Assaying Lakatos's philosophy of mathematics

Studies in History and Philosophy of Science Part A

Volumn 28, Issue 1, 1997, Pages 99-121

  Corfield, David ^a  

^a UNIVERSITY OF LEEDS   (United Kingdom)

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EID: 0001802342     PISSN: 00393681     EISSN: 18792510     Source Type: Journal    
DOI: 10.1016/S0039-3681(96)00002-7     Document Type: Article

References (94)

1. I. Lakatos, Proofs and Refutations. The Logic of Mathematical Discovery (Cambridge, 1976).
2. W. V. Quine, Review of Proofs and Refutations, British Journal for the Philosophy of Science 28 (1977), 81-82.
3. P. Davis and R. Hersh, The Mathematical Experience (Boston: Birkhaüser, 1981), p. 347.
4. One attempt that should be mentioned here is M. Hallett, 'Toward a theory of mathematical research programmes', British Journal for the Philosophy of Science 30 (1979), 1 25, 135-159. I intend to explore the possibility of a theory of mathematical research programmes in a later paper.
5. Op. cit., note I, pp. 127-128.
6. Teun Koetsier has shown that even the material in the footnotes is historically inaccurate, T. Koetsier, Lakatos' Philosophy of Mathematics: A Historical Approach (Amsterdam: North-Holland, 1991), Sec. 1.3.3. This, however, has no bearing on the philosophical purpose of rational reconstruction. I thank one of the referees for clarifying this point.
7. E. Palmer, 'Lakatos's "internal history" as historiography', Perspectives on Science 1 (1993), 603-626; E. Glas, 'Mathematical Progress: Between Reason and Society', Zeitschrift für allgemeine Wissenschaftstheorie 24 (1993), 43-62, 235-256.
8. E. Palmer, 'Lakatos's "internal history" as historiography', Perspectives on Science 1 (1993), 603-626; E. Glas, 'Mathematical Progress: Between Reason and Society', Zeitschrift für allgemeine Wissenschaftstheorie 24 (1993), 43-62, 235-256.
9. I. Lakatos, Mathematics, Science and Epistemology, Philosophical Papers Vol. 2, J. Worrall and G. Currie (eds) (Cambridge, 1978), 93-103.
10. It is very easy to misread the paragraph on p. 96 in which he discusses this stage as identifying the Popperian method with the method of proofs and refutations. The editors falsely remark in a footnote that this stage is described in Chapters 1 and 2 of Proofs and Refutations, which comprise the bulk of the book. However, a close reading of this paragraph in conjunction with pp. 6 and 7 (especially the footnotes) of Proofs and Refutations shows that Lakatos intends us to think of the first stage as the production of a formula that works for a range of common polyhedra. Indeed, he tells us of Euler, prior to any proof, testing the formula against various prisms. Polya, who suggested this case study to Lakatos, used this process of guessing and testing as an example of inductive reasoning in mathematics.
11. The expression appears to be used synonymously with the 'method of proofs and refutations'.
12. Ibid., p. 96.
13. Ibid., p. 99.
14. Ibid., p. 68. The account Lakatos presents here runs against an indication he gives elsewhere that Proofs and Refutations describes a research programme, see I. Lakatos, Philosophical Papers, Vol. 1, J. Worrall and G. Currie (eds) (Cambridge, 1978), p. 52n. In his article, op. cit., note 6, Glas rightly underlines the similarities between the methodology of proofs and refutations and the methodology of research programmes and would have the 'idea that the relationship V-E+F=2...expresses some fundamental feature' belong to the hard core (p. 49). This I would agree is the correct way to proceed and below I sketch the hard core of modern algebraic topology in terms of an aim, but it contradicts Lakatos who resorted instead to hidden lemmas become axioms as the constituents of the hard core.
15. Ibid., p. 68. The account Lakatos presents here runs against an indication he gives elsewhere that Proofs and Refutations describes a research programme, see I. Lakatos, Philosophical Papers, Vol. 1, J. Worrall and G. Currie (eds) (Cambridge, 1978), p. 52n. In his article, op. cit., note 6, Glas rightly underlines the similarities between the methodology of proofs and refutations and the methodology of research programmes and would have the 'idea that the relationship V-E+F=2...expresses some fundamental feature' belong to the hard core (p. 49). This I would agree is the correct way to proceed and below I sketch the hard core of modern algebraic topology in terms of an aim, but it contradicts Lakatos who resorted instead to hidden lemmas become axioms as the constituents of the hard core.
