Wittgenstein pp 83-111 | Cite as

# Mathematics: An Anthropological Phenomenon

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## Abstract

Wittgenstein had been a schoolteacher, and when he spoke of the ‘foundations’ of mathematics he gave the word an appropriately robust meaning. Elementary instruction is the foundation for later learning. The foundations of mathematics are the psychological, social and empirical facts upon which the structure of knowledge is actually raised. A child has got to the bottom of arithmetic when he has learned how to apply numbers, ‘and that’s all there is to it’ (*LFM*, p.271).^{1} This is a deliberate rejection of the idea of ‘foundations’ as it is used by logicians such as Russell and Frege. They say that logic is the foundation of mathematics. They want to start with logically primitive concepts and exhibit an unbroken chain of deduction that leads up to mathematics proper.^{2} But, asks Wittgenstein, ‘Why hanker after logic?’ (*LFM*, p.271). Mathematics and logic are two language-games on a par with one another: both equally the product of instinct, training and convention. One is no more a foundation for the other ‘than the painted rock is the support of a painted tower’ (*RFM*, V,13).^{3} To look at mathematics through the eyes of the logician will be to overlook all the techniques of inference that are special to it, and all the differences between those techniques (*RFM*, IV,24).

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## Notes and References

- 1.Cf.
*OC*, 47.Google Scholar - 2.See, for example, B. Russell,
*Introduction to Mathematical Philosophy*, London, Allen & Unwin, 1919Google Scholar - G. Frege,
*The Foundations of Arithmetic*, trans. J. Austin, Oxford, Blackwell, 2nd rev. edn, 1959Google Scholar - S. Körner,
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*LFM*, pp.43 and 271.Google Scholar - 4.G. Hardy,
*A Mathematician’s Apology*, Cambridge, Cambridge University Press, 1967 (first printed 1940) pp.123–4.Google Scholar - 5.K. Gödel, ‘What is Cantor’s Continuum Problem’, in P. Benacerraf and H. Putnam (eds),
*Philosophy of Mathematics, Selected Readings*, Englewood Cliffs, Prentice-Hall, 1964, pp.271 and 272.Google Scholar - 6.Cf.
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*Studies in the History and Philosophy of Science*, vol.4, no.2, 1973, pp.173–91.CrossRefGoogle Scholar - 8.‘However queer it sounds, the further expansion of an irrational number is a further expansion of mathematics’ (
*RFM*, IV, 9).Google Scholar - 9.That this is the analogy Wittgenstein had in mind when he spoke of a move into another ‘dimension’ is clear from the fact that in
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*Portraits from Memory and Other Essays*, London, Allen & Unwin, 1956, p.116.Google Scholar - 13.G. Frege,
*The Foundations of Arithmetic*, trans. J. Austin, Oxford, Blackwell, 2nd rev. edn 1959, sections 7, 9, 10. Frege refers rhetorically to Mill’s ‘gingerbread or pebble arithmetic’ (p.vii).Google Scholar - 14.For a detailed discussion of how this theory can be supplemented by sociological considerations and thereby meet Frege’s requirements of objectivity, see D. Bloor,
*Knowledge and Social Imagery*, London, Routledge & Kegan Paul, 1976, esp. ch. 5, ‘A Naturalistic Approach to Mathematics’.Google Scholar - 15.C. Boyer,
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*Isis*, vol.10, 1928, pp.466–84, p.481.CrossRefGoogle Scholar - 42.See, for example, the interesting discussions in: K. Mannheim, ‘On the Interpretation of “Weltanschauung”’, in
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