Full Idea
Foundationalists (e.g. Quine and Lewis) have shown that mathematics can be rendered in theories other than the iterative hierarchy of sets. A dedicated contingent hold that the category of categories is the proper foundation (e.g. Lawvere).
Gist of Idea
Mathematical foundations may not be sets; categories are a popular rival
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.87
A Reaction
I like the sound of that. The categories are presumably concepts that generate sets. Tricky territory, with Frege's disaster as a horrible warning to be careful.