16. Ibid., p. 68. The account Lakatos presents here runs against an indication he gives elsewhere that Proofs and Refutations describes a research programme, see I. Lakatos, Philosophical Papers, Vol. 1, J. Worrall and G. Currie (eds) (Cambridge, 1978), p. 52n. In his article, op. cit., note 6, Glas rightly underlines the similarities between the methodology of proofs and refutations and the methodology of research programmes and would have the 'idea that the relationship V-E+F=2...expresses some fundamental feature' belong to the hard core (p. 49). This I would agree is the correct way to proceed and below I sketch the hard core of modern algebraic topology in terms of an aim, but it contradicts Lakatos who resorted instead to hidden lemmas become axioms as the constituents of the hard core.
17. M. Steiner, 'The Philosophy of Imré Lakatos', Journal of Philosophy 80 (1983), 502-521. S. Feferman, 'The Logic of Mathematical Discovery Vs the Logical Structure of Mathematics', in P. D. Asquith and H. E. Kyburg (eds), PSA 1978, Vol. 2 (1981), pp. 309-327.
18. M. Steiner, 'The Philosophy of Imré Lakatos', Journal of Philosophy 80 (1983), 502-521. S. Feferman, 'The Logic of Mathematical Discovery Vs the Logical Structure of Mathematics', in P. D. Asquith and H. E. Kyburg (eds), PSA 1978, Vol. 2 (1981), pp. 309-327.
19. Ibid., p. 310.
20. Ibid., p. 310.
21. M. Steiner, Mathematical Knowledge (Ithaca: Cornell University Press, 1975).
22. I. Lakatos, op. cit. (1976), p. 6n.
23. Ibid., p. 136.
24. I. Lakatos, op. cit. (1978), p. 66.
25. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology (Princeton University Press, 1952). Results announced in S. Eilenberg and N. Steenrod, 'Axiomatic Approach to Homology Theory', Proceedings of the National Academy of Sciences, U.S.A. 31 (1945), 177-180. Leaving aside problems with the idea of characterising a branch of mathematics in terms of its axioms, which I shall discuss later, it is worth briefly remarking that by 1952 homotopy theory had become just as important a part of algebraic topology as homology and cohomology theory, marking its difference by its failure to satisfy Eilenberg and Steenrod's 'Excision axiom'. In their book (p. 49) Eilenberg and Steenrod date the axiomatisation of the homotopy groups to a paper of 3. Milnor later published in 1956.
26. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology (Princeton University Press, 1952). Results announced in S. Eilenberg and N. Steenrod, 'Axiomatic Approach to Homology Theory', Proceedings of the National Academy of Sciences, U.S.A. 31 (1945), 177-180. Leaving aside problems with the idea of characterising a branch of mathematics in terms of its axioms, which I shall discuss later, it is worth briefly remarking that by 1952 homotopy theory had become just as important a part of algebraic topology as homology and cohomology theory, marking its difference by its failure to satisfy Eilenberg and Steenrod's 'Excision axiom'. In their book (p. 49) Eilenberg and Steenrod date the axiomatisation of the homotopy groups to a paper of 3. Milnor later published in 1956.
27. J. Dieudonné, A History of Algebraic and Differential Topology: 1900-1960 (Boston: Birkhaüser, 1989), p. 16.
28. shall follow Feferman's enumeration of his questions and criticisms from (i) to (x), op. cit., note 12, pp. 316-320.
29. I. Lakatos, op. cit. (1976), pp. 6-7.
30. I. Lakatos, op. cit. (1978), p. 70.
31. This is not quite correct. The proof given in Proofs and Refutations involves choosing coefficients for the chains from the field of integers mod 2, a technique introduced by Tietze only in 1908.
32. H. Poincaré, 'Analyse de ses travaux scientifiques', Acta Mathematica 38 (1921), 323.
33. H. Poincaré, 'Analysis situs', in his Oeuvres, Vol. 6 (Paris: Gauthier-Villars, 1953), pp. 193-288.
34. Op. cit., note 24, p. 323.
35. M. Kline, Mathematical Thought from Ancient to Modern Times (Oxford University Press, 1972), see pp. 655-665.
36. The concept of a multiple of a variety was left vague. It was to be thought of as a set of varieties slightly deformed from each other.
37. It was only proved much later that for differentiable manifolds this is no restriction.
38. Donald Gillies informs me that after his escape from Hungary to England, Lakatos made a careful study of Russell's writings and that it was there that he found the concept of 'quasi-empiricism'.
