Some Remarks on Wittgenstein’s Philosophy of Mathematics
- 1 Honorary Research Fellow, Swansea University, Swansea, UK
Abstract
Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers.
- Ambrose, A. (1986a). Wittgenstein on Some Questions in Foundations of Mathematics. In J. V. Canfield (Ed.), The Philosophy of Wittgenstein: Volume 11, Philosophy of Mathematics (Chapter 2, pp. 21-38). London: Garland.
- Ambrose, A. (1986b). Proof and the Theorem Proved. In J. V. Canfield (Ed.), The Philosophy of Wittgenstein: Volume 11, Philosophy of Mathematics (Chapter 6, pp. 99-109). London: Garland.
- Apostol, T. M. (1957). Mathematical Analysis: A Modern Approach to Advanced Calculus. Reading, MA: Addison-Wesley.
- Bangu, S. (2018). Ludwig Wittgenstein: Later Philosophy of Mathematics. In Internet Encyclopedia of Philosophy. http://www.iep.utm.edu/wittmath/
- Black, M. (1964). A Companion to Wittgenstein’s Tractatus. Cambridge: Cambridge University Press.
- Courant, R., & Robbins, H. (1958). What Is Mathematics? London: Oxford University Press.
- Gerrard, S. (1991). Wittgenstein’s Philosophies of Mathematics. Synthese, 87, 125-142. https://doi.org/10.1007/BF00485331
- Gerrard, S. (1996). A Philosophy of Mathematics between Two Camps. In H. Sluga, & D. G. Stern (Eds.), The Cambridge Companion to Wittgenstein (pp. 171-97). Cambridge: Cambridge University Press. https://doi.org/10.1017/CCOL0521460255.006
- Goodstein, R. L. (1986). Wittgenstein’s Philosophy of Mathematics. In J. V. Canfield (Ed.), The Philosophy of Wittgenstein: Volume 11, Philosophy of Mathematics (Chapter 12, pp. 269-284). London: Garland.
- Math-Reddit (2013). What Do Mathematicians Think about Wittgenstein? https://www.reddit.com/r/math/comments/1dcrx3/ what_do_mathematicians_think_about_wittgenstein/
- Moore, A. W. (2011). Wittgenstein and Infinity. In O. Kuusela, & M. McGinn (Eds.), The Oxford Handbook of Wittgenstein (Chapter 5). Oxford: Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199287505.003.0006
- Moore, A. W. (2019a). The Infinite (3rd ed.). London: Routledge. https://doi.org/10.4324/9781315145921
- Moore, A. W. (2019b). Language, World and Limits: Essays in the Philosophy of Language and Metaphysics. Oxford: Oxford University Press.
- Potter, M. (2011). Wittgenstein on Mathematics. In O. Kuusela, & M. McGinn (Eds.), The Oxford Handbook of Wittgenstein (Chapter 6). Oxford: Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199287505.003.0007