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by bheadmaster
998 days ago
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> Getting 'f' vs 'f(x)' right mostly is important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category Mathematicians deal with higher order functions all the time, e.g. in functional analysis. |
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OTOH, the lack of rigor is definitely one of the problems of contemporary math. Many years ago, when I was a student, I studied one paper, coauthored by 2 people - call them X and Y. X was a very established mathematician, Y was a relative newcomer. There was one (set-theoretical) argument I couldn't understand, so I asked Y (he was my advisor's friend) about it. He told me "yeah, X asked me this, too, and I told him to use Zorn's lemma, and after a moment of thinking, he said, «yeah, that would work»". I'm not set theorist myself, but it smelled suspicious to me, so I asked another friend, who knew much more about set theory than me. He smiled and said "of course it's wrong, it's a very common mistake".
Had X and Y written out the argument more rigorously, we'd have one less published result with no correct proof...
And I have quite a few other anecdotes like this, unfortunately.
One professor at my former faculty once told how he approaches refereeing papers. "For the first 30 minutes, I try to prove the main result myself. If I don't succeed, I spend the next 30 minutes trying to find a counterexample. This way I write most reviews in half an hour."
A few years ago I coauthored a book about non-linear analysis. Quite a few quite interesting topics. One of the coauthors insisted on writing out proofs in detail and rigorously, and now we joke that our book is the first one where some (quite established and known in this field) theorems are proved correctly for the first time. (And that includes proofs with gaps/mistakes in both research papers and monographs, btw.)