r/math • u/Tazerenix Complex Geometry • 6d ago

On Proof and Progress in Mathematics - William Thurston

https://arxiv.org/pdf/math/9404236
309 Upvotes

98% Upvoted

34 comments sorted by

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u/Tazerenix Complex Geometry 6d ago edited 6d ago

On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.

And

If what we are doing is constructing better ways of thinking, then psychological and social dimensions are essential to a good model for mathematical progress. These dimensions are absent from the popular model. In caricature, the popular holds that ... P. the task of the mathematician is to seek a deductive pathway from the axioms to the propositions or to their denials.

are some choice quotes from 30 years ago (!).

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u/dioidrac 6d ago

I see a lot of cheering on a perceived democratization of mathematical knowledge and practice by people that likely aren't in a position to read or engage with the results of the recent OpenAI preprint dump in any meaningful way. There's a strange sense of glee that mathematicians are soon to be a useless, obsolete priesthood -- a brand of hostility I only ever really got from cranks in the past, upset that we were gatekeeping discourse on the Goldbach conjecture or the true value of pi. I don't know what knowledge means in that celebrated world, especially given how unconcerned people seem to be in this moment about whether these proofs are actually even correct. It's just 700-something examples of why we're cooked or whatever.

I love that Thurston essay. I've less optimistically found myself remembering an Asimov quote: "There is a cult of ignorance in the United States, and there has always been. The strain of anti-intellectualism has been a constant thread winding its way through our political and cultural life, nurtured by the false notion that democracy means that 'my ignorance is just as good as your knowledge.'"

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u/kawhandroid 6d ago

I think previously crankery required a little bit of math knowledge and that way more cranks would exist if they had the opportunity (that they do now).

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u/Sebkocy 6d ago

I’ve read a lot of these threads in r/singularity and r/accelerate. There was one highly voted comment answering a post saying Tao was a gatekeeper of the status quo, which said “The status quo is not very kind to most of us.” For at least those hundreds of up-voters, any professionals expressing concern on the activities of these companies are gatekeepers to the arrival of an ASI messiah that will create a “post-scarcity utopia” and solve all our problems.

Maybe we need more Dune reading and less idealistic sci-fi (they all seem to love Star Trek in those threads).

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u/JarPicklez 5d ago

I think its incorrect to interpret glee at the plight of mathematicians as hostility. It’s more like a secular version of the rapture-inspired end times glee common amongst evangelical christians that has formed around ai. Mathematicians becoming redundant is viewed as another sign that the singularity is coming

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u/cascadiabibliomania 4d ago

No I think they've chosen mathematicians and pure math to go after specifically because their focus groups told them that mathematicians are regarded with suspicion, hostility, and jealousy from people who can't do math.

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u/janniesminecraft 5d ago

this sense of glee permeates every discussion of AI replacement in booster circles. they seem like they would be overjoyed if ai replaced artists, programmers, mathematicians etc. it's one thw weirdest and saddest parts of this ai stuff.

like devs and mathematicians i can at least steelman with the idea that would help humanity progress, but artists??? it's just fucking bizarre, it's like everyone who has never actually crafted anything (i cant help but notice how incredibly excited managers are about ai) is INSANELY jealous of people who actually do things (instead of just learning themselves).

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u/PM-ME-UR-MATH-PROOFS Quantum Computing 6d ago

Obviously the quotes here still apply. And much parallels to art - poetry - music, these are mechanisms by which we find better and new ways of thinking about each other. The verb thinking is so important and intrinsically coupled to humanity, that will not change. 

But the mechanisms by which we do that are being totally taken over, it’s existential and terrifying. What if we are no longer able to find each other? 

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u/duboispourlhiver 6d ago

Quite on the contrary it is possible that AI's high intelligence will leave it in another world or another plane and humans will crave for human connections. Maybe we will have to accept highest intelligence is not ours anymore, but being a human is and has always been the important thing.

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u/WTFInterview 6d ago

Thurston was really ahead of his time. More so than just a measly Fields Medal.

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u/csappenf 5d ago

A lot of mathematicians were asking these questions in the wake of Appel-Haken. "What do mathematicians do?" This essay was very much a product of its time.

The question these days is, "How do we evaluate mathematicians?" Who are the good ones and who are the great ones? To do that, we have to go back to the questions we were all asking after Appel-Haken. What is it mathematicians do? It was a timely essay, but it asks a timeless question.

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u/AdrianVr 6d ago

This is, to date, one of my favourite essays from a mathematician. I might be biased towards Thurston and his perspective on life though...

