Abstract
Lakatos liked to view his work in the philosophy of mathematics against the background of the traditional epistemological battle between dogmatists and sceptics. Dogmatists are those who hold A) that we can attain truth and B) that we can know that we have attained truth. Sceptics are those who hold A) that we cannot attain truth, or at least B) that we cannot know that we have attained truth. Lakatos himself represented a form of mitigated scepticism (often called critical fallibilism). Like the sceptics, he held A) that we cannot attain truth, or at least B) that we cannot know that we have attained truth, but he held in addition — and in this respect he distinguished himself from extreme sceptics — C) that we can improve our knowledge and know that we have improved it.
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Notes
Imre Lakatos, “Infinite Regress and the Foundations of Mathematics”, in: John Worrall and Gregory Currie (eds.) Imre Lakatos, Philosophical Papers,Vol. II, Cambridge [Etc.]: Cambridge University Press 1978, pp. 3–23. I will denote the paper as “Infinite Regress”
Blaise Pascal, Penseés, Texte établi par Léon Brunschvicg, Paris: Garnier-Flammarion 1976, pp. 172–173.
Richard H. Popkin, The History of Scepticism from Erasmus to Spinoza, Berkeley: University of California Press 1979.
Unlike many other sceptics, Hume also explicitly held sceptical views about mathematics. He held, in Popkin’s words, “that we can never have adequate grounds for maintaining that any mathematical proposition is true”, Richard A. Watson, James E. Force (eds.), The High Road to Pyrrhonism, papers by Richard Popkin, San Diego: Austin Hill Press, Inc. 1980, p. 143.
Karl R. Popper, Objective Knowledge, An Evolutionary Approach, revised edition, Oxford: Clarendon Press 1979, p. 99.
In J. Worrall and G. Currie (eds.) Imre Lakatos, Philosophical Papers, Vol. II, Cambridge University Press 1978, pp. 61–69.
Corfield rightly criticises Lakatos for underestimating the fertility and the importance of quasi-formal mathematics in David Corfield, “Assaying Lakatos’ Philosophy of Mathematics”, Studies in the History and Philosophy of Science 28, 1997, pp. 99 121.
Imre Lakatos, “Proofs and Refutations”, British Journal for the Philosophy of Science 14, 1963–64, pp. 1–25, 120–139, 221–245, 296–342. References are to the reprint in John Worrall and Elie Zahar (eds.), Imre Lakatos, Proofs and Refutations, The Logic of Mathematical Discovery, Cambridge [Etc.]: Cambridge University 1976.
Op. cit., Appendix 2, p. 146. There are some interesting remarks on the influence of Hegel on Lakatos in John Kadvany, “The Mathematical Present as History,” The Philosophical Forum 26, pp. 263–287. John Kadvany, Imre Lakatos and the Guises of Reason,Duke University Press, 2001, appeared after this paper was written. See also the paper by M. Motterlini in this volume.
I am referring here to the end of the original 1963–64 paper.
Teun Koetsier, Lakatos’ Philosophy of Mathematics, A Historical Approach, Amsterdam: North-Holland 1991.
Written in 1970–71 and published in 1974; references are to the reprint in J. Worrall and G. Currie (eds.) Imre Lakatos, Philosophical Papers, Vol. 11, Cambridge University Press 1978, pp. 139–167.
I will not go into the question of whether the MSRP is a satisfactory answer to an extreme sceptic. It is very difficult to satisfy an extreme sceptic. For a more general defence of critical fallibilism against the sceptics, I refer to Alan Musgrave, Common Sense, Science and Scepticism, A Historical Introduction to the Theory of Knowledge, Cambridge: Cambridge University Press 1993
Elsewhere, I modified the MSRP in order to be able to use it in a more precise way to describe some developments in mathematics. I called the result the Methodology of Mathematics Research Traditions. Cf. Koetsier, op. cit.. For a different view and a detailed discussion of the application of the MSRP to mathematics I refer, in particular, to Ladislav Kvasz’ contribution in this volume. For the general problem of the applicability of the MSRP to mathematics see also E. Glas, “Testing the Philosophy of Mathematics in the History of Mathematics”, Studies in the History and Philosophy of Science 20, 1989, pp. 115–131 and pp. 157–174, and D.D. Spalt, Vorn Mythos der Mathematischen Vernunft, Darmstadt: Wissenschaftliche Buchgesellschaft 1981.
Bertrand Russell, The Principles of Mathematics, Cambridge University Press, 1903 ( Second edition: London, 1937 ).
Bertrand Russell, My Philosophical Development, London: Unwin Books 1975, p. 162.
Paul J. Hager. Continuity and Change in the Development of Russell’s Philosophy, Dordrecht [Etc.]: Kluwer Academic Publishers 1994.
Here I follow Gödel’s 1944 paper “Russell’s Mathematical Logic”, reprinted in P. Benacerraf and H. Putnam, Philosophy of Mathematics, Selected Readings,Second Edition, Cambridge [Etc.]: Cambridge University Press 1983, pp. 447–485.
