Proofs and Prompts

Opinions against the PhD, from an incoming PhD applicant

Sirawit Pongnakintr, MSc student in Mathematics at ETH Zurich

Mathematics is beginning to face a turbulent age. A recent Summit on PhD Math Education in the Age of AI (CMSA) discussed recommendations on the adaptations for the Math PhD programs. I appreciate the efforts and the opinions of the experts involved, and I agree with the main points, but my major objection is precisely at the defining characterization of a PhD in mathematics.

The summit’s report cites the AMS Policy Statement on Ethical Guidelines: “a Ph.D. degree is a certification of both mathematical knowledge and independent achievement in mathematics. Institutions are responsible for ensuring both sufficient knowledge by the recipient of important branches of mathematics outside the scope of the thesis and the high level and originality of the Ph.D. dissertation work.”, and acknowledges that “In the age of AI, we need to reevaluate what mathematical knowledge and independent achievement mean, how those goals can be attained, and how we assess them.”, but does not provide a full resolution to this problem. Furthermore, number 8 of the report’s top recommendations (page 2) reads “A thesis should be an original scholarly contribution. This should be understood broadly; for instance, the thesis need not record the first proof of a statement.”

I strongly disagree with this position. More precisely, I want to convey two opinions in this post:

  1. I do not think an original scholarly contribution is possible in the near future.
  2. Even if it is possible, and a PhD in mathematics can still be well-defined, I think it is still unfair to the incoming PhD students to evaluate the PhD upon original contribution.

On the first point, I argue that it has become increasingly clearer and clearer that the current AI systems can solve open problems very efficiently. Navier–Stokes was solved in 88 hours of computation (perhaps with or without partial contribution of Alpöge–Buckmaster; OpenAI did not rule out this possibility). My point isn’t that every open problem will be solved very soon (there will be no paper if no one pushes the buttons), but rather that “when anyone wishes to find an answer to a certain question, the AI system will act as if it’s a Hilbert’s dream oracle (Entscheidungsproblem – Wikipedia), and spits out the answer instantly”. In this respect, the proof is already there; we only need to press a few buttons. Even if the report’s recommendation says that “This should be understood broadly; for instance, the thesis need not record the first proof of a statement.”, I’d still argue that this model will absolutely break, sooner or later. There is nothing new to be done. Even if we give up the dream of proving statements, new expositions would still be written more quickly, more efficiently, and more correctly, by AI systems. Asking for an original contribution from a PhD student is nothing but asking a mere mortal to race against a superintelligent machine. I don’t think this should be the goal of a PhD anymore.

On the second point, assuming that an original contribution and all that is still possible, we’re still admitting that we’re living in a very turbulent time of mathematics. If a PhD advisor comes to me in the next year, handing me a research program Q with a specific research problem P, which the AI can’t solve yet (for example, Q := “Langlands Program”, and P := a specific conjecture there), there is still absolutely no guarantee that AI cannot solve P in the next year. An example scenario is that the advisor sets up a plan for me to study Q for 18 months, then begin attempting on attacking P in the next 18 months. In these turbulent times, perhaps only 9 months have passed since I started the PhD, and the AI becomes strong enough to solve the whole Q. Does this count as failure as a PhD? How will the committee allow me to branch out into other possible paths to complete the PhD? What if the whole program Q is so big that by having such an AI solving the whole Q, Q is basically “killed” (Thurston, S0273-0979-1994-00502-6.pdf, page 173) in the process? Of course, this situation is not entirely new, compared to how Thurston killed the theory of foliations, and to how Grothendieck killed functional analysis, but my point is that this situation is becoming increasingly riskier and riskier when AI capabilities are unpredictable. I think it would therefore be more reasonable if PhD programs can provide a failsafe for PhD students. If someone works for years and then tomorrow AI solved the problem completely, I argue that the person should still qualify for a PhD, even if no original contribution comes out.

In this sense, due to the mentioned two points, I’m arguing for a radical change in the definition of a PhD: future math PhDs should NOT expect original contributions anymore (since it is neither plausible nor safe to do so), and this should be explicitly stated. In particular, I’m currently considering not studying in a PhD program that evaluates my worth based on my output—it is too risky and too costly for me to bear such expectation.

My position is therefore very clear, and I’m willing to state it clearly in PhD applications (if no one can convince me otherwise in the near future), that I do not expect myself to arrive at an “original scholarly contribution” as it is too heavy to bear that expectation. I currently consider applying to PhD programs as a commitment to study certain topics deeply and perform original attempts seriously, while improving my communication skills in collaborations and seminars, and performing teaching services to the community simultaneously. I already consider this plan to be “enough for a PhD”.

