8 ms·

A beginning for mathematics

266 points by robinhouston 12d ago

gleezard 11d ago
Mathematics is a solved problem.
ksd482 12d ago
> I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied. I think this is a refreshingly forward looking idea and I agree with it 100%, especially the the "rigorous defense" part. That is a good measure of how well the topic has been researched and understood by the researcher. This is where the humans can be "in the loop". > How different would this look from current PhDs? I think students would still meet with an advisor, who might suggest a topic. That topic could be explored with AI assistance, or not, but the student would be responsible for understanding it; it might be much more open-ended and larger than the typical PhD is currently. Interesting point about "more open-ended" and "...larger than the typical PhD". I think the author has a point. Earlier, the bottleneck was the candidate's/researcher's understanding and knowledge. Now with AI tools, it is so much easier to zero in to relevant knowledge, get your questions answered quickly which might lead to understanding more quickly. For e.g., before the advent of public libraries and printing press, the knowledge was inaccessible and guarded. So that was the bottleneck. Then books became ubiquitous and the bottleneck to knowledge and understanding was people's motivation AND knowledge of WHAT books and topics to research. Then came the internet and free PDFs of books and research articles. Now, the bottleneck was still people's motivation and a mild version of what books and topics to research. I say "mild" because one can lookup articles and newsletters, and book reviews and come up with a list of reading. Now comes AI and it looks like the only bottleneck is people's motivation. I believe there was also a silent, yet potent, bottleneck all along which is also removed by AI: personal tutor/coach/teacher/professor etc. Let's say if I am reading a textbook on manifolds or some research paper and I have a question about a specific theorem or even a mathematical operator being used. Before AI my only way to get my questions answered was to read more books (PDFs or print), or ask on math exchange or math overflow and wait for someone to answer, or to ask a professor. This could take up to a week. Now all of that has been cut down to 1 hour or less with an interactive chatting session. !!!!! So....the only bottleneck is people's motivation! QED Exciting time!
- emil-lp 12d ago
  > the "rigorous defense" part In my country, that's exactly how it is. Yes, you need to have a thesis to defend, but ultimately it all comes down to the (oral and live) defense/disputation.
- robotpepi 12d ago
  I'd say this is the most optimistic scenario. there are really difficult problems to be solved in terms of access to AI.
- gowld 12d ago
  Why is "Doctor of Philosophy" the correct certificate of "becoming expert in a topic"? That's a radical departure from "PhD" being a certificate that someone is qualified to produce new research. What you describe is more like a Masters Degree.
  - breezybottom 12d ago
    I'm not sure why an advanced degree is necessary for that at all, besides the pride of a vanity title. It's pretty much what Bill Nye does for science.
  - emil-lp 12d ago
    I think you have misunderstood. PhD has nothing to do with expertness. If you have a PhD, you have completed some kind of research training. That's all there is. Says nothing about knowledge or whether or not you're a genius. You cannot conclude anything else, and nobody claims that you can. If someone has a PhD, they have some training in doing research.
    - ksd482 12d ago
      Looks like @gowld is agreeing with you.
- kurthr 12d ago
  The idea that most any modern "interesting" aspect of mathematics is going to be understood (or often even explained in enough detail to reveal what is interesting) in an hour is pretty rare. Either the student's aptitude, the tutorial, or the mathematics are unique. There is a reason that these are PhDs and not undergraduate HW sets. I think we often delude ourselves as to how well we understand problems and their solutions. Some instructors even make you feel that you understand better than you do by pointing to a few approximations or simple solution spaces that obscure the larger complexity. Just looking in wonder at the many categories of three-body solutions (currently on hnews) is enough to remind me of this.
bobajeff 12d ago
The more I see these posts about mathematics institutions reforms and challenges from AI advancements the more it looks like they may need to go through a death. Or to put it another way they may need to start again from first principles. If math is truly about spreading intuition and understanding then our institutions have dropped the ball decades ago and have not been able to grab hold of it since (if they ever had it to begin with)
- deleted 12d ago
  [deleted]
- omnicognate 12d ago
  Not sure why you're down the bottom when the current top post says pretty much the same thing. I agree that the reevaluation and refocusing that is being forced by AI is one the maths establishment could fruitfully have had a long time ago.
hintymad 12d ago
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Jun8 12d ago
Excellent optimistic post in a sea of negativity, and with actual suggestions, too. After reading, my mental image is this: think of Olympiads in Ancient Greece. * A weightlifter was only awarded a laureate if he were able to lift a heavy stone (have no idea what they were lifting, for illustrative purposes only :-) * Along comes Archimedes who invents what we would call an exoskeleton. Now any regular guy can lift twice as much as last year’s athlete. * What to do? You can cancel the Olympiads, but they are actually useful as training, motivation, etc So now you have to give the prize on other factors, eg how well he can lift, has he opened a gym in the city, etc BTW, physics and bio are not exempt, so those researchers better read and try to stay ahead.
- boccaff 12d ago
  I don´t think so. For programming agents can run code, check compiler output, etc. For mathematics, it is almost the same once you factor in the usage of lean. For the reality, you can´t close the loop that fast, or with that precision. You will have to slow down by several orders of magnitude.
  - Jun8 12d ago
    Doesn’t have to. There are petabytes of experimental physics data that can be fed to AI to extract additional insights. The only holdup is this is slightly harder to than math. With bio, you’re right, generally designing and conducting an experiment goes hand in hand and theoretical biologist is not a common label.
    - lolakutty 11d ago
      >There are petabytes of experimental physics data.. There is one thing you are missing. Math is precise. Physical measurements are arbitrary imprecise...
vld_chk 12d ago
