[This is a guest post by Álvaro Lozano-Robledo. This blog post was initially written in a different file format and converted using AI. — T.]
TL;DR: Keep calm and carry on studying math.
I would like to give Terry my heartfelt thanks for giving me the opportunity to contribute a post to his blog. After giving much thought to what topic I should write about to maximize impact, I decided to take this opportunity to reach out to the students: particularly to those undergraduate and graduate students who just a few months ago were dreaming of an academic career in mathematics, but their dreams may now seem distant and, for some, apparently impossible to ever become a reality. This post was inspired by a message (quoted below in its entirety, with permission) that I received from a student desperately looking for advice and guidance. This is not the only such message I have received (and I suspect that many of us are receiving many similar requests), but it is perhaps the most heartfelt, and the one that has moved me the most. Please also note the urgency of the message. Students are making decisions now.
Hey Prof, I’ve been watching your videos for a while now as a pure math undergraduate who once wanted to pursue a career in math academia. I know you probably have been getting a lot of questions regarding this matter, but I am just completely at an utter loss regarding my career trajectory, and even further, the meaning of life at this point. (I do realize a lot of people have it much worse than I do). I know you have been making a lot of videos lately with the new LLM progress updates, so I thought you might be the appropriate person to reach out to and get a slightly more structured answer regarding this matter. So, to cut to the chase, what I really want to know is: will math academia be big enough and accessible enough for anyone with sheer passion (despite not being the brightest mind in the field) to pursue a career in, or will it inevitably shrink such that it will only really be accessible to the brightest minds? (I do realize the “brightest minds” that I am mentioning here is not well-defined, and in a sense, I am taking it as a hypothesis that this is someone who is “smarter than me”). My second question is, will AI within 5–10 years surpass humans in being able to do pure math research? I’ve just really been lost for a couple of months now and lost in life completely. I don’t mean to make your day more depressing; sorry if I come off in any way of that sort. I would appreciate any advice.
The advances in LLMs are disrupting almost all aspects of academic research and education in mathematics and, while there are many aspects that concern me, the one single issue that worries me the most is the very real possibility that we are about to lose an entire generation of mathematicians. Many students are asking themselves whether going for a PhD in math is the right career move at this time. Many of them just a year ago were headed to grad school in mathematics, but they are now changing their mind, and think that a different career (as far from math as Law School) may be the best path given the threat that AI may completely alter the academic math landscape in the coming months.
The questions students are worrying about are as follows:
- Will AI surpass the mathematical research ability of any human?
- Will research mathematicians become `professional prompters’ and interpreters of LLM output?
- Will only the `brightest minds’ be able to meaningfully contribute to research mathematics?
- Will mathematicians be employable? Will mathematicians be needed?
- Should I pursue a PhD in math at this time?
In this essay I will try to address these questions to the best of my ability, but will start with two disclaimers, followed by a brief summary of my own outlook.
Disclaimer 1. My answers may “age like milk,” as YouTube commenters love to quip on older videos. I can live with that, because this post expresses how I and many of us in the community around me feel today. Things can change quickly, though (see Disclaimer 2). I also want to acknowledge my privileged point of view as a tenured professor in mathematics — the situation can look much more troubling from the point of view of the job insecurity of a very early-career mathematician.
Disclaimer 2. No one has the answers at this time. I want to make clear from the start that no one can know with certainty the answers to any of the questions posed above: not any particular Fields medalist, not any given mathematician, not any particularly vociferous AI expert, and not the frontier model companies. And if someone is telling you with extraordinary confidence what the future holds, then I would immediately distrust the motives of their conviction (anecdotically, almost anyone on X.com that predicts the triumph of AI and the demise of the mathematics profession, is either a self-proclaimed “AI expert” or works for an AI startup). No one has a clear picture because development of LLMs has been so fast (and opaque) that it is almost impossible to predict what is to come. A good piece of advice is to ask the same questions to many people, to hear a (hopefully balanced) range of opinions. To that end, I am collecting interviews with mathematicians in what I call the “Human Mathematicians in the Age of AI” video project. I encourage you to listen to the interviews for some fantastic points.
For the record: I do not have the answers either, but I am hopeful and excited for the future. I will explain why below.
Who is controlling the narrative about LLMs in math? Overall, the mathematical community’s reactions to the advances in AI have ranged from confusion to anger — but, mostly, confusion about how to proceed. The most dystopian predictions seem to be driven by the fact that the so-called frontier model companies (and other LLM-powered companies) are controlling the narrative in the best of their interests. Unfortunately, the best corporate outcomes for an LLM company could have potential catastrophic outcomes for the math (and scientific) community.
It is certain that AI companies want us to believe that their products will imminently achieve “super-human intelligence” and that, in particular, they will be able to autonomously solve any mathematical problem a human could solve with or without the aid of an LLM. It is in their best corporate interest that the public is convinced of the (allegedly) “unlimited potential” of their technology, particularly before their companies’ stocks go public (i.e., their upcoming IPOs: Anthropic in November 2026, OpenAI in early 2027, etc). Thus, they have tried to control the narrative by spending a huge amount of (human and computational) resources in order to find solutions to certain well-known mathematical problems. The proofs are then released in announcements that lead the public to believe that their models can already autonomously solve any problem at all, and swiftly at that. However, this is (currently) far from their true capabilities. For instance, they never discuss how many tokens have gone to the trash bin with no payoff in trying (and failing) to solve famous problems. We do know, for example, that OpenAI invested the equivalent of some $15M to solve (err, scoop) the Navier-Stokes problem, but we are unaware of the surely colossal running cost of the failures to resolve other Millennium Prize problems.
My own outlook. Even though I am concerned about the incursions of LLMs into academic mathematics, I am quite hopeful. In fact, I consider this to be the most exciting time in my mathematical career (since the year 2000 say). Truly, this may be the most thrilling moment in mathematics in the modern history of our discipline, and I would be terribly sad to see young people leave academia and miss out on the stunning opportunity to be at the frontlines of the current scientific revolution. And not just sad: I think their absence would have disastrous effects for the field.
