Reasoning In Mathematics and Machines: The Place of Mathematical Logic in Mathematical Understanding

摘要

Mathematical logic and mechanical reasoning have turned out to be largely irrelevant to the practice of mathematics, and to our philosophical understanding of the nature of that practice. My aim is to understand how this can be. We will see that the problem is not merely that the logician formalizes. Nor even is it, as Poincare argues, that logicians replace all distinctively mathematical steps of reasoning with strictly logical ones. Instead, as will be shown by way of a variety of examples, the problem lies in the way the symbolic language of mathematical logic has been read.