39. M. Steiner, The Philosophy of Imré Lakatos', Journal of Philosophy 80, pp. 502-521, see p. 505.
40. I. Lakatos, op. cit. (1978), p. 42.
41. M. Steiner, op. cit. (1983), p. 504.
42. I. Lakatos, op. cit. (1978), p. 118.
43. Ibid., p. 120.
44. Ibid., p. 116.
45. Ibid., p. 120.
46. Ibid., p. 122.
47. Ibid., p. 66.
48. See ibid., p.67: 'Does all this mean that proof in a formalized theory does not add anything to the certainty of the theorem involved? Not at all...if we manage to formalize a proof of our theorem within a formal system, we know that there will never be a counterexample to it which could be formalized within the system as long as the system is consistent...if formalization...conforms with some informal requirements, such as enough intuitive counter-examples being formalized in it and so on, we gain quite a lot in the value of proofs'.
49. M. Steiner, op. cit. (1983), p. 521.
50. I. Hacking, 'Lakatos's Philosophy of Science', British Journal for the Philosophy of Science 30, 381-410. Is this difficulty sufficient to account for the fact that we find such disagreements between reviewers of his work as 'Lakatos...espouses a correspondence theory of truth', M. Steiner, op. cit. (1983), p. 508, and 'Lakatos's problem is to provide a theory of objectivity without a represen-tational theory of truth', I. Hacking, op. cit. (1979), p. 384. Hacking does, however, go on to note (ibid., p.385n) a doubt expressed by Feferman about his claim that Lakatos is not primarily concerned with characterising knowledge by how well it represents reality. Steiner, meanwhile, draws support for his claim that Lakatos was a realist from the following quotation: 'I think that the bulk of logic and mathematics is God's doing and not human convention...But in consequence I am a fallibilist not only in science, but in mathematics and logic as well', I. Lakatos, op. cit. (1976), p. 127. But then, if he is right about Lakatos's realism, they have further grounds for agreement.
51. I. Hacking, 'Lakatos's Philosophy of Science', British Journal for the Philosophy of Science 30, 381-410. Is this difficulty sufficient to account for the fact that we find such disagreements between reviewers of his work as 'Lakatos...espouses a correspondence theory of truth', M. Steiner, op. cit. (1983), p. 508, and 'Lakatos's problem is to provide a theory of objectivity without a represen-tational theory of truth', I. Hacking, op. cit. (1979), p. 384. Hacking does, however, go on to note (ibid., p.385n) a doubt expressed by Feferman about his claim that Lakatos is not primarily concerned with characterising knowledge by how well it represents reality. Steiner, meanwhile, draws support for his claim that Lakatos was a realist from the following quotation: 'I think that the bulk of logic and mathematics is God's doing and not human convention...But in consequence I am a fallibilist not only in science, but in mathematics and logic as well', I. Lakatos, op. cit. (1976), p. 127. But then, if he is right about Lakatos's realism, they have further grounds for agreement.
52. I. Hacking, 'Lakatos's Philosophy of Science', British Journal for the Philosophy of Science 30, 381-410. Is this difficulty sufficient to account for the fact that we find such disagreements between reviewers of his work as 'Lakatos...espouses a correspondence theory of truth', M. Steiner, op. cit. (1983), p. 508, and 'Lakatos's problem is to provide a theory of objectivity without a represen-tational theory of truth', I. Hacking, op. cit. (1979), p. 384. Hacking does, however, go on to note (ibid., p.385n) a doubt expressed by Feferman about his claim that Lakatos is not primarily concerned with characterising knowledge by how well it represents reality. Steiner, meanwhile, draws support for his claim that Lakatos was a realist from the following quotation: 'I think that the bulk of logic and mathematics is God's doing and not human convention...But in consequence I am a fallibilist not only in science, but in mathematics and logic as well', I. Lakatos, op. cit. (1976), p. 127. But then, if he is right about Lakatos's realism, they have further grounds for agreement.