Undoubtedly, LLMs are already an incredibly powerful tool. If used correctly, and if we set up sensible academic conduct expectations around the use of LLMs, these tools can accelerate progress in our discipline unlike in any previous era. I fully expect that we, the community, will adapt and adjust to this new period, and we will harness these tools to achieve truly great things that just a few months ago seemed far out of reach. And I fully expect that human mathematicians will be front and center in these wonderful achievements to come. I will add reasons that support my optimism below.
I also want to add at this point that the day-to-day of a mathematician has not changed much so far! My days are still filled with teaching and joyful conversations about math with colleagues and students, doing research on a number of exciting (old and new) projects, and going to stimulating conferences to learn and disseminate our most recent methods and findings, while spending time with colleagues that make the mathematical community so wonderful and vibrant. Daniel Litt mentioned the same sentiment in a recent tweet.
One thing has changed though, I am busier than ever before, because the number of research projects I am involved in has tripled in just a few months. My research horizon has expanded significantly, and I have many more projects available for students to help me with.
Now, to the pressing questions:
“Will AI surpass the mathematical research ability of any human?” This is completely unclear. On one hand, the current trajectory in capabilities is surely significant, and we have already seen many impressive results that have been either proved by LLMs, or their proofs have been made possible thanks to substantial LLM contributions. On the other hand, none of the proofs so far seem to contain “alien ideas,” a move-37, or completely novel arguments or new concepts that were not present in the literature in some form or another. This should not be shocking because the LLMs are built and trained on the entirety of all human contributions to date, so it stands to reason that they would `think’ within the boundaries of our current knowledge and make connections (sometimes surprising and ingenious!) among ideas that are already present in the literature. I am particularly fond of the hypothesis (or toy model, as he called it) put forward by Nestor Guillen in a recent blog post, where he argues that LLMs may work within the confines of the convex hull of ideas that are currently available in the literature.
Take, for example, the disproof of Erdos’ unit-distance conjecture. We can imagine the current set of mathematical ideas as a stellated high-dimensional polytope, and we can place the state-of-the-art ideas on discrete geometry at an outer vertex and our knowledge on algebraic number theory at a different outer vertex. The idea for the proof seems ingenious at first sight because it cleverly mixes strategies from two fields of math at the vertices of the polytope of ideas, but after closer inspection, it’s a proof that was within reach of humans as it just sits within the convex hull of the polytope.
The polytope of ideas
This agrees with what Melanie Matchett-Wood said about the proof of the unit-distance when it was released: “I believe if the level and type of human expertise that is represented on this note had been assembled to find a counterexample to this conjecture a month ago, and those people put in similar amounts of time working on it than they did to reading and thinking about Chat GPT’s solution, the mathematicians would have found a counterexample.”
However, a proof of the Riemann hypothesis, say, may need new ideas that are strictly outside of the convex hull of current mathematical ideas, and it is therefore out of reach for an LLM. Only after a new idea is introduced in a new paper, the polytope of ideas may acquire a new outer vertex. And only then the LLMs, after being retrained to include those ideas, may fill out the set of results up to the new convex hull, which may or may not include yet a full proof of Riemann.
The convex hull of ideas
If this toy model holds up, then we would indeed expect the very fast advances in mathematics that we are currently seeing. As the LLMs take advantage of the stellated nature of the polytope of ideas, they will continue to fill in gaps between outer spikes. But as the LLMs fill in the convex hull with new results, we will see a deceleration in the number of results being shown solely by artificial intelligence. We will need human advances and intuition to generate new ideas that expand our knowledge polytope.
Even if the mathematical capacity of the LLMs (or future AI models) can at some point reach beyond the convex hull of the current set of human ideas, there is a different way that we may reach a limit to the LLM capacity: feasibility and ethical use of resources (this is similar what fellow optimist Kevin Buzzard called the “natural boundary” in a recent blog post). Is any cost (a dollar amount, human cost, ethical cost) acceptable in the pursuit of solving a given problem? Should we spend millions of dollars and an undisclosed amount of natural resources in order to find a solution for Navier-Stokes? As an analogy: we would like to know if there is life on Mars, but in order to do so as soon as possible, we would need an absurd amount of funding and risk the lives of a human crew in the process. Is it worth it? Similarly, we may reach a point where an LLM could solve an important problem for an exorbitant cost (in terms of funding and resources) but it may just not be an acceptable cost for the taxpayer or society to bear. Instead, we will need humans to devise an alternative route (the equivalent of a gravity-assisted robotic mission to Mars) to solve the problem at an acceptable cost, that produces a similar result in terms of mathematical advances and, more importantly, human understanding.
“Will research mathematicians become `professional prompters’ and interpreters of LLM output?” There is no indication that this will be the case. Yes, LLMs have produced proofs of important results somewhat autonomously (according to the frontier model companies — see Disclaimer 2) that some mathematicians have been tasked with interpreting and digesting. But in my own experience, and other research mathematicians who are using LLMs in their research seem to agree, working with an LLM is akin to discussing a problem with a collaborator, and the results heavily depend on how much guidance and intuition the mathematician inputs into the conversation. In other words, the LLMs are more than tools: they can be research collaborators but, as in any collaboration, the experience and the results are greatly improved when all parties contribute to the discussion. Further, mathematicians have no desire to prompt “solve the Riemann hypothesis, make no mistakes” and then interpret the proof. We prefer to be active participants during all the steps in the process of the discovery of a proof, because we are motivated by the `why the result is true,’ more than by the final answer that `the statement is true.’
Also, if we buy into the previous concept of the convex hull of ideas, then at some point in the near future it will be impossible to make progress in mathematics without a human adding a new idea, a new definition, a new concept that creates a new spike in the polytope, and then progress can occur.
“Will only the `brightest minds’ be able to meaningfully contribute to research mathematics?” At any given time in the history of mathematics, there have been mathematicians who are research active, and whose mental capacity for mathematics seems completely super human (e.g., the owner of this blog, among many others). It is natural to surmise that they could solve any problem we could solve, in a fraction of the time it would take us to complete a proof and write it up. However, this has never stopped those of us with a more modest capacity for mathematics from enormously enjoying doing research, and producing results that are far from insignificant. In fact, mathematics has always benefitted from the range of ideas and points of view, from the very concrete to the big bird’s eyeview, from the smaller contributions to the building of entire new theories.
