> Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research. This expert knowledge is essential, both to effectively use mathematics, and to continue to articulate new and significant research questions. In a word, the job of the mathematics department is not only to produce mathematics, but mathematicians. Similarly, the output of programming is not only a program, but also a programmer. It is you. Outsourcing the work deprives you of who you become by writing it.

> Outsourcing the work deprives you of who you become by writing it. Just because AI can do something that resembles work should not mean outsourcing work to it. Mathematicians should not outsource their work to AI just like programmers should not outsource programming to AI. Humans working with AIs in a tight loop means intellectual work becomes more high-level and creative, but a human should always own the work, validate it and stake their reputation to it. Simply ban any humans who produce low quality work using AI.

> By that definition nothing should ever be automated. Many things shouldn't. Understanding is one of them.

To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect. Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable. Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.

> much more interesting than the problem’s elementary statement might lead one to expect This is reinforced by the immediate (human) use of the idea to resolve in the negative another significant problem, the sum-product conjecture on reals. Explanation of what was involved: https://www.erdosproblems.com/forum/thread/blog:6

I am not a mathematician and did not read the unit distance solution too carefully, but my impression was that it used a variation of a known technique to solve the problem. And that makes perfect sense to me, there are a lot of techniques and lot of less relevant problems, I am not surprised that one can solve some of them with known techniques that just nobody has tried [hard enough] before. I am much more sceptical when it come to the important unsolved problems where every known technique has probably been tried several times over. In those instances it will probably take a true leap in understanding to solve them and I am sceptical that large language models are well suited for that because of the way they work.

We're very fortunate to have had some very eminent mathematicians backfill the OpenAI proof with history, context, and a literature review [1]. Ideas behind the proof seem to have been "in the air". Indeed, looked at certain point of view, the OpenAI construction can be viewed as a high-dimensional generalization of a known low-dimensional one. In this vein see the remarks of Gowers, Sawin and Tsimerman in [1]. Are LLMs capable of "true leap[s] in understanding"? I have absolutely no idea. But LLMs keep surprising me. [1] https://arxiv.org/html/2605.20695v1

Do you not think that solutions to erdos problems might end up stepping stones to other important problems? Either by introducing new tools, or by proving things that were previously unproven that end up helping in unexpected ways? That's often how math goes, isn't it?

I think nuance gets lost in these conversations. Your distinction between the practical and the theoretical is important. Practicality is important - everything we do is a matter of practicality of means or method , even how we pursue theoretical ends - but two points. First, there is more to life than the practical. Some truths are known for their own sake, even if they also tell us about still more profound truths (also known for their own sake) or may have incidental practical relevance and consequences in some other context. Second, while the theoretical terminus is the truth for its own sake, the practical terminus is always something other than itself. Well, what is that "something else"? You can't have an infinite regress of practicality. The meaning of a proximate, practical end is always other than itself. The practical requires an end beyond itself to justify it. I agree that most people don't seem to inquire much about such ultimate ends. Their thoughts are confined to the proximate. Of course, how have they determined what the proximate should be? Something for people to contemplate. Where science is concerned, it depends. On the one hand, there are fields that are certainly more theoretically oriented. It's not "the game" that motivates theory - that would make it mere recreation, with the truth taking a backseat - but the truth. (For this reason, I hesitate to call Erdos theoretically motivated. AFAICT, he was motivated by the challenge of problem solving and not the truth, insight, and understanding to be gained which would have been merely incidental and instrumental for him.) However, I would also say a good chunk of science is motivated by a background motivation of technology production and the mastery of nature. Think Francis Bacon who viewed science as an instrument of power and showed a preference for the "how" over the "what" (τόδε τι) or the "why" (τὸ διότι). This set the tone for a great deal of modern science. A great deal does less explaining and more predictive modeling, because predictive modeling can be sufficient for control. Indeed, a truly theoretical causal account and understanding of a thing's nature can be less useful as a practical instrument than a merely predictive model. Now, AI is a practical tool. I think they can be enormously useful as research aids, even in theoretical contexts, provided that one 1. understands their nature; 2. understands the purpose of the theoretical activity undertaken. What is their nature? Well, they're statistical models that can unearth interesting and useful correlations and patterns. But they are not reasoning and knowing things. Their results are generated mechanically and mindlessly. Knowing this means taking their results with a healthy skepticism and a critical eye. What about the purpose of theory? By analogy, think of a student in school who uses AI to complete all his assignments. Has he satisfied the purpose of those assignments? No, because the purpose of the assignments isn't to produce the effect - the solutions - per se, but to learn something. Theoretical work is like that; it's purpose is to understand and to grasp some truth. An AI can be used to assist this process, just as a calculator or a search engine can, but if you use it in a manner that circumvents that purpose instead of supporting it, then you're not achieve that purpose and wasting your time. What's the point?

