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Human mathematicians are being outcounterexampled

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Re: Human mathematicians are being outcounterexampled

#81

Post by AlotOfReading »

hgoel wrote: post #42
zeroonetwothree wrote: post #13

Earlier quoted context omitted.

Mathematics only really matters insofar as humans can understand it.

Aren't deep learning models themselves a case where we have hints of some deeper underlying logic to why some things are more effective than others, but we lack the mathematical tools to properly work it out for anything of practical size? All we're able to do is apply flawed analogies, generic information theoretical models, trial and error, post-hoc rationalizations and benchmarks without really understanding why .

You don't need anything as recent as deep learning for that. Look at economics or social sciences, which have been influencing national politics for well over a century now.

Re: Human mathematicians are being outcounterexampled

#82

Post by lioeters »

dzdt wrote: post #40

I suppose it will fall to AI as well to compose the mathematical equivalent of The Ballad of John Henry. Who will be the human champion, the last great hero who can deliver proofs "from the book" that a machine cannot outperform? [1] https://en.wikipedia.org/wiki/John_Henry_(folklore) [2] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

"Gonna Die With My Hand-Written Proof in My Brain" - Recorded in 2027 and compiled in the Anthology of American Folk Mathematics (2052)

Re: Human mathematicians are being outcounterexampled

#83

Post by ChadNauseam »

SpicyLemonZest wrote: post #49
mb7733 wrote: post #43

Earlier quoted context omitted.

Those are all good questions, but I don't really understand what alternative you or the OP are looking for actually resolving an untrue conjecture, besides a counter example.

I would frame it differently. The existence of compact counterexamples to a true-seeming conjecture suggests that there’s some deeper understanding waiting to be discovered. Fuzz testing for theorems, if that makes sense. I hope mathematicians in 2036 will be able to explain in detail why the Jacobian conjecture was false and identify which similar, true conjectures the community’s intuition was pointing towards.

We can take a simpler example. Let's say someone conjectures that all linear maps are isomorphic if they have the same domain and codomain*. A counterexample is easy to find, but true insight would be to notice that all linear maps with the same domain and codomain that are not isomorphic map some non-zero elements to zero. That is much more interesting than just finding a counterexample. Although, that isn't to say that finding a counterexample is not very interesting.

*statements only apply to maps whose domain is finite-dimensional

Re: Human mathematicians are being outcounterexampled

#84

Post by throwaway676712 »

parpfish wrote: post #11
satvikpendem wrote: post #5

That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.

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

Counterexamples are literally the only way to show a "for all" statement is false. (Non-constructive proofs by contradiction work by showing a counterexample must exist.)

Also, 'brute-force' style attacks where one simply feeds the input into the computer and it yields a solution are nothing new and certainly predate LLMs: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec...

Hell, one could even go further into the past and refer to the thankless work of pre-computer era mathematicians who sweated over manual calculations in order to disprove various prime related conjectures: https://en.wikipedia.org/wiki/Mersenne_conjectures

You seem to have an objection to non-intutionist mathematics in general, a position that was once held by many an illustrious mathematician but is relatively fringe in the contemporary academic community. Mathematical facts don't have to be intellectually satisfying or make sense to you, the human, rather it is up to you to wrap your mind around discovered mathematical facts.

Re: Human mathematicians are being outcounterexampled

#85

Post by linzhangrun »

parpfish wrote: post #11
satvikpendem wrote: post #5

That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.

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

> 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

Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.

Re: Human mathematicians are being outcounterexampled

#86

Post by angry_octet »

jandrese wrote: post #61
angry_octet wrote: post #55

Earlier quoted context omitted.

A recording of a car crash: discovering on live radio/podcast that the central tenet of your book is wrong, and amateurishly so. Naomi Wolf 'death recorded' on BBC[1], skip to 5:51. 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. [1] https://www.bbc.com/news/av/world-us-canada-48639663

[flagged]

No it's not that they don't check facts, it's that facts don't matter. She found a community where poor scholarship and vibes was celebrated.

Re: Human mathematicians are being outcounterexampled

#87

Post by angry_octet »

drnick1 wrote: post #70
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.

A Ph.D. is usually a collection of papers/chapters, so even if a fatal flaw is found in one of the chapters, it does not normally result in "failing the Ph.D." It just means that that particular paper isn't publishable. The bar for getting a Ph.D. is actually quite low; what is truly difficult is clearing the bar in terms of publications for academic positions and later tenure.

You would have to go back and do more work though, and if you couldn't fix it you would have to do another defence. But yes, it is a different situation to actually getting a good post doc.

Re: Human mathematicians are being outcounterexampled

#88

Post by learningstud »

angry_octet wrote: post #57

I wish I had LLM-built Lean formalisations in university, so much of the math in the slides had errors, and some professors are very bad and ungracious admitting it, while simultaneously rejecting requests for clarifications by saying "the proof is in the slides". Of course Lean proofs are rarely a good way to understand proofs, but hopefully they can be used to generate more human understandable arguments.

Yes, or to settle dispute and remove doubt once and for all, i.e. the Leibniz way.

Re: Human mathematicians are being outcounterexampled

#89

Post by vlovich123 »

adw wrote: post #80
vlovich123 wrote: post #71

> A few days earlier I had got an email from a professor in the maths department here at Imperial, expressing surprise that some of our graduate students were paying $200 per month to access models such as Sol and Fable. He said that he thought that these people were crazy. I did not immediately respond. But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who wa…

The living-costs stipend for an EPSRC PhD student is around £20k so it's about ten percent of that... big commitment for a student to make!

Which is why the school should be covering it.

Re: Human mathematicians are being outcounterexampled

#90

Post by adw »

vlovich123 wrote: post #89
adw wrote: post #80

Earlier quoted context omitted.

The living-costs stipend for an EPSRC PhD student is around £20k so it's about ten percent of that... big commitment for a student to make!

Which is why the school should be covering it.

They’re skint too. It’s something you would really like the research council to take on.
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