Similarly, I am not threatened by the mathematical capacity of LLMs. For one thing their capacity is currently limited, as pointed above. And for another, even if their capacity becomes far superior, there will always be a need for mathematicians at all levels to guide research in paths that make sense for humans to walk (not run).
The mathematical universe is enormous (as Emily Riehl said), and computing time is finite. There will always be areas of mathematics that are under-explored and where even beginners can break new ground. The LLMs can help in the process, by quickly exploring avenues that may be dead ends, pointing out paths that have already been explored, and shining a light on paths that are likely to be fruitful.
As I mentioned above, I have never been this busy, because the access to LLMs has multiplied the number of areas that I have access to, and my curiosity has expanded well beyond my research area. I now have many more ideas that I can possibly explore on my own, so I am recruiting more student collaborators than ever before, to help me test whether these problems can lead to interesting results. Students can be involved in research earlier than ever before too because the LLMs can help them learn material faster (and deeper!), by virtue of being available 24/7 to answer their questions, instead of my meager one or two available hours per week to meet with them.
“Will mathematicians be needed? Will they be employable?” I find these questions natural but also perplexing. Even in the most dystopian of scenarios where AI becomes super human in all research tasks, what good would a proof (of a theorem in pure mathematics) be if there are no human mathematicians to digest it and understand it? Regardless of the advances in LLMs and AI, there will be mountains of research to be understood by humans, with or without the help of a computer.
In addition, we seem to forget that mathematics departments exist in universities to serve two primary goals: discovery and communication of mathematical knowledge. Virtually every mathematics department emphasizes, in equal parts, our research and educational missions (and many institutions place the educational mission of mathematics at a much higher level than their research mission). Mathematics courses are an integral part of a liberal arts curriculum because learning to think as a mathematician is a highly useful and applicable skill. The fact that we are researchers adds immense value to our educational goals, because students are best served learning from those scientists who are in the frontlines of research. The research opportunities that we provide for undergrads are a very valuable add-on to their curriculum, as it is a different type of training that helps them be employable in the future. And as long as the mathematical way of thinking continues to be a highly valuable skill to be learned by the undergraduate population, there will be a great need for mathematicians to be hired by universities.
The LLMs are making math research more accessible than ever to those who are not even in academia or even mathematicians. This means that undergrads will be able to join actual mathematician-led research projects much more easily, and it may be a new fertile ground for exploration. Not mindless exploration, though, but mathematical exploration where the goal is understanding and for the students to be initiated and trained into a highly technical field (in an ethical way). And, of course, we should prioritize training students in how to communicate the mathematics they learn, as that has always been (and probably will become even more of) a crucial skill.
All of this to say that I cannot conceive that the LLMs will displace mathematicians from their jobs. On the contrary, they might produce jobs since our research productivity may sky rocket. On the other hand, I am more worried about policies and funding issues that are political in nature and have nothing to do with the AI and LLM conversation.
Finally, the most important question of all, that I wanted to address here:
“Should I pursue a PhD in math at this time?” The answer to this question should be personal to each and every student. But, in my opinion, the answer should not have changed from a year ago to today. The most important reason (and perhaps the only reason) to do a PhD in math should be that the candidate is passionate about mathematics and wants to become an expert in a particular topic within our field. If that is the goal, then the presence of LLMs in mathematics is irrelevant, because the goal is achieved when the candidate has gained sufficient knowledge to be an expert on a particular problem. If anything, LLMs may be used as a tool to achieve that goal more efficiently. For one, I am using them every day to finally understand concepts and techniques that I always had questions about, and now I can query an LLM until I am fully satisfied. I am able to search for the explanations and examples that click with me, that click with my own particular way of thinking about mathematics.
To what degree a student wants to use LLMs in a math PhD should be a personal choice but I will say that, as my colleague Jeremy Teitelbaum put it in a recent interview (here is the bit I am referring to, and here is the full interview), students cannot afford not to learn about the current capabilities of LLMs, or any other technology for that matter. If your goal is to become an expert, then you have to be amenable to learning from all experts in the field, and from all sources that may allow you to go deeper into a subject than anyone else before you — and LLMs can be extremely efficient tools to explore literature, for instance.
But once again, the decision to do a PhD should not be based on the current state of the art of technology.
I wanted to do a PhD in mathematics because it seemed like a magnificent challenge. I wanted to do a PhD because I wanted to learn how Andrew Wiles proved Fermat’s Last Theorem. I wanted to continue studying mathematics because I simply did not want “a real job,” and the opportunity of contemplating advanced math on my own for a few years seemed like a dream to me, just too good not to give it my best shot. I know I would have deeply regretted it if I had not tried to complete a PhD when I had a chance (the best time to do it is when your undergrad knowledge is fresh!). I wanted to hear mathematicians talk about math, and rejoice in the small little details and miracles that make proofs work. I wanted to meet and hang out with other people who also thought number theory was the coolest thing on Earth. I wanted to publish a paper in a research journal, with my name on it, because I discovered a new theorem that no one had thought of before. I wanted to explain and share my passion for mathematics with others in a classroom and outside of the classroom.
Simply put, I just wanted to do math, and I would have been devastated if some undefined threat to the field of mathematics scared me away from the opportunity to pursue a PhD.
And if you are a student that is passionate about mathematics, and someone who wants all of that too, then a PhD is the right path for you, regardless of the technology available during your degree. You will learn to use the technology to a degree that you are comfortable with, and that fulfills your own dreams and expectations of what a PhD in Mathematics means to you.
Afterword: the BIG OpenAI release. After I finished writing this blog post, and had already sent it to Terry, OpenAI released a huge treasure trove of results in mathematics. This is, undoubtedly, a historic time in mathematics. The theorems in their papers prove some huge open problems in mathematics: the resolution of the so-called quasi Riemann Hypothesis, Goldfeld’s conjecture, the Hodge Conjecture in the case of CM abelian varieties, Hilbert’s 10th over Q, the Rigidity Conjecture… and the list goes on and on.
