This includes if you define "proof" in a way that implies that, say, Gauss never wrote a proof in his life
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"Gauss wrote 8 proof-sketches of quadratic reciprocity. If only Coq had been released back then, he might have written 8 proofs"
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What if your main thrust is that "what mathematicians do" is becoming increasingly problematic and divorced from genuine mathematical practice, eg Kevin Buzzard asking "Is this even mathematics?"
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I think the philosophy of anything doesn't define the subject but creates some sort of adjacent interface with it in order to answer its own questions. Just like mathematicians do what suits them, a philosopher of math uses mathematics in a way to answer their own questions.
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It seems strange for somebody from a subject that is decidedly not empiric to only allow for empiric opinion when treating that subject philosophically.
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There's a big merit to changing the definition to something easier accessible than "what mathematicians do", because "what mathematicians do" is only understandable in consensus of what constitutes, well, doing mathematics and what doesn't.
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So a proper philosophy of mathematics must in your view be time-dependent?
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Isn't all human activity time-dependent? (No trolling intended)
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I am of the opinion that we don't really write formal proofs so much as we write meta arguments that say it is possible to write a formal proof.
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What mathematicians "do" is take older ideas, models, proofs, formulae, etc and combine them to create new ideas. models, proofs, formulae, etc, much in the same way a computer programmer takes older code modules to create newer code. 1/n
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The difference between a mathematician and a programmer is that one runs the math on a computer, and the other is running the code on either their own or other people's minds. However, that distinction is being eroded now by HoTT and trinitarianism. 2/2
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