Categories and Images of Mathematics

The first part of this book described the development of ideal theory from Dedekind to Noether. This account was meant to illustrate, from the perspective offered by that specific theory, the rise of a central image of twentieth-century mathematical knowledge according to which mathematics is seen as a science of structures. The second part analyzed three reflexive attempts to elucidate that image of knowledge by means of formal mathematical theories. Ore’s theory of structures had already disappeared from the world of mathematical research in the early fifties. Bourbaki’s work at large constituted a mathematical achievement of outstanding value and reached an enormous, worldwide popularity, but the actual contribution of the theory of structures to the mathematical elucidation of the idea of structure was rather marginal, and certainly below the expectations of its creators. Category theory, in its turn, emerged after the sixties as an autonomous mathematical discipline. It developed its own particular tools, created its own sub-disciplines, and defined its urgent open problems. The theory afforded a relatively successful alternative approach to several open foundational issues in mathematics. At the same time, some of its limitations became increasingly manifest over the years. But how did the developments related to these three reflexive theories concomitantly affected the images of mathematics? This is discussed in the present, concluding chapter of the book.