Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.

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_alternator_ 2 hours ago | parent | next [-]

Yes, and they will. But what's happening here is that the system that cultivates mathematics (and mathematicians) is recieving likely the biggest shock of its history. How do you reward merit and identify talen when people can't absorb the number of proofs being generated, much less understand them? Perleman's proof of the Poincare conjecture took several years for the mathematical community to digest; the proof of Navier-Stokes will probably take a similarly long time. In the mean time, it looks like all open problems will be solved (or proved that they can't be solved).

It's not that the horizon is expanding because of this. It's more like a forest getting clear-cut.

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layman51 an hour ago | parent | next [-]

This reminds me of the time an AI was taught how to play a racing sim game (Gran Turismo if I remember correctly). The AI was able to race its car very well, but it took a lot of risks that a human player probably would not. A human player might be able to copy the approach the AI took, but they would probably crash.

Going back to chess, I think the situation is similar where you can’t expect an amateur player to get better by trying to play like a strong engine. I think even professional chess players mainly use engines to prepare or memorize variations that are counterintuitive for their opponent. In other words, getting into situations that look wild, but that part of one player’s preparation.

I’m not sure how it is in math, but in chess, it seems like top players can play just like engines when they are in “normal” positions, so that is where I get a bit confused as to where the direction of insight is coming from because it’s been my view that AI is able to make leaps that we would never think of taking and I’m not sure that anyone could actually learn how to do that on their own unless they were willing to keep failing over and over.

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johnsmith1840 an hour ago | parent | prev [-]

So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before?

Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?

If this is V0.5 of AGI/ASI then by V1 the only system that will be understanding any of this is the AI itself. If AI creates a new field of mathematics month 1, then solutions to new problems in month 2, then another field of mathematics on top of that at month 3 there's no human who will ever keep up with that.

Or the alternative is a flattening of abilities, the AI cannot proceed further than the collective intelligence of humans and in that case this is correct. We'd be in a future where nobody wants to work in a field with an AI dominating it and when AI hits the limit of no useful training data input we'd have this giant gap of nobody know wtf it's done for years and nobody willing to figure it out and advance it.

Ooo here's a dytopian story: - AI gets better at everything humans do - humans stop trying - AI cannot improve anymore than its input data + human support - AI slowly degrades itself (model collapse) for decades, it slowly hallucinates little by little until its hallucinating entire scientific fields losing quality over time - there's a mass population of people in the future who never learned to do anything and now have to relearn and figure out the equivalent of 100k years of AI work in order to prevent its slow degredation while all the systems they've come to rely on start failing around them. The AI has solved every problem but every real solution is saturated with 1000 false ones. - humanity starts from scratch?

I love the idea of an archive of every solution to every problem existing but it's impossible to figure out the correct one. Infinite library like!

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qlte 6 minutes ago | parent | next [-]

Pure mathematics (defined by anything without a known application) exists not to "solve problems" in the real world, but by whatever mathematicians find interesting or lacking in current knowledge. Based on the agreed set of rules formed over time that ensure rigor.

It just so happens that even bizarrely esoteric math can later turn out to have some extremely useful and economically valuable applications. And even more useful to have mathematicians available who already understand that specific math.

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keithnz 34 minutes ago | parent | prev | next [-]

there's a book I read "The Practice Effect" such that technology becomes super advanced based on using something, it gets better and better, but the people regress and become more like a medieval society as they just care that using things improves them.

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SpicyLemonZest an hour ago | parent | prev [-]

> Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understand proofs if they are correct?

Most modern mathematical problems are sufficiently abstract that their proofs or disproofs have no direct application. There's no problem you can fix or invention you can build based solely on OpenAI's construction, because analytic solutions to the Navier-Stokes equations are not used for practical purposes in fluid dynamics. The problems and their proofs are only interesting to the degree that they help us better understand how the math works.

IIUC the Navier-Stokes proof is understandable by human beings, but if it weren't it would be no more useful than a proof that 3 dimensional florg-complete entry seams have no durdle-nodes.

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johnsmith1840 an hour ago | parent [-]

So...why can't an AI do the exact same thing? Make AI so it understands math better for future math to understand more math.

Unless your argument is that mathematicians are effectively useless?

I am assuming that's not your point though.

	▲	gpm 30 minutes ago \| parent [-]
		> Unless [...] mathematicians are effectively useless? It's always been a bit bizarre that this isn't the case. Mathematicians are almost always working on problems that there is no good reason to expect to have utility in the real world... problems they selected because of their elegance or whatever... yet there is a strong historical trend of their work having huge importance after the fact. Sometimes in fields that weren't even invented yet at the time of the work. There's something to be said for the idea that disrupting a system that is working well for no apparent reason is a bad idea.

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lalalanananana 2 hours ago | parent | prev | next [-]

It's possible but the approaches these tools take are usually verbose and strange. Think about it like anything else llms do. Even when the picture is right and there are only 5 digits on each hand all the textures are off and so is the lighting and postures. Or in code, the code is always way larger then it needs to be and tightened up strangely with weird loose ends. Or in writing weird idioms, words, structure, and a weaselly way to turn 3 sentences into 8 paragraphs.

People usually use these tools in math and science to find an answer. Then often they will work it back using more sane or human pathways. So it's shareable or even beautiful.

Knowing the answer has value. But, often in math the best thing was how someone got there.

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

Pure math is practiced mostly for the intellectual thrills and peer recognition among a very small group of peers. There's little else to it. You don't become rich, you don't become a celebrity. You teach students, write papers, and probably know most other people who work in the same subfield as you. Now, Tao is a sort of a celebrity of the quarter on HN, but I promise you that outside this forum, almost no one has ever heard of him.

If you take that away and turn math into a less fulfilling pursuit where you mostly try to make sense of the output of an LLM, and it's "Astra's theorem #18398" and not "John Doe's last theorem", I'd wager that far fewer people will have any interest in the field.

This is really not unique to math, by the way. AI is undermining a lot of creative work. Why blog when you have much better odds of making it to the top of HN with autogenerated blog-slop? Why write books when many nonfiction categories on Amazon are now dominated by AI? The list goes on.

There's plenty of people on HN who think it's nothing new, ignoring the huge change in scale. And those who think this is good because there's no inherent value to human creativity if we can get the same content faster and for less. I disagree.

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ReflectedImage 2 hours ago | parent | prev [-]

Well no because it works by joining together existing novel insights.

	▲	roywiggins an hour ago \| parent [-]
		Today, maybe. Where's the law of nature that says it won't be generating novel insights in two years? Five? Ten?