My general opinion on the ontological-foundational question in the philosophy of mathematics at this point is something like: structure is an unformalizable extramathematical term referring to what we abstract out from things, whether the things we’re abstracting are concrete or abstract themselves. It’s fine that this is informal because there’s no reason to expect that we’d be able to fully characterize mathematics within mathematics. Nor can we fully characterize mathematics without mathematics; studying axiom systems in the abstract is an important part of the endeavour. I’m not sure how minimal of ontological commitments we can get away with here but they’re not going to be any worse than ZFC or whatever.
Connections between different areas of mathematics really are “coincidence” in the sense that they occur when structures coincide, but not in the sense that there’s no reason they happened; where structures coincide it’s generally for some sort of reason, but we don’t need a global reason for all coincidences of structures (see also: don’t need a formal notion of “structure”, have no “set of all sets”), just reasons for any particular coincidence. formalism cannot explain this. intuitionism cannot explain this. ordinary mathematical platonism cannot explain this. (structuralism can, and i think this applies in both platonist and antiplatonist varieties.)
What we call a “foundation” is a framework in which to do mathematics which best possesses the appropriate philosophical qualities to be foundational. These qualities may include simplicity, minimal ontological commitments, canonicity, ability to theoretically cover mathematics, closeness to real practice, and so on; how much we care about different ones depends on our general philosophical stance. Some approaches:
- A default position in foundations is a sort of platonism that’s very pragmatic about its ontological commitments but impractical in its usage: we must have objects which we could hypothetically do all of mathematics with, but having established that it’s possible to do it there’s no reason to actually bother. So after that let’s just try and be as simple as possible.
- Frege was somewhat platonist and very strongly committed to canonicity not just in a mathematical but in an ontological sense. This commitment backfired so badly that strict formalism was a respectable response.
- Do formalists care about foundations? are we allowed to interpret their statements about what they do and don’t care about?
- Ultrafinitism takes minimality of ontological commitments to its ultimate extent, consigning most mathematical practice to the fire.
- Type theory is generally associated with a greater concern for computability, canonicity, and closeness to actual reasoning (i think this often comes from a structuralism), at the expense of some simplicity and (to a platonist) more ontological commitments than necessary. Type theory isn’t exclusively constructive but insofar as it is (equally, insofar as it’s computable) it gives up on some amount of existing mathematical work.
Finally: foundations are not as central as they’re made out to be. I think a more important question is what makes things interesting or not.