Abstract
Although Lakatos’Proofs and Refutations (henceforth PR) is one of the classics of twentieth century philosophy of mathematics, few attempts have been made to build on its ideas.1 Rather, quite often PR has become an object of respectful reference and detailed exegetical efforts. Of course, Lakatos is credited for having taught philosophers that philosophy of mathematics has to take into account the history and practice of mathematics. This orientation towards the history and practice of the discipline certainly represents progress beyond the ahistorical traditions of logicism, formalism, platonism, and intuitionism. It should help to get rid of the most pernicious vice with which the philosophy of mathematics is plagued; to wit, elementarism, which considers elementary arithmetics of natural numbers or Euclidean geometry as typical for all of mathematics. I think, however, that there is more to learn from Lakatos than just a general orientation towards the history and practice of real-life mathematics. As I want to show in this paper, these insights concern the role of inventing and varying mathematical concepts. More precisely, I want to use Lakatos’ ideas of “concept-formation” and “concept-stretching” to sketch an evolutionary theory of mathematical knowledge, which takes axiomatic variation of concepts as the fundamental driving force of the ongoing evolution of mathematics (PR, 83ff.).2
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References
Bloor, D., 1991 (1976) Knowledge and Social Imagery, Chicago, The University of Chicago Press.
Brandon, R.N. (1996) Concepts and Methods in Evolutionary Biology, Cambridge, Cambridge University Press.
Cohen, H. 1902 (1977) Die Logik der reinen Erkenntnis (Hermann Cohen Werke 6 ), Hildesheim and New York, Olms.
Corfield, D. (1997) Assaying Lakatos’s Philosophy of Mathematics, Studies in History and Philosophy of Science 28, 99 121.
Edalat, A. (1997) Domains for Computation in Mathematics, Physics and Exact Real Arithmetics, The Bulletin of Symbolic Logic 3, 401–452.
Elstrodt, J. (1996) MaIf-und Integrationstheorie, Berlin, Springer.
Ernest, P. (1997) The Legacy of Lakatos: Reconceptualizing the Philosophy of Mathematics, Philosophie Mathematica 5, 116–134.
Fine, A. (1981) Conceptual Change in Mathematics and Science: Lakatos’ Stretching Refined, Proceedings of the 1978 Biennial Meeting of the Philosophy of Science Association (PSA), vol. 2, 328 341.
Hamilton, W.R. (1967) The Mathematical Papers of Sir William Rowan Hamilton, Vol. 3, Algebra, Cambridge, Cambridge University Press.
Hahn, Hans, 1997 (1914) Über Annäherung an Lebesguesche Integrale durch Riemannsche Summen, in Gesammelte Abhandlungen 3, ed. by L. Schmetterer and K. Sigmund, Wien, Springer, 17–48.
Hahn, Hans, 1997 (1915) Über eine Verallgemeinerung der Riemannschen Integralfunktion, inGesammelte Abhandlungen 3, ed. by L. Schmetterer and K. Sigmund, Wien, Springer, 49–64. Hawkins, Th. (1970) Lebesgue’s Theory of Integration, Its Origins and Developments, Madison,The University of Wiscons in Press.
Henstock, R. (1991) The General Theory of Integration,Oxford, Clarendon Press.156 THOMAS MORMANN
Koetsier, T. (1991) Lakatos’ Philosophy of Mathematics. A Historical Approach, Amsterdam, North-Holland.
Lakatos, I. (1976) Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge, Cambridge University Press.
Lakatos, I. (1978) Philosophical Papers, Volumes 1 and 2, Cambridge, Cambridge University Press.
Larvor, B.P. (1997) Lakatos as Historian of Mathematics, Philosophia Mathematica (3) 5, 42–64. Natorp, P. (1910) Die logischen Grundlagen der exakten Wissenschaften, Leipzig und Berlin, Teubner.
Pickering, A. (1995) The Mangle of Practice, Time, Agency, and Science, Chicago, The University of Chicago Press.
Rota, G.C. (1997) Indiscrete Thoughts, ed. by F. Palombi, Boston, Birkhäuser.
Toulmin, St. (1972) Human Understanding I. General Introduction and Part I: The Collective Use and Evolution of Concepts, Princeton, Princeton University Press.
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Mormann, T. (2002). Towards an Evolutionary Account of Conceptual Change in Mathematics: Proofs and Refutations and the Axiomatic Variation of Concepts. 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_9
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