The dogma that mathematical proofs could, in principle, be turned into logical proofs remains a conjecture; it is not feasible in practice. Existing proof systems are not up to the job; can they be beefed up sufficiently? Research community disconnect!https://xenaproject.wordpress.com/2020/02/09/where-is-the-fashionable-mathematics/ …
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Replying to @Meaningness
Do you think that there is actual dogmatic belief in the idea that it should be possible to turn mathematical proofs into logical ones? If so, I would be curious to see why.
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Replying to @rice_cooker_
Widely held in the philosophy of mathematics. Seems to be informally believed by many/most mathematicians? I don’t know.
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Replying to @Meaningness @rice_cooker_
Certainly I'd say the goal of a proof is to write things you're confident you _could_ turn into formal logical statements if you needed to. We're too lazy to do the dereferencing, but we're pretty sure we could.
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Of course, we could easily be wrong about that, and almost assuredly are wrong at least some of the time. But our intention is to write only things that we could turn into logical proofs.
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(I suspect a lot of mathematicians are uninterested in formal proof systems because it feels like they're a long and tedious way to do something basically trivial in principle.)
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Yes, that’s my impression! I hope computer proof systems can eventually be routinely useful, but it seems still far off
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