I am not a mathematician, but I can’t see how we are going to address the problem which we already see in coding: Impossibility to independently validate all AI results And in math it goes even worse. In coding code reviews are typically still the form of action you do within days. In math, historically, the lifecycle of proof is months if not years. Take as an example Millennium problems. They require at least two years of validity after publishing. Two years! In modern times with amount of output AI can produce, it feels like infinity. We are inches close if not at the moment already when humans can’t reliable validate proofs and mathematics produced by AI. Then next research will be based on this AI-written-no-human-in-the-loop results. And we will end up in just few years in a world where novel and frontier problems will be articulated by AI and proven by AI based on AI results and humans will be incapable of understating the mere nature of the solution.
- robinhouston 12d ago
  I don't think that's actually the real problem. Along with the progress in answering mathematical questions, recent progress on AI-powered autoformalisation has been astonishing. All the recent AI discoveries have been accompanied by Lean proofs. And, yes: that doesn't absolutely guarantee correctness. The Lean kernel has had soundness bugs, and may have some still. But it's pretty strong evidence of correctness nevertheless. The concern among mathematicians is not mainly that they doubt the correctness of any of these discoveries, but that human understanding may be devalued.
  - vld_chk 12d ago
    I am not that worried, but rather just observing. Humanity is about to enter the phase when we will be using things based on ideas no human ever properly understands. This thought … disturbing, somehow? It is perfectly valid counterpoint to say that we already do it. We everyday use myriad of things, tools, and software we have 0 clue how it operates. But for us as humans it was reassuring that we know that at least there are a few other alive humans who know it, who create it and who can explain it. With AI soon that comforting zone will be gone.
esafak 12d ago
> I think so. This machine might produce answers we value, but it would not, in itself, produce human understanding of those answers. It's nice that the author is optimistic, but won't the AI be best placed to dumb down its increasingly complex proofs into a language us lowly humans can understand? To keep thinking until it can refactor complex proofs into ones from 'the book'?
- magicalist 12d ago
  As they go on to explain, a human understandable proof is different than a human actually understanding the proof. That actual human understanding (like, in a brain of a human) is one of their stated goals. Producing human understandable proofs is possibly a job best for humans today, but the author appears to agree with you that this is probably fleeting (and argues that even if you disagree, it should probably be treated as if it is fleeting when planning for the future): > Right now AI systems arguably underperform us at theory-building, asking questions, exposition, … so we could prioritize and reward those skills. I think this is unwise: compare the speed at which the academy adapts to the speed at which model capabilities improve. We need to consider the endgame. If the models remain incapable in some domain, we can adjust later.
- vouaobrasil 11d ago
  AI will never be able to dumb down a proof to a level simple enough for someone to understand who has never studied math and put a lot of effort into it. Some concepts just need time and effort to absorb no matter how relatively simply they are phrased.
  - esafak 11d ago
    Sure, but some proofs are so complex that even experts can't follow them. As AI progresses, this may constitute an increasing fraction of proofs. I meant that AI could work to find the simplest possible proof.
    - vouaobrasil 10d ago
      > Sure, but some proofs are so complex that even experts can't follow them. If that's the case then I don't see much reason for them to exist in the first place except mental one-upmanship.
Bluestein 12d ago
I love how this is (without slighting the problems entailed) coming at it from a perspective of infinitude and abundance (we will always have more problems to solve) - which is the correct framing, particularly when dealing with ideas, or fields in the which ideas are the driver/product/output/material, and ideas themselves, the field itself, are infinite.- PS. The validation problem, being one.-
waynecochran 12d ago
As someone who has a degree in math, I still can't help but think mathematicians are getting a little bit of a comeuppance. In a lot of areas of mathematics there had been little effort to make the work understandable and leaves numerous folks who could benefit from the knowledge on the outside looking in. Now AI comes along and do the same to mathematicians. Makes me chuckle a little bit.
- chneu 12d ago
  Lowering the bar for the masses isn't always a good thing
  - PeterWhittaker 12d ago
    Examples, preferably academic?
    - applfanboysbgon 12d ago
      Half of modern academia, for one. Universities have turned into degree mills with the expectation that >50% of the population requires a college degree, regardless of whether they actually have any interest or need for one beyond doing it because it's a prerequisite for a good career, independent of whether said career actually uses the knowledge in any way. Not that I actually agree that math was at the right level of gatekeeping. It definitely feels intentionally opaque beyond reason, and I think it's why LLMs are able to cut through the obfuscation and solve problems that maybe wouldn't actually have been considered quite so hard if mathematicians did a better job of making their work accessible.
      - JoeAltmaier 12d ago
        <pedant>28% of adult Americans</pedant>
        applfanboysbgon 12d ago
        66% in Japan. Who said anything about America?
jkhdigital 12d ago
> resulting in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications. I hope, from the depths of my soul, that the static typeset report format for transmitting knowledge and understanding will finally die and be laid to rest.
- emil-lp 12d ago
  As a researcher in theoretical computer science, I love PDFs more than any other format when it comes to mathematics. There simply is no contender to LaTeX and PDFs. Lucky for you, almost all research in math, cs, and physics, are put on arxiv, where you can download the source code (.tex) as well as get an HTML render.
- warkdarrior 12d ago
  What would you prefer instead of "static typeset report"?
  - Jblx2 12d ago
    Not the OP, but how about things like: https://ciechanow.ski/archives/ https://ciechanow.ski/archives/ ...for starters?
wrs 12d ago
