Bourbaki and foundations

[Monroe Eskew]

> On 14.05.2022, at 22:24, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> 
> More likely, an example of an (allegedly) "inherent limitation of set theory" would be the fact that in ZFC, everything is a set.  So if you firmly believe that (say) topological spaces and groups are fundamentally different types of mathematical objects, then you may chafe at the requirement to define both topological spaces and groups in terms of sets. You may feel that doing so creates the misleading impression that topological spaces and groups are fundamentally "made of" the same kind of stuff, just packaged slightly differently.

I don’t see how concerns about the “true nature” of mathematical objects leads one to assertions that set theory is “full of theoretical holes.”  What are the holes? I can only think of incompleteness as the thing being referred to here, which is of course a very naive reason to prefer an alternative foundation. 

When I first started to learn logic, I was not satisfied with reducing numbers to a series of nested set constructions with “nothingness” (the empty set) at the base. But after seeing what the theory can do, I began to think of these ontological worries as somewhat frivolous and not really the getting at the most philosophically interesting issues. It seems to me that one who embraces structuralism as a motivation for category theory could also use this philosophy as a way to not worry about the ontology of ZFC and just look at the interesting structures that arise from the theory, which are quite rich and not beset with “limitations.”