¹ Lakatos, Imré, Proofs and refutations: the logic of mathematical discovery, Cambridge, 1976.CrossRefGoogle Scholar

² Douglas, Mary, Natural symbols: explorations in cosmology, Harmondsworth, 1973.Google Scholar

³ Op. cit. (1), pp. 74–5.Google Scholar

⁴ Ibid. pp. 139, 144–6.

⁵ Ibid. chapter II.

⁶ Op. cit. (1), p. 34.Google Scholar

⁷ Ibid. p. 36.

⁸ Ibid. p. 13.

⁹ Ibid. p. 15.

¹⁰ Ibid. pp. 15–21.

¹¹ This technical term is used here in the sense given it by Hesse, Mary in her The structure of scientific inference, London, 1974, chapter VIII.Google Scholar

¹² Op. cit. (1), p. 87.Google Scholar

¹³ Ibid., p. 151.

¹⁴ Ibid., p. 86.

¹⁵ Ibid., p. 93.

¹⁶ Ibid., p. 102.

¹⁷ My interpretation of Lakatos's ideas has been influenced by the similarity that they bear to Mary Hesse's general account of scientific concepts given in her op. cit. (11). The negotiability of all predicates or classifications is central to her ‘network model’. Lakatos's rejection of a perfectly understood vocabulary of simple terms is essentially the same as Hesse's rejection of the pure ‘Observation language’ of empiricism. The general structural features of our network of knowledge, Hesse's ‘coherence conditions’, are at least in part provided by Lakatos's strategies for responding to counter examples.

To see Lakatos as arguing that all classifications and hence all theorems and proofs are negotiable is a reading of Lakatos that has been challenged by, of all people, his own editors. This important matter is discussed in the appendix at the end of this paper.

¹⁸ Op. cit. (1), p. 29.Google Scholar

¹⁹ The classic statement is Durkheim, E. and Mauss, M., Primitive classification, London, 1963Google Scholar (first published 1903). For two detailed studies in this tradition see Bulmer, R. ‘Why is the cassowary not a bird? A problem of zoological taxonomy among the Karam of the New Guinea highlands’, Man, 1967, 2, 5–25CrossRefGoogle Scholar, and Tambiah, S. J., ‘Animals are good to think and good to prohibit’, Ethnology, 1969, 8, 424–59.CrossRefGoogle Scholar

²⁰ R. Horton argues that in primitive societies social structures provide models for nature because social knowledge is the most stable intellectual resource available. See his ‘African traditional thought and western science’, Africa, 1967, 37, 50–71, 155–87.Google Scholar For a mathematical example in which the social is appealed to as a metaphor, see Forman, Paul, ‘Weimar culture, causality, and quantum theory, 1918–1927: adaptation by German physicists and mathematicians to a hostile intellectual environment’, Historical studies in the physical sciences, 1971, 3, 1–115.CrossRefGoogle Scholar Section II.4 on intuitionism vividly illustrates the way images of a cultural crisis were used to characterize and create a sense of mathematical crisis. This episode can also be read as an example of concept-stretching, where a deliberate restriction in the scope of accepted mathematical procedures is advocated in order to produce counter-examples and anomalies where previously none had existed.

²¹ The original theory, that the abominations were category-violators was proposed by Douglas, Mary in Purity and danger: an analysis of concepts of pollution and taboo, London, 1966.CrossRefGoogle Scholar A developed version of the theory which describes the historical circumstances which gave special significance to eating pork is to be found in her Implicit meanings: essays in anthropology, London, 1975Google Scholar, chapter XVII.

²² See Douglas, , op. cit. (2), p. 87.Google Scholar

²³ Op. cit. (1), p. 14.Google Scholar

²⁴ Ibid., p. 21.

²⁵ Ibid., p. 17.

²⁶ Ibid., p. 16.

²⁷ Ibid., p. 14.

²⁸ Ibid., p. 19.

²⁹ Ibid.

³⁰ Ibid., p. 30.

³¹ Ibid., p. 24.

³² Ibid., p. 39.

³³ Ibid., p. 26.

³⁴ Ibid., p. 24.

³⁵ Ibid., p. 36.

³⁶ Ibid., pp. 79, 97.

³⁷ Ibid., p. 81.

³⁸ Ibid., p. 75.

³⁹ Ibid., p. 23.