The author argues for evaluating Ph.D. candidates based more on the oral thesis defense than on the actual thesis. By essentially the same reasoning, I’ve been arguing for prioritizing in-person design/code reviews over code-only async PR comments. The important thing is to verify that the human has a coherent design in mind and can demonstrate that it got implemented, regardless of who or what was at the keyboard. “I dunno, I guess Claude thought this was a good idea” is not a coherent design.
- getnormality 12d ago
  I think upweighting the live components of academia is inevitable in the age of automatically produced writing, but I also find it depressing that people think so little of writing that they imagine it obsolete because of AI. AI writing is aggressively, aggressively mediocre. It is only good for how cheap it is. If you're indifferent to how much better good human writing is than AI writing, you should not be the one to evaluate human writing.
- Frost1x 12d ago
  I can see how this trend is going to go. Manager: “Why isn’t feature X available?” Person B: “key pieces are delayed due to the developer not understanding all of the LLM doesn’t and implementation.” Manager: “does it work? What are the risks?” Person B: “well yes it works for now but we’re accumulating tech debt due to a lack of understanding and potential flaws that haven’t been thought out yet” Manager: “they want feature X, ship it, we can deal with it later, I don’t care if it’s not coherent as long as it works.” How many decades at this point has these been a push for functionality over everything at all costs? And you have a mechanical snow plow now. Most businesses don’t care about later risk or any future planning beyond the quarter horizon, they’re not concerned about how it will effect their performance in 3 quarters or lead to instability or issues, those are future problems for a future person and we’re here for money now.
  - AdieuToLogic 12d ago
    The simulated dialog you have provided is very much representative of what many of us have heard first-hand. However, I disagree with: Most businesses don’t care about later risk or any future planning beyond the quarter horizon, they’re not concerned about how it will effect their performance in 3 quarters or lead to instability or issues, those are future problems for a future person and we’re here for money now. Businesses care about "later risk" and what it implies. Individuals within an organization do not unless it will specifically affect their bonuses/promotions.
theodorewiles 12d ago
Yes the fascinating thing is: 1. It will take much longer to understand the output of the machine that it takes to prompt and create it. 2. The only? best? one? way to /verify/ that you /in fact/ understand the output of the machine is to explain it to someone else. So there will be a machine generating koans which need to be meditated upon and discussed with human social back-pressure validating understanding. I think this could be much more cooperative and at a minimum this will be a way different math social construct.
wcfrobert 12d ago
I like the quote from Hilbert that was brought up in the article: "we must know, we will know". With AI, it might be the case that we don't know, we won't know, but the machine does. The central question, namely whether humans should be in the loop, will be repeated again and again in the years to come for all industries, starting with mathematics.
wanderingmind 12d ago
Better plan is to shut down PhD and make students take a oral thesis at bachelor and master level and help them become a productive economic participant as soon as possible.
ComplexSystems 12d ago
I admit I just don't understand this attitude. People had the same complaints that the code produced by early coding models was messy, lazy, poorly commented, had terrible architecture and so forth. The central complaint was that it was just too difficult for humans to review. The answer is just to improve the models and move on. Similarly now we're getting AI doing math. The proofs compile but are a mess. So just make the models better at writing clean proofs and explaining what they're doing to humans. That's the end of it. Rather than just go on and on about how it's the end of the world if we don't do this, why don't we just do it?
- JW_00000 12d ago
  There's a fundamental difference between the goal of code and math. The goal of code is to produce software that does something useful. As long as the code does what it's supposed to do, arguably, it's good to ship. (As you imply, we want the code to be good enough to also be reasonably certain there are not too many bugs, that it is maintainable and can be extended etc. This is what early models failed at but now seems broadly fine.) But for math: what is the point of a proof if no one will read it and no one uses its result? To quote the article, "AI systems will [...] result in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications." But if there's no one reading the PDFs, what's the point - no matter how good your AI model. The point of math is understanding. So mathematicians should feel free to use AI as much as you want, but in the end, they should've gained some understanding on what happened.
  - ComplexSystems 12d ago
    > But for math: what is the point of a proof if no one will read it and no one uses its result? What is the point of writing software if nobody will run it? > So mathematicians should feel free to use AI as much as you want, but in the end, they should've gained some understanding on what happened. So they ask the AI to explain the proof.
  - kenjackson 12d ago
    If the point of math is understanding then maybe our incentive structure is wrong. Maybe it should be teaching the concepts to as many people as possible rather than just continuing to write papers that 10 people in the world understand, which is the current state of a lot of math.
bonoboTP 12d ago
> We already interview faculty hires; we must now do the same for graduate admissions They hire PhD students without hearing them give a talk and then doing interviews? In Germany, the applicant gives a talk (30-40 min) to the research group they want to join, usually presenting their master's thesis, engage in discussion, often share lunch with the group, then do 1 on 1s with individual members of the group and a longer one with the PI. Obviously this can vary within Germany too, but I couldn't imagine hiring someone without something like this.
- bobmarleybiceps 12d ago
  1. US a lot of international PhD applicants, so traveling before even being accepted is difficult, and 2. lots of people don't do a masters. when I applied to PhD programs (not in math) it was basically CV + personal statement + recommendation letters + short chats with interested faculty :shrug: Maybe it was because my CV was "strong" but the chats were more see if interests were aligned, rather than actually interviewing me.
  - bonoboTP 12d ago