I do not know the opinions of other MSc/PhD students, and I believe that math departments will survive regardless, but I still strongly urge math departments to rethink PhD students and the risks we’re confronting. And, if possible, let the future generations grow safely, without this worry about the impending doom in this turbulent time of mathematics. I believe a lot of students (including me) are still willing to do mathematics, even without recognition, even without credits, even without the chance to discover new theories. Just for the sake of understanding today what I did not know yesterday. That is enough a reason for me to continue studying mathematics.

Received 25 September 2026.

40 responses to “Opinions against the PhD, from an incoming PhD applicant”

  1. Denys D. Avatar
    Denys D.

    I am writing this as an Associate Dean of Graduate Studies. A ‘failsafe’ guaranteeing a degree is simply not going to happen for the straightforward reason that not every student admitted to a graduate programme deserves a PhD. Earning a doctorate requires demonstrating genuine scholarly competence, not merely putting in time. Unfortunately, from where I sit, I see instances where that standard is simply not met. So, my friend, you will have to work hard, very hard, to get there. But that is precisely what makes the degree worth having.

    Reply
    1. Anon Avatar
      Anon

      It seems to me that this comment misses the point of the post. The post argues that genuine scholarly competence can no longer fairly be measured simply by checking if the PhD student produces novel research — it is entirely possible that AI solves the student’s problem before the student does, rendering their work unoriginal, despite the student demonstrating scholarly competence deserving of a PhD.

      I also don’t think that the post argues for a failsafe that unilaterally guarantees a degree to every student admitted to a graduate program, rather just some mechanism that protects a PhD student that is otherwise competent but was not able to keep up with the speed and power of AI.

      Reply
      1. Denys D. Avatar
        Denys D.

        I understood the point of the post perfectly; it seems you have missed the central point of my comment. The premise that AI introduces an entirely unprecedented unfairness by “scooping” a student’s problem is simply historically inaccurate. Long before LLMs, researchers and doctoral students regularly faced the risk of another group publishing a solution at mid-thesis. When that happens, normally constituted scholars do not throw up their hands and ask for a safety net. Real researchers adapt: candidates explore different methods, offer complementary perspectives, analyse edge cases, or pivot. Navigating that friction is precisely the precious training, because ultimately, the primary product of a PhD is the researcher themselves.

        My point remains straightforward: no matter what tools emerge or how the landscape shifts, the degree cannot be awarded based on good intentions, coursework, or effort alone. A doctoral degree fundamentally demands substantial, verifiable proof of work and completed scholarly output. Any “failsafe” that removes that standard ceases to be a safeguard. It becomes a lowering of the bar. To earn the degree, you will simply have to work hard, adapt to the reality of the discipline, and deliver undeniable results.

        Reply
        1. Marcin Kotowski Avatar
          Marcin Kotowski

          Can you be SPECIFIC about what you propose as the solution? The problem of a PhD student being scooped exists, but has always been relatively marginal. The risk is definitely not marginal now and cannot be mitigated by the advisor’s wisdom and networking, choosing the right problem etc. Anyone, anywhere can be scooped, potentially even by a middle school kid with some money to burn.

          Reply
        2. Anon Avatar
          Anon

          I see now after reading your comment and rereading the post that the post can be interpreted as asking for a stronger change in what should constitute a doctoral degree than I originally thought. Given this, let me rephrase and refine what I meant to say.

          I completely agree that the primary product of a PhD is the researcher themselves, and that a doctoral degree demands significant effort, dedication, and proof of work. However, I feel that this is at odds with the requirement that a PhD _only_ be awarded if original research has been conducted. This is what I meant to say in my comment (and what I think the post is arguing for) — especially given the unpredictability and pace of the development of AI, originality of output produced during the 3-5 years of a doctoral program is not a sufficient metric to judge a doctoral candidate’s ability to carry out research, and it may be time to search for better metrics.

          Regarding scooping: while this problem is certainly not historically unprecedented, I don’t think scooping at this scale and speed has been possible until now. I’d like to ask you for your thoughts on the example scenario in the original post, where AI grows strong enough to conduct the entire planned research program of a doctoral student. Depending on the depth and speed of the AI’s work, it may not be possible to adapt or pivot in the time allotted to complete a doctoral degree, and it doesn’t seem to me that an inability to pivot in this scenario is a meaningful reflection of the candidate’s ability to do research.

          Reply
          1. Sirawit Pongnakintr Avatar
            Sirawit Pongnakintr

            Replying to Anon and Denys D., thanks for your comments. My concerns are less about scooping or having others finishing one’s work before one’s own. My concerns are more like: imagine we have an AI system and you want to work on problem X. Working on your own would take 3 years to solve X. Pushing buttons take 3 minutes to solve X. In that sense, even if no one would steal your work, why would anyone spend 3 years on X only to produce something worth 3 minutes?

