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Human mathematicians are being outcounterexampled

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Re: Human mathematicians are being outcounterexampled

#131

Post by fuglede_ »

remus wrote: post #39
parpfish wrote: post #11

Earlier quoted context omitted.

proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians

I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.

Right; see Lakatos. In its roughest form, you study the structure of whatever counterexamples you find, add those as (negated) preconditions to your proposition, rinse and repeat until you have a true statement. If the proposition remains useful, you now have a new definition.

Re: Human mathematicians are being outcounterexampled

#132

Post by kqr »

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…

> 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.

This kind of professor/researcher/teacher needs more praise. One of the first engineering courses I took when I started out in higher education was taught by such a person.

Maybe it's just me, but I never felt so welcomed and included during my time in higher education as when that lecturer told a bunch of first-year students "here are some things we haven't figured out which you can help with, let me know if you come up with something". It was inspiring and a great introduction to what's otherwise a rather dull first couple of years of academia.

Re: Human mathematicians are being outcounterexampled

#133

Post by SiempreViernes »

OscarCunningham wrote: post #126
veunes wrote: post #118

Earlier quoted context omitted.

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

They're trained on human data. I would expect them to emulate human biases as closely as possible.

Is it? I'd expect most of the training set to be synthetic data extrapolated from a small set of human authored texts.

Re: Human mathematicians are being outcounterexampled

#134

Post by jibal »

kqr wrote: post #132
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…

> 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. This kind of professor/researcher/teacher needs more praise. One of the first engineering courses I took when I started out in higher education was taught by such a person. Maybe it's just me, but I never felt so welcomed and included during my time in higher education as when t…

https://en.wikipedia.org/wiki/George_Dantzig

> During his study in 1939, Dantzig solved two unsolved problems in statistics due to a misunderstanding. Near the beginning of a class, Professor Neyman wrote two problems on the blackboard. Dantzig arrived late and assumed that they were a homework assignment. According to Dantzig, they "seemed to be a little harder than usual", but a few days later he handed in completed solutions for both problems, still believing that they were an assignment that was overdue.[4][6] Six weeks later, an excited Neyman eagerly told him that the problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[7] This story spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time, some facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the 1997 film Good Will Hunting.[6]

Re: Human mathematicians are being outcounterexampled

#135

Post by lou1306 »

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…

> he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.

For a more extreme (although somewhat inverted) version of this, see Zeeman. He spent years trying to find a knotted sphere in a 5D space. Then realised this was impossible and got a proof for it in a few hours. [1]

[1] https://ima.org.uk/28009/sir-erik-christopher-zeeman-the-mat...

Re: Human mathematicians are being outcounterexampled

#136

Post by jibal »

parl_match wrote: post #62
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…

> I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour. He spent an entire weekend before having the wisdom to pause, and let someone else contribute their time to finding a counter.

> having the wisdom

Ahem.

> 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 was a parallel effort ... we don't know how many people were working on it that weekend. And since the professor wanted it to be true and presumably believed that it was true, why the heck should he wait for students of unknown number and ability to find a counterexample that he didn't think existed?

Re: Human mathematicians are being outcounterexampled

#137

Post by jibal »

vintermann wrote: post #114
angry_octet wrote: post #54

Earlier quoted context omitted.

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…

There are mathematical statements that have been shown to be undecidable (no proof for or against is possible) and such a demonstration is a huge deal.

Re: Human mathematicians are being outcounterexampled

#139

Post by elias_t »

I wonder if at some point mathematicians will be over-flooded with proofs to check and eventually some over confident false claim will make it into math.

Maybe in the future the work of Mathematicians will be like the ones of SWEs with AI, check thousands of lines of AI generated proof and find the subtle errors

Re: Human mathematicians are being outcounterexampled

#140

Post by jobigoud »

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…

How interesting I was just reading yesterday his paper "An enduring error" about how we have been miscounting the Archimedean solids for two thousand years.

But also, for this conjecture to be wrong is quite surprising to me. Intuitively I would think any convex polygon to be topologically equivalent to a circle, and any convex n-gon should be deformable into its regular version, then back to the other one…

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