I cant speak for engineers, but as a mathematician I wholeheartedly disagree with everything you claim in your comment. Almost none of the mathematicians that I know care about the optimization aspect of mathematics: the pursuit of optemizing constants in theorems and providing minor technical improvements is mostly seen as pointless unless there are significant new mathematical insights that fuel the improvement. I think most mathematicians rather build their identity around providing actual understanding of problems using mathematics and improving society's understanding of mathematical problems. Of course AI threatens this too, but the threat is of a much lesser degree. One could even argue that AI is helpful here with getting mathematicians to the 'frontier of knowledge' as AI is usually good in combining ideas from different fields.

Much like for many the point of chess is that it's played by humans, with truly superhuman AI relegated to a training aid, mathematics is in many ways about human comprehension. You can use AI to find and proof new theorems. But if you get to the point where humans can't understand it, is it even still math?

Neural networks are already systems of linear algebra that are beyond human understanding. Most humans could probably grok a 1 or 2 dimensional slice of a network, but the latent vector space is completely beyond the human brain. We have to use tools to analyze neural networks piecemeal in exactly the same way that we analyze any other higher-dimensional construct. Few humans are truly capable of reasoning in 4+ dimensions, that doesn't make string theory "not math". Nor does a trillion-dimension vector space of an LLM make it "not programming". Humans by themselves invented mathematical concepts beyond human understanding a long time before we invented neural networks.

Perhaps P=NP. The new algorithms are handed down to us. We can apply them without fundamentally understanding why P=NP.

Surely such AI would also be able to reduce and simplify the math for human understanding. Which is what mathematicians do all the time, from turning base 60 cuneiform into modern number systems to simplifying Maxwell's equations for the students.

So that is kind of the point of studying maths right? Why something in unsolvable or undecidable can be as important as the output of a theorem. Questions like these, fields medal level problems or Karp’s 21 NP-complete problem are problems working mathematicians are interested in. Will LLMs help as an human assistant in the future? Probably. Will LLMs answer these questions themselves, provide insights and bounds to these new mathematics and teach other mathematicians why this new math they create is true? Will these models have phds and take candidates teaching them how to apply and think about the maths problems they are interested in?

> However, the declaration argues math is more than a machine for producing correct answers. There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds.

The most important part of math is advancing human understanding. A correct answer by itself is not as important as understanding why it is correct.

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for humans to identify what is meaningful and what is not in the slop.

Do AIs produce answers whose work is incomprehensible to humans? It seems like you could just have the AI elaborate multiple times until you were satisfied with the explanation and documentation of what went into figuring out the answer. It’s not like the AI is one shotting the answer in a single opaque query anyways.

> Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. > Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand. This feels like a little bit of a jump to me. AIs arent actually alive so of course someone is going to have to pose the question. They arent going to just do stuff on their own. And of course mathmaticians are going to need to interpret the results if we are to glean anything beyong if the conjecture is true or false. But you seem to be suggesting that mathematicians will have to micromanage every step. That seems like a bit of a jump which i dont see much evidence for.

Excluding supergeniuses, pure mathematics—even at a very basic, undergraduate level—simply can't be understood passively. Even with an infinitely patient AI teacher who could answer any question on-demand, it'd still require a massive amount of work to actually understand anything in research-level mathematics. Basically every single word in a mathematical definition is a term of art, and (IME) if one doesn't grok each of those words at a fairly deep level, the new definition never really makes too much sense. And this applies recursively: each of the words has some thoroughly inscrutable definition of their own. Of course it'd be super helpful to have, say, a teacher who could tailor explanations to anyone's precise background (e.g. where possible, using examples that come from the student's field of study when explaining some abstract concept). Or, if some definition comes with some precondition that has no obvious purpose, perhaps an omniscient teacher could explain why it's there with concrete counterexamples.[0] But even granting all this, I think that mathematical intuition is necessarily based on a lot of hard work actually exploring definitions on one's own, with pencil-and-paper and a lot of thought. That is to say, even though the process could probably be sped up a lot with a nigh-omniscient teacher[1], I doubt that a student wouldn't still need years of training to even have a clue what's going on. (I'm saying all this, by the way, as someone who is terrible at all this and has very little mathematical maturity[2]—I'm speaking from my own frustrating experience....) [0] c.f. Lakatos' excellent book Proofs and Refutations [1] without the "curse of knowledge," or else we're back to square one of "answers that are correct but useless" [2] e.g. the "post-rigorous stage" described in https://terrytao.wordpress.com/career-advice/theres-more-to-...

