The origin of mathematics

Abstract

1. The two traditions. There are two great traditions, easily discernible, in the history of mathematics: the geometric or constructive and the algebraic or computational. If it could be shown that each of these has a single source and there are many rather familiar facts that suggest that this is so and if, moreover, in both cases the sources turn out to be the same, it would be plausible to claim that we have found the unique origin of mathematics. That is what I propose to do. The student who has taken our usual high-school courses in mathematics will recognize the two traditions easily enough. In the second year he studies plane geometry, where he encounters something quite different from his studies in algebra. Previously he computed, whereas now he constructs perpendiculars, angle bisectors, etc. ; and he deals with theorems and proofs. In algebra he learned that "minus times minus is plus," but this wasn't proved or even called a theorem. At best there was a rough explanation, or more probably he was simply told that that's the way it is. From a strictly mathematical point of view there can hardly be any reason why proof should enter geometry and not algebra, and one might speculate that the notion of proof first arose in the geometric tradition (and then passed to the algebraic). Such a guess will hardly come as a surprise to the readers accustomed to look to Classical Greece for the origin of the constructive tradition.