    Remote talks are also a common alternative in such cases. Zoom etc. I also know that another major difference is that American universities tend to hire without a professors involvement, into a generic "program", then the PhD student seeks an advisor after being accepted, so applicants have to woo some unconnected committee pursuing various goals misaligned from the PIs instead of convincing the PI. In much of Europe a professor basically "owns" a chair and really is boss and decides hiring pretty much alone. Anyways, if they don't have a masters' yet, they could present their bachelor thesis. But I already think it's a bad model to combine the masters (courses) and the phd (research) into this American hybrid that's the direct-bachelor-to-PhD jump, but that is somewhat unrelated.
    - bobmarleybiceps 12d ago
      I agree the combined masters + phd is weird tbh. I think almost everyone treats it like "just focus on research and spend as little time on course work as possible." I would prefer it if courses were more flexible. I'm not a big fan of the US application setup. IDK how it is in Europe, but in the US, it feels like there's a lot of not-very-meritocratic "secret" stuff you need to know to up your chances.
youoy 12d ago
As with the rest of the domains, AI/LLMs will do syntax and search better than any human. In code, any developer whose differentiaton was clean code and knowledge of different technologies is now average. In math, any mathematitian whose differentiation was to manipulate formal systems and know tricks of different domains will be average. Fortunately, humans do more than syntax and search. The bad news for developers is that if you know what the output of your program should be (which happens most of the time), almost all of the job is syntax and search to build the code that reproduces the output. The good news for mathematitians is that for the majority of problems you never know the output, or you just know the output is either "True" or "False". There are some cases where you need something else, for example "a solution that blows up in finite time". For those cases AI will outperform you easily (see new Navier-Stokes solution) So if as a mathematitian you were doing more than syntax and search, then keep doing that and use AI just for what its best.
jplusequalt 12d ago
Another mathematician waxing poetically about a future that will not come to happen. Call me a pessimist if you'd like, but capital has no incentive to ensure mathematicians maintain their current status in society. If you're a mathematician you are in the same boat as the software engineer, and the Dodo. Better learn a trade buddy /s
jsrozner 12d ago
"A computer or monkey could easily start at the axioms of ZFC and iteratively apply deduction rules....simply conjecture all mathematical propositions in alphabetical order...The prospect of automating mathematics by enumerating all conjectures, and all proofs of ZFC, is probably not so disturbing to you." I thought we were going to get at least some brief comment on Godel here?
- anyfoo 12d ago
  Gödel effectively says ZFC must be incomplete, otherwise it would not be sound, but does that stop you from listing all mathematical propositions it can generate in some well-defined order?
  - jsrozner 12d ago
    That's right. You could list all those propositions and search for proofs of them. In fact this is very similar to Hilbert's very program to which Godel's First Incompleteness Thm was a response (https://en.wikipedia.org/wiki/Hilbert%27s_program https://en.wikipedia.org/wiki/Hilbert%27s_program). Godel showed that there are true statements that cannot be proven, and also that among the unprovable statements from within the system is the consistency of the system itself. As I understand, mathematicians are still trying to figure out how much this matters. One of the best examples of its mattering is may be the Continuum Hypothesis: CH is consistent with ZFC, and ~CH (not CH) is also consistent with ZFC. In other words, you have enough flexibility in constructing your ZFC world such that in some ZFC-consistent worlds CH is true, and in others CH is false. Litt's statement is not wrong; it's just that what he wrote sounds so much like Hilbert's program, that I'm surprised we didn't get some even minor comment on what kinds of truths we could reach if we embarked on such an effort.
    - Chinjut 12d ago
      People always bring up the Continuum Hypothesis as though it has something to do with Gödel's incompleteness theorems, but it doesn't really. The Continuum Hypothesis being neither proven nor disproven by some particular axioms is a similar phenomenon as that the group axioms neither prove nor disprove commutativity, the ordered field axioms neither prove nor disprove the existence of a square root of 2, etc. There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is. Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent. But the inability of ZFC to consistently decide the Continuum Hypothesis is established in a completely different manner, with the Continuum Hypothesis not consistently decided by ZFC + Con(ZFC) either, or any such thing.
bonoboTP 12d ago
I think this goes in the right direction. You have to rethink the role of human work in math, can't put your head in the sand and cling to your comfy institutions and system just because you got to know it's ins and outs and just want it to be like that forever. But the bigger picture is: while I understand the author know his field best and wants to keep the post focused, the same issue will hit many more fields. We need to also have a broader discussion that involves more fields of knowledge work, largely academic scholarship but also regular office work, then it will come to engineering design, medicine, it's already coming for 3d modeling and vfx, software dev, it will come for a lot of middleman services. Not at the same rate, but we have to understand that it's not just that math will change. Change will be the default. Everything will change. It will be much easier to make math change because all things will change. You shouldn't worry and imagine that funding criteria will be like today or that journals or academia or politicians expectations will be like today. No, everything will adjust with some timing differences of course but it won't be a static world and then math changing and having to justify and fight to explain the change to other actors who are baffled. They won't be baffled they will themselves have to change. The world will transform as much as it did when society moved from feudal agrarian to urban capitalist industrial, or from the vast majority doing physical labor to a service economy with a huge amount of desk jobs. I can't tell how it will change exactly but it will be bigger than what we have seen in the last couple of generations or maybe more.
eliauelkouby 12d ago
[flagged]
asa123 12d ago
