Gödel's Mathematics of Philosophy

Abstract

Mathematicians with an interest in philosophy, such as the present author, find an interest in the latter when they see it as a metaphor – or, even better, an inspiration – for the former. Godel provides a study case here because he is well known to have declared that (part of) his mathematical work was a direct consequence of his philosophical assumptions. If one takes this to mean Godel's own philosophy in an academic sense, for example, as crystallized in the Nachlass , then it is difficult to make precise sense of his remark, as some scholars have experienced. The following observations intend to show that the difficulties disappear if one interprets Godel's “philosophical assumptions” in a more popular sense, as meaning assumptions of philosophers whose thought he happened to know and find interesting. I claim that some of Godel's main results can be interpreted as being mathematically precise formulations of intuitions of Aristotle, Leibniz, and Kant. We know of a direct cause-and-effect connection only in some of the cases, but this is not the point. What we really care about, as nonprofessional readers, is to show that (part of) philosophy can be reinterpreted as asking questions and suggesting answers that mathematics makes precise. To put it in a more general slogan, in intellectual history, everything happens twice: first as philosophy, then as mathematics. The Arithmetization Method (1931) According to Gerald Sacks (1987), Godel said that he got the idea of arithmetization from Leibniz. Sacks seems to be skeptical about the causal effect and suggests that Godel may have only thought of it after the fact.