Shapiro in Thinking About Math talks about this. For a brief period it probably was the case that philosophy determined who studied what. And in some sense it is probably still around today. But the effect was probably highest in the past. Poincare rejected any actual infinite, whereas Godel said just because we can’t construct it doesn’t mean it doesn’t exist platonically.
Here’s a quote of Godel found in Shapiro’s Thinking About Math, “If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence of totalities containing members.” Pg. 10
If there really are mathematical objects independent of us, we can use things like impredicative definitions. Poincare rejected any actual infinity, independent of us or otherwise. Therefore he abstained from certain definitions Godel had no problems with. Things like defining objects by using a collection that already contains it. Without actual infinities in some sense, these definitions are “viciously circular” to Poincare.
And for similar reasons, intuitionist mathematics has to reject things like the law of excluded middle.
BUT, Shapiro is quick to show, these are isolated moments in the history of mathematics. The dominance of accepting things like the law of excluded middle, impredicativeness, axiom of extensionality, have proven too useful and successful for natural science and studying further mathematics.
These temporary disagreements of a philosophy-first mathematic rapidly gave way to objects and definitions mathematicians “could not help using…and with hindsight we see how impoverished mathematicians would be without them”. “These nuisances proved to be artificial and unproductive”.
So now there is a complete retreat of philosophy-first mathematics and more of a sense of following what is successful.
That isn’t to say everyone works on the exact same stuff and no one has any disagreements. Joel David Hamkins does say mathematician’s “beliefs” about realism, multiverse, which extensions of set theory, etc do affect what they study to some degree.
But the language of mathematics can still be used and understood regardless of philosophy. That is kind of a miracle of language in a way.