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Locke and the Intuitionist Theory of Number

Published online by Cambridge University Press:  25 February 2009

Richard Aaron  and
Philip Walters
Richard Aaron
Affiliation:
University College of Wales, Aberystuyth
Philip Walters
Affiliation:
University College of Wales, Aberystuyth
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Extract

The Purpose of this paper is to ask how far Locke can be said to have anticipated modern theories of number, particularly the intuitionist theory of Brouwer and Heyting. It has in mind Mr Edward E. Dawson's statement that Locke's account of number was not merely ‘a good effort in his own day’ but that ‘what Locke had to say really was quite fundamental, and a good deal of modern mathematics assumes his position, either explicitly or implicitly’. Mr Dawson thinks that some of the central notions of the intuitionist theory are already present in the Essay Concerning Human Understanding, II, xvi, ‘Of Number’. We should like to examine the view.

Type
Articles
Information
Philosophy , Volume 40 , Issue 153 , July 1965 , pp. 197 - 206
Copyright
Copyright © The Royal Institute of Philosophy 1965

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References

page 197 note 1 Philosophical Quarterly, 1959, p. 302.Google Scholar

page 197 note 2 First edition; Second edition, p. 165-6.Google Scholar

page 197 note 3 p. 304.

page 198 note 1 II, xvi, 1.

page 198 note 2 II, xvi, 2.

page 198 note 3 ib., p. 304.

page 199 note 1 Heyting, A., Intuitionism, An Introduction, Amsterdam, 1965, p. 13.Google Scholar

page 200 note 1 It may be written formally thus: [{P(o)&(n){P(n):⊃P(n + l)]]⊃(n)P(n)].Google Scholar

page 200 note 2 cf. Smith, D. E.: Source Book in Mathematics, p. 73 (Proof of Corollary 12).Google Scholar

page 201 note 1 Eves, H.: An Introduction to the History of Mathematics, p. 261.Google Scholar

page 201 note 2 Oeuvres (ed. Tannery et, Henry) Vol. 2, p. 431, Fermat to Carcari, 1659.Google ScholarKneale, W. (Probability and Induction, p. 37) thinks the principle was formulated by Fermat, and no doubt has this passage in mind.Google Scholar

page 202 note 1 The Foundations of Arithmetic (tr. b y Austin, J. L.), § 31.Google Scholar

page 202 note 2 ib., §§ 37-9.

page 202 note 3 ib., § 39.

page 202 note 4 ib., § 39.

page 203 note 1 ib., 304.

page 203 note 2 intuitionism, p. 13.

page 203 note 3 ib.., p. 15.

page 205 note 1 intuitionism, p. 2.