Outlook

Abstract

Abstract The central concern of this book is the epistemic status of non-finitary mathematics. Epistemology is not the only concern in foundational studies, though it has been dominant. The nature and intrinsic organization of mathematics has also been a major concern. Later developments in mathematics show that set theory is not the only basis for this kind of inquiry. Of course, those who think that true mathematics must be constructed will reject not only classical set theory but also the nineteenth-century mathematics out of which it grew. In this regard the development of constructive analysis can be regarded as partial fulfilment of an alternative foundational programme.I am not able to evaluate the success and significance of this programme, and perhaps it is too early anyway. For those who accept classical mathematics, category theory has been offered as an alternative to set theory for its catholic reach. Mathematics is definitely not just logic, not just higher-order logic, not just set theory. The old picture of a single fundamental theory to which all else must be reduced has faded. If pure mathematics is the study of abstract structures, set theory is just one framework among others for thinking about that subject matter, and it may not be the best. Universes of sets are themselves structures, and these may be instances of something more general, as is suggested by topos theory. In addition, topos theory sheds new light on the intrinsic organization of mathematics, revealing a surprising unity between apparently disparate fields, topology and algebraic geometry on the one hand, and logic and set theory on the other.