            My own answer is “because I want to learn and gain my research competence”. I think I can put in the effort and dedication, and I might be likely to produce X. But the main point I disagree is that if you’re going to judge a person based on X alone, you’ll be more likely to approve button pushers of a PhD than approve a serious student that wishes to gain research competence. Of course, if I put in the effort and dedication and at the end am still unable to produce X, I agree that I should not receive the PhD. (and I think everyone agrees at this point)

            I think the summit (CMSA) argued for “judging based on oral defenses rather than purely on the thesis itself”, and I agree with this, but in my opinion I think it’s not enough–people can demonstrate strong understanding and subject competences on oral defenses, but that doesn’t mean they have a strong research competence. My argument is something like: AI is making these two axis more “orthogonal” than is usually coupled–one can understand without producing contribution, and one can produce contribution without understanding, and when it’s the time we choose, I urge the PhD to rely on understanding more than on contribution.

            My own personal view (which might be unpopular) is that in the near future (i.e. in 1 year) AI will have the capabilities to solve each of the Millennium Prize Problems in an hour, and I was writing this post under the position of “when that time comes, what should a PhD means”. And after some thinking, I’ve arrived at the position that “providing an original contribution” is as meaningless as “pushing buttons”, and the meaningful part is just the human individual understanding and interpersonal connection.

            Reply
            1. Denys D. Avatar
              Denys D.

              Dear Sirawit, You wrote: “why would anyone spend 3 years on X only to produce something worth 3 minutes?” Here is the fundamental answer: the product of a PhD is not the proof. The product of a PhD is the human being. A doctorate has never been about awarding a prize to an equation on paper. It exists to train a person to think critically, navigate uncertainty, master a body of knowledge, and develop the taste to ask meaningful questions. Working for three years on a hard problem is not a waste of time just because a computer can calculate the answer in minutes because the computer did not gain judgment, intuition, or understanding during those three minutes. You did. A committee of serious mathematicians will never award a PhD to a “button-pusher” who merely submits raw outputs they cannot conceptualise, interrogate, or defend. That is why the thesis defense and oral exams exist.You do not go to the gym to see weights lifted; you go so that you become stronger. Solving X is just the weight you lift. The goal has always been your training, your expertise, and your transformation into an independent researcher.

          2. Denys D. Avatar
            Denys D.

            Thank you for the thoughtful follow-up. To answer your question directly: if an AI solves a research program faster than expected, it does not mean research is dead. It means the starting line/boundary has moved forward. In mathematics and the sciences, an answer is rarely the end of the story. If a machine produces a 500-page calculation or an unexpected proof, the immediate scientific questions become: Why does it work? Can it be simplified? What is the conceptual insight behind it? Many famous theorems have ten different proofs, and each proof opened up entirely new fields. Providing that understanding, verification, and conceptual clarity is itself original research. A PhD student is not expected to be a calculator racing against machine. They are trained to be a thinker. If we remove the requirement of original contribution or tangible output, what remains? Passing exams and reading the literature? That is already certified by a Master’s degree. The PhD exists specifically to certify that a person can step into the unknown and contribute something meaningful of their own. This is my vision.

            Reply
            1. Sirawit Pongnakintr Avatar
              Sirawit Pongnakintr

              Thanks for letting me hear about your vision! I’d also want to reply to your other sub-comment here simultaneously, so let me write it here.

              I strongly agree that “Solving X is just the weight you lift. The goal has always been your training, your expertise, and your transformation into an independent researcher.”, but on “If a machine produces a 500-page calculation or an unexpected proof, the immediate scientific questions become: Why does it work? Can it be simplified? What is the conceptual insight behind it?”, I’d reply with “I can work 3-5 years trying to understand it further, and I’m willing to work on it, but this AI can give you the description you want in 3 minutes”.

              In this sense, the point of my original post is not to convince the committees to construct a “failsafe” for incoming PhD students. The point is that I’m trying to convince the committees that “when that time comes, when AI can discover the proofs faster than you, can understand the proofs more deeply than you, and can even explain and simplify the work better than you, all of this with a much cheaper cost than hiring you”, I urge the committees to retain a safe place for PhD students. Perhaps the committees decide that “we want to train people to think to become an independent researcher, and using AI will not going to help it and will make it more harmful to the students”, then we should agree to let the student solve X without using AI, even if it’s slower, even if it’s not making an original contribution.