While I broadly agree with the premise of re-directing the “purpose” of math, I quite detest the idea that judgement might be primarily based upon some in person discussion, or oral presentation, and the claim that written mathematics that is not orally communicated might be less worthwhile in some sense (i know this isn’t the exact statement of the authors proposition). There are a good deal of people, whom, falter much more in oral discussions, whether this be for a psychological thing, stage fright, or difficulty explaining things on the spot. There are also certainly brilliant people, who can’t give an informative, discussion inviting talk to save their lives, but given enough time, can formalize their thoughts in writing at the highest levels of their field, and that writing is likewise very enlightening (sometimes). It’s not clear to me, that, AI as is, could not pose successfully in an oral discussion of a topic. I mention this because it seems that one implication of the article is that AI might write things that are logically correct, but devoid of understanding. I suggest rather that 1) it is not extremely improbably that AI is incapable of generating mathematics that furthers human understanding and if 2) it is indeed highly likely that they cannot generate mathematics that furthers human understanding in a textual format, then surely one could also differentiate between human and AI on a textual level, and judge the contribution of a human, without the need of oral discussion? I suppose another aside is, one might claim that the existence of AI means people have much much more text to filter for, and so, it becomes difficult to find one person’s good writing amidst a sea of, logically correct, yet understanding devoid textual content. But by and large much or mathematical academia certainly operates off of some reputation/vouching system presently anyways, that already serves as a “filter” in some sense. Perhaps the existence of such a system/culture is not a good thing, but oral discussions/seminars certainly aren’t immune from such predilections. Perhaps I’m babbling like an idiot, but the entire and sole purpose of this comment is just to say: for the love of god please don’t let the standard be judged by oral presentation
augment_me 12d ago
What is the incentive for a person to sit though seminars and evaluations? People already hate redundant meetings. What is the incentive to change the system from the existing one to one that rewards this verification somehow? Who benefits from this change?
sabujp 12d ago
[flagged]
soundworlds 12d ago
I was involved in this research a few years ago: https://metamiditoolkit.com/research-paper/ https://metamiditoolkit.com/research-paper/ One of the things that struck me was that most people have an area of their workflow that they would be happy to hand off to an AI, so that they can focus on the thing they love. However, that area is different for each person. One person's grind-work is another person's love-work. This is why it's so hard to build consensus around where the "red lines" are in this space.
gulugawa 12d ago
Misanthropic and Open Artificial Ignorance are misleading the public. They are leeching off the work of mathematicians and pretending that new math was invented.
Jblx2 12d ago
Does anyone have a rough estimate of how many research mathematicians there are in the U.S. or world?
sashank_1509 12d ago
I’m not a mathematician, but if I were I would imagine the main attraction would be the act of thinking hard and solving problems. Maths felt like the ultimate profession for someone who loved puzzle solving. If you delegate the thinking, any percent of it to an LLM, I don’t think it’s maths anymore. Might as well join any other job and earn better money now. A completely ludicrous proposition, that maths will no longer involve mathematical problem solving.
- vouaobrasil 11d ago
  Modern incentives, unfortunately, reward producing solutions rather than going after the joy of solving problems. And it's the latter that inspires others, not the former.
GMoromisato 12d ago
My worry is that the frontier of math is too far away for most humans to reach. AI is only going to make that worse. If today it takes twenty years of math study to reach the frontier (in a narrow field), what's it going to be like when it takes forty years? Or four-hundred years? Will the fields just get narrower and narrower to accommodate finite human intelligence?
- bell-cot 11d ago
  > If today it takes twenty years of math study to reach the frontier ... Are you counting from when a kid first learn 1 + 1 = 2? That seems a misleading metric. The rate at which kids are learning math in the first half to 2/3 of their schooling is very low. Or are you asserting that Ph.D.'s don't reach the frontier until age 40+? Actually, the frontier of math is too far away for most humans to reach because >99% of humans don't have the drive and talent to study and learn enough advanced math. Similar barriers exclude >99% of people from Olympic-level athletics, or being famed violinists, or being important architects, or being US Senators, or being successful ancient historians, or many other things.
MarceliusK 11d ago
The key shift seems to be from "can you produce mathematics?" to "do you understand mathematics?"... Those used to be correlated strongly enough that a thesis or paper could serve as evidence of both. If AI breaks that correlation, evaluating people through defenses, discussion and live problem solving suddenly makes a lot more sense.
Kvarnek 11d ago
Always fascinated by how foundational math concepts built up. Euclid's Elements still blows my mind for its logical rigor.
Tbarlow 11d ago
Pretty neat how these abstract origins still shape our world, especially in CS. Kinda makes you appreciate the shoulders we stand on.
dropshade77 11d ago
As a recent CompSci graduate and software engineer, I can sympathise with the author’s position. At the very least, these times are highly unsettling.
samayashar 11d ago
> AI does not care if you are anti-AI. This >>>
econ 11d ago
Imagine, eventually, a large ever growing library full of unread books. One could roam around forever and discover new things. Might be possible to generate a 3d world using some taxonomy estimate the amount of titles and generate the books as the area is explored. Like the manuscripts in Sakya Monastery in Tibet and like the millions(?) of cuneiform clay tablets. Vast amounts of data that may contain information.
darkstarsys 11d ago
This is an excellent essay, and especially relevant when read together with the MIT report on the future of education in an AI era: https://aiandeducation.mit.edu/report/ https://aiandeducation.mit.edu/report/ (and my own https://oberbrunner.com/blog/ai-future-of-mathematics https://oberbrunner.com/blog/ai-future-of-mathematics). I'm heartened to see this kind of thoughtful, reasoned discussion around long-needed reforms in the educational community, with AI as the catalyst. We certainly have interesting times ahead.

