The Journal of Philosophy

Volume 113, Issue 5/6, May/June 2016

Philosophy of Logic and Mathematics

W. W. Tait
Pages 261-273

Kant and Finitism

An observation and a thesis: The observation is that, whatever the connection between Kant’s philosophy and Hilbert’s conception of finitism, Kant’s account of geometric reasoning shares an essential idea with the account of finitist number theory in “Finitism” (Tait 1981), namely the idea of constructions f(X) from ‘arbitrary’ or ‘generic’ objects of various types (triangles, natural numbers, etc.). The thesis is that, contrary to a substantial part of contemporary literature on the subject, when Kant referred to number (as a common noun) and arithmetic, he was not referring to the natural or whole numbers and their arithmetic, but rather to the real numbers (as then understood) and their arithmetic. (This thesis owes, and will receive, some account of Kant’s discussion of number as the schema of magnitude.)