"The point is simply that when explaining the general notion of structure and of particular kinds of structures such as groups, rings, categories, etc., we implicitly presume as understood the ideas of operation and collection." (Solomon Feferman, "Categorical foundations and foundations of category theory", 1975)
"A theory is a category with certain operations (defined up to isomorphism) [...] The notion of theory is thus 'intrinsic', i.e., independent of a particular presentation via formal languages and axiomatic systems. In this sense, categorical logic may be viewed as 'synthetic' or 'intrinsic' logic by opposition to the usual 'analytic', 'formal' logic." (André Joyal & Gonyalo E Reyes, "Forcing and generic models in categorical logic", 1977)
"But the question automatically arises as to exactly why, in introducing ordinary mathematical notions into the theory, one must make a detour through the somewhat opaque notion of discrete category. It is difficult to see how this can be explained except by appeal to the notion of 'unstructured' category, i.e. set." (John L Bell, "Category theory and the foundations of mathematics", The British Journal for the Philosophy of Science 32(4), 1981)
"But this seems to me highly dubious, for it is surely the case that the unstructured notion of class is epistemically prior to any more highly structured notion such as category: in order to understand what a category is, you first have to know what a class is." (John L Bell, "Category theory and the foundations of mathematics", The British Journal for the Philosophy of Science 32(4), 1981)
"[…] it would be technically possible to give a purely category-theoretic account of all mathematical notions expressible within axiomatic set theory, and so formally possible for category theory to serve as a foundation for mathematics insofar as axiomatic set theory does." (John L Bell, "Category theory and the foundations of mathematics", The British Journal for the Philosophy of Science 32(4), 1981)
"It is a remarkable empirical fact that mathematics can be based on set theory. More precisely, all mathematical objects can be coded as sets (in the cumulative hierarchy built by transfinitely iterating the power set operation, starting with the empty set). And all their crucial properties can be proved from the axioms of set theory. (. . . ) At first sight, category theory seems to be an exception to this general phenomenon. It deals with objects, like the categories of sets, of groups etc. that are as big as the whole universe of sets and that therefore do not admit any evident coding as sets. Furthermore, category theory involves constructions, like the functor category, that lead from these large categories to even larger ones. Thus, category theory is not just another field whose set-theoretic foundation can be left as an exercise. An interaction between category theory and set theory arises because there is a real question: What is the appropriate set-theoretic foundation for category theory?" (Andreas Blass, "The interaction between category theory and set theory", 1983)
"We consider forcing over categories as a way of constructing objects by geometric approximation, including a construction of a generic model of a geometric theory as its special case." (Andrej Scedrov, "Forcing and Classifying Topoi", Memoirs of the American Mathematical Society Vol. 48, 1984)
"Symmetries of a geometric object are traditionally described by its automorphism group, which often is an object of the same geometric class (a topological space, an algebraic variety, etc.). Of course, such symmetries are only a particular type of morphisms, so that Klein’s Erlanger program is, in principle, subsumed by the general categorical approach." (Yuri I Manin, "Topics in Noncommutative Geometry", 1991)
"[…] categorical logic is, to a great degree, autonomous, even in matters syntactical. ([203], 54.)" (Michael Makkai, "Generalized sketches as a framework for completeness theorems", Journal of Pure and Applied Algebra, 115(1), 1997)
No comments:
Post a Comment