think that we may need to move towards a notion of an object of a theory.

17 There are technical problems in defining identity conditions for patterns. E.G., should orientation count?

18 In the case at hand, the discrimination can be made via Godel's β-function. This suggests that where a theory does not quantify over functions, sets and relations defined on its objects ? as in elementary number theory ? the functions, sets and relations could be viewed as linguistic devices for describing operations on its pattern.

19 Categorical theories treat a single pattern. I am not sure about the others. In a sense group theory studies a single pattern ? the group pattern ? but not all groups are isomorphic. Here we have patterns within patterns just as we have the same conditional statement form in both the forms ‘if A then B' and ‘if A and B then C or E'.

20 Due to William J. Thomas.

21 I would like to thank Richard E. Grandy, Douglas Stalker, and Paul Ziff for their comments and criticisms of this paper. I am also indebted to Anthony Coyne for many discussions which prompted the ideas developed here.