> 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.

The foundations of a great our mathematics came from landed gentry who enjoyed the work and were not paid for it. Today we prosper in our technologies and algorithms (not talking about recsystems) because of them. If math should once again become the pusuit of the curious and free but with even greater "skills", that does not seem to be an issue for math, or for those of us who benefit from math. An issue for people as a concept and for mathemeticians, yes. But that's a separate topic, like the anguish of not getting into the Premier league.

Sure, the world is a better place with fewer serfs in it, but what exactly is of "higher value" than being a research mathematician? It's already a profession that consists essentially of exercising our highest and most distinctly human capacities: creativity, abstract reasoning, and passing the results of those on through a distinctive language and culture. I don't think the comparison with serfs is useful. I'm sure most research mathematicians would like more freedom from some of the drudgery of their work (grading, admin, etc.), just like the rest of us. But we should be aiming for a world that allows more people to become mathematicians, not fewer.

This is actually a very old AI insight, acknowledged at least as early as the 80s, let me see if I can find the quote. Found it: > Rodney Brooks explains that, according to early AI research, intelligence was "best characterized as the things that highly educated male scientists found challenging", such as chess, symbolic integration, proving mathematical theorems and solving complicated word algebra problems. "The things that children of four or five years could do effortlessly, such as visually distinguishing between a coffee cup and a chair, or walking around on two legs, or finding their way from their bedroom to the living room were not thought of as activities requiring intelligence. Nor were any aesthetic judgments included in the repertoire of intelligence-based skills.

> In the past "computer" was a job description and mechanical power came from serfs. Serfs, all right, but in what world do you live where "computers", people who did manual computing (i.e. mechanical additions/multiplications/... with very large numbers) are the same as actual research mathematicians, who are basically pure logicians? The only perspective where it makes sense to root for mathematicians to go away is if you're a misandrist that thinks humanity should be replaced by robots (for reasons...). Or isn't logic something that's a defining human trait, and one of the main reasons we became the dominant species on the planet?

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?

It worked for humans. It took a lot of us, but eventually we accepted zero as a number. Then negative numbers. Then "imaginary numbers" as a useful trick, and then as meaningful. In our case, hundreds of millions, but we got there.

Ehh, I've had the opposite experience, with lots of writers and artists in my circle. The markets that have replaced writers and artists with slop never valued them in the first place, and the markets that do will never replace them with AI, and I say this as an AI engineer. Writing movies, writing theater, creating clearly original illustrations for various purposes, these are all tasks AI will never threaten, because there is just no point. And also, the market sizes for this kind of thing are a rounding error compared to say coding or back office automation which is incidentally the bulk of the token spend right now, confirming all this.

My impression is that most developers are motivated to create a finished product for a variety of reasons. If what someone enjoys is the craft purely for its own sake then isn't the resulting situation exactly the same as for artists assuming a hypothetical future where the core activities of both are largely automated? Such people can still engage in the craft purely for their own enjoyment just as anyone who wants to is free to write assembly by hand today.

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?

In general, most humans top out at 3D visualization, and instead rely on crude mathematical tools to work with higher dimensions. Every so often, people like Euler or Leibnitz pops up to give people new methods for blind men with a cane to explore the unseen yet knowable world(s). Scientific work is not normally naturally statistically salient for LLM observational data inferences. =3

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?

No, it doesn’t sound like you get it. It has nothing to do with the properties of LLMs and everything to do with the complexity of mathematics. 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. My point is the human is a critical piece to the puzzle, but not just any human, a career mathematician.

Sorry but I couldn't agree less. Deep esoteric research and trivial looking boring research can be as useful as state of the art trending areas. "Jobs for nerds" as has been stated, has given surprising and unexpected advances, or leveraged incredible advancements. An standard and boring bacteria in a specific Spanish biome, gave us CRISPR-Cas. There ar hundreds of examples. True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society.

> Sorry but I couldn't agree less. > … > True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society. Unless I misunderstand, it sounds like you do agree? My point is that without human mathematicians LLM output is meaningless, and without human mathematicians holding the reins, LLMs would probably quickly devolve into “proving” things that are not only completely unintelligible by humans, but have no utility. Your examples of esoteric mathematical concepts are anecdata. The vast majority of esoteric mathematics does not have utility. Mathematics is an incredibly large space of concepts. Consider the number of provable theorems in number theory alone, perhaps even related to specific subsets and sequences of numbers. The vast majority of the findings in that domain will not be isomorphic to some real world problem, they will be trivia. We will need mathematicians to separate the signal from the noise.

Wonder what Frank Herbert would have to say about letting machines do the thinking for us.

Boole was trying to improve logic for humans, "The Laws of Thought". So it has a connection to human problems, and eventually to practical matters. He could instead have been working on something much more abstract and much less useful. By which I'm trying to make an abstract point about the inevitability of staying somewhat down to earth. I mean "pure" curiosity is great, except it isn't ever really pure, and abstract mathematics isn't ever totally abstract, it's just sort of meta in relation to practical things that humans care about.

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.

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

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?

Spoken like someone that has no idea why mathematicians are important.

The struggle itself is virtuous.

i haven't read their memo, but, the article talks about math being something deeply human and the AI taint. I think it's a bit of both.

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

Yea, math is a crazy thing. just "a tool to solve problems" is a wild take. On one extreme, there's an edgy but logical / plausible hypothesis that we live in a universe of mathematical objects, and at the other, math also discovers a lot of questions, the exact opposite of solving problems.

