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 was the sort of thing that he really wanted to be true; he liked things smooth and beautiful. I, on the other hand, hoped it was false as I like the weird and exceptional in mathematics. 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.

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.

My single (quite small) contribution to mathematical research was a counterexample because it was all I could do. The story does illustrate that it can be helpful to have people with different tools, hopes, and motivations working on a problem, though. I was not, and will never be, even a shadow of that great mathematiciam I studied under, but on that occasion, I had reason to look in a different direction than he did.

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bananaflag 18 hours ago | parent | next [-]

> 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 frequently is beyond my comprehension.

But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.

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Dove 12 hours ago | parent | next [-]

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 practical for a young grad student to look for a clever misbehaving polygon than to try to prove something about all of them at once.

	▲	jobigoud 7 hours ago \| parent \| next [-]
		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…
	▲	bananaflag 10 hours ago \| parent \| prev [-]
		Thanks! It makes sense.

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veunes 9 hours ago | parent | prev [-]

I think the asymmetry depends on the representation

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veunes 9 hours ago | parent | prev | next [-]

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

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tantalor 4 hours ago | parent | next [-]

That's not why.

It's because counterexamples are easy compared to proofs which require new mathematics.

GenAI is great at combining existing things in new ways (interpolation). It's terrible at creating new things from scratch (extrapolation).

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jebarker an hour ago | parent | next [-]

This is the wrong way to think about mathematical (or any other kind of) creativity in my opinion. In the extremely high-dimensional space of "ideas" (whatever that means) there are almost certainly profound ideas that are the interpolation of existing knowledge, i.e. the curse of dimensionality. It's not at all clear that you need to extrapolate from existing knowledge to be creative.

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vonneumannstan an hour ago | parent | prev [-]

>GenAI is great at combining existing things in new ways (interpolation). It's terrible at creating new things from scratch (extrapolation).

I think you believe a fallacy about how human cognition works if you think we actually do something different than interpolation

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datsci_est_2015 an hour ago | parent [-]

I think this is somehow related to GenAI’s issue of “confident incorrectness”. I’ve been trying to prompt GenAI with some highly challenging prompts recently, especially terse ones:

> Which Anjunabeats and Anjunadeep compilations have the most breakbeat tracks?

GenAI was able to give me breakbeat tracks that were on the Anjuna labels, but it completely flubbed on whether and which those tracks were on compilations. It was very confidently, mostly incorrect.

I think this is somehow isomorphic to the “interpolation vs extrapolation” issue. GenAI is forced to generate an answer, and there’s no mechanism by which we can interrogate the model for its confidence (at least, that’s my understanding).

And yet, I asked “How confident are you in your answer along different dimensions, ie whether those tracks are breakbeat, whether they are Anjuna tracks, or whether they are on those compilations?”

And its answer was surprisingly satisfying, it only gave 30% confidence that the tracks were on the compilations, which is where it flubbed.

Anyway, personal observations, GenAI is meant to be interacted with, not just a single prompt black box.

	▲	gwerbin 3 minutes ago \| parent [-]
		Out of curiosity which model & interface did you use? I'm starting to think that, for IR tasks like this, the number one differentiator among models and harnesses is the ability for the model to look at its available evidence and conclude that it doesn't know the answer. It depends a lot on how the chain of thought goes. I have seen a lot of newer models try to do this in their thinking traces, and it seems very hit or miss. Even Opus 4.8, GPT 5.5, and Gemini 3.1 Pro (haven't tried 5.6 yet) confidently make up BS sometimes and needed to be reminded to verify thnings instead of inferring or relying on "memory" from training data. IMO harnesses need some kind of built in "are you sure about that?" loop checkpoint that pauses the main interaction and asks the LLM to evaluate the strength of evidence for a claim it's about to make. It seems almost unreasonable to expect an LLM to next-token-predict its way to such a conclusion, reliably, without prodding. That or we just need another year of RLHF and another 250 billion parameters, IDK.

