“Good general theory does not search for the maximum generality, but for the right generality.”

An analog clock reading 8:082012-09-05 / 2012-W36-3T20:08:55-05:00 / 0x5047f7a7

Categories: math

This comes from Saunders Mac Lane in his textbook Categories for the Working Mathematician (p. 108, emphasis mine), but I contend it applies to more than just mathematics:

One may also speculate as to why the discovery of adjoint functors was so delayed. Ideas about Hilbert space or universal constructions in general topology might have suggested adjoints, but they did not; perhaps the 1939–1945 war interrupted this development. During the next decade 1945–55 there were very few studies of categories, category theory was just a language, and possible workers may have been discouraged by the widespread pragmatic distrust of “general abstract nonsense” (category theory). Bourbaki just missed [. . .] Bourbaki’s idea of universal construction was devised to be so general as to include more–and in partcular, to include the ideas of multilinear algebra which were important to French Mathematical traditions. In retrospect, this added generality seems mistaken; Bourbaki’s construction problem emphasized representable functors, and asked “Find $F,x$ so that $W(x, a) \cong A(F,x, a)$”. This formulation lacks the symmetry of the adjunction problem, “Find $F,x$ so that $X(x, G,a) \cong A(F,x, a)$”—and so missed a basic discovery; this discovery was left to a younger man, perhaps one less beholden to tradition or to fashion. Put differently, good general theory does not search for the maximum generality, but for the right generality.