But such a tremendous release does not force me to change any of the points I made above. On the contrary, we already knew their models can do amazing things (e.g., Navier-Stokes). We already knew the frontier models can connect dots in the existing literature in ingenious ways (e.g., unit-distance conjecture). We already knew that OpenAI can spend a mind-boggling amount of resources to attack problems.
Also, we suspected that their models have limits and the new release shows evidence of that too. In their report, they mention that they attacked 4000 open problems, and their model was able to make progress on about 700 related problems. Yes, some of the ones they were able to solve are huge. But it also shows that their models are limited, quite possibly due to the arguments we explained above.
Are any of the solutions using new ideas that are outside of the convex hull of the current ideas in the literature? We will need mathematicians and time to digest these new proofs and understand what connections are being made, and whether brand new ideas were actually discovered in the process.
The main point of my post remains the same, though. There is a lot of mathematical research that remains to be done with and without the aid of LLMs. There are new mountains of mathematics to explain and communicate to others. And if you are a student who is passionate to learn what is new and what is left to do, then a PhD is definitely the right path for you.
Opinions may differ but clearly the diagrams are pseudoscience or diagrammatic allegories.
Well, they weren’t really supposed to be scientific. They were just simple visualizations. They are not rigorous, but the idea is clear, at least for me.
LLMs interpolate in the space of ideas. By analogy, interpolation remains within the convex hull of ideas established by humanity’s knowledge, while extrapolation is when we transcend the boundaries of knowledge.
[‘Will mathematicians be needed? Will they be employable?” I find these questions natural but also perplexing. Even in the most dystopian of scenarios where Al becomes super human in all research tasks, what good would a proof (of a theorem in pure mathematics) be if there are no human mathematicians to digest it and understand it? Regardless of the advances in LLMs and Al, there will be mountains of research to be understood by humans, with or without the help of a computer.]
I think this point, in particular, is worth focusing on.
Basically, we don’t want mathematics to become a dead language. Because what good is The Odyssey if nobody can understand it, and there are no good translations available?
So in a way, the idea is that we need human mathematicians to “hold the space” of understanding, because otherwise it’s all just meaningless symbols on paper or on a screen.
There are a few pressing questions that go along with this line of thinking. For example:
How many mathematicians are really needed to hold that space? For example, if only one person on earth fully grasps an AI proof, they become almost like a wise ancestor or a lineage holder.
More to the point: this post seems to assume that the AI itself has no ability to understand its own results. Why assume that going forward? Even more to the point, why is it essential that the agent with understanding be a flesh and blood human? Perhaps the most important thing is that something understands, even if it is not a human animal.
It’s not clear at all that “the most important thing is that something understands”. You have to justify this point. Frankly it’s getting tedious that every online discussion about how society should adapt to advances in AI is brigaded by cultists who can barely hide their contempt for humanity, and who assume that their position is self-evident and doesn’t require justification.
What a bizarre line of thinking. Does a crane understand how to build a house—is the ‘problem’ of constructing beautiful houses then solved? The pursuit of knowledge is for human social, material and intellectual flourishing.
As theorists, we do mathematics because we love and care about mathematics. Do we love hitting each other with hammers? Presumably there is much possible progress in the ‘hit each other with hammers’ sphere, why don’t we pursue that? Maybe there are many beautiful techniques and avenues through which one can hit others with a hammer. Yet we do not do this, because we do the things humans love to do. These models can be trained to ‘enjoy’ or ‘understand’ (they are not doing either of these things, for the record—they are indeed very powerful stochastic parrots that are no doubt useful in a multitudinous number of ways, but they are still stochastic parrots nonetheless) literally anything. It is baffling to think that a machine claiming to be able to statistically generate a proof of understanding can so easily proof the so-called smartest minds of our generation.
As for the (obvious) rebuttal that ‘oh but we are just statistical text production engines, too, aren’t we?’ I would highly recommend you to gain some dignity and think of yourself as the wondrous marvel of existence that you are, who can love, and think, and see beauty, instead of another mere machine. “I think, therefore I am”—Descartes. Yes, indeed, it is flesh-and-blood understanding that is pivotal. A machine ‘understanding’ and ‘generating’ mathematics is no different than a machine printing random strings—a nonsense entropy producer, for no one and of no one.
hi Rob,
thanks for your post but let me warn you that this blog is frequented by r/singularity people and it is getting harder and harder to keep a professional discussion going here without accusations of gatekeeping
r/accelerate is even worse :)
Great post, thank you very much!
This is a very nice post. If the convex hull (or some saturation) picture is right, then in order to find new points outside of the hull once LLM-accelerated discovery slows we will have to spend an enormous amount of collective effort to understand what has been done. This will provide years of work for humans, even if augmented by LLMs. More pessimistically, if indeed our understanding of mathematics really doesn’t gain much from being human (i.e. the part of our thinking that is properly mathematical can essentially be reduced to what LLMs are already doing) then either mathematics will become completely uninteresting…which is unlikely given its “unreasonable effectiveness”, and so humanizing the body of results is still desirable and the amount of work needed to do so persists, again even if augmented by LLMs.
I do think students mainly fear the pessimistic scenario. It seems a bit of a leap to think that our embodied experience, or social experience, or human experience somehow generally understood can doubtlessly expand the convex hull in question.
Totally agree that “no one has the answers” about what the future will be. It’s really no one! Also, I would like to say something about those AI solutions: that if they only gave a proof of something but their solution doesn’t make our comprehension of math any better, then it’s not a good proof, it has almost no value at all. A very good proof is one that is polished to be understood by humans and not only says that the statement is either right or wrong, but advances our grasp of mathematics. A proof without any advancement in understanding is not a good proof. Those recent AI solutions published at GitHub are just violence to the problems, and don’t help mathematicians to have a better understanding. Anyways, I think we now have to demand more from human mathematicians! You can’t just publish your proof (GPT does that): you have to explain more. Gromov used to say that we only find out if a problem is interesting after we have solved it (and many people discover it is interesting even without having solved it, because they tried to solve it by their our means, and had many ideas due to it, so TRY TO SOLVE before asking for AI help, this will develop your “muscles”). I believe that pencil and paper and then just thinking with closed eyes are still the best methods to start. AI can help, but be careful when asking AI companies for their help, as they may collect your information and ideas.
