Notes towards my perspective on the philosophy of mathematics

As the sage once said, “always historicize”. An account of the nature & subject matter of mathematics must be compatible with its history. It doesn’t need to fully explain it — if it could, it’d just be overfitting a theory to the contingencies and ambiguities of history — but it should be capable of interfacing with the history of mathematics and also with historical views of the nature of mathematics. (This doesn’t rule out all platonisms but it does kill the “ontologically minimal” ones.)
Mathematics is a broad category of human activity and we should strive to understand all of it instead of declaring that some subcategory is the true essence. The framing of models (which can be v useful!) as “reductions” of one field to another, the philosophical exaltation of formalized proof at the expense of the informal, the neglect of the real process of coming to definitions and proofs: all mistakes.
The ontological question is overemphasized. However, every answer to the question has to account for human thought about mathematics which means going through a sort of psychologism or intuitionism even if it’s a derived phenomenon and not a foundation; this is where every actual consequence of the ontology will manifest, so it should be more of a focus. Traditional intuitionism is too restricted & built on a shaky foundation. Also, platonism is silly but hard to talk around.
Per above: I think mathematics consists of a process of mentally abstracting consistent structures — from real things or other abstractions — and applying various practices to understand them and keep our thoughts about them consistent. (semi)formal proof is one of the most crucial of these practices but not the only one
Connections between fields of mathematics occur where the structures they study overlap or coincide. Neither side has to be “primary” or “foundational” — rather, a connection opens up the use of practices developed for the study of one field to the study of another, and sometimes both. Cartesian geometry allows both the use of algebraic techniques to construct and study geometric objects and the use of geometric interpretations & techniques to study systems of numbers (calculus, phase space, and so on).
The striking similarities between independent historical developments of mathematics are because the same abstract structures (numbers, geometry, polynomials, etc) are useful enough to appear in these different contexts. A conclusion about what’s true of a structure is not contingent on anything but the structure; but which structures are studied, and which statements about them are considered relevant, is purely contingent on the world, historically variable, and sometimes ambiguous. (Contingent on which elements of the world?)
The use cases which originate a field are often not the ones which drive it internally.