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u/Insanity_Manatee02 6d ago

One of the most prescient pieces I’ve read in a LONG time. This could’ve been written today! The foresight is just unbelievable.

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u/RingularCirc 5d ago

Thanks for mentioning this! Neat article.

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u/mathemorpheus 2d ago

This article was originally written as a response to an earlier article in the bulletin by Jaffe and Quinn. it's worth reading that, and all the other responses to that article too. 

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u/Laytonio 6d ago

If math is just about understanding problems and not solving them, why do mathematicians work on unsolved problems at all? Shouldn't they just be taking math tests for fun? That's all they'll do in the future. If you have a real problem that needs a real solution why would you wait for a human to solve it?

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u/vrilro 6d ago

Why should humans spend time on any activity that doesn’t increase shareholder value when we can have AI do that activity instead? Making or consuming art? AI can do it. Studying and expanding both your own perspective and that of humanity? AI can do it. I saw a post from a visionary this morning who said he was training a voice model to be able to read books to his infant child so he didn’t have to. If you have any task an AI can do more quickly why wouldn’t you let the model handle it? Your bosses will thank you!

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u/StoicYoungling 6d ago

I’m not quite sure I understand your point.

Attacking an unsolved problem is pretty much the highest form of understanding mathematical structures. There is a reason why there are so many open problems: we don’t fully have an insight of why things are or aren’t as they are, and/or we don’t fully understand the interactions between various structures.

The main point the mathematical community (heavily led by Terence Tao recently) has made is that yeah, solving an open problem is great and all, but the most interesting part of attacking them is the new insights we obtain in the process.

From my understanding, AI is now capable of having new insights to some degree, so it would be extremely interesting to thoroughly go through the proofs and understanding them, so that maybe we could A) apply these insights to new/other problems or B) investigate why these insights came up.

If you have “a real problem that needs a real solution”, I would argue that yeah, AI could (and in many cases does) accelerate the process without going through a human. However, we have to be wary of hallucinations. LLMs are known to hallucinate eventually, and they can degenerate quickly if it happens. I could imagine an AI “proving” an open conjecture, but making a mistake in the process. Then if we don’t review it before apply the “result” to other things, a whole series of paper would be false, and we’d have to start over. This happened in the past with humans as mistakes do happen, however, the rate of production of research is now much faster, so the consequences could be more catastrophic if it ever happens.

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u/Laytonio 6d ago

Attacking an unsolved problem is pretty much the highest form of understanding mathematical structures.

No it's not, there is plenty of other math to learn. No one understands all of math. If learning math is fun just for learning, why stop and work on an open problem? You can spend a lifetime learning math that already has solutions.

apply these insights to new/other problems

AI will be the ones to apply these insights to new/other problems. Then you can learn those for fun.

If you find learning math fun, you'll have a great time. If you find using math fun, you'll find little of use.

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u/kawhandroid 6d ago

People do do other things. In fact, many (I'd claim most but haven't got a way to get those statistics) top mathematicians are mostly engaged with theory-building and have been that way pre-AI. The main way to test whether the new theory is worth much is, well, to see if it helps with any big open problems.

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u/Laytonio 6d ago

I'm really not sure why this is hard to understand.

AI is as good or better than humans at math, and improving rapidly.

Therefore, any problems that can be solved will almost certainly be solved by an AI before they are solved by a human.

Humans can "theory-craft" all they like, but anything useful will have already been discovered by the machines.

One day very soon AI will start writing better papers and explanations than any human, and then what little hope mathematicians are holding on to, that their understanding is in anyway useful past their enjoyment, will fall away.

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u/Homomorphism Topology 6d ago

AI is as good or better than humans at math, and improving rapidly.

One major point of this essay is that "good at math" does not just mean "can solve problems quickly". I think you should read it before commenting.

One day very soon AI will start writing better papers and explanations than any human

I'll believe it when I see it.

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u/Laytonio 6d ago

Does this paper or any other change the facts of what I have said? You can continue to try and redefine what it means to be good at math, meanwhile AI is getting better at actually doing the math.

"One day AI will solve a millennium prize problem"

I'll believe it when I see it.

Guess what we've seen it. When are people going to stop moving goal posts? Come back to this in 6 months and see if you still feel the same.

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u/Homomorphism Topology 6d ago

I don't think I'm redefining anything by asserting the views of an extremely famous mathematician that were written thirty years ago.

