Kantian Journal

2015 Issue №1(51)

Back to the list Download an article

Kant and new mathematics 100 years later

DOI
10.5922/0207-6918-2015-1-1
Pages
7-16

Abstract

Cassirer’s critique of Russell’s philosophy of mathematics and the Neo-Kantian philosophy of science and mathematics as a whole is of special relevance in the context of modern mathematics and mathematical physics. The fact that the modern standard axiomatic architecture of mathematical theories does not take into account the object-based character of mathematical knowledge, which was stressed after Kant by Cassirer, complicates the application of new mathematical theories in natural sciences and technology. In particular, this can explain why modern physical string theory is empirically unverifiable; it can be adjusted to accommodate a wide range of possible outcomes of observations and experiments. At the same time, there are reasons to believe that certain recent ap-proaches in foundations of mathematics such as category theory, topos theory, and Univalent Foundations may help to improve the situation in the near future. The problem of applicability of new mathematical knowledge in science and technology shows that the Kantian approach in phi-losophy of mathematics is at least partly relevant to today’s mathematics.

Reference

1.    Арнольд В. И. О преподавании математики // Успехи математических наук 1998. № 1 (53). С. 229—234.
2.    Бонола Р. Неевклидова геометрия. М., 2010.
3.    Кантор Г. Основы общего учения о многообразиях // Труды по теории множеств. М., 1985. С. 63—106.
4.    Родин А. Программный реализм в физике и основания математики // Вопросы философии. 2015. № 4—5 (в печати).
5.    Родин А. Теория категорий и поиск новых математических оснований физики // Вопросы философии. 2010. № 6. С. 67—82.
6.    Cassirer E. Kant und die moderne Mathematik // Kant-Studien. 1907. № 12. S. 1—40.
7.    Friedman M. Ernst Cassirer and Contemporary Philosophy of Science // Angelaki. 2005. № 10. P. 119—128.
8.    Friedman M. Kant and the Exact Sciences. Cambridge, 1992.
9.    Heis J. Ernst Cassirer's Neo-Kantian Philosophy of Geometry // British Journal for the History of Philosophy. 2011. № 4(19). P. 759—794.
10.    Heis J. “Critical philosophy begins at the very point where logistic leaves off”: Cassirer’s Response to Frege and Russell // Perspectives on Science. 2010. № 4(18).
P. 383—408.
11.    Jonson A. K. Neo-Kantianism // Internet Encyclopedia of Philosophy. URL: http://www.iep.utm. edu/neo-kant/ (дата обращения: 19.01.2015).
12.    Pulkkinen J. Thought and Logic: The Debates Between German-Speaking Philoso-phers and Symbolic Logicians at the Turn of the 20th Century. P. Lang, 2005.
13.    Rheinberger H.-J. On Historicizing Epistemology. Stanford University Press, 2010.
14.    Rodin A. Axiomatic Method and Category Theory // Synthese Library. Springer, 2014. Vol. 364.
15.    Russell B. Principles of Mathematics. L., 1903.
16.    Russell B. A Critical Exposition of the Philosophy of Leibniz. L. ; N. Y., 1996.
17.    Schreiber U. Classical field theory via Cohesive homotopy types. arXiv:1311.1172 (2013) (дата обращения: 19.01.2015).
18.    Smolin L. The Trouble With Physics. Houghton Mifflin Harcourt, 2006.
19.    Voevodsky V. et al. Homotopy Type Theory: Univalent Foundations of Mathemat-ics. Princeton, 2013.
20.    Wigner E. The unreasonable effectiveness of mathematics in the natural sciences // Commun. Pure Appl. Math. 1960. № 13. P. 1—14.

Reference

[