Abstract

Mathematical structures are identified with classes (in naive set theories, which were based on Comprehension Principle), or with sets (in axiomatic set theories, which adopted the principle in its restricted form). By analyzing abstraction operators and the set-theoretical diagonal arguments, the author indicates both classes and sets could be best regarded as certain self-applicable functions, treated as objects on their own. On the other hand, contemporary higher-order type theories, guided by the Curry-Howard Isomorphism, identify mathematical structures with propositions. According to this conception, formal derivation of a judgment counts both as the proof of a proposition and as the construction of a structure. The author examines its significance to the study of how language contributes to the construction of mathematical structures.