266 points by robinhouston 12d ago

Mathematics is a solved problem.

> I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied. I think this is a refreshingly forward looking idea and I agree with it 100%, especially the the "rigorous defense" part. That is a good measure of how well the topic has been researched and understood by the researcher. This is where the humans can be "in the loop". > How different would this look from current PhDs? I think students would still meet with an advisor, who might suggest a topic. That topic could be explored with AI assistance, or not, but the student would be responsible for understanding it; it might be much more open-ended and larger than the typical PhD is currently. Interesting point about "more open-ended" and "...larger than the typical PhD". I think the author has a point. Earlier, the bottleneck was the candidate's/researcher's understanding and knowledge. Now with AI tools, it is so much easier to zero in to relevant knowledge, get your questions answered quickly which might lead to understanding more quickly. For e.g., before the advent of public libraries and printing press, the knowledge was inaccessible and guarded. So that was the bottleneck. Then books became ubiquitous and the bottleneck to knowledge and understanding was people's motivation AND knowledge of WHAT books and topics to research. Then came the internet and free PDFs of books and research articles. Now, the bottleneck was still people's motivation and a mild version of what books and topics to research. I say "mild" because one can lookup articles and newsletters, and book reviews and come up with a list of reading. Now comes AI and it looks like the only bottleneck is people's motivation. I believe there was also a silent, yet potent, bottleneck all along which is also removed by AI: personal tutor/coach/teacher/professor etc. Let's say if I am reading a textbook on manifolds or some research paper and I have a question about a specific theorem or even a mathematical operator being used. Before AI my only way to get my questions answered was to read more books (PDFs or print), or ask on math exchange or math overflow and wait for someone to answer, or to ask a professor. This could take up to a week. Now all of that has been cut down to 1 hour or less with an interactive chatting session. !!!!! So....the only bottleneck is people's motivation! QED Exciting time!

emil-lp 12d ago
> the "rigorous defense" part In my country, that's exactly how it is. Yes, you need to have a thesis to defend, but ultimately it all comes down to the (oral and live) defense/disputation.
robotpepi 12d ago
I'd say this is the most optimistic scenario. there are really difficult problems to be solved in terms of access to AI.
gowld 12d ago
Why is "Doctor of Philosophy" the correct certificate of "becoming expert in a topic"? That's a radical departure from "PhD" being a certificate that someone is qualified to produce new research. What you describe is more like a Masters Degree.
- breezybottom 12d ago
  I'm not sure why an advanced degree is necessary for that at all, besides the pride of a vanity title. It's pretty much what Bill Nye does for science.
- emil-lp 12d ago
  I think you have misunderstood. PhD has nothing to do with expertness. If you have a PhD, you have completed some kind of research training. That's all there is. Says nothing about knowledge or whether or not you're a genius. You cannot conclude anything else, and nobody claims that you can. If someone has a PhD, they have some training in doing research.
  - ksd482 12d ago
    Looks like @gowld is agreeing with you.
kurthr 12d ago
The idea that most any modern "interesting" aspect of mathematics is going to be understood (or often even explained in enough detail to reveal what is interesting) in an hour is pretty rare. Either the student's aptitude, the tutorial, or the mathematics are unique. There is a reason that these are PhDs and not undergraduate HW sets. I think we often delude ourselves as to how well we understand problems and their solutions. Some instructors even make you feel that you understand better than you do by pointing to a few approximations or simple solution spaces that obscure the larger complexity. Just looking in wonder at the many categories of three-body solutions (currently on hnews) is enough to remind me of this.

bobajeff 12d ago

The more I see these posts about mathematics institutions reforms and challenges from AI advancements the more it looks like they may need to go through a death. Or to put it another way they may need to start again from first principles. If math is truly about spreading intuition and understanding then our institutions have dropped the ball decades ago and have not been able to grab hold of it since (if they ever had it to begin with)

deleted 12d ago
[deleted]
omnicognate 12d ago
Not sure why you're down the bottom when the current top post says pretty much the same thing. I agree that the reevaluation and refocusing that is being forced by AI is one the maths establishment could fruitfully have had a long time ago.

