The Method of Analysis in Mathematics

Abstract

Analysis is the reduction of an unknown to the known. Analysis in early chemistry, for example, was the process of determining the constituents — i.e. the known elements — of an unknown chemical substance. In this paper, I shall discuss two examples of analysis in mathematics, namely two different attempts to prove the Pythagorean theorem — one enormously successful, the other less successful and eventually discarded. In the more successful example of analysis, Euclid’s proof of the Pythagorean theorem, the unknown is the Pythagorean theorem, the knowns are the Euclidean axioms, and the method of reduction is to deduce the Pythagorean theorem from the axioms. In the earlier and less successful analysis, the unknown is the Pythagorean theorem, the knowns are the whole numbers, and the method of reduction is to show that geometrical figures (in this case right triangles) can be represented by whole numbers. As I shall explain, the Euclidean method of analysis, i.e. of reducing the unknown to the known, was much more successful than the Pythagorean.