Mathematicians and scientists have vague folk theories of what math and science are that both are blurred ancestral memories of pre-WWII logical positivism.
These theories are totally wrong, but do little *direct* harm because they are mainly ignored in practice.
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It's easy to find old, old examples of this. Consider Hilbert's published "proof" of the Continuum Hypothesis. Or any one of thousands of similar (though usually less spectacular) examples.
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So published proofs are sometimes wrong. So what? This doesn't seem especially notable. It is, of course, nice to have fairly reliable process for telling what's well established from what's wrong, but publication of a paper has never been more than a small part of that.
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The lack of replication in Math is not about proofs. It's about the processes leading to the proofs. A replication would be rederiving a proof with the newly gained understanding from the paper. If it can be rederived by other people, then it replicates.
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Yes, I think the point
@XenaProject was making is that this is often not actually feasible in practice. - 2 more replies
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A much better source than polemicists (which you might have heard of) is William Thurston, himself at the forefront of a recent mathematical renaissance, https://arxiv.org/abs/math/9404236 …. It still feels true today
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Thurston’s paper is a classic, yes. Yes, it’s obvious that The Ideal Mathematician is satirical, but it points out what seem to be real phenomena as well.
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There's obviously a lot of room for abuse, but I think this is rather one-sided and obscures the important fact that mathematics isn't performed in a directed, linear way. It's absolutely normal to try to nibble away at a question from both ends.
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Frequently, in doing so, you reach a point where you've reduced the problem from both premise *and* conclusion to a 'smaller' question which, you notice, other people have also noticed. This is a signal that the 'smaller' question is *interesting*, even if it looks strange.
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@estnihil I know basically nothing about math but you might find this thread interesting? -
hmmm don't think i agree at all tbh most disagreements in maths are about which axioms to take. but wrt 'rigour' maths is as rigorous as it gets, unlike science which doesn't state axioms and can't solve the problem of induction, underdetermination.
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