hintymad 12d ago

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Jun8 12d ago

Excellent optimistic post in a sea of negativity, and with actual suggestions, too. After reading, my mental image is this: think of Olympiads in Ancient Greece. * A weightlifter was only awarded a laureate if he were able to lift a heavy stone (have no idea what they were lifting, for illustrative purposes only :-) * Along comes Archimedes who invents what we would call an exoskeleton. Now any regular guy can lift twice as much as last year’s athlete. * What to do? You can cancel the Olympiads, but they are actually useful as training, motivation, etc So now you have to give the prize on other factors, eg how well he can lift, has he opened a gym in the city, etc BTW, physics and bio are not exempt, so those researchers better read and try to stay ahead.

boccaff 12d ago
I don´t think so. For programming agents can run code, check compiler output, etc. For mathematics, it is almost the same once you factor in the usage of lean. For the reality, you can´t close the loop that fast, or with that precision. You will have to slow down by several orders of magnitude.
- Jun8 12d ago
  Doesn’t have to. There are petabytes of experimental physics data that can be fed to AI to extract additional insights. The only holdup is this is slightly harder to than math. With bio, you’re right, generally designing and conducting an experiment goes hand in hand and theoretical biologist is not a common label.
  - lolakutty 11d ago
    >There are petabytes of experimental physics data.. There is one thing you are missing. Math is precise. Physical measurements are arbitrary imprecise...

vld_chk 12d ago

I am not a mathematician, but I can’t see how we are going to address the problem which we already see in coding: Impossibility to independently validate all AI results And in math it goes even worse. In coding code reviews are typically still the form of action you do within days. In math, historically, the lifecycle of proof is months if not years. Take as an example Millennium problems. They require at least two years of validity after publishing. Two years! In modern times with amount of output AI can produce, it feels like infinity. We are inches close if not at the moment already when humans can’t reliable validate proofs and mathematics produced by AI. Then next research will be based on this AI-written-no-human-in-the-loop results. And we will end up in just few years in a world where novel and frontier problems will be articulated by AI and proven by AI based on AI results and humans will be incapable of understating the mere nature of the solution.

robinhouston 12d ago
I don't think that's actually the real problem. Along with the progress in answering mathematical questions, recent progress on AI-powered autoformalisation has been astonishing. All the recent AI discoveries have been accompanied by Lean proofs. And, yes: that doesn't absolutely guarantee correctness. The Lean kernel has had soundness bugs, and may have some still. But it's pretty strong evidence of correctness nevertheless. The concern among mathematicians is not mainly that they doubt the correctness of any of these discoveries, but that human understanding may be devalued.
- vld_chk 12d ago
  I am not that worried, but rather just observing. Humanity is about to enter the phase when we will be using things based on ideas no human ever properly understands. This thought … disturbing, somehow? It is perfectly valid counterpoint to say that we already do it. We everyday use myriad of things, tools, and software we have 0 clue how it operates. But for us as humans it was reassuring that we know that at least there are a few other alive humans who know it, who create it and who can explain it. With AI soon that comforting zone will be gone.

esafak 12d ago

> I think so. This machine might produce answers we value, but it would not, in itself, produce human understanding of those answers. It's nice that the author is optimistic, but won't the AI be best placed to dumb down its increasingly complex proofs into a language us lowly humans can understand? To keep thinking until it can refactor complex proofs into ones from 'the book'?

magicalist 12d ago
As they go on to explain, a human understandable proof is different than a human actually understanding the proof. That actual human understanding (like, in a brain of a human) is one of their stated goals. Producing human understandable proofs is possibly a job best for humans today, but the author appears to agree with you that this is probably fleeting (and argues that even if you disagree, it should probably be treated as if it is fleeting when planning for the future): > Right now AI systems arguably underperform us at theory-building, asking questions, exposition, … so we could prioritize and reward those skills. I think this is unwise: compare the speed at which the academy adapts to the speed at which model capabilities improve. We need to consider the endgame. If the models remain incapable in some domain, we can adjust later.
vouaobrasil 11d ago
AI will never be able to dumb down a proof to a level simple enough for someone to understand who has never studied math and put a lot of effort into it. Some concepts just need time and effort to absorb no matter how relatively simply they are phrased.
- esafak 11d ago
  Sure, but some proofs are so complex that even experts can't follow them. As AI progresses, this may constitute an increasing fraction of proofs. I meant that AI could work to find the simplest possible proof.
  - vouaobrasil 10d ago
    > Sure, but some proofs are so complex that even experts can't follow them. If that's the case then I don't see much reason for them to exist in the first place except mental one-upmanship.

Bluestein 12d ago

I love how this is (without slighting the problems entailed) coming at it from a perspective of infinitude and abundance (we will always have more problems to solve) - which is the correct framing, particularly when dealing with ideas, or fields in the which ideas are the driver/product/output/material, and ideas themselves, the field itself, are infinite.- PS. The validation problem, being one.-

waynecochran 12d ago

As someone who has a degree in math, I still can't help but think mathematicians are getting a little bit of a comeuppance. In a lot of areas of mathematics there had been little effort to make the work understandable and leaves numerous folks who could benefit from the knowledge on the outside looking in. Now AI comes along and do the same to mathematicians. Makes me chuckle a little bit.

