Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Human mathematicians are being outcounterexampled

xenaproject.wordpress.com

Post reply on HN
Print view View original

41–50 of 265 posts

Re: Human mathematicians are being outcounterexampled

#41

Post by chowells »

taneq wrote: post #22
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

Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.

It's elegant if all you're concerned with is whether a conjecture is true or false. Answered, move along!

But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that?

That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding?

Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.

Re: Human mathematicians are being outcounterexampled

#42

Post by hgoel »

zeroonetwothree wrote: post #13
criddell wrote: post #12

Earlier quoted context omitted.

For now. I wonder if we will ever get to the point where the computer starts doing mathematics that we just can't understand. Surely there must be some limit to what we can understand (like how a gorilla will never understand prime numbers, there are probably limits to our intelligence as well).

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.

Re: Human mathematicians are being outcounterexampled

#43

Post by mb7733 »

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…

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.

Re: Human mathematicians are being outcounterexampled

#44

Post by delecti »

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

You could spend the rest of your life coming up with conjectures that look elegant but are ultimately false. Disproof by counterexample only works if it's false, and we shouldn't be satisfied with a false conjecture to begin with.

Re: Human mathematicians are being outcounterexampled

#46

Post by moralestapia »

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

Not much worth in understanding a statement that is wrong and has been shown wrong.

Unless you want to spend time "proving" that 2 * 2 = 1.

Re: Human mathematicians are being outcounterexampled

#47

Post by OG_BME »

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…

ChatGPT's idiomatic translation of the poem:

Yu Xin’s was a life of utter desolation; in old age, his poems and rhapsodies stirred the riverlands.

Re: Human mathematicians are being outcounterexampled

#48

Post by skinner_ »

paulpauper wrote: post #3

mathematicians have been using computers for well over half a century, but this was after "bounding" the problem first and then running through the cases with a computer. Now AI is doing the first part. However, mathematicians are still needed at crafting prompts, and knowing where to look, still. The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinat…

Okay, I'm not sure about the original one, but here is the prompt of a successful reproduction:

https://aaronlou.com/jacobian_counterexample_prompt.pdf

Obviously it is not random, but it's very generic. No mention of search space or how to reduce it.

Re: Human mathematicians are being outcounterexampled

#49

Post by SpicyLemonZest »

mb7733 wrote: post #43
SpicyLemonZest wrote: post #32

Earlier quoted context omitted.

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…

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.

Re: Human mathematicians are being outcounterexampled

#50

Post by jameshart »

jknoepfler wrote: post #10
skinner_ wrote: post #7

Earlier quoted context omitted.

> The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials. Maybe the prompt contained a part like this: "the search space is too big to just try all the combinations of 3 variable polynomials, so be clever about it". Or maybe this part was omitted from the prompt, because modern LLMs are smart enough to figure this out wit…

If someone has written that in a paper or article they've ingested, as they no doubt have, then sure.

The ability of LLMs to solve problems is not confined to the training data they ingested. Claude knows how to read mathematical papers because of its training data, but it can and will pull in literature relevant to a specific problem into its context.

We really need to stop thinking about LLM training data as the knowledgebase they build from and instead consider it more the skillset they start with.

Post reply on HN