Rigor and structure

We are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure. But what do such formulations really mean? And how are they related? This book offers an account of how the requirement of rigor (that however one discovers a conjecture, it is not justified as a theorem until one has given a proof) historically has been and currently is understood. This account frankly acknowledges the degree to which there are still disagreements among leading mathematicians about such matters, and the extent to which such agreement as does at present obtain is at some risk of being upset by ongoing technological developments. The book also advances the suggestion that the much-discussed "structuralist" orientation prevailing in contemporary mathematics is perhaps better viewed, not in the way that many philosophers would view it, as a manifestation of the peculiar ontological nature of mathematical entities, but rather as an artifact of the way in which the requirement of rigor is enforced: The way in which mathematicians, while being required to be very careful about deriving new results from old, are permitted to remain largely indifferent as to how the old results were derived from first principles. Along the way, the question of what the first principles themselves should be taken to be is also addressed: The role of set theory as a so-called foundation is explained, and the rival foundational aspirations of category theory evaluated.