Reason's nearest kin. Philosophies of arithmetic from Kant fo Carnap (Q2760422)

The author discusses the philosophies of arithmetic of Kant, Frege, Dedekind, Russell, Wittgenstein, Ramsey, Hilbert, Gödel and Carnap. According to Kant, arithmetic concerns our intuition. The temporal structure of sensibility is essentially implicated in counting; this fact ensures the applicability of arithmetic to the world. Kant felt that arithmetic needed no axiomatic foundation. In search of a more precise foundation Frege put forth logicism: the view that arithmetic, as `reason's nearest kin', can be based on logic. Frege saw logic as dealing with the way we think; that is why arithmetic is applicable to everything we can think. Technically Dedekind's and Frege's treatment of arithmetic are very similar. Their accounts, however, go beyond logic in the modern sense of the word. Russell's work most clearly shows the problems logicism runs into. Russell viewed logic as an empirical science dealing with the logical aspect of the world. This enabled him to introduce the axiom of reducibility on empirical grounds. Why would we, however, believe the axiom of reducibility? Hilbert attempted a foundation of meta-arithmetic along Kantian lines in our intuitions of finite arrangements of concrete objects. However, soon on the technical level, Gödel's incompleteness theorems demonstrated the limits of formalism. This illuminating book shows how in the period between the publication of Frege's Grundlagen in 1884 and World War II our technical and philosophical insight in the foundations of arithmetic grew considerably, while at the same time, the philosophical problems concerning the existence and properties of the natural numbers were not solved.