and Tierney recognized that axiomatizing the subobject classifier yielded a more general kind of topos, now known as an elementary topos, and that elementary topoi were models of intuitionistic logic. In addition to providing a powerful way to apply tools from logic to geometry, this made possible the use of geometric methods in logic. Spaces and structure According to Kevin Carlson, Neither of these words ["space" and "structure"] have a single mathematical definition. The English words can be used in essentially all the same situations, but you often think of a "space" as more geometric and a "structure" as more algebraic. [...] So you could think of "structures" as places we do algebra, and "spaces" as places we do geometry. Then a lot of great mathematics has come from passing from structures to spaces and vice versa, as when we look at the fundamental group of a topological space or the spectrum of a ring. But in the end, the distinction is neither hard nor fast and only goes so far: many things are obviously both structures and spaces, some things are not obviously either, and some people might well disagree with everything I've said here. Nevertheless, a general definition of "structure" was proposed by Bourbaki; it embraces all types of spaces mentioned above, (nearly?) all types of mathematical structures used till now, and more. It provides a general definition of isomorphism, and justifies transfer of properties between isomorphic structures. However, it was never used actively in mathematical practice (not even in the mathematical treatises written by Bourbaki himself). Here are the last phrases from a review by Robert Reed of a book by Leo Corry: Corry does not seem to feel that any formal definition of structure could do justice to the use of the concept in actual mathematical practice [...] Corry's view could be summarized as the belief that 'structure' refers essentially to a way of doing mathematics, and is therefore a concept probably just as far from being precisely definable as the cultural artifact of mathematics itself. For more information on mathematical structures see Wikipedia: mathematical structure, equivalent definitions of mathematical structures, and transport of structure. The distinction between geometric "spaces" and algebraic "structures" is sometimes clear, sometimes elusive. Clearly, groups are algebraic, while Euclidean spaces are geometric. Modules over rings are as algebraic as groups. In particular, when the ring appears to be a field, the