              Maybe my hypothetical is an outlier position, and might be too far away from your vision, but I believe that in the near future, AI will not only be able to solve thing faster, but more like “whatever Y you think of doing it in the next 3 years, AI can finish it in 3 minutes” and “Y” is arbitrary here: you can replace Y with “solving an open problem X”, “understand how the work Z is done”, or “simplify this explanation into an easily digestible paper”. My hypothesis is that all of them will become cheap and abundant, to the point that evaluating someone based on some “Y” will become absurd. As the point of a PhD is to train people to become competent, such evaluation should therefore not be based on “original scholarly contribution”–any paper/solution/exposition will all become cheaply producible by AI, even if you argue “but then there’s this next thing”, I’d argue back that “this next thing is also cheaply producible by AI”. That’s why the main point of my post is the objection to this criteria that we still use it as the principal characterization of a PhD.

              Whatever criterion “Y” you would judge a person whether they will receive a PhD or not, if “Y” is *materialistic*, then I argue that AI will be able to produce “Y” more cheaply and better than the PhD candidate. My conclusion is therefore, “such Y must not be materialistic” in the sense that “it shouldn’t even be concretely measurable–since all measurable things are replaceable by AI”. And therefore “PhD should ask for competence and mastery of the student, not for the objective contribution”.

              I understand now that my hypothetical is pretty far away from the mainstream, and therefore I could see why other people could misinterpret my position. I’ll keep this in mind and try to communicate better next time. Thank you!

  2. Michel Schellekens Avatar
    Michel Schellekens

    I think that your worries are understandable. In my experience, smarter people can have a tendency to worry more (as a general rule, not an absolute). This is an awkward time to start on a PhD but we will all adapt and learn to carry out mathematics in new ways. I do not believe this is game over for young mathematicians or for mathematics. The speed of progress is daunting, but recursive self improvement is unlikely to be a quick next step. See the recent MIT tech review on that front. It is a difficult and fascinating period of transition. Pick a great supervisor, a topic you love and discuss these worries with your supervisor and a plan forward. There won’t be a quick institution-wide solution that will be adopted everywhere, but a good supervisor will have your back and will adjust your study program according to how matters develop. If you did great work prior to being beaten by an AI, the supervisor will know and steer you in a new direction while taking into account the point you arrived at towards your PhD.

    Reply
  3. Hume Chang Avatar
    Hume Chang

    I think the solution is to allow survey papers or textbooks count toward doctoral thesis. My argument is that regardless of AI’s endgame capability, there are always marginal benefits of having one more textbook. Take linear algebra as an example. It’s considered a solved field. Any professional mathematician is more than capable of solving any standard linear algebra problems. But just count how many textbooks are written after Halmos’ Finite Dimensional Vector Space. Each of the new books, Hoffman & Kunze, Friedberg et al, Strang, Axler, brought unique perspectives to the table and has taken on a life of its own. Writing a good textbook has always been a contribution in the pre-AI era regardless whether the topic is trivial to professional mathematicians. Thus, it doesn’t matter if AI’s knowledge is above and beyond your field of interest or even if you can prompt AI to write a textbook. As long as you write a textbook that brings your unique perspective and personal taste, it’s a contribution. And, given people’s prediction of theorem abundance, there’ll be a huge demand of new textbooks to canonize AI’s findings.

    Reply
  4. lohn_jennon Avatar
    lohn_jennon

    “My position is therefore very clear, and I’m willing to state it clearly in PhD applications (if no one can convince me otherwise in the near future), that I do not expect myself to arrive at an “original scholarly contribution” as it is too heavy to bear that expectation. I currently consider applying to PhD programs as a commitment to study certain topics deeply and perform original attempts seriously, while improving my communication skills in collaborations and seminars, and performing teaching services to the community simultaneously. I already consider this plan to be “enough for a PhD”.”

    Let me try to convince you otherwise. Graduate school admissions are extremely competitive, far more competitive than most people realise. In undergrad, at my institute, I talked to my the head of the graduate admissions committee for tips on applying to grad school and they told me ‘grad admissions are basically looking for reasons to refuse you’.

    So I think if you are applying (at least in the case of US institutions), then you have preemptively given them a reason to not consider you very seriously.

    It is indeed true that hiring at both PhD and more senior levels will need to be revamped, but it is unlikely that this will be done in this admissions cycle. My strong suggestion to you is to send out your applications as if is business as usual. When you get in somewhere AND get an advisor, you should then discuss the expectations for a PhD, which I think will change substantially by Fall 2027.

    Best of Luck!

    Reply
  5. Yemon Choi Avatar
    Yemon Choi

    First of all, you have my sympathies, and I don’t have easy answers, nor can I claim that your concerns are misplaced. However, since mathematics should still involve attention to details, I want to comment that one should learn not to casually repeat this “Grothenieck killed functional analysis” line (due to Dieudonne? Cartier) which has not been borne out by events at all.