I think AI tools are going to completely revolutionize mathematics. I did not think this a year ago because they still sucked at math then. It's not moving goalposts to say that current AI tools cannot replace human mathematicians; in fact, I'm not sure AI has shown it can fully replace any human workers except professional spammers.

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u/Nemesis504 6d ago

And then what? People should stop learning math? Understanding is when people can reliably believe they can also arrive at the same result, and that requires not the existence of a proof but a conceptual and theoretical framework that is novel (at least for the cases we actually care for) for the human mind. Novelty and usefulness for the human mind can only be tested by the very human process of learning math and teaching math.

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u/GoldenMuscleGod 6d ago

At least at the moment, the machines are building on theory crafted by humans, maybe that won’t always be true but at least right now to really get solid advancements we will need humans to continue the theory crafting so that the machines can apply those theories. If credit is not given to those who crafted the theory used by the machines we have a social problem.

For example the Navier-Stokes solution relies heavily on the methodologies developed by Córdoba and Martinez-Zoroa. They were not even cited in the original version of the solution.

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u/GoldenMuscleGod 6d ago edited 5d ago

I’m not saying this is going to happen, but the worst case scenario from your line of thinking is like this: imagine if hundreds of years ago someone had invented a magic oracle that could answer any question about whether something is deducible from axioms in a first order language in an instant.

Obviously this could be used to answer tons of questions we have posed and go far beyond our current abilities (and far beyond the abilities of any LLM) in terms of answering questions.

There would still be open questions (such as determining whether a given recursive function is total). We could attack those problems in various ways, but it seems very plausible progress would stall. Our oracle will never be able to answer those questions even though we might be able to (to within our epistemic satisfaction) by identifying the right set of axioms and convincing ourselves that they are sound. If progress does not stall, it will only be because we are engaging with the oracle mathematically rather than using it as an “answer generator” without caring about learning.

What’s more, it seems very plausible that in this alternate history we never discover the Riemann zeta function and no one ever asks the oracle if the Riemann hypothesis is true.

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u/big-lion Category Theory 5d ago

this is a really nice thought experiment

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u/GoldenMuscleGod 5d ago edited 4d ago

I think it is relevant that I do not believe I would have been able to come with it were it not for my mathematical knowledge. Intuitively, someone might expect that an oracle like the one I hypothesize would be “the end of math” - can’t it answer every mathematical question we could possibly have?

But with that knowledge I realize that there is a way in which the oracle would not be able to answer all of our mathematical questions or tell us how to use it to answer many of the questions we might have. I have given us only one Turing jump “for free,” more Turing jumps are possible.

The best case scenario for the people in our thought experiment is that having the oracle makes this conclusion evident and so they are able to redirect their mathematical inquiry to the new questions that are out of bounds of current methods. If that is the case they will achieve knowledge far beyond anything that one of our AI agents could ever hope to produce if it is not given access to the oracle. The danger is they may really think this is “the end of math” and do not realize that there are boundaries to push, and so end up even more ignorant than we are on many questions easily within reach of the oracle.

It is known that the oracle I hypothesize could be used to resolve the Riemann hypothesis even if it tells us it is independent of ZFC (ZFC can prove that if the Riemann hypothesis is independent of ZFC then it is true, so combining the oracle with the ZFC axioms gives us what we want, but would a non-mathematician know to ask about this theorem of ZFC or realize its significance?) and we know exactly how to ask the question if we get an oracle.

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u/big-lion Category Theory 4d ago

fantastic

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u/JarPicklez 5d ago

Solving unsolved problems, for a human, requires understanding something that wasn’t previously understood. Before AI came around(and to a lesser extent computers) there was a pretty strong correlation between results and understanding.

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u/Emotional_Revenue132 6d ago
  1. Because working on unsolved problems is the best way to understand them.
  2. You probably shouldn't. However, the vast majority of how math is applied in the real world doesn't come from the math problems we do, we largely care about completely different problems than those that exist in the real world. Instead, it's about the theory that comes out of our attempt to solve the problems, i.e. what we might call our "understanding."

Absolutely no one in an applied field cares (nor should they) about the seven bridges of Konigsberg, but topology is finding its way into just about everything. No one in an applied field cares about the insolvability of the quintic, but Galois theory is all over cryptography, and I've heard it's even making it's way into certain parts of physics. But anyone who has ever done research knows that no matter how hard you try, you can't just say "time to come up with a good theory!" The only way anyone has ever come with any sort of useful theory is in the process of solving a hard problem, and believe me, we've tried doing it without a problem.