The standard "foundation" for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions. — Saunders Mac Lane Copy Share Image
... the notions category and functor were not formulated or put in print until the idea of a natural transformation was also at hand. — Saunders Mac Lane Copy Share Image
. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation… — Saunders Mac Lane Copy Share Image
The standard "foundation" for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of sets but… — Saunders Mac Lane Copy Share Image
. . . in one sense a foundation is a security blanket: If you meticulously follow the rules laid down, no paradoxes or contradictions… — Saunders Mac Lane Copy Share Image
