The Nature of Progress in Mathematics: The Significance of Analogy

Abstract

Mathematicians of all stripes, whether intuitionists or structuralists, have acknowledged the fundamental role played by analogy in mathematical invention. Thus for Poincaré analogy is the inventor’s principal “guide” (Poincaré 1900, 127). He himself tells us that he was able to find the representation of a category of Fuchsian functions in terms of a series, because he was guided by an analogy with elliptic functions. Consequently, analogy plays an essential role in progress, or in “the future of mathematics, ” as he put it in a paper delivered at the International Congress of Mathematicians in 1908. Virtually every breakthrough relies on analogies, either within a given field of mathematics or between different fields. Indeed, the analogy that presents itself between a given problem and a more extended class of other problems opens up a generalization of the terms as well as of the solution of the given problem. Poincaré went on to point out that a generalization “is not a new result, it is rather a new force” (Poincaré 1908, 169). It is a force primarily because it provides an economy of thought. Every mathematician knows the importance of being able to encompass a large set of facts and results that are apparently different from one another, or that belong to distinct domains of mathematics, in one single glance. Hilbert’s school too emphasized the fruitfiilness of “übersichtlichkeit.” Artin, for instance, has written that the true happiness of the mathematician is not to hold on to the logical sequence, but to “see at one glance the whole architecture in every direction.”¹