    Believing that Grothendieck killed functional analysis, just because some people said so, is to accept hearsay rather than attempting to gather and interpret empirical evidence. If a PhD is to have any future, it should involve learning to wean oneself away from this tendency.

    Reply
    1. Orr Shalit Avatar
      Orr Shalit

      I approve! (But maybe we can graciously think of Dieudonne as a millenial – saying the Grothendieck “killed it!” just means he’s done a very nice job 🙂

      Reply
    2. Sirawit Pongnakintr Avatar
      Sirawit Pongnakintr

      Sorry, perhaps it’s my bad on the writing part. I was not meaning to literally assert that “Grothendieck killed functional analysis”. I was simply trying to say that some people say that “Grothendieck killed functional analysis”, and I was trying to convey that some whole theory can be disrupted entirely from an individual contribution. I do not intend this “disruption” to be interpreted as “being broken”, but I’m trying to say that a whole community can change its trajectory and beliefs based on some specific event.

      I agree with Orr Shalit’s interpretation.

      I appreciate your [Yemon Choi] advice, and I will be more careful when citing something vaguely.

      Reply
  6. anonymous Avatar
    anonymous

    “current AI systems can solve open problems very efficiently. Navier–Stokes was solved in 88 hours of computation”

    a $10 million dollar solution is efficient?

    Reply
    1. Sirawit Pongnakintr Avatar
      Sirawit Pongnakintr

      It mainly depends on your view, and I can understand if you think “no”.

      But for me, yes, a $10 million dollar under 88 hours is **extremely efficient**. If it wasn’t happening I would imagine that it will take at least $1 billion dollar and at least 1 month of computation to solve a Millennium Prize Problem using AI.

      Reply
      1. anonymous(op) Avatar
        anonymous(op)

        Could you explain why do you find it efficient?

        Reply
        1. Sirawit Pongnakintr Avatar
          Sirawit Pongnakintr

          Because I was expecting the Millennium Prize Problems to be much harder, almost unreachable by today’s mathematics, and I wasn’t even expecting one to be solved within the year 2100. Maybe it’s my biased belief, but this is basically my belief according to how I (very vaguely) know about the problems.

          Reply
          1. anonymous Avatar
            anonymous

            Yeah this is not a convincing argument to me and arguably doesn’t make sense to argue that something is *efficient* because of *your* *expectations*

            Reply
            1. Sirawit Pongnakintr Avatar
              Sirawit Pongnakintr

              Efficiency is subjective. Unless you have rigorously defined an underlying axiom or definition of “efficiency”, I have the rights to say that “I interpret this as efficient” even when “[you] don’t interpret this as efficient”.

              I’m not here to change your opinion or coerce you in any sense, and I think we can agree to disagree. Thanks.

    2. technicallybeard69aeef4272 Avatar
      technicallybeard69aeef4272

      It was only $15 million.

      Reply
      1. IS Avatar
        IS

        I mean sure $10 or $15 million is quite a lot of money and is more than the $1 million dollar prize.

        I think one point is that the goalposts for AI constantly shift. In the recent past, it would have been absurd to suggest that AI will make any relevant contribution to Navier Stokes. Now the claim is that “well yeah we won’t comment on the fact that AI can contribute to Navier Stokes, but it did cost $10 million dollars”.

        Furthermore, “training” a human is expensive. You have to provide food, shelter, healthcare etc until the person is 18 without factoring in any higher education expenses whatsoever. Getting someone from undergrad level to the point where they can contribute to math at anything resembling Navier Stokes probably takes at least another number of years. Once the person is in the phd and beyond, they have to be paid wages and benefits. If you add all of these things up, I’d assume that $10 million is enough to train maybe like 5-10 humans from birth to the level when they can contribute to something at the level of Navier Stokes. Plus, this doesn’t include all the people who were trained in mathematics at the university level that never reach the level where they are doing math at the level to contribute to Navier Stokes. And AI doesn’t die, so once you train it, it exists forever.

        And this is assuming that the cost of AI remains high, but I suspect that the cost will only go down as time passes further.

        Reply
        1. anonymous Avatar
          anonymous

          I’ve seen this argument before and I don’t think it holds up because of a simple reason: humans do much more than prove a single big result, they make new humans (aka babies), they contribute in other spheres of life and society, they teach other people, they contribute to mathematics in more ways than proving a millennium problem (we assume that we are talking about a person that’s in position to prove one)… I could go on for a while

          And this has nothing to do with moving the goalpost, we can have a discussion on efficiency of something without talking about the ability of the ai, the ability is impressive and it has been said by everyone a million times, I don’t see a reason for that amount of apologeticism that I should preface every discussion with “it’s impressive but…”

          Reply
          1. IS Avatar
            IS

            I mean the AI corporations aren’t designing the models to do math as their main goal, they are trying to do a bunch of other things, including replace a load of jobs, make a load of money, and apparently some want to do crazy stuff like merge with AI or whatever. So yes people don’t have kids or design AI models just to do math, but it this does not actually change the cost for either an AI or a human to do math.