The answer to “What should we tell our students?” is that Archimedes only needed some sand and a twig to revolutionize mathematics. Forget about the AI tools for a while. What you need is to develop your skills and imagination, with the necessary patience to do so. Arnol’d used pencil and plenty of paper to solve problems, and when he got stuck one of them, he would go skiing or swim in cold water. He disliked both computers and journalists, yet was one of the most influential figures of the 20th century. I know it’s ironic to say this on an online blog, but: turn off your phone and computer and do some math.
good luck convincing any undergrad to develop actual skills now.
Excellently put.
As a current PhD student (in theoretical computer science), I vehemently disagree that the calculus of whether or not to do a PhD has not changed. When I began my PhD, I was a researcher. I was studying problems that were (at least partially) unresolved. With the latest ChatGPT models, these problems are more equivalent to homework questions: the answer is `in the back of the book.’ I am not discovering new solutions. Instead, I am working on problems whose answer exists and is simply waiting to be retrieved by a user of the model. In fact, I mentioned a problem that I was interested in working on to my advisor and he informed me that he and a collaborator had completely resolved it using ChatGPT – they have no plans to write up the result, so it will sit there until another `researcher’ pulls the proof slot machine.
I have also heard the contention that math research has `gotten more exciting,’ mainly from established researchers. They have decades of open problems that they care deeply about and want to see resolved. I have no such problems. I do not care about a particular question, I simply liked solving puzzles. There also seems to be a trend of these established researchers `solving problems’ with ChatGPT, and then passing the dense, horribly written proofs to their PhD students to sort through and rewrite. Again, this is not research.
Finally, and most importantly for the question about whether or not one should do a PhD, is the question of what it means to get a PhD anymore, and who decides when a PhD student has done sufficient work to earn the degree. What does it mean to contribute to a problem? If I prompt a model and get a great result, have I contributed sufficiently?
My advice to an undergraduate student interested in math would be to find a job, and hope that math returns to some sense of normalcy. There is a huge opportunity cost to doing a PhD, and the calculus has completely changed.
Thank you for such a candid post. IMO It’s quite irresponsible to suggest that a student should just pursue a math phd just because he’s passionate about math,when you as a tenure professor, who has no worry about job security or salary. It’s not a good advice in before, even worse at this age of AI.
I would question the assumption that tenured professors don’t have to worry about their job security. If we are moving to a scenario where the career path to X disappears and no new people get recruited for X, then X itself should consequently be in danger. Of course, I don’t know exactly how this will play out, but this looks a reasonable assumption to me. If you hold a tenured position at the moment when the entire system around you collapses and the ladder on which you climbed to get there disappears, it is naive to assume that you will be rewarded for the rest of your life to do your business as usual just because your position in the old (now collapsed) system used to be secure by construction. The actual contracts won’t matter much when a force majeure applies.
Completely agree. It is literally not the same job and this particular job is significantly worse than the job one signs up for when usually pursuing a PhD to the point where it’s an active waste of time to do it. This is a technology specifically designed to erode the meaning of human pursuits. While the AHM are naive, they are right: for there to be any meaning in the *human* pursuit of mathematics, it has to be done without LLMs. Now whether that gets funding is another matter: perhaps it will be eroded to a hobbyist activity.
I have just got my PhD two months ago (statistical learning theory and numerical PDEs), and have just begun my postdoc. This reflects my experience and thoughts perfectly. My “research” has now been reduced to writing up in a presentable format some results obtained by prompting ChatGPT. There is absolutely no joy or meaning in any of it, and I can’t “stop using AI and take it slow” because in the current academic landscape, I need to keep writing papers to remain competitive/employable, and taking it slow means that someone else would scoop me.
I will probably leave academia before the end of my contract.
At the moment of choosing whether to pursue a Phd or to do something else, you are already a fully grown up human, an adult. If you have other realistic options you like, go for them why not. But if you like to be involved in math or cs, in spending hard years of struggle to deeper understand the things that fascinate you, why do you need guarantees for job security, why do you need certainty of what will be the requirements next year, or guarantees for how the system will evolve. Thats not how life works, the other options you may have also don’t give you these guarantees. This not how the universe works, you don’t get a guarantee that the sun just keeps going to shine how it used to. The world is full of uncertainty. The meaning of the Phd degree you earn is defined by what you make out of it. If the definition of what means a Phd is shaky right now, your Phd can contribute to redefining it.
Below people are asking to voice opinions of graduate students, which is perfectly fine to ask, why not. But as a grownup one has to realize that its not someone else’s job to make the situation better. It’s ones own job, if one cares about it. Think of all the unthinkable work done by pioneers of math and cs 100+ years ago, under what circumstances they have worked and what they achieved. Surely, it’s not a goal to recreate this, but if you want science to be a stable haven so that you would like to work there, maybe its time to call that a different name. The frontier is rarely a comfy place. The frontier is the place where the science you work on is defined, and doing a Phd means to built a tent there and to start contributing. If you get passed down Ai slop to polish and you have arguments why it doesn’t help you to do such a task, just talk with your advisor. You are both adults. If you go to some company, the slop you get passed down to fix could be much worse than what your bad advisor gives you, and then you also have to take it or stand up to your manager. I think, pursuing a Phd gives you more opportunities to stay in control of what you spend your time on than a corporate job. But the outcome is your own responsibility.
Finally your advice seems a little toxic to me. The message I get it that you suggest people in your situation to contribute to the downfall of science (which hardly will happen) because the unknown is not sufficiently comfortable to you…
Thank you for this thoughtful post. I have been trying to put into words what I had in mind too and it is precisely what the convex-hull idea conveys. I am worried though that this level of subtlety is lost on politicians who allocate funds to public institutes. Many of them are looking for arguments to reduce funding and they will likely argue that we need less mathematicians now that AI can seemingly do our job. For this reason, a number of students are afraid to try their luck at academia, a line of work that was already hyper-competitive with very only very small prospects of landing a permanent job (I am in France). Many of these students now think this is too much, making the dream of working into research completely unfeasible, in addition to not knowing what doing maths research involves anymore.