⁴⁰ Ibid, pp. 4, 5, 37.

⁴¹ Ibid., p. 68.

⁴² Ibid.

⁴³ Ibid., pp. 30, 37.

⁴⁴ These dimensions were first introduced in Douglas, , op. cit. (2)Google Scholar, chapters IV and VI. The form of the diagram used on p. 129 will be followed here rather than the earlier, somewhat more complicated, figures.

⁴⁵ This theory has been applied to interview data from contemporary scientists by C. Bloor and D. Bloor ‘Does grid and group apply to industrial scientists? An empirical test of Mary Douglas's theory’ (forthcoming, 1978. Paper read at Professor Douglas's seminar, University College, London, 23 November 1976). Here empirically-based grid, group ratings are constructed for individual scientists, and checked against their attitudes to science. This provides a detailed illustration of the concepts of the grid, group theory in action. The most detailed statement of the various ‘cosmologies’ generated by different grid, group positions is to be found in Douglas, Mary, Cultural bias (Occasional papers of the Royal Anthropological Institute of Great Britain and Ireland, No. 34), London, 1978.Google Scholar

⁴⁶ Op. cit. (1), p. 84.Google Scholar

⁴⁷ The biologist Bateson used to say that plants were not matter, but systems through which matter passed; cf. Coleman, W. ‘Bateson and chromosomes: conservative thought in science’, Centaurus, 1970, 15, 228–314.CrossRefGoogle Scholar I have used this striking idea in the above paragraph. For a valuable discussion of ‘structural’ explanations in sociology, and a comparison with such explanations in other branches of science, see Barnes, S. B.Interests and the growth of knowledge, London, 1977, chapter III.Google Scholar

⁴⁸ Op. cit. (1), p. 48.Google Scholar

⁴⁹ Ibid., p. 55.

⁵⁰ Ibid., p. 81.

⁵¹ Ibid., p. 131.

⁵² Ibid., p. 136.

⁵³ Ibid.

⁵⁴ Turner, R. Steven, ‘The growth of professorial research in Prussia, 1818–1848—causes and context’, Historical studies in the physical sciences, 1971, 3, 137–82CrossRefGoogle Scholar; ‘University reformers and professorial scholarship in Germany, 1760–1806’, in Stone, L. (ed.), The university in society, Oxford, 1975, ii, 495–531.Google Scholar

⁵⁵ Turner's account may be compared with Ben-David, J.'s earlier treatments of the same episode, summarized in The scientist's role in society: a comparitive study, Englewood Cliffs, 1971, chapter VII.Google Scholar The two accounts agree on the main point that a competitive system was established around the middle of the nineteenth century.

⁵⁶ Poggendorff, J. C. (ed.), Biographisch—literarisches Handwörterbuch zur Geschichte der exakten Wissenschaften, Leipzig, 1863, ii. 896–7.Google Scholar

⁵⁷ See for example the description of the ‘Big-men’ societies of New Guinea in Mary Douglas, op. cit. (2).

⁵⁸ For example Frankel, Eugene in ‘Corpuscular optics and the wave theory of light: the science and politics of a revolution in physics’, Social studies of science, 1976, 6, 141–84CrossRefGoogle Scholar, describes how a power struggle between supporters and opponents of Laplace in the Paris Académie des Sciences precipitated a crisis in the corpuscular theory of optics. His account would appear to fit well with the description of monster-barring versus theorem-barring in high group, low grid communities. Another example is provided by the same institution a little later, only this time the monster-barrers won. This was when Pasteur disposed of the ‘incompetent’ experiments of Pouchet who believed in spontaneous generation. See Farley, J. and Geison, G., ‘Science, politics and spontaneous generation in nineteenth-century France: the Pasteur-Pouchet debate’, Bulletin of the history of medicine, 1974, 48, 161–98.Google ScholarPubMed

⁵⁹ This does not mean that the theory can never contribute towards such an understanding, or that such subjects fall outside the scope of sociology; far from it. Proof procedures may often arise by an extension of existing ones that have become customary, or they may be explained by social interests highlighting empirical processes which may be made the basis of new procedures. For example see the accounts of how eugenic concerns influenced the growth of Galton's and Pearson's statistical ideas in Cowan, Ruth S., ‘Francis Galton's statistical ideas: the influence of eugenics’, Isis, 1972, 63, 509–28CrossRefGoogle ScholarPubMed, and MacKenzie, D., ‘The development of statistical theory in Britain, 1865–1925: a historical and sociological perspective’, University of Edinburgh PhD thesis, 1978.Google Scholar

⁶⁰ Notice for instance the lack of unifying theory in Barnes, S. B., Scientific knowledge and sociological theory, London, 1974Google Scholar, and Bloor, D., Knowledge and social imagery, London, 1976.Google Scholar

⁶¹ The problems I have in mind are, for example: how do the concepts of grid and group relate to the idea of role? Can there really be a unique grid, group rating for an individual as he moves from role to role? Or take the case of the followers of Weierstrass represented as low grid, low group. They were put here because of their dialectical methodology and their initial rejection of monster-barring, etc. But their attitude to counterexamples produced later by Cantor's work was precisely to resort to monster-barring (Lakatos, , op. cit. (1), p. 50)Google Scholar. This subtle pattern of different methodologies being adopted in different circumstances, governed by the varying demands of expediency, is a problem which awaits the theory. Cf. also the way Jonquières used ‘monster-barring against cavities and tunnels but monster-adjustment against crested cubes and star-polyhedra’, (Ibid., p. 38). But only those who want to see the theory fail will assume at this stage that these problems are refuting instances.

⁶² All page references in the appendix are to Lakatos, , op. cit. (1).Google Scholar

⁶³ Russell, Bertrand, Mysticism and logic, London, 1963 (first published 1910), p. 59.Google Scholar

⁶⁴ In 1935 Tarski posed the problem of how to demarcate logical words from others. Popper thought that he had solved the problem in 1947, but is now of the opinion that he failed, and is sceptical about future success. Cf. Popper, Karl, ‘Logic without assumptions’, Proceedings of the Aristotelian Society, 1947, 47, 251–92CrossRefGoogle Scholar, and ‘Replies to my critics’ in Schilpp, P. A. (ed.), The philosophy of Karl Popper, La Salle, 1974, p. 1096.Google Scholar The fact is that logical words are stretchable. For example they had to be stretched to cope with the introduction of the truth table method.

⁶⁵ For a clear description of ‘if-thenism’ see Musgrave, Alan, ‘Logicism revisited’, British journal for the philosophy of science, 1977, 28, 99–127.CrossRefGoogle Scholar On p. 123 Musgrave suggests that Lakatos's account of mathematics rests on a naive Platonism.

⁶⁶ J. S. Mill's despised ‘psychologistic’ theory of mathematics has recently beeen defended and developed in Bloor, D., op. cit. (60), chapter V.Google Scholar

⁶⁷ For Popperian rhetoric against the threatening ‘tide of subjectivist relativism’ see Alan Musgrave, ‘The objectivism of Popper's epistemology’, in Schilpp, P. A. (ed.) op. cit. (64), p. 588.Google Scholar

⁶⁸ Instead of retreating back into formalism the editors might have helped rather than hindered the development of Lakatos's research programme by drawing attention to other work in logic and mathematics of a similar tendency. For instance, those impressed by Lakatos's achievement would have much to learn from Alfred Sidgwick's direct and forceful attack on formal logic in his The use of words in reasoning, London, 1901.Google Scholar He too had reached the conclusion that ‘To offer proof is to offer definite points of attack’ (p. 82). Sidgwick shares Lakatos's attitude to counterexamples: ‘The difference of method proposed is not that between attending only to rules and attending only to exceptions, but between avoiding and welcoming the discovery of exceptions to rules’ (p. 130). But if there is any book that deserves detailed comparison with Proofs and refutations it is Wittengstein's Remarks on the foundations of mathematics, Oxford, 1964.Google Scholar Both writers reject the usual view of mathematics having its 1foundations' in a trivial logical starting point; both are aware of the contrived and distorted effect of translating living and growing concepts into the impoverished apparatus of formal logic; both are masters of the art of spotting alternatives to steps in reasoning which look ‘compelling’, or conclusions which look ‘inevitable’; both have, in one sense of the word, a ‘finitist’ picture of mathematical proof; both are profound critics of the glib ‘Platonism’ or ‘Realism’ so prevalent in logic and mathematics. Indeed, what gives their work its force is the fact that both men are deeply responsive to the social dimension of knowledge.