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Wittgenstein pp 83-111 | Cite as

Mathematics: An Anthropological Phenomenon

  • David Bloor
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Part of the Theoretical Traditions in the Social Sciences book series

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. 1.
    Cf. OC, 47.Google Scholar
  2. 2.
    See, for example, B. Russell, Introduction to Mathematical Philosophy, London, Allen & Unwin, 1919Google Scholar
  3. G. Frege, The Foundations of Arithmetic, trans. J. Austin, Oxford, Blackwell, 2nd rev. edn, 1959Google Scholar
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  5. 3.
    Cf. LFM, pp.43 and 271.Google Scholar
  6. 4.
    G. Hardy, A Mathematician’s Apology, Cambridge, Cambridge University Press, 1967 (first printed 1940) pp.123–4.Google Scholar
  7. 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
  8. 6.
    Cf. RFM, I, 130.Google Scholar
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    D. Bloor, ‘Wittgenstein and Mannheim in the Sociology of Mathematics’, Studies in the History and Philosophy of Science, vol.4, no.2, 1973, pp.173–91.CrossRefGoogle Scholar
  10. 8.
    ‘However queer it sounds, the further expansion of an irrational number is a further expansion of mathematics’ (RFM, IV, 9).Google Scholar
  11. 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 RFM he said the move was ‘as it were from the line into a surrounding plain’ (RFM, IV, 11).Google Scholar
  12. 10.
    In Wittgenstein’s text the figures are not numbered.Google Scholar
  13. 11.
    Cf. LFM, p.98.Google Scholar
  14. 12.
    B. Russell, Portraits from Memory and Other Essays, London, Allen & Unwin, 1956, p.116.Google Scholar
  15. 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
  16. 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
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    C. Boyer, The History of the Calculus and its Conceptual Development, New York, Dover, 1959, pp.143–4.Google Scholar
  18. 16.
    Ibid, p.143.Google Scholar
  19. 17.
    Ibid, p.143.Google Scholar
  20. 18.
    The quotation continues, ‘It does not establish that they are there; they do not exist until it makes them.’Google Scholar
  21. 19.
    Throughout the remainder of this section I will be following the valuable historical work of Joan Richards J. Richards, ‘The Reception of a Mathematical Theory: Non-Euclidean Geometry in England, 1868–1883’, in B. Barnes and S. Shapin (eds), Natural Order: Historical Studies of Scientific Culture, London, Sage, 1979, pp.143–66.Google Scholar
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  26. 23.
    Quoted by Richards, ‘Reception of a Mathematical Theory’, p.150, from Whewell’s Of a Liberal Education in General.Google Scholar
  27. 24.
    Quoted in Richards, ‘Reception of a Mathematical Theory’, p.154, from H. Helmholtz, ‘The Origin and Meaning of Geometrical Axioms’, Mind, vol.1, 1876, pp.301–21 and 304.CrossRefGoogle Scholar
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    Richards, ‘Reception of a Mathematical Theory’, p.159, from W. Jevons, ‘Helmholtz on the Axioms of Geometry’, Nature, vol.4, 1871, pp.481–2.CrossRefGoogle Scholar
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    W. Cayley, Presidential Address, Report of the British Association for the Advancement of Science, 1883, pp.3-37.Google Scholar
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    Quoted in Richards, ‘Reception of a Mathematical Theory’, p.159, from Cayley, Report of the British Association, p.9.Google Scholar
  31. 28.
    Quoted in Richards, ‘Reception of a Mathematical Theory’, p.162.Google Scholar
  32. 29.
    The point about conceivability and consistency proofs is made by Richards, ibid, pp.154-5.Google Scholar
  33. 30.
    ‘One would like to say: the proof changes the grammar of our language, changes our concepts. It makes new connections, and it creates the concept of those connections’ (RFM, II, 31).Google Scholar
  34. 31.
    This is assuming that the relation between their variables is one of linear regression.Google Scholar
  35. 32.
    K. Pearson, ‘Notes on the History of Correlation’, in E. Pearson and M. Kendall (eds), Studies in the History of Statistics and Probability, London, Griffin, 1970, pp.185–205, first published in Biometrika in 1920.Google Scholar
  36. 33.
    Strictly speaking the variables in equation (5) refer to errors, not measurements.Google Scholar
  37. 34.
    Pearson, ‘Notes on the History of Correlation’, p.188. Pearson is here quoting his own words from 1895.Google Scholar
  38. 35.
    Ibid, p.191.Google Scholar
  39. 36.
    Ibid, p. 187.Google Scholar
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    Ibid, p.191.Google Scholar
  41. 38.
    For a clear account of these matters, see D. MacKenzie, Statistics in Britain, 1865–1930. The Social Construction of Scientific Knowledge, Edinburgh, Edinburgh University Press, 1981, esp.ch.3 and appendices 3 and 4. I have greatly benefited from MacKenzie’s book and have relied on it throughout the whole of the present section.Google Scholar
  42. 39.
    Ibid, esp.p.65.Google Scholar
  43. 40.
    Ibid, p.71. The suggestion here is that Bravais and Galton were working within different ‘paradigms’ in Kuhn’s sense. Bravais was doing ‘normal science’ within the tradition of error-theory.Google Scholar
  44. 41.
    ‘Bravais... remained blind to the stupendous idea in whose vicinity his mind was hovering... he might, with one leap of creative imagination, have pounced squarely upon this conception.’ H. Walker, ‘The Relation of Plana and Bravais to the Theory of Correlation’, Isis, vol.10, 1928, pp.466–84, p.481.CrossRefGoogle Scholar
  45. 42.
    See, for example, the interesting discussions in: K. Mannheim, ‘On the Interpretation of “Weltanschauung”’, in Essays on the Sociology of Knowledge, London, Routledge & Kegan Paul, 1951, ch.II, esp. pp.55–63Google Scholar
  46. H. Garfinkel, Studies in Ethnomethodology, Englewood Cliffs, Prentice-Hall, 1967, p.40.Google Scholar
  47. 43.
    Pearson, ‘Notes on the History of Correlation’, p.189.Google Scholar
  48. 44.
    See MacKenzie, Statistics in Britain, ch.3. To make sense of these facts MacKenzie appeals to the notion of social interest. Eugenics, he argues, represented a middle-class ideology. The explanation has been refined in a valuable paper by Searle, who increases the magnification and looks at which segments of the professional middle class resisted eugenics. It transpires that it was rejected by those, like social workers, whose interests were not helped by Galton’s programme. See G. Searle, ‘Eugenics and Class’, in C. Webster (ed.), Biology, Medicine and Society, 1840–1940, Cambridge, Cambridge University Press, 1981, pp.217–42.Google Scholar

Copyright information

© David Bloor 1983

Authors and Affiliations

  • David Bloor
    • 1
  1. 1.University of EdinburghUK

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