53. I. Hacking, 'Lakatos's Philosophy of Science', British Journal for the Philosophy of Science 30, 381-410. Is this difficulty sufficient to account for the fact that we find such disagreements between reviewers of his work as 'Lakatos...espouses a correspondence theory of truth', M. Steiner, op. cit. (1983), p. 508, and 'Lakatos's problem is to provide a theory of objectivity without a represen-tational theory of truth', I. Hacking, op. cit. (1979), p. 384. Hacking does, however, go on to note (ibid., p.385n) a doubt expressed by Feferman about his claim that Lakatos is not primarily concerned with characterising knowledge by how well it represents reality. Steiner, meanwhile, draws support for his claim that Lakatos was a realist from the following quotation: 'I think that the bulk of logic and mathematics is God's doing and not human convention...But in consequence I am a fallibilist not only in science, but in mathematics and logic as well', I. Lakatos, op. cit. (1976), p. 127. But then, if he is right about Lakatos's realism, they have further grounds for agreement.
54. I. Hacking, 'Lakatos's Philosophy of Science', British Journal for the Philosophy of Science 30, 381-410. Is this difficulty sufficient to account for the fact that we find such disagreements between reviewers of his work as 'Lakatos...espouses a correspondence theory of truth', M. Steiner, op. cit. (1983), p. 508, and 'Lakatos's problem is to provide a theory of objectivity without a represen-tational theory of truth', I. Hacking, op. cit. (1979), p. 384. Hacking does, however, go on to note (ibid., p.385n) a doubt expressed by Feferman about his claim that Lakatos is not primarily concerned with characterising knowledge by how well it represents reality. Steiner, meanwhile, draws support for his claim that Lakatos was a realist from the following quotation: 'I think that the bulk of logic and mathematics is God's doing and not human convention...But in consequence I am a fallibilist not only in science, but in mathematics and logic as well', I. Lakatos, op. cit. (1976), p. 127. But then, if he is right about Lakatos's realism, they have further grounds for agreement.
55. S. MacLane, 'Concepts and Categories in Perspective', in P. Duren (ed.), A Century of Mathematics in America, Part I (Rhode Island: American Mathematical Society, 1988), p. 325.
56. S. MacLane, Mathematics: Form and Function (New York: Springer-Verlag, 1986), p. 378.
57. I. Lakatos, op. cit. (1978), p. 66.
58. Ibid., p. 68.
59. Ibid., p. 96. ((1978), p. 68).
60. J. van Mill and G. Reed (eds), Open Problems in Topology (Amsterdam: North Holland-Elsevier, 1990).
61. I. Lakatos, op. cit. (1978), p. 100.
62. Ibid., p. 62.
63. W. Thurston, On Proof and Progress in Mathematics', Bulletin of the Amererican Mathematical Society 30 (1994), 164.
64. P. Johnstone, Topos Theory (London: Academic Press, 1977), p. xvi.
65. In point of fact, one of Lawvere and Tierney's axioms was later found to be redundant.
66. G. Kreisel and A. MacIntyre, 'Constructive Logic Versus Algebraization I'. in A. Troelstra and D. van Dalen (eds), The L. E. J. Brouwer Centenary Colloquium (Amsterdam: North-Holland (1982), p. 233.
67. Ibid., p. 232.
68. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology (Princeton University Press, 1952), pp. x-xi.
69. H. Weyl, 'A Half Century of Mathematics', American Mathematical Monthly 58, printed in Weyls Gesammelte Abhandlungen, Band IV (Springer-Verlag, 1951), p. 465.
70. H. Weyl, 'A Half Century of Mathematics', American Mathematical Monthly 58, printed in Weyls Gesammelte Abhandlungen, Band IV (Springer-Verlag, 1951), p. 465.
71. Op. cit., note 3, p. 354.
72. I. Lakatos, op. cit. (1976), p. 138.
73. Indeed, we find Hersh comment elsewhere: 'we can see the reason for the "working mathematician's" uneasy oscillation between formalism and Platonism. Our inherited and unexamined philosophical dogma is that mathematical truth should possess absolute certainty. Our actual experience in mathematical work offers uncertainty in plenty'. R. Hersh 'Some Proposals for Reviving the Philosophy of Mathematics', Advances in Mathematics 31 (1979), 38, author's emphasis.
74. P. Davis and R. Hersh, op cit. (1981), p. 354.
75. Ibid., p. 347
76. I. Lakatos, op. cit. (1978), pp. 62-63.
77. We could attempt to make a distinction between the use of the terms 'axiomatisation' and 'formalisation', but need not in this discussion as Lakatos claims to use the terms interchangeably (see ibid., p. 67).