chneu 12d ago
Lowering the bar for the masses isn't always a good thing
- PeterWhittaker 12d ago
  Examples, preferably academic?
  - applfanboysbgon 12d ago
    Half of modern academia, for one. Universities have turned into degree mills with the expectation that >50% of the population requires a college degree, regardless of whether they actually have any interest or need for one beyond doing it because it's a prerequisite for a good career, independent of whether said career actually uses the knowledge in any way. Not that I actually agree that math was at the right level of gatekeeping. It definitely feels intentionally opaque beyond reason, and I think it's why LLMs are able to cut through the obfuscation and solve problems that maybe wouldn't actually have been considered quite so hard if mathematicians did a better job of making their work accessible.
    - JoeAltmaier 12d ago
      <pedant>28% of adult Americans</pedant>
      - applfanboysbgon 12d ago
        66% in Japan. Who said anything about America?

jkhdigital 12d ago

> resulting in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications. I hope, from the depths of my soul, that the static typeset report format for transmitting knowledge and understanding will finally die and be laid to rest.

emil-lp 12d ago
As a researcher in theoretical computer science, I love PDFs more than any other format when it comes to mathematics. There simply is no contender to LaTeX and PDFs. Lucky for you, almost all research in math, cs, and physics, are put on arxiv, where you can download the source code (.tex) as well as get an HTML render.
warkdarrior 12d ago
What would you prefer instead of "static typeset report"?
- Jblx2 12d ago
  Not the OP, but how about things like: https://ciechanow.ski/archives/ https://ciechanow.ski/archives/ ...for starters?

wrs 12d ago

The author argues for evaluating Ph.D. candidates based more on the oral thesis defense than on the actual thesis. By essentially the same reasoning, I’ve been arguing for prioritizing in-person design/code reviews over code-only async PR comments. The important thing is to verify that the human has a coherent design in mind and can demonstrate that it got implemented, regardless of who or what was at the keyboard. “I dunno, I guess Claude thought this was a good idea” is not a coherent design.

getnormality 12d ago
I think upweighting the live components of academia is inevitable in the age of automatically produced writing, but I also find it depressing that people think so little of writing that they imagine it obsolete because of AI. AI writing is aggressively, aggressively mediocre. It is only good for how cheap it is. If you're indifferent to how much better good human writing is than AI writing, you should not be the one to evaluate human writing.
Frost1x 12d ago
I can see how this trend is going to go. Manager: “Why isn’t feature X available?” Person B: “key pieces are delayed due to the developer not understanding all of the LLM doesn’t and implementation.” Manager: “does it work? What are the risks?” Person B: “well yes it works for now but we’re accumulating tech debt due to a lack of understanding and potential flaws that haven’t been thought out yet” Manager: “they want feature X, ship it, we can deal with it later, I don’t care if it’s not coherent as long as it works.” How many decades at this point has these been a push for functionality over everything at all costs? And you have a mechanical snow plow now. Most businesses don’t care about later risk or any future planning beyond the quarter horizon, they’re not concerned about how it will effect their performance in 3 quarters or lead to instability or issues, those are future problems for a future person and we’re here for money now.
- AdieuToLogic 12d ago
  The simulated dialog you have provided is very much representative of what many of us have heard first-hand. However, I disagree with: Most businesses don’t care about later risk or any future planning beyond the quarter horizon, they’re not concerned about how it will effect their performance in 3 quarters or lead to instability or issues, those are future problems for a future person and we’re here for money now. Businesses care about "later risk" and what it implies. Individuals within an organization do not unless it will specifically affect their bonuses/promotions.

theodorewiles 12d ago

Yes the fascinating thing is: 1. It will take much longer to understand the output of the machine that it takes to prompt and create it. 2. The only? best? one? way to /verify/ that you /in fact/ understand the output of the machine is to explain it to someone else. So there will be a machine generating koans which need to be meditated upon and discussed with human social back-pressure validating understanding. I think this could be much more cooperative and at a minimum this will be a way different math social construct.

wcfrobert 12d ago

I like the quote from Hilbert that was brought up in the article: "we must know, we will know". With AI, it might be the case that we don't know, we won't know, but the machine does. The central question, namely whether humans should be in the loop, will be repeated again and again in the years to come for all industries, starting with mathematics.

wanderingmind 12d ago

Better plan is to shut down PhD and make students take a oral thesis at bachelor and master level and help them become a productive economic participant as soon as possible.

ComplexSystems 12d ago

I admit I just don't understand this attitude. People had the same complaints that the code produced by early coding models was messy, lazy, poorly commented, had terrible architecture and so forth. The central complaint was that it was just too difficult for humans to review. The answer is just to improve the models and move on. Similarly now we're getting AI doing math. The proofs compile but are a mess. So just make the models better at writing clean proofs and explaining what they're doing to humans. That's the end of it. Rather than just go on and on about how it's the end of the world if we don't do this, why don't we just do it?

JW_00000 12d ago
There's a fundamental difference between the goal of code and math. The goal of code is to produce software that does something useful. As long as the code does what it's supposed to do, arguably, it's good to ship. (As you imply, we want the code to be good enough to also be reasonably certain there are not too many bugs, that it is maintainable and can be extended etc. This is what early models failed at but now seems broadly fine.) But for math: what is the point of a proof if no one will read it and no one uses its result? To quote the article, "AI systems will [...] result in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications." But if there's no one reading the PDFs, what's the point - no matter how good your AI model. The point of math is understanding. So mathematicians should feel free to use AI as much as you want, but in the end, they should've gained some understanding on what happened.
- ComplexSystