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Wholes, Parts, and Infinite Collections

Published online by Cambridge University Press:  30 January 2009

P. O. Johnson
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Extract

In his book, The Principles of Mathematics, the young Bertrand Russell abandoned the common-sense notion that the whole must be greater than its part, and argued that wholes and their parts can be similar, e.g. where both are infinite series, the one being a sub-series of the other. He also rejected the popular view that the idea of an infinite number is self-contradictory, and that an infinite set or collection is an impossibility. In this paper, I intend to re-examine Russell's wisdom in doing both these things, and see if it might not have made more sense, and caused his enterprise fewer problems, if he had simply stuck to our commonplace ideas. To this end, I shall also be considering his treatment of certain paradoxes that he claims can only be resolved by the abandonment of the above notions, as well as certain others which his theories appear to have generated.

Type
Articles
Information
Philosophy , Volume 67 , Issue 261 , July 1992 , pp. 367 - 379
Copyright
Copyright © The Royal Institute of Philosophy 1992

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References

1 Russell, B. A. W., The Principles of Mathematics (London: George Allen and Unwin, 1985), Ch. XLIII, sections 339–341, pp. 357–360.Google Scholar

2 Op. cit. Ch. XVII, section 140, pp. 143–144.

3 Kline, M., Mathematics in Western Culture (USA: Oxford University Press, 1953), p. 444Google Scholar. Unfortunately, Kline gives no specific reference for this quotation.

4 Op. cit., 446.

5 Ibid.

6 Ibid.

7 Ibid., 449.

8 Russell, B. A. W., op. cit. Ch. XLIII, sections 339–341, pp. 357–360.Google Scholar

9 Op. cit. Ch. XLIII, section 340, p. 358.

10 Op. cit. Ch. XLIII, sections 340–341, pp. 358–360.

11 Ibid.

12 , B. A. W.Russell, My Philosophical Development (London: George Allen and Unwin, 1959), 58–59.Google Scholar

13 Op. cit. 59.

14 Russell, B. A. W., ‘Mathematical Logic as Based on The Theory of Types’, Logic and Knowledge: Essays 1901–1950 (London: George Allen and Unwin, 1956), 59–61.Google Scholar

15 Op. cit. 63.

16 Op. cit. 63 (footnote).

17 Earlier in The Principles of Mathematics, Russell in fact grants that the distinctive feature of a collection is that it is defined by the enumeration of its terms. See Ch. VI, section 71, p. 69.

18 Russell, B. A. W., My Philosophical Development, 58.Google Scholar

19 Ibid.