The Logical Structure of Computer-Aided Mathematical Reasoning

Abstract

1. NEW MATH. Over the past decade or so, the professional mathematician has changed from being a person who sits at a desk working with a paper and pencil to a person who spends a lot of time sitting in front of a computer terminal. The paper and pencil are still there, but a lot of the mathematician's activities now involve use of the In particular, powerful computer packages like Mathematica and Maple advertise themselves as systems for math on a computer. This rapid transformation of mode of working has changed the nature of doing mathematics in a fundamental way. Mathematics done with the aid of a computer is qualitatively different from mathematics done with paper and pencil alone. The computer does not simply 'assist' the mathematician in doing business as usual; rather, it changes the nature of what is done. In particular, the logical structure of mathematical reasoning carried out with the aid of an interactive computer system is different from the structure of the more traditional form of mathematical reasoning. This paper, in part survey, in part a presentation of some new ideas (for none of which the author claims priority or unique observation), compares the old and the new from a logician's standpoint. Strictly speaking, many of the points raised about computer-aided mathematics are not restricted to mathematics done on a Most of the points apply, to some extent, to mathematics done in the fashion familiar to the ancient Greeks. However, the introduction of computer-aided techniques in mathematics has made those points far more salient, indeed unavoidable.