People talk about research mathematics being a science or an art, but it's also a sport. AI will kill the sport aspect of it. The art will survive. The science will thrive.

Math for non mathematicians is a tool. Math for mathemeticians is an art in the same way an artisan takes pride in his work. That's why there's a disconnect when you go from math for engineers to the stuff above it. It feels less useful and very different

If spending millions of hours rote memorizing formulas and rules like a robot is "art" then sure

It’s a sad state of affairs when so many think math is about “rote memorizing formulas and rules like a robot”. That’s how math is taught through freshman or sophomore material for a somewhat ‘general’ audience. But real math is nothing like that - it requires far more creativity. You need to discover formulas and rules. You invent new rules and see what the consequences are. This all requires a great deal of creativity. Nothing “rote” about it. If you don’t believe me, crack open a text on something like graph theory (that’s pretty accessible, and if you’re a programmer, you’re familiar with graphs) and read through some proofs. Or better yet, try to prove some theorems yourself. No amount of rote memorization of formulas or rules will replace the creativity needed to write these proofs. Doubly so for discovering the facts in the first place.

This seems like an intentionally reductive view on mathematics, however it is true it can be taught like so. If you are interested, perhaps check out 3Blue1Brown on youtube, they manage to show some of the (very) real beauty in mathematics! Edit: Also, theoretical computer science is a subset of mathematics, and considering where we are on the internet, I get the feeling you like computer science.

>and the pursuit of knowledge for its own sake Except when someone hands you a magic button that just gives you knowledge?[at least in the framing of this "warning"] Then it's about peoples' livelihoods, about "culture", etc? "Computer" used to be a job. Did science on the whole lose or gain by making these clerks obsolete?

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.

Sorry did you read the article or just the headline? The theme is mathematics is a human-endevor and automation undermines that, particularly the ones starting out. It risks killing the culture entirely. Some other key points: - AI-generated papers could overwhelm peer-review systems with low-quality work. - It may become difficult to assign proper credit for discoveries. - Researchers who choose not to use AI tools could be disadvantaged. - There are ethical concerns about mathematical work being used to train AI for military and surveillance purposes.

>mathematics is a human-endevor Just like numbers and logic, it isn't and never was reserved only for humans. They need to adapt.

Idk man creativity is something pretty human. insufferable people are very fine if, for example, music & art is outsourced to AI so they can make a some $$. But those things are meant to be enjoyed, not consumed.

This feels like a troll comment, but I'll bite. Mathematics requires substantial creativity at every level. There is problem selection, conjecture formation, proof strategies, definitions, models, and explanations. Yes, it's constrained and guided by logic and rigor but having logic won't give you creativity. > Music is a human endeavor and musical recordings hurt musicians, bands, orchestras, etc. Especially those starting out. The medium it is recorded on has no bearing on what composed the music. If people don't get rewarded for composing they won't. Same with mathematics. If people don't get paid for being creative they just won't be creative. I am not saying I agree with everything in the article. OP of this thread just made a low effort comment that was addressed in lengths during the article.

Mathematics is not inherently a human endeavor and claims such as those are why the GOP voters are fine with cutting research funding so heavily. Even if you think it's true you probably shouldnt write think pieces that say so because it's bad politics.

Did you mean to say "culture"? > Culture holds value until it does not.

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

""" 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. Those values often clash with the incentives driving AI development. “The tech industry proceeds in accordance with commercial logic, which is antithetical to the values of mathematics,” declaration co-author Michael Harris of Columbia University told The New York Times. """ I mean, what field doesn't? Everyone works to make money. Slightly unrelated, but, their website " https://leidendeclaration.ai/ " itself gives an eerie feeling of being built by Sonnet. That color scheme and the layout is what Sonnet chooses by default most of the times.

This is all contingent on AI forays into mathematics being slop and low quality. However it's clear that recent AI models are capable of genuine mathematical achievements which surpass the frontier of what humans are able to accomplish (wrt the unit distance Erdos problem). The issue is, how is a group of intellectuals, whose identity derives from their ability to do something rare, useful, and requires many years to get good at, react when a machine can produce all of their useful output nearly automatically, can verify its own outputs, and is getting better exponentially? It is the complete annihilation of one's sense of value and purpose when the binding element to your culture is commodified. I think there will be a lot of arguments trying to claim that the point of mathematics is curiosity, or that there is always some ineffable human element that AI can't replicate, but I fail to see how somehow these wishy-washy human centered values somehow mean anything compared to the amoral pursuit of mathematical truth, which has nothing to do with humans. It's just that we humans happened to be the only beings in the universe good at math until ~2025. Now there is another species which can do many of the things we do, and it is not bound by the size of the human brain, our short term memories, or the architectural limits of biological computation. To imagine that humans would retain supremacy in this very un-human like discipline seems like wishful thinking.

AI is the interpolation of the human corpus. Is it suprising AI recombines sucessfully where human attention has not explored all plausible solutions? N,o not especially. The key fallacy is that AI is other than human. This is really no different from computer proofs, e.g. 4 color theorem. The fact the prompt is not linked to the solution by individual human intention alone does not make the solution less human in origin.

I will argue that AI and flood of low quality slop makes genuine human work more valuable, not less. The ability to clearly outmatch trillion dollar machines is a very unique satisfaction. I even write ordinary internet comments with an intention to make them clearly better and more fun to read than boring Claude output.

You will never be able to match insanity of human mind embodied in a walking sack of meat that was shot out of another.