            And yes people contribute in other ways. But people aren’t getting paid to “contribute to society” in some abstract way. Look at music or arts departments at universities. They contribute to humanity just as much as anyone. But they get absolutely shafted funding wise.

            My point is that the AI might already be cheaper if you consider how many resources it takes to get a human to Navier Stokes level research territory. If one doesn’t agree, I wouldn’t be surprised if the models get more efficient pretty soon. The point about the goalposts was that the debate used to be about whether AI produced accurate mathematics. We have now silently moved on to debating whether or not AI is “efficient”.

            Reply
            1. anonymous Avatar
              anonymous

              We are talking about a very specific case, the case of N-S, which costed $10-$15 millions; your (and others I’ve seen) say that it would take the same or a larger amount of money to get a human(s) to the level of proving N-S. I’m saying that you have to take into account everything else humans do, since all those resources that get invested into a human don’t only go into solving a single problem N-S, like that was case the with the ai: all of that money went *only* into N-S. Arguably all the other stuff people do can even be measured with money. Go ahead and try isolate the (investment into a human)/(contribution to a millenium prize) proportions

              There’s no goalposts being moved, talking about efficiency was always present, that whole discussion is in a completely different category from ai capabilities. If you think goalposts are being moved here, can you then say which ones? Like, how was it not from the start the expectation from everyone that this thing should be reasonably efficient? Reasonable efficiency is not some new criteria… Perhaps the unprecedented amount of compute spent on a single problem was what triggered more discussions about efficiency.

              And btw, im just addressing the ops statement on efficiency specifically, I didn’t pull that card just to say “ai expensive -> ai bad” which seems how you are reading this here; I was just curious how he reconciles that statement with money spent.

            2. IS Avatar
              IS

              I don’t know if this reply will get placed under what I’m trying to reply to.

              “There’s no goalposts being moved, talking about efficiency was always present, that whole discussion is in a completely different category from ai capabilities. If you think goalposts are being moved here, can you then say which ones?”

              So what I’m trying to get at is people used to make the argument that AI doesn’t meaningfully contribute to research mathematics. If the AI doesn’t actually help do math, then efficiency is irrelevant because an efficient AI with no meaningful capability is not useful at all.

              When I say the goalposts have moved, I mean that people have silently stopped making the argument that AI doesn’t work and have started saying that even if it works it is too “inefficient”. I suspect this group of people are the types that say “well AI can do math but it is cheaper to hire humans still so all should be fine”. If you are not part of this group, my apologies for implying that you are.

              I soon think that the inefficiency argument will become outdated too. I’m assuming that it is far easier to reduce the cost involved with AI systems than to build the capability in the first place.

              “Perhaps the unprecedented amount of compute spent on a single problem was what triggered more discussions about efficiency.”

              So I think that $10 million (which is a drop in the bucket for OpenAI) is actually quite a cheap and efficient way for OpenAI to advertise which is the actual goal of doing NS. What better way would there be to promote to politicians, investors, the general public that your product is great then by showing that you can do stuff the best human mathematicians had not been able to do? There is also the added PR benefit that you can claim AI advances science instead of just creating data centers, causing job loss, and destroying the environment.

              Plus, It seems like they have produced impressive research math for far cheaper on less famous problems. Apparently the ten problem release costed like 200$ a problem or something. I don’t remember the number but it was low. In my experience, what I am working on right now could probably be reproduced with the fancy AI in a far shorter period of time and for far less money. I can’t be certain because I haven’t used the fanciest models, but it seems like a reasonable guess from trying the $20 models. So I honestly don’t really care if NS was more expensive than researcher wages because I don’t do research at that level, I do “normal working mathematician” level research. Of course there is no way to measure this, but I wouldn’t be surprised if researchers working towards something NS related have been paid more than $10 million in wages. It just seems plausible given the problem has been open for so long and has so much interest.

              By the way too, I think AI totally sucks for mathematicians and I think political organization is the only answer really. I just think making claims that automation and job loss aren’t something to worry about because math has X esoteric property that absolutely requires humans or that current AI systems are too inefficient/can’t do X yet are a bad strategy because they prevent effective political organization.

  7. 12 axes Avatar
    12 axes

    The objection to defining the PhD as certification of independent achievement is thoughtful, especially when the CMSA summit report itself admits that AI forces a rethink of what independent achievement means. Recommendation 8 on what counts as an original contribution seems like the crux. I run 12 axes, a free political values test, and how institutions should adapt to AI is becoming a real dividing line in education debates.

