Human mathematicians are being outcounterexampled • Hacker News Board

Re: Human mathematicians are being outcounterexampled

#111

Post by bananaflag » Tue, Jul 21, 2026, 6:22 AM UTC

Dove wrote: post #95

bananaflag wrote: post #38

Earlier quoted context omitted.

> It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample. Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequent…

We were studying geometry - my adviser was the great Branko Grünbaum: https://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaum The conjecture had to do with whether one convex polygon could be continuously deformed into another while remaining convex, under certain conditions and constraints. The answer turns out to be no, but surprise and disappointment are understandable reactions to that outcome. It was indeed much more…

Thanks! It makes sense.

Re: Human mathematicians are being outcounterexampled

#112

Post by murphyslaw » Tue, Jul 21, 2026, 6:27 AM UTC

AndrewKemendo wrote: post #75

zahlman wrote: post #63

Earlier quoted context omitted.

Is the video available not through a proprietary player? > After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies. That's quite an extreme shift considering she had previously been a leading figure in third-wave feminism and an OWS activist.

I couldn’t get the video to work. Here’s the relevant clip I found on youtube. https://youtu.be/3uRCcEoGWxs

BBC iPlayer is not available outside the UK.

Re: Human mathematicians are being outcounterexampled

#113

Post by murphyslaw » Tue, Jul 21, 2026, 6:33 AM UTC

SpicyLemonZest wrote: post #32

mb7733 wrote: post #25

Earlier quoted context omitted.

Are you thinking of proof by contradiction, which is rejected by constructionism? [Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?

It’s very straightforward, but it often doesn’t (and here didn’t) fully satisfy the curiosity that was embedded in the original problem. Why did the Jacobian conjecture seem to be true? Is there some underlying symmetry that’s very slightly broken? Or perhaps there’s all kinds of counterexamples, and the intuitive pattern is only real for certain kinds of functions which happen to predominate in our intuition. Then h…

Isn't the fact that you now know that the conjecture is false a huge help? At least it will help convince people to look at the conjecture more closely, no?

Re: Human mathematicians are being outcounterexampled

#114

Post by vintermann » Tue, Jul 21, 2026, 6:38 AM UTC

angry_octet wrote: post #54

simonreiff wrote: post #31

Earlier quoted context omitted.

That sounds like the worst "exam nightmare" scenario imaginable, but did the student get the PhD in the end?

Proof fix-ups are quite common, but if it was not possible, then no, not for that topic.

I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".

Probably there are understandable reasons for that... But I think "negative science" is really important, and soft results like "we tried that for a long time and it wasn't very fruitful" are actually very important for progress even if they're poorly attested in the written record. I guess in mathematics they come informally from your thesis advisor...

Re: Human mathematicians are being outcounterexampled

#115

Post by CurtMonash » Tue, Jul 21, 2026, 6:39 AM UTC

tchalla wrote: post #24

hintymad wrote: post #14

> The Jacobian Conjecture Interestingly, Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up…

I was once in a presentation for a math PhD thesis. During the thesis, the evaluator of the thesis noticed a flaw in their proof. The student understood and then asked “What now?” The evaluator prof simply shrugged.

I first heard that kind of story about a thesis defense at Princeton. The twist was that they had been short one person to judge it, so they roped in a professor who was available at that moment ... and he found a counterexample on the fly.

The professor was John Milnor.

Re: Human mathematicians are being outcounterexampled

#116

Post by veunes » Tue, Jul 21, 2026, 6:42 AM UTC

[flagged]

Re: Human mathematicians are being outcounterexampled

#117

Post by CurtMonash » Tue, Jul 21, 2026, 6:43 AM UTC

A large fraction of the problems assigned in the Ross Program were of the form "Prove or disprove, and salvage if possible." The rest were usually a calculation, meant to motivate a general proposition you would encounter soon after.

Re: Human mathematicians are being outcounterexampled

#118

Post by veunes » Tue, Jul 21, 2026, 6:46 AM UTC

Dove wrote: post #15

When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute. On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It…

This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly

Re: Human mathematicians are being outcounterexampled

#119

Post by veunes » Tue, Jul 21, 2026, 6:47 AM UTC

bananaflag wrote: post #38

Dove wrote: post #15

When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute. On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It…

> It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample. Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequent…

I think the asymmetry depends on the representation

Re: Human mathematicians are being outcounterexampled

#120

Post by reinitctxoffset » Tue, Jul 21, 2026, 6:47 AM UTC

recursivecaveat wrote: post #92

abdullahkhalids wrote: post #51

Earlier quoted context omitted.

Mistakes in proofs are relatively common, but such a mistake doesn't automatically mean that the proof is entirely wrong. Many times, the mistake is just in the exposition, and can be fixed easily. Other times, the mistake is fixable and the fix is apparent. Perhaps the author forget to treat a relatively trivial edge case. In the first two cases, the student would likely just pass with minor corrections to be submit…

A few months after Andrew Wiles presented the proof for Fermat's last theorem, some mistakes were found that subsequently took him over a year to patch over. Imagine being 12 months into trying to put the pieces back together, after all the years of work and announcing your finding publicly.

I can recommend without reservation the book by Simon Singh, at least for a lay audience (don't know how an expert would experience it).

It's understandably the natural place to get the dramatic tension from what at least purports to be a sober account with some momentum. It's reasonably consistent with the way it's treated on the lectures I've seen from Sir Andrew Wiles on YouTube.

It sounds like a nightmare. He had intentionally set the proof up in a high stakes way, working in private, very quiet on what he was really doing. I gather this is not the done thing, eccentric at best kind of vibes, and I think by that point it was almost crackpot adjacent to even try seriously. His advisor forcefully pushed him off even trying earlier. So it's compound on the stakes. The beg reveal is intentionally as dramatic as possible. And then the questions start to land, most are minor clarifications, notation deficits, but one is sticky, one won't go away.

Kept me up late reading about it.