78. Other commentators have noted this aspect of Lakatos's thought. Eduard Glas describes Felix Klein's programme as 'the best example of progress through generalization and consolidation' likening it to Lakatos's methodology, but he then adds that 'proofs and refutations typically belong to propositions, and Klein was not primarily concerned with the proofs of propositions but with the development of models', E. Glas, Testing the Philosophy of Mathematics in the History of Mathematics, Part II: The Similarity Between Mathematical and Scientific growth of Knowledge', Studies in the History and Philosophy of Science 20 (1989), 157-174, see p. 168. Kitcher's view of explanatory progress leads him to recognise 'the need to break away from concentration on accepted statements (a feature of logical empiricism that survives in Lakatos and Laudan) and to focus on the ways in which statements are used in answering questions', P. Kiteher, The Advancement of Science (Oxford University Press, 1993), p. 112n (author's emphasis). There was always the fear in Lakatos's mind that too many concessions on this issue would lead to the introduction of something like Polanyi's 'know-how', which Lakatos felt to be irrelevant to his programme of explaining the development of theories in the Third World.
79. Other commentators have noted this aspect of Lakatos's thought. Eduard Glas describes Felix Klein's programme as 'the best example of progress through generalization and consolidation' likening it to Lakatos's methodology, but he then adds that 'proofs and refutations typically belong to propositions, and Klein was not primarily concerned with the proofs of propositions but with the development of models', E. Glas, Testing the Philosophy of Mathematics in the History of Mathematics, Part II: The Similarity Between Mathematical and Scientific growth of Knowledge', Studies in the History and Philosophy of Science 20 (1989), 157-174, see p. 168. Kitcher's view of explanatory progress leads him to recognise 'the need to break away from concentration on accepted statements (a feature of logical empiricism that survives in Lakatos and Laudan) and to focus on the ways in which statements are used in answering questions', P. Kiteher, The Advancement of Science (Oxford University Press, 1993), p. 112n (author's emphasis). There was always the fear in Lakatos's mind that too many concessions on this issue would lead to the introduction of something like Polanyi's 'know-how', which Lakatos felt to be irrelevant to his programme of explaining the development of theories in the Third World.
80. Op. cit., note 56, p. 12.
81. This is one of the interesting points made by Kreisel and Maclntyre, op. cit. (1982). They recommend (p. 236) the perusal of essay reviews or lectures to learned societies by mathematicians as an antidote to misconceptions on this score.
82. R. Hersh, 'Mathematics has a Front and a Back', Synthese 88 (1991), 127-133, see pp. 131-132.
83. S. MacLane, Mathematics: Form and Funcation (New York: Springer-Verlag, 1986), p. 441. A similar sentiment to the one lying behind this avoidance of the word 'true' is expressed by another mathematician, Gian-Carlo Rota: 'We shall establish a distinction between the concept of truth as it is used by mathematicians and another, superficially similar concept, also unfortunately denoted by the work 'truth', which is used in mathematical logic and has been widely adopted in analytic philosophy'.
84. A. Jaffe and F. Quinn, ' "Theoretical Mathematics": Towards a Cultural Synthesis of Mathematics and Theoretical Physics', Bulletin of the American Mathematical Society 29 (1993), 1-13.
85. See ibid., p. 8.
86. Op. cit., note 19, p. 36.
87. A. Jaffe and F. Quinn, op. cit. (1993), p. 7.
88. J. Dieudonné, op. cit. (1989), p. 15.
89. It seems fair to say that the geometric branches of mathematics which have taken longest to 'settle down' are the ones most likely to bring about reconceptualisations of mathematics. It was algebraic topology which led to the formulation of the first ideas in category theory and it was in algebraic geometry, for which Dieudonné dates the advent of rigour as late as 1950, that the notion of a topos first emerged.
90. P. Alexandrov 'Poincaré and Topology', Russian Mathematics Surveys (1972), reprinted in F. Browder (ed.), The Mathematical Heritage of Henri Poincaré, Vol. 2 (Rhode Island: American Mathematical Society, 1983), pp. 245-255.
91. P. Alexandrov 'Poincaré and Topology', Russian Mathematics Surveys (1972), reprinted in F. Browder (ed.), The Mathematical Heritage of Henri Poincaré, Vol. 2 (Rhode Island: American Mathematical Society, 1983), pp. 245-255.
92. C. C. Gillispie (ed.), Dictionary of Scientific Biography (New York: Charles Scribner's Sons, 1975), p. 52.
93. H. Breger, 'A Restoration that Failed: Paul Finsler's theory of sets', in D. Gillies (ed.), Revolutions in Mathematics (Oxford University Press, 1992), pp. 248-264.
94. Ibid., p. 256.