On Topos Theory
Grothendieck introduced toposes as a notion both of space and of categories of space, a notion which permits spatial intuition to be applied to widely different parts of mathematics. It is a huge conceptual advance on Hausdorff's notion of topological space.
Lawvere and Tierney extracted elementary axioms of toposes which also served as an algebra of higher-order predicate logic, as boolean algebras are an algebra of propositional logic through Lawvere's discovery that both higher-order and quantifiers arise as adjoints.
Grothendieck's work lead, for example, to Deligne's proof of the Weil conjectures used in such concrete results as Zhang's theorem on gaps in the primes.
Lawvere and Tierney's work lead to a revolution in logic, but also to the possibility of a further development of Grothendieck's notion of space for teaching and progress in geometry.
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