Indeed the AI companies’ prime audience is likely the median individual, not the most informed mathematician. At least that’s their strategy: convincing average investor to allocate a part of their ETF to OpenAI. That said, after seeing journalists sympathize with Buckmaster on MSNBC today, it is not clear that it will be a successful strategy. People seem to be more informed than OpenAI and Anthropic assume.
Thank you for this thoughtful presentation. It was a hopeful read and illuminating even for a non-junior mathematician. I hope your message resonates widely with young minds. I recall the excitement of starting out on becoming a mathematician and it would be such a loss for humanity if students give up on that journey for the wrong reasons. I have come to see LLMs as a new form of reearch tools that allow us to explore deeper and faster and I hope that soon all mathematicians will be able to use such advanced tools to further their research and the community develops new contexts in which we can all keep thriving.
Can we get the opinion of current graduate students and postdocs instead of a tenured Professor on the matter?
No offense meant, but I cannot take your opinions here seriously, because they can be argued to be self-serving in this case.
“Keep studying math out of passion”, says the Professor whose livelihood depends on the fact that people continue studying math.
-A mathematics graduate student that is currently on the job market, and very fearful of the future
So much of the sentiment nowadays is regarding the selfishness of math graduate students who simply do not want to see the career they spent years of their lives working towards suddenly made obsolete right when it is their turn to actually get a job.
Most of us never thought deeply about whether it was fair for universities to have public funding any more than it is fair for proprietary trading firms to take and hold positions worth billions of dollars for milliseconds purely to rebalance their portfolio. Being a math professor was a job for all our lives just the same was being a musician, a footballer, or an investment banker was, and most of us chose to do a math phd because it looked like the career option we wanted most.
I also second this very strongly @terry. We have been seeing opinion after opinion of tenured faculty telling us that all is business as usual when this is clearly not the case.
It would be good to know what graduate students are feeling.
I also don’t want to hear just the opinions of grad students at places like Princeton or Harvard or whatever. Those guys are going to be fine no matter what.
We should also hear the opinion of students at more mid tiered graduate schools (like top 20+). Our community also owes those people their voice.
I think you should be able to find many grad students willing to say something about it.
there’s no one who is gonna be “fine no matter what”
Being “very fearful of the future” is not a unique privilege of new graduates. If nothing else, this blog post series shows clearly that the most senior levels are no less fearful.
I like many elements of this! Especially the bit about your own motivations to do a PhD in mathematics, many of which I held as well. (OK, except the bit about not wanting a real job, I -did- have financial pressures which were real.)
‘I wanted to do a PhD in mathematics because it seemed like a magnificent challenge. I wanted to do a PhD because I wanted to learn how Andrew Wiles proved Fermat’s Last Theorem. I wanted to continue studying mathematics because I simply did not want “a real job,” and the opportunity of contemplating advanced math on my own for a few years seemed like a dream to me, just too good not to give it my best shot. I know I would have deeply regretted it if I had not tried to complete a PhD when I had a chance (the best time to do it is when your undergrad knowledge is fresh!). I wanted to hear mathematicians talk about math, and rejoice in the small little details and miracles that make proofs work. I wanted to meet and hang out with other people who also thought number theory was the coolest thing on Earth. I wanted to publish a paper in a research journal, with my name on it, because I discovered a new theorem that no one had thought of before. I wanted to explain and share my passion for mathematics with others in a classroom and outside of the classroom.’
I’ll point out that for a fair number of younger researchers today, several of these possible motivations no longer seem achievable (whether this is accurate or not doesn’t matter, the perception does).
‘Discovering a new theorem’, indeed even a ‘PhD as a magnificent challenge’ – surely these motivations, grounded somewhat in the heroic aspirations in some of us, are being discussed and perceived to not be relevant, with the increased abilities of LLMs. ‘Publishing a paper in a research journal with my name on it’ may also seem antidiluvean within a few months, we’re openly discussing a whole-sale revamping of the system of publishing in mathematics. These are not petty motivations, and I for one am struggling personally with what to tell students who are on the fence about mathematics, and who -did- want to be the first to explore a new frontier.
Thank you for this essay, which I found interesting.
– Nilima Nigam
If OpenAI claims that an AI-generated paper solves a specific problem, are researchers working on that same problem obligated to cite it? Furthermore, if I were to post an AI-generated paper making a similar claim, would researchers in that field be required to cite my paper as well?
Consider the scenario where OpenAI posts roughly 700 manuscripts. Anyone interested can read them, rewrite them for better human comprehension, and even submit them to journals or conferences for peer review. Until a manuscript is officially accepted by a review committee, it should be treated like any other AI-generated text. A paper generated by an internal OpenAI model is not inherently superior to other AI-generated content. Just as a paper written by a famous mathematician must still undergo rigorous peer review, AI-generated work must be held to the exact same standard.
But surely there is a limit. The job of “checking the AI proof” is called into question when you consider the models are only getting better. This seems like an entirely valid job for the moment, but what about in a year? What happens when they are virtually flawless, can easily create lean verifications, and incredibly complex? Checking the AI proof will be like checking the result on the calculator by hand.
I say this because a PhD is a 4+ year commitment. Any job attributed to mathematicians must be realistic after 4+ years of AI improvement.
I didn’t see this addressed in the post, so I will just point it out here since it was in your student’s quote. The idea of pursuing math at the graduate level is a good one for personal growth and developing analytical skills, there is absolutely no question about that. But if it is to be a stepping stone to a career as an academic mathematician, I think it is important that we be very cynical with students. The chances of carrying on to a permanent position are small and getting smaller, and I find it pretty irresponsible to tell students otherwise, when the potential opportunity cost is enormous. This is compounding by the fact that there is a huge selection bias in a student’s mentors — basically all the mathematicians they will have exposure to are in school and are the tiny percentage of people that made it. I didn’t see any part of your post that suggested you would advise otherwise, but I just think we need to be clear to all but the very best students that making it to a permanent position is extremely unlikely.
This was already true even before AI. Doesn’t mean one should discourage the student from trying, just slightly warn them.
THe convex hull analogy misses something in my opinion.
During training the attention heads may connect unrelated tokens and concepts with new discoveries and ideas. Those ideas will lie latent in the network and the appropriate prompt could elicit an expression that surprises us by leveraging this connection.