How do you make use of something that you don't understand?

"What is the answer to the Ultimate Question of Life, the Universe, and Everything" 42

Once you now something is correct, with a proof. It is MUCH easier to understand why it is correct. Than to start from a slate that you don't even know whether something is correct or not. In that sense AI that can just solve high level math problems is immensely useful. It allows a mathematician to explore ideas at a much more rapid pace.

The value of human understanding just cratered because we have machines to understand for us now.

Not quite. You have machines that can aid and expand your (human) understanding greatly, that wasn't possible without machines. Machines don't think. They aren't human. They have no soul, agency/free will, self-reflection/awareness, moral imperatives or ethics. You've been watching too much Terminator, son.

> So it has a connection to human problems, and eventually to practical matters. But in relation to parent post I was replying to, it did not provide an answer or solution to anything. It has much closer relation to philosophy than anything. Focusing on only ‘solutions’ in any field is shortsighted because you can’t know how the dots will connect. Someone’s seemingly pointless curiosity or experiment can unlock something unexpected, just like Boole

For most engineers a mathemetician is a machine for producing correct algorithms, like a chef is a machine for producing tasty food. In both cases that overlooks the human element, but that's a critical skill for a limited mind with finite resources to grok infinite complexity. You can read that as permission to be an asshole or a neccesary compromise.

No, it's not the most important part. It can be argued that most important part is asking the right questions

Assume someone solves P=NP Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?

That's a bit too simplistic -- if there is a small group that really pushes things forward in a big way, then maybe not, but if this result builds upon decades of prior work, then Cook and Levin might be equally or even slightly more famous than the solver group after the dust settles. But it is a moot point anyway. Cook and Levin are very well known already in TCS, and credit is not directly enumerable like money, so "more than a lot of credit" doesn't make too much sense. For this problem in particular, asking the right kind of question was really important for the field and led to a lot of discoveries even before it will be answered.

Depends on the solution. If the solution is that P≠NP, the concept of NP-completeness remains the bigger contribution, unless the proof techniques are particularly interesting and lead to other major results. The same applies if P=NP but the proof does not ultimately lead to a practical algorithm. If we get a practical algorithm, the answer is more valuable than the question.

Do questions like goldbach conjecture, fermats last theorem, etc deserve more credit than whoever answers them?

Fermat's last theorem probably doesn't. While attempts to solve it have led to many mathematical discoveries, the theorem itself feels little more than a piece of trivia. I'm less familiar with the implications of the Goldbach conjecture. In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.

If the problem resolves to P=NP, that result would probably be more celebratee than being able to formulate the problem, but being able to formulate the problem and get people interested in it is probably worth more than the average primal dual trick to prove a polylog integrality gap for some integer linear program.

>> However, the declaration argues math is more than a machine for producing correct answers. > There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds. I can produce an infinite number of verifiably correct papers, if that's all that matters. 1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 1 + 4 = 5 1 + 5 = 6 Shall I continue? Or do you think that choosing which questions to answer might have some level of importance, in addition to getting correct answers?

This reminded me of my 11 yr old who, when I give her math problems to solve, is too focused on “getting the right answer”. I’ve told her plainly, I don’t care if you get the right answer right now, I want to see your reasoning. She has yet to understand this.

A statement that some proposition is true or false is usually less useful than a new framework for understanding the class of problem. A machine that takes longer and longer to prove propositions in ever more inscrutable ways is hardly useful at all. The machine too needs to produce more generalizable and comprehensible systems, for it to scale up its own conceptualization. Needing to load all the new mathematics in the context window won't be great either.

Even from the most purely instrumental perspective, what we care about is our ability to make use of correct answers, which is quite distinct from the possession of correct answers. There are many theorems that aren't directly interesting, but whose proof requires techniques that are of substantial further interest, that lead to new domains, and/or new practical applications. Simply being handed a proof for those theorems isn't enough--we require the ability to apply those techniques in the real world, or discover further areas of mathematical research that build on that proof or its techniques. It may be that AI can build on its own work for the long-term, but so far, AI does best at exploration in areas that have precisely specified and measurable goals. Actually creating understanding, and making use of mathemtical results outside of pure mathematics is more challenging than simply creating proofs. I think the field will figure out how to make use of AI, and it will be better off for it. But that is not the same as just saying "answers good, grog want more answers."

