All Questions

As per the title, I'd be curious to know if there have been attempts at constructing a foundation of mathematics taking, somehow, purely the notion of "structure" as primitive, maybe via a system of ...

Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The ...

There are categorical foundations for mathematics axiomatizing the category of sets (Lawvere's ETCS), cartesian closed categories (type theory), and the category of spaces (homotopy type theory). ...

I can't find a statement about the axiom of regularity anywhere in treatments of ETCS. Perhaps this is due to the unfortunate clash of terminology with 'foundations'.

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...

Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set ...