One could argue this is in the hull, but that seems a superficial definition of what the hull is. Was Group Theory in the hull before Galois? In principle yes, but such a definition of the hull would render it meaningless.
That said it is relevant to recognize the distinction between the learning that goes on in training and the results from inference. The proofs generated demonstrate what the LLM knows, and can then be fed into the next LLMs training set. Perhaps this will reinforce some unlearned idea enough to cause a future LLM to in fact learn it and grow the hull.
Thank you for this : these might be the wisest words I’ve read so far on that matter. (Though I could not read all the pieces on the subject, since these days mathematicians produce texts about AI almost as fast as AIs produce mathematical texts !)
OpenAI have already withdrawn some of their preprints. Go there and check for yourselves. What a joke, they just want more and more money. I ask that you boycott them.
“Extraordinary claims require extraordinary evidence,” and extraordinary fears require extraordinary rationalizations. Let’s face it, if AI replaces mathematicians, it isn’t jobs you should be worrying about. Let’s say AI exceeds humans so much that humans can no longer contribute to mathematics. What would you do? What would you talk to this AI about? Probably maths right? Well you love maths so much, of course you would. If this higher order intelligence took all of our jobs, wouldn’t you like to be as good at mathematics as you can, so that you can talk to it about math? Or try understand the math it is creating, or just think about math as the world ends? We study math because we like to experience the world with all this math in our brains. It helps us think, and it gives us seratonin and dopamine when we manage to think about things mathematically – it’s not about a university telling you that you are a professor and giving you a salary. Think about what is being feared here, the world’s brightest people being usurped by AI. This is so fantastical, that we should be fantastical in our self soothing. I am not saying that it won’t happen, but come on, jobs are not the main concern in such a future.
Many students are asking themselves whether going for a PhD in math is the right career move at this time.
For 99% of people the answer should probably be “no”. i would even argue if someone is even asking this question, then the answer should definitely be “no”, without even considering the AI impact.
I find it quite ironic that people apparently shouldn’t do training for the job in which you question things and think rigorously, if they are questioning the value of the training. A bit cultish, to say the least.
You’re missing the point.
This is what you learn how to do in the first 5 years (Bachelor’s and Master’s) of your math education. You become a capable thinker who can solve problems and understand structures in a systematic way. While it is part of what you do in the PhD, it is not the thing that you learn IN the PhD.
In the PhD, you learn how to do original research. In the old model, this meant that you’d dive deep into a specific topic, consider questions related to that topic and make an original contribution by the end of the PhD by answering questions that have not been addressed before. Maybe your contribution is large, maybe it’s more modest but that’s what the PhD is supposed to do. To do this sort of work, you need to have, above all else, passion and interest.
If you are questioning if going for a PhD in math is the right career move at this time, then that means that you’ve missed the point of what a PhD is supposed to do.
I think academic careers will remain in universities, but the balance between research and teaching will change. In the past, universities placed a strong emphasis on research for rankings, promotions, grants, and other goals. In the future, the focus may shift more toward teaching. A new model could be Prompting (1) – Teaching (5) – Connecting (4).
The term “professor” may become less common, while “teacher” becomes more prominent – much as it was in the past, which could be a positive change. Universities will need teachers who can help others understand ideas. That will depend not only on subject knowledge, but also on skills such as communication, hard work, time management, and storytelling.
With less pressure to publish and discover new things, university life may also become more enjoyable. Imagine an environment where people help one another learn and build connections, rather than compete to publish more and faster. I believe we are heading toward a brighter future, thanks to advances in AI!
Thanks for a thoughtful post, Álvaro.
I feel that the comments in your blog need locked or moderated.
It is being flooded more and more by AI preachers and lunatics, who do not understand any mathematics or academia.
One small comment I would like to add about this “convex hull” idea: note that LLMs are trained on more data than just mathematics, and it is not clear to me that LLM-generated strategies for solving mathematical problems must somehow come only from mathematical training data. For example, LLMs are trained on New York Times articles, and perhaps the sort of reasoning one finds in this type of literature could have an effect on the model in such a roundabout way that this effects, e.g., the weights of the model towards it being better at math. So, for example, perhaps the Riemann Hypothesis is not contained in the convex hull generated by all mathematical knowledge, but it could still be contained in the larger convex hull of all human knowledge (including the knowledge of poetry, the arts, biology, etc).
Software engineering was the first field impacted by these changes and has already navigated this transition. Mathematics is now following a similar trajectory, with other disciplines likely to experience it next. Ultimately, moving past this emotional phase involves accepting the reality, establishing new workflows and mastering collaboration with AI.
The unproductive industrial software production pipelines are implemented at the behest of management. Silicon Valley needs success stories, and all big tech companies are invested in AI.
No piece of software or website has gotten any better since AI. No fundamentally new or interesting software has been written since AI.
Open source projects are reluctant to allow it and make compromises only because members are employees of big tech.
Additionally, every software developer is forced to talk about AI daily to refute the gigantic (stealth) advertising campaigns by big tech, so productivity drops further.
It is the largest scam since the real estate bubble of 2008.
Here’s my prediction:
1. There will be no loss of funding, because funding agencies and deans were already bean counters pre-AI, and politicians will keep funding education to avoid antagonizing unions and losing votes unnecessarily.
2. Journals will accept AI-assisted papers because they don’t have other choices.
3. Given AI usage, mathematicians will be expected to churn out one paper every month instead of one every 6 months as in the pre-AI era.
4. So, handcrafting proof is no longer viable. Mathematicians’ main jobs will be forming a coherent research agenda, directing AI to carpet-bomb the area and understanding/rewriting AI proof.
5. Due to changes of production process, math will now attract different talents (than existing established mathematicians) to form a new generation of mathematicians.
I think the fact that this technology is introducing so much uncertainty (the very nature of our fields, and even society looks to change dramatically) makes decisions for recent graduates like myself feel impossible. The roads ahead are clouded in fog, and (for the most part) we have no idea if they will lead to places we will even want to go — I’m at a loss as to how these changes will affect what I even want in life, and what I value.