I imagine the concern is more towards using LLM's to create proofs rather than using them to understand things.

Understanding is and always has been the "hard" bottleneck. In programming work, if one drops understanding and eg let's an agent write code with only superficial human review or none at all, I believe that they can easily get 100x fast or more, the main question being whether the process collapses some point due to sloppy code. In research fields like mathematics, skipping understanding is not something that can be done without a radical reconstruction of what mathematics (as a process/activity/field) is. It sounds plausible that LLMs help generate insights that humans have missed. But there are many open questions, eg the rate of generating insightful vs uninsightful but plausible statements, which can affect how useful they will be, and of course "open"ai has no incentive to share how much effort/cost (tokens and/or human-review) had been put into investigating erdos problems before coming up with this solution.

It's also a way to model the world and produce new useful abstractions. It's not just about solving problems.

You're right, but what I'm saying is that solving the problem isn't necessarily the primary goal and these new abstractions can be valuable in their own right.

I think the bigger issue is that mathematicians historically invented the abstractions to make maths easier to understand for humans. With LLMs, will we get abstractions that only computers can understand?

This is false on so many levels. Is it ragebait? Both mathematics and art are comprised of two phases, the first, technical one, where the novice grinds the skill and the second, the creative one which can only be achieved if you have the means (skill) to express yourself. What you described is the technical phase, not the creative one. There is intrinsic value to it that has nothing to do with money or cleverness, something that if you ever experienced it yourself even once, wouldn't need to be explained to you. Only people who never reach phase two have your stance. Artists and mathematicians who pick academia didn't exactly have great commercial prospects before AI was a thing, yet they still chose those paths because that's what having a real passion looks like. >They like people to think it all came naturally and that its genetic and that they are special snowflakes. No, they don't. Most of them are the humble people that know the value of cultivating a skill and when they do pride themselves it's precisely because they know the staggering amount of hard work and commitment they invested. Most of them are worried for unemployment and don't want all their work to be reduced to training data and on top of that not be given well-deserved credit for it. The only thing being exposed here, is how much AI in its current form was being underestimated and constantly labeled as "not real/good enough intelligence". This was and still is a shared sentiment even among tech people. Can't really blame them for going through a bargaining or acceptance stage. And since you also sound like the kind of person who thinks prompting can replace the "robotically spending millions of hours" of practice, I've got news for you: it cannot. You are about to learn the hard way the value of skill and human understanding because as much as capitalism rewards "impact" and "results", the market never values easy things.

Why is it wrong to expect humans (mathematicians) to adapt here? AI is already producing solutions to problems that humans could not find. Culture holds value until it does not.

So your downside here is problems get solved faster?

Isn't the whole point of the field of mathematics in a theoretical sense the pursuit of formal solutions? So, why would they be advocating for limitations on arriving at solutions?

It's more nuanced than this. Peter Scholze said in response to this declaration: > The goal of mathematical research is human understanding of mathematics, and so mathematics can only thrive in a community of human mathematicians. It is crucial to preserve this communal spirit. [0] Terence Tao has also talked about the requirement for a mathematical proof: along with generation and formal verification, there is an important step of "proof digestion" > understanding the essence of a solution, placing it in context with previous literature, summarizing and explaining it effectively, and gaining insights on other related problems and topics [1] [0]: https://siliconreckoner.substack.com/p/the-leiden-declaratio... [1]: https://mathstodon.xyz/@tao/116450581967483825

Solve it and understand it; seems intuitive to me. I don't understand how that contradicts my question.

From the article: > However, the declaration argues math is more than a machine for producing correct answers. The discipline, its authors believe, is a deeply human endeavor built on creativity, understanding, collaboration, and the pursuit of knowledge for its own sake. Generation X was the last generation that had 'general knowledge', as in an abundance of fairly useful information stored in 'grey matter' that could be recalled quickly. When search engines came along there really wasn't much need to know anything since most things could be looked up. However, you still had to think. With LLMs, thinking is kind-of optional. This really is an existential threat to our intelligence since 'use it or lose it applies'. I am glad these mathematicians are doing their duty as canary in the coal mine.