Alfred North Whitehead and Bertrand Russell, Principia Mathematica,Vol. I, Cambridge: University Press 1950 (reprint of the second 1925 edition), p. xiv.
My description of Brouwer’s programme is based upon Teun Koetsier and Jan van Mill, “General Topology, in Particular Dimension Theory, in the Netherlands: the Decisive Influence of Intuitionism”, in C.E. Aull and R. Lowell (eds.), Handbook of the History of General Topology, Vol. I, Dordrecht: Kluwer Academic Publishers 1997, pp. 135–180.
Luitzen Egbertus Jan Brouwer, Over de Grondslagen der Wiskunde,Amsterdam, 1907. The references will be to the English translation: L.E.J. Brouwer, “On the Foundation of Mathematics”, in: Arend Heyting (ed.), L.E.J. Brauwer Collected Works, Vol. 1, Philosophy and Foundations of Mathematics,Amsterdam [Etc.]: North-Holland 1976, pp. 15–101.
Walter P. van Stigt, Brouwer’s Intuitionism, Amsterdam: North Holland 1990, pp. 320–321.
D. Hilbert, “Über die Grundlagen der Geometrie”, Mathematische Annalen 56, 1902. Reprinted as an appendix in D. Hilbert, Grundlagen der Geometrie, 4th edition, Leipzig, 1913, pp. 163–218.
L.E.J. Brouwer, “De onbetrouwbaarheid der logische principes”, Tijdschrift voor Wijsbegeerte 2, 1908, pp. 152–158. References are to the English translation: L.E.J. Brouwer, “The unreliability of the logical principles”, in: L.E.J. Brouwer Collected Works, Vol. 1, Philosophy and Foundations of Mathematics,Amsterdam [Etc.]: North-Holland 1976, pp. 107–1 1 1.
An abbreviated version of the obituary is included as an appendix of Constance Reid, Hilbert, Berlin [Etc.]: Springer-Verlag 1970, pp. 245–283
David Hilbert, Gesammelte Abhandlungen,Zweiter Band, [Repr.] New York: Chelsea Publ. Corp. 1965, p. 401 (italics is mine T. K.).
Weyl wrote: “There could not have been a more complete break than the one dividing Hilbert’s last paper on the theory of number fields from his classical book, Grundlagen der Geometrie,published in 1899” (Reid, op. cit., p. 264). This statement does not take into account that in his number theoretic papers Hilbert does not deal with foundational matters. Grundlagen der Geometrie and the number theoretic papers have in common that the problem that Hilbert intends to solve is solved by considerations that are structural, and more abstract than many of his contemporaries were capable of exploiting. It is only natural that concentration on the foundations leads to a more explicit formulation of that structural point of view.
See p. 197 of Paul Bernays, “Hilberts Untersuchungen über die Grundlagen der Arithmetik”, in David Hilbert, Gesammelte Abhandlungen, Dritter Band, [Repr.] New York: Chelsea Publ. Corp. 1965, pp. 196–216.
J. Van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 18791931, Cambridge: Harvard University Press 1967, pp. 129–138.
Hilbert acted in different ways. The confrontation of the two mathematicians reached a climax in 1928 when Hilbert was ill and worried that after his death his fellow-editor of the Mathematische Annalen, Brouwer, would gain too much influence. Hilbert dismissed Brouwer from the board of the Annalen. For an extensive treatment of the ‘Crisis of the Mathematische Annalen’ we refer to D. van Dalen, “The War of the Frogs and the Mice, or the Crisis of the Mathematische Annalen”, The Mathematical Intelligenter 12, Number 4, 1990, pp. 17–31. A Lakatosian approach requires that one concentrates on the internal history. That is what I am doing in this paper. Brouwer’s dismissal as editor of the Annalen is part of the external history.
Stephen G. Simpson, “Partial Realisations of Hilbert’s Program”, The Journal of Symbolic Logic 53, 1988, pp. 349–363.
Russell, 1975, op. cit., p. 61.
An important book about the competition between pre-twentieth century mathematics and modern structuralist mathematics is Herbert Mehrtens, Moderne Sprache Mathematik, Frankfurt am Main: Suhrkamp 1990. Mehrtens describes the transition as a very complex process. From my point of view, he insufficiently emphasises the fact that, at heart, the structuralist programme came to dominate because of the large number of results it produced and the ease with which it incorporated other results.
Imre Lakatos, “A Renaissance of Empiricism”, in: J. Worrall and G. Currie (eds.) Imre Lakatos, Philosophical Papers, Vol. II, Cambridge University Press 1978, pp. 24–42.
David Hume, Dialogues Concerning Natural Religion (edited with commentary by Nelson Pike), Indianapolis and New York, 1970, p. 113 footnote.
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Koetsier, T. (2002). Lakatos’ Mitigated Scepticism in the Philosophy of Mathematics. In: Kampis, G., Kvasz, L., Stöltzner, M. (eds) Appraising Lakatos. Vienna Circle Institute Library, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0769-5_11
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