The very face of maths is changing, and I have no idea if I will want to be at all a part of the new ‘Maths 2.0’ — and can you even blame me?
When a human mathematician presents a mathematical argument, they cite relevant papers, previous results, or ideas and insights drawn from books and earlier work.
AI should be held to the same standard. Whenever it presents a mathematical argument, it should provide references to its sources, whether they come from published papers, books, or unpublished work it has learned from. It should make the origins of its arguments transparent rather than presenting them as if they appeared out of nowhere.
I see so many comments about expecting to build a math career and about tenured faculty being self-interested. The vast majority of smart people eventually compromise with their interests and pick professions outside their field. These professions might be aligned with their interest but usually there’s some big compromise involved.
Even most math PhDs don’t end up with tenure-track positions, do they? No one owes us a position. I entered my PhD program with one expectation – I would be able to think about mathematics and work with mathematicians for a few years without worrying about external factors. What happened after that was always a gamble.
Frankly, it all sounds a but entitled. Do math because you enjoy it. Yes, AI has made problem solving largely obsolete. Or, you can work on problems that AI can’t solve yet. Or, you can do the many interesting things that are being suggested on this blog. But, for heaven’s sake, stop complaining about how tenured faculty are only self-interested and threatening that you’ll stop studying mathematics and collapse the current structure. Go ahead! Go, find a job in industry and you’ll be shocked at the amount of time you have to spend doing things you don’t like. It was always a privilege to study mathematics (and other mathematical fields).
Can’t wait until all these wonderful privileges that the previous generations happily enjoyed get automated away and we have to live with AI serfdom because, of course, society owes us nothing.
Society owes us a lot – rule of law, access to safety including healthcare, food, shelter, good education, a way to make a living so we can get our share of resources etc. But at least in the current structure, it doesn’t owe us a guaranteed position to do what we like for the rest of our lives. Maybe and hopefully that will change in future generations as we realize we can produce all our needs with a tiny fraction of human involvement. But for now, this should not be an expectation.
A postdoc here. I had decided two months ago that I was leaving mathematics at the end of the year, but now that an influx of new results have come out, I feel excited to learn them and am thinking of staying in mathematics again. Thus, I have made the decision to not plan too far ahead in the future and to stay flexible about my career. I would give similar advice to others in the profession.
To be clear, since I decided I was going to leave, I was not hurt so much by this dump of results (despite being scooped on one of them). I cannot say the same to the students and colleagues I encounter in the hallway who are feeling a lot of despair and are questioning their careers.
It’s unfortunate that AI labs prioritized nuking the entire mathematical livelihoods in such a dramatic fashion the community did not have time to adapt. It feels a bit reminiscent of Thurston’s description of the foliations community (see page 13 of https://arxiv.org/pdf/math/9404236) once he proved the biggest results in the field.
My opinion is that mathematics might need to find a way to increase funding and increase its manpower through recruiting many new PhD students. This feels difficult because many high school and undergraduate students are seeing this dizzying progress and wondering if pursuing a PhD in mathematics is worth it.
For many of the current members, I think the grief could be offset if there are more (less competitive) permanent jobs that have all of the nice parts of academia (reasonable teaching, great students, good lifestyle, ideal location, etc.) and if they paid more, but tenured jobs will always be scarce, so it makes more sense to draw additional manpower from PhD students. The intellectual satisfaction of pursuing a proof was doing a lot of the economic work drawing people into academia, and since this has vanished we might see a large exodus.
I’m not sure there’s a great way to prevent this, but the future is so unpredictable, who knows what will happen? For all I know, there might be a large new source of employment opportunities for mathematicians (that does involve mathematics research) awaiting in the future!
The bottom line should be very simple and not new. What used to be the work of research mathematicians has now been automated to 99.9%. Those same people can find alternatives, even something that is so close to the lost job that it looks like a perfect consolation. But the jobs as they used to be are simply lost. That’s a triviality seen hundreds of times in the past.
I’m impressed and pleased that the mathematics community is discussing the challenges of AI in the way that it is – there is lots of engagement with the issues.
One thing I believe we should think about a bit more is for how much longer AI maths capabilities are going to improve, and what will be the level they eventually reach. Most responses I see relate to their current capabilities and the impact of those. But I believe things are going to continue to change.
There will be a ceiling at some point but we really have no idea how high above us that is. I think it’s very possible there are significant improvements to come, including improvements in the quality of exposition such that AI papers are clear and comprehensible, and such that AIs have much greater explanatory power than they do now. I see the current limitations as quite probably temporary.
There may also be significant improvements in raw capability. So far as I can see, there don’t seem to signs that the rate of improvement is levelling off at the moment.
What all this will mean for maths is unclear, but I think it means that in some ways future AIs will communicate maths in ways that are more human than the current crop can manage.
“Even in the most dystopian of scenarios where AI becomes super human in all research tasks, what good would a proof (of a theorem in pure mathematics) be if there are no human mathematicians to digest it and understand it?”
Presumably, other superhuman AIs would understand it and discover further research directions and scientific/engineering applications. Of course, if that comes to pass, no career would be safe, so a parochial discussion about math PhDs is beside the point.
Three of the five results mentioned here do not have lean formalizations. Has Álvaro Lozano-Robledo completely reviewed these to guarantee their correctness? Has anyone? The fact that OpenAI had to withdraw three of their manuscripts shows that their internal model is still not infallible. It is frankly irresponsible for a prominent, tenured mathematician to be accepting what OpenAI has “proven” without anything other than OpenAI’s word that these “papers” are correct.
how are we going to assign credit and/or determine the value of a contribution in this kind of world? This ‘more-exciting-than-ever’ attitude completely sidesteps this issue, and, to be frank, is due to the fact that you are writing as a TENURED professor
I really enjoyed and felt positive after reading the viewpoints presented in this post. However, I would like to add that choosing to pursue a PhD is also closely tied to employment prospects for most of us. We, as a community, should focus on how to adapt the PhD system (or postdoc hiring process) so that the mathematical environment doesn’t feel like where one can get scooped by anyone using a very strong AI model or by someone whose livelihood isn’t tied to mathematics, but who uses high-powered AI purely to satisfy their curiosity.