Abstract
The arguments in this book depend more on the study of history than on the study of logic; logic plays a role, but it is subordinate or subsidiary. Philosophers are now used to the idea that history is central to philosophy of science, but its pertinence to the philosophy of mathematics seems to need renewed defense. Thus I rehearse the helpful argument of the philosopher of history, W. B. Gallie, who shows that there can be no Ideal Chronicle, which in turn helps me to contest Philip Kitcher’s early and influential position in The Nature of Mathematical Knowledge, where he invokes the history of mathematics but in a manner that finally leaves mathematics once again ahistorical. Thus I turn to the French philosopher of mathematics Jean Cavaillès whose sense of history was more refined, make some remarks about reference, and take up briefly Wiles’ strategy in his celebrated proof of Fermat’s Last Theorem, in order to carry out a thought experiment involving the ‘Math Genie’ that shows that proofs are embedded in history, like all human actions.
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References
Aristotle. (1947). Introduction to Aristotle. R. McKeon (Ed.). New York: Random House.
Benacerraf, P. (1965). What numbers could not be. Philosophical review, 74(1), 47–73.
Breger, H. (2000). Tacit knowledge and mathematical progress. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 221–230). Dordrecht: Kluwer.
Brown, J. (2008). Philosophy of mathematics. Oxford: Routledge.
Cavaillès, J. (1937). Réflections sur le fondement des mathématiques. In Travaux du IVe Congrès international de philosophie, VI/535. Paris: Hermann.
Cavaillès, J., & Lautmann, A. (1946). La pensée mathématique. Bulletin de la Société française de philosophie, 40(1), 1–39.
Cellucci, C. (2013). Rethinking logic: Logic in relation to mathematics, evolution and method. Dordrecht: Springer.
Danto, A. (1965). The analytical philosophy of history. Cambridge: Cambridge University Press.
Darmon, H., Diamond, F., & Taylor, R. (1997). Fermat’s last theorem. In Conference on Elliptic Curves and Modular Forms (pp. 1–140), Dec. 18–21, 1993. Hong Kong: International Press, 1997.
Della Rocca, M. (2016). Meaning, the history of philosophy, and analytical philosophy: A parmenidean ascent. In M. van Ackeren (Ed.), Philosophy and the historical perspective. Proceedings of the British Academy. Oxford: Oxford University Press.
Frey, G. (1986). Links between stable elliptic curves and certain diophantine equations. Annales Universitatis Saraviensis, 1, 1–40.
Gallie, W. B. (1964). Philosophy and the Historical Understanding. New York: Schocken.
Gardiner, P. (1961). The Nature of Historical Explanation. Oxford: Oxford University Press.
Gillies, D. (1993). Philosophy of science in the twentieth century. Oxford: Blackwell.
Hersh, R. (1999). What is mathematics, really?. Oxford and New York: Oxford University Press.
Ippoliti, E. (2008). Inferenze ampliative. London: Lulu.
Kitcher, P. (1983). The Nature of Mathematical Knowledge. New York: Oxford University Press.
Kitcher, P. (1988). Mathematical progress. Revue Internationale de Philosophie, 42/167(4), 518–531.
Kripke, S. (1980). Naming and Necessity. Princeton: Princeton University Press.
Li, W-C. (2001, 2012, 2013, 2014). Class notes, recorded by Emily Grosholz, Pennsylvania State University.
Lucretius. (2007). The nature of things (A. E. Stallings, Trans.). London: Penguin.
Maddy, P. (2000). Naturalism in mathematics. New York: Oxford University Press.
Plato. (1961). The collected dialogues of Plato including the letters. New York: Bollingen Foundation.
Putnam, H. (1975). Mind, language and reality. New York: Cambridge University Press.
Quine, W. V. O. (1953/1980). From a logical point of view. New York: Harper.
Ribet, K. (1990). On modular representations of Gal (Q/Z) arising from modular forms. Inventiones Mathematicae, 100, 431–476.
Ribet, K. (1995). Galois representations and modular forms. Bulletin of the American Mathematical Society, 32(4), 375–402.
Russell, B. (1905). On denoting. In Mind, 14(56).
Sinaceur, H. (1994). Jean Cavaillès, Philosophie mathématique. Paris: PUF.
Sinaceur, H. (2013). Cavaillès. Paris: Les Belles Lettres.
Soames, S. (2007). Reference and description: The case against two-dimensionalism. Princeton: Princeton University Press.
Wagner, R. (2016). Making and breaking mathematical sense: Histories and philosophies of mathematical practice. Princeton: Princeton University Press.
Wiles, A. (1995). Modular elliptic curves and Fermat’s last theorem. Annals of Mathematics, 141(3), 443–667.
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Grosholz, E.R. (2016). Philosophy of Mathematics and Philosophy of History. In: Starry Reckoning: Reference and Analysis in Mathematics and Cosmology. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-46690-3_2
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