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refactor_master 5 hours ago | parent | prev | next [-]

This reminds me of the Go Grandmaster speaking out after losing to AlphaGo, that the model has no sense of "aesthetic play", as long as it would lead to a win within the rules.

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randusername 3 hours ago | parent | next [-]

I thought there was a lot of buzz about AI creativity after the infamous move 37 in that series?

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voxic11 3 hours ago | parent [-]

Yes but it was so shocking because it was such an inhuman, "unaesthetic" play. It was considered to be "creative" in that no human would have thought of making the move, so it can't simply be copying human play.

	▲	jebarker an hour ago \| parent [-]
		Indeed. To get "aesthetic" play you would have to mimic real human play for an interesting reason: the space of possible Go configurations and games is so mind-bogglingly vast that all of human history has only ever seen a infinitesimal sample. So any sense of aesthetics is just a consequence of chance and memetics. AlphaZero is effectively exploring a new branch of Go history so its aesthetics are completely different. Although maybe that just shows that the idea of aesthetics here isn't very meaningful.

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nh23423fefe 3 hours ago | parent | prev [-]

if humans could do it, it would be called beautiful. defining something to be ugly is cope

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OscarCunningham 9 hours ago | parent | prev [-]

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

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avianlyric 6 hours ago | parent | next [-]

This is where harness, and the fact that a machine can be endlessly prompted to try again comes in.

Even if an LLM starts by pursuing things that follow human bias, continuous failures and re-prompting to try something different will eventually force it to consider things outside of what ever biases it has.

You can do the same thing to a human. But most people would consider it unethical to lock someone in a box and force them to keep trying to solve the same problem over and over again until they figure it out.

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2b3a51 6 hours ago | parent | prev | next [-]

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.

	▲	sebastiennight 4 hours ago \| parent [-]
		> 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? You might be fascinated when you read the story of Golden Gate Claude: https://www.anthropic.com/news/golden-gate-claude

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catigula an hour ago | parent | prev | next [-]

No, they are not. Specifically, this was a method based specifically on learning from scratch, like most modern AI models.

Why do you think it's called Alpha ZERO?

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voidUpdate an hour ago | parent [-]

Programmers love to give things names that they think sound cool

	▲	catigula an hour ago \| parent [-]
		That's nice. It's called Alpha Zero specifically because it was trained from scratch - zero, not on human data.

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SiempreViernes 8 hours ago | parent | prev | next [-]

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

	▲	TeMPOraL 5 hours ago \| parent [-]
		Most of the training set is half of the Internet. LLMs are pre-trained on general set of human biases and patterns of thinking.

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epolanski 4 hours ago | parent | prev [-]

It's more complex than that, especially as post training is often goal based.

	▲	OscarCunningham 4 hours ago \| parent [-]
		I wouldn't have expected that there was post training specifically on the issue of looking for proofs vs counter examples. But it might be that other post training has a side effect of making AIs better at looking for counter examples. I wonder if these agents are overall less biased and more rational than humans. Can you expand on what you mean by goal based training?

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kqr 8 hours ago | parent | prev | next [-]

> 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.

	▲	jibal 8 hours ago \| parent [-]
		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]

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codemog 17 hours ago | parent | prev | next [-]

There’s a story in How to Solve It that’s basically the same.

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lou1306 8 hours ago | parent | prev | next [-]

> 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...

	▲	gowld 3 hours ago \| parent [-]
		Trying and failing to prove something tells you quite a lot about what a counterexample would look like.

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parl_match 16 hours ago | parent | prev [-]

> 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.

	▲	Dove 12 hours ago \| parent \| next [-]
		This was back when the internet was mostly chain emails and personal web pages, being unreachable once you went home for the weekend was perfectly normal and expected, and automatically thinking the worst of people was not a common form of public performance art. ;)
	▲	jibal 8 hours ago \| parent \| prev [-]
		> 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?