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The Nature of Mathematics

Published online by Cambridge University Press:  14 March 2022

Harold Jeffreys
Harold Jeffreys*
Affiliation:
St. John's College, Cambridge, England
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Extract

There is a considerable divergence of opinion about the meaning of mathematics, and it is only with hesitation that I, as a theoretical physicist concerned mainly with geophysics, venture to discuss a matter that professional logicians differ about. Nevertheless I am concerned with the problems of the acquirement of knowledge by scientific methods, and the solutions of these problems involve pure mathematics, the validity of which one is usually willing to take for granted. But the logical schools are not so willing, and aim at demonstration. Logic must be true in all possible worlds, and all empirical propositions must be rigorously excluded. A scientist, on the other hand, is interested mainly in empirical propositions, and mathematics (with logic) is primarily a tool for investigating their relations. The proof of mathematics would therefore have interest for science, provided that the proof is sufficiently general to cover scientific requirements. If it is not, and the subject matter of the proof is restricted so that it fails to cover the scientific use of mathematics, then in its scientific application it is merely an analogy, and we are no better off than we should be if we simply took the validity of mathematics as it stands as a postulate.

Type
Research Article
Information
Philosophy of Science , Volume 5 , Issue 4 , October 1938 , pp. 434 - 451
Copyright
Copyright © Philosophy of Science Association 1938

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References

Notes

1 Introduction to Mathematical Philosophy, 1924, p. 18.

2 The Foundations of Mathematics, 1931, 60.

3 x ≠ y means “x is not identical with y.”

4 Wittgenstein, Tractatus Logico-Philosophicus, p. 139.

5 Loc. cit. p. 149.

6 A practical computer checks his work by working out his results by different methods. Thus induction is used to test the correctness of what is meant to be deduction.

7 Jeffreys, Proc. Aristot. Soc., 1937, 61-70; Nature, 141, 1938, 672-676, 716-719.

8 Jeffreys, Proc. Roy. Soc. A. 162, 1937, 487-491.

9 Wrinch and Jeffreys, Phil. Mag. 38, 1919, 715-731; Jeffreys, Scientific Inference. 1937, 218-222; Phil. Mag. 32, 1936, 337-359.

10 Proc. Roy. Soc. A., 165, 1938, 190-192.

11 Lewis, Mind 21, 1912, 522-531; 23, 1914, 240-247; Survey of Symbolic Logic, 1918. Moore, Proc. Aristot. Soc. 20, 1919, 40-62; see also D. J. Bronstein, Mind 45, 1936, 157-180.

12 ‘Deducible’ should, I think, be read as ‘deducible by a perfect reasoner.’ The actual reasoner, deductive or inductive, differs to some extent from perfection. The difference can be illustrated by the propositions ‘the properties of numbers entail Fermat's last theorem’; ‘Mult. Ax. entails C = N1.’