    Reply
  8. Orr Shalit Avatar
    Orr Shalit

    Dear Sirawit, it has happened before LLMs that as some PhD student was working in one place struggling with their problem, some other researcher in another place solved the problem…. That’s not the nicest thing that could happen but it also didn’t render any PhD worthless. [And sometimes a problem was already solved when the student started working on it (this, at least, is much less likely to happen today)]. In some cases this led to a joint publication, in some cases one could find two papers oblivious of one another, or with a note: “as we were preparing this manuscript, we found that [17] …”. There is usually some difference of perspective or methods or scope that make the works complementary. To some extent, you could only be sure of not being scooped if you worked on a problem that nobody cared about but you.

    I think that striving for producing/discovering original results in a PhD is a good ideal goal to aim at, an exercise in scholarship and research. As many have said, the product of the PhD is you.

    Reply
    1. Sirawit Pongnakintr Avatar
      Sirawit Pongnakintr

      Dear Orr Shalit, thanks for your comment! I understand the case with having someone else solving the problem one’s been working with for some time. My worry is something beyond being “stolen” or “raced”, but is something like “having an AI oracle that can generate solutions faster than finding one on one’s own”. I’ve attempted to describe my view in another comment, but let me reproduce a part of it here:

      > My own personal view (which might be unpopular) is that in the near future (i.e. in 1 year) AI will have the capabilities to solve each of the Millennium Prize Problems in an hour, and I was writing this post under the position of “when that time comes, what should a PhD means”. And after some thinking, I’ve arrived at the position that “providing an original contribution” is as meaningless as “pushing buttons”, and the meaningful part is just the human individual understanding and interpersonal connection.

      I think, for now, I will still “strive for producing/discovering original results in a PhD” (if I were to happen to be admitted to a PhD position at some point). But my main worry is that this could no longer be the case in the near future (i.e. the supervisor will have to explicitly choose whether to “let me use AI” and the work is completed instantly and then let me aim at understanding those results, or to “forbid me from using AI” and then I could spend the main effort in trying to come up with an original result).

      Reply
  9. mhairer Avatar
    mhairer

    You say that you intend to “[commit] to study certain topics deeply and perform original attempts seriously, while improving [your] communication skills”. It seems to me that this commitment is almost certain to result in an original scholarly contribution, so please don’t torpedo your chances to be admitted at a graduate program by claiming that you do not have such an expectation.

    Reply
    1. Sirawit Pongnakintr Avatar
      Sirawit Pongnakintr

      Thank you for your advice. I will consider this.

      Reply
  10. Nilima Nigam Avatar
    Nilima Nigam

    Dear Sirawit,
    this essay is particularly poignant in the wake of OpenAI’s release of a large number of results last night. [I personally think of the announcement as an act of drive-by vandalism, but that’s me.]

    You’re entirely correct that (a) the standard for ‘original research in mathematics’ is unclear at least for the near future, and (b) given this uncertainty, there’s a great deal of unfairness at play when it comes to doctoral students.

    I see all manners of analogies out there trying to explain the ai-in-math era, including one about the Library of Babel. This is a darker vision in some ways – the Borgesian wonder was notably and functionally entirely useless to humans [accurate texts formed a set of measure zero amidst the unreadable or the pointless, if I recall].

    One way the norms on ‘original contribution’ could shift is towards that in the humanities. This is what I make of the whole ‘we’re moving to an era of understanding’ perspective. I don those fields, we don’t expect doctoral students to produce high works of art, a student isn’t expected to write ‘Pride and Prejudice’. Instead, you’re supposed to engage with a primary text in a way that’s novel, examining it from a novel view [historical, cultural, political….]. Or you might help readers of the primary text see new details, or find connections to other works. This is very different from science and mathematics, where originality has meant producing the primary work itself.

    Maybe we’ll move towards math-as-engineering, finding ways to turn AI-generated theorems into better consumer products. Or maybe we’ll move towards art. I don’t know for sure. We barely have a clear consensus on what it means to ‘mathematically understand’ something, let alone evaluate it fairly.

    So. You write clearly, you are committed to learning and thinking. As others suggest, I hope you’ll still consider pursuing higher studies in mathematics, for the sake of the learning. The goalposts are indeed being shifted, so it’s important you find an advisor and program that’s supportive. Perhaps a math department that sits in a faculty of Arts and Science [rather than just in Science] will be better able to pivot its expectations on originality. In all of this upheaval, it’s the fate of young scholars such as yourself – and all those to come – that occupies most of my thinking time these days. Please know that at least to some of us you’re far, far more important and consequential than problem #129.2 solved by ‘chat, j’ai pété’ release N.

