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Human mathematicians are being outcounterexampled

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Re: Human mathematicians are being outcounterexampled

#141

Post by NooneAtAll3 »

somenameforme wrote: post #97
globnomulous wrote: post #58

Earlier quoted context omitted.

This is astonishing. How could a person go to the lengths of writing an entire book without ever looking up this sort of thing?

The term she mistook for meaning a death/execution had been recorded was "death recorded". So it's a mistake that could be made similarly to how one might confidently assume that a "public school" is a school which is a school that is either government funded or open to anybody in the local area. More generally I think people are inappropriately confident in general. When you go through history just about every centu…

...what is public school then?

Re: Human mathematicians are being outcounterexampled

#143

Post by somenameforme »

NooneAtAll3 wrote: post #141
somenameforme wrote: post #97

Earlier quoted context omitted.

The term she mistook for meaning a death/execution had been recorded was "death recorded". So it's a mistake that could be made similarly to how one might confidently assume that a "public school" is a school which is a school that is either government funded or open to anybody in the local area. More generally I think people are inappropriately confident in general. When you go through history just about every centu…

...what is public school then?

A fee charging privately operated institution that does not receive public funding. [1] Brits have a way with words.

[1] - https://en.wikipedia.org/wiki/Public_school_(United_Kingdom)

Re: Human mathematicians are being outcounterexampled

#145

Post by ykonstant »

elias_t wrote: post #139

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

>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.

That point had come some time ago. Nowadays the literature is both enormous and littered with false proofs and an unknown, but nonzero, number of false published results.

Re: Human mathematicians are being outcounterexampled

#146

Post by ykonstant »

blueTiger33 wrote: post #142

as someone who loves math, I want to collaborate with mathematicians to solve some hard problems

This comment gives a sliver of hope for us currently outside academia and in low income academic institutions: enthusiastic laypeople with access (read: funds) to the SOTA models can collaborate with destitute researchers to produce research the academic alone could not.

Plus, as a non-expert, you will naturally want to understand more about what you are proving together with the expert. LLMs can help there, too, by carving a path from elementary mathematics to the research problem more efficiently and in a more targeted manner than a generic exposition or survey.

That could give birth to a beautiful research-exposition pair that can benefit both academics and interested laypeople, who have (rightfully, but inevitably) felt excluded from the insights of high level research. I have long hoped for something like that. There, however, the academic must watch the LLM like a hawk, because expository interpretation of results is prime ground for hallucinations, and adversarial agents will not have much effect in improving it.

Re: Human mathematicians are being outcounterexampled

#147

Post by 2b3a51 »

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.

Your comment stopped me in my tracks a little bit.

Is a 'bias' in a piece of writing generally a property of word to word choice and sentence to sentence construction or is it something more nebulous? Especially in terms of the appreciation of mathematics and someone's hesitance about publishing a mathematical argument they think is ugly or brute forced in some way.

Re: Human mathematicians are being outcounterexampled

#148

Post by SonOfLilit »

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

The counterexample in the news cycle today helps better understand how the math works. I can't think of one that doesn't.

Re: Human mathematicians are being outcounterexampled

#149

Post by t-3 »

somenameforme wrote: post #143
NooneAtAll3 wrote: post #141

Earlier quoted context omitted.

...what is public school then?

A fee charging privately operated institution that does not receive public funding. [1] Brits have a way with words. [1] - https://en.wikipedia.org/wiki/Public_school_(United_Kingdom)

The US still somewhat has a similar distinction with public and private colleges. Private vs public funding rather than private vs public admission.

Re: Human mathematicians are being outcounterexampled

#150

Post by bananaflag »

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…

> 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".

Indeed there isn't such tradition. I have one or two results like that -- proofs that some proof strategies cannot work because some object does not exist, but since that object would not be interesting for anyone not trying that particular proof strategy for that particular (already obscure) problem, one cannot publish.

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