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comment Negative impact of wrong or non-rigorous proofs
Come to think of it, I heard not that long ago about an entire field full of what you all are talking about, a field from which any sensible young person would flee, even if they love the subject. Forgive me. I'm getting old and my memory is terrible.
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awarded  Popular Question
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comment Negative impact of wrong or non-rigorous proofs
user36931, I agree.
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comment Negative impact of wrong or non-rigorous proofs
Let me add that I don't see evidence for the claim that people doing the "cleaning up" receive little credit. If "cleaning up" means filling in routine details, then it's generally acknowledged that the original proof is in fact a rigorous one. If, on the other hand, "cleaning up" requires the introduction of new ideas or techniques, then it appears to me that the people doing this are given a lot of credit and the fact that their work provides rigorous proofs of the "theorems" of a prominent mathematician makes it look only better and not worse.
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comment Negative impact of wrong or non-rigorous proofs
S. Carnahan, that's a good point. I'm willing to entertain rumors about this really happening.
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revised Negative impact of wrong or non-rigorous proofs
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comment Negative impact of wrong or non-rigorous proofs
Lee, thanks. I agree.
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comment Negative impact of wrong or non-rigorous proofs
Yemon, can you cite a specific example where you believe this happened?
Jul
19
accepted Point of maximal distance on a non-positively curved PL surface
Jul
19
comment Negative impact of wrong or non-rigorous proofs
quid, thanks! I had seen that already. That does seem rather damaging, but I'm wondering whether it was even worse than that. My impression is that Severi published several very interesting but wrong theorems, which must have misled others working in the subject. This definitely qualifies as negative impact. But I was also wondering if there are examples of people using Severi's work or ideas to compound the problem, publishing results that were wrong because they relied on Severi's work.
Jul
19
comment Point of maximal distance on a non-positively curved PL surface
It occurs to me that I think I know the answer. I imagine that an extreme point would be a vertex just like the extreme vertices of Connelly's flexible polyhedron. One where there are several edges (which should be viewed as hinges) meeting a vertex and the faces "zig zag" around the vertex, which allows them to be flexible under bending along the hinges. The curvature, I would guess, is zero at the vertex.
Jul
19
comment Negative impact of wrong or non-rigorous proofs
Ben, that sounds pretty interesting. I would love to learn about an explicit example of that.
Jul
19
comment Negative impact of wrong or non-rigorous proofs
André, thanks. This, I believe, is the same as what Jaffe and Quinn said, citing the work of people like Gromov, Sullivan, and Thurston. It appears to be difficult to establish or debunk this assertion, but I'm looking for an explicit example where this negative impact might have occurred. I have the impression that the contributions of Gromov, Sullivan, and Thursday have had the exact opposite effect of stimulating a lot of activity clarifying their ideas, and I'm looking for counterexamples to my belief.
Jul
19
comment Negative impact of wrong or non-rigorous proofs
Ben, thanks again. I posted my comment before seeing yours.
Jul
19
comment Negative impact of wrong or non-rigorous proofs
Thanks, Ben. Are you're saying that Newton and Leibniz's approach to calculus did not lead to many harmful consequences, because people used a well-defined set of logically consistent axioms (which we now like to call nonstandard analysis) that we now know always produces correct results?
Jul
19
comment Negative impact of wrong or non-rigorous proofs
NAME_IN_CAPS, I'm a little confused by what you're saying. Are Gromov and Thurston's "proofs an example of what Darij is talking about or not?
Jul
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Jul
18
comment Negative impact of wrong or non-rigorous proofs
All of the examples I know are either isolated theorems that had little impact on the rest of mathematics or important theorems where the first proofs were incomplete or wrong but the theorem is close enough to being right (largely because, I think, no one found any logical inconsistencies that would invalidate the theorem) that most if not all of the important consequences also remained essentially correct after the proof was repaired. I'm particularly curious about whether Newton's calculus is a counterexample to my belief about this.
Jul
18
comment Negative impact of wrong or non-rigorous proofs
Thanks, Will. But did this do any damage?
Jul
18
comment Negative impact of wrong or non-rigorous proofs
For various reasons, including the one you cite, I'm wary about what one person says about a theorem or proof. I'm more interested in situations where there is a consensus among a group of well-regarded mathematicians that something is flawed.