    Reply
    1. Sirawit Pongnakintr Avatar
      Sirawit Pongnakintr

      Thank you for this comment. In my understanding this is one of the rare comments that accept my hypotheticals and engage with the scenario explicitly. Perhaps my original post wasn’t unclear on the hypotheticals and I apologize for that.

      This:
      > the Borgesian wonder was notably and functionally entirely useless to humans [accurate texts formed a set of measure zero amidst the unreadable or the pointless, if I recall].

      is my assumption, and I’m sincerely surprised that a lot of people seem to not be holding this view. However, as I’ve explained and as you’ve clarified it explicitly, under this zero marginal material benefit to the society, I’ve decided to **still go and learn mathematics**. My main message is therefore that I want to ask for PhD committees to carve out a safe sanctuary for those of us who’d still walk this path. In particular, I’m more worried about PhD committees being damaged from cognitive dissonances, and by “keep asserting that the student produce an original contribution” to their deaths, more than I’m worried about being able to find a PhD position at all. Committees are free to reject me and I respect that, but if the committees don’t even considering this risk and protecting us, then who will? At that point I’ll probably decide to withdraw from PhD applications and rather go home reading books on my own. That may be better and perhaps a more peaceful path.

      For now I don’t particularly have to make that decision (yet), so I’ll be willing to apply for PhD programs with the strong and sincere dream of learning, thinking, attempting original research, communicating, and teaching. If the time comes, I’ll eventually have to choose what works best for me. Thank you for your comment.

      Reply
      1. Nilima Nigam Avatar
        Nilima Nigam

        Dear Sirawit,
        ‘Committees are free to reject me and I respect that, but if the committees don’t even considering this risk and protecting us, then who will? At that point I’ll probably decide to withdraw from PhD applications and rather go home reading books on my own. That may be better and perhaps a more peaceful path.’

        I concur. If your future supervisor/committees aren’t willing to address these very real questions and concerns in a meaningful, actionable way then it would be foolhardy to pursue doctoral work with them. As an immediate corollary – I hope you (and other) applicants are able to have some honest conversations with prospective supervisors. Each of you will have different goals and aspirations and views around AI, as will the supervisors. Even more than before, the question of intellectual and ethical ‘fit’ cannot be ignored.

        My sincere best wishes to all of you.

        Reply
  11. Craig Avatar
    Craig

    It seems to me, ego is one of the biggest drivers of mathematics research progress. Being first to solve a problem and to let the world know you’ve solved it by publication is what motivates most research mathematicians. Take this possibility of being first away and also their bragging rights away and most mathematicians will quit doing research. Understanding math and experiencing the beauty of math is also a big motivator, but it’s not the primary motivator. Since AI has shown itself to be better than human mathematicians, I don’t see much future for math research. University mathematicians will still have jobs, but their jobs will be primarily as teachers.

    Reply
    1. Random mouse Avatar
      Random mouse

      “Take this possibility of being first away and also their bragging rights away and most mathematicians will quit doing research” Reducing the career of mathematician as a matter of bragging and pride is ignoring the huge sacrifice that comes with that choice. Maybe next time don’t vomit out your uncooked prejudices?

      Reply
  12. Thorbjorn Frommelt Avatar
    Thorbjorn Frommelt

    My utmost respect to the author for addressing this matter publicly.

    This is not a comment on the body or your post but on the (a) philosophy behind it.

    The townsquare should, hopefully increasingly, have its fair share of voices from the youngest of the community be heard. The fear of mass exodus from the field, as many posts here allude to, increasingly seems an inevitability.

    Speaking from the same vantage point as the author, the view is bleak. As we’ve been often told, the competition is ferocious and the burdens are many to accept.

    “As it has always been!”, we are reminded. That statement is then increasingly bitterly heard as the situation continues to unfold.

    A response to the situation which does not take into account the concerns from the youngest (and most vulnerable) members of the community, is no response at all.

    Among my circle of talented, eager and dedicated “baby” mathematicians, many have pivoted paths as a direct consequence of the instability and loss of soul of, what we see, as the profession of a professional mathematician.

    “As it always happens”! Perhaps. Or perhaps there is a tipping point.

    Reply
  13. James Eldred Pascoe Avatar
    James Eldred Pascoe

    Probably just means we need way more PhD students to be able to find the people who have really original ideas. (Not that the funding for that will materialize.)

    Reply
  14. anon Avatar
    anon

    Well written and I’m impressed by your ability to respond graciously to those who (in my opinion) do not understand the situation nearly as well as you do.

    If I may, it’s worth reflecting on all the ways AI will transform the world outside of mathematics. Very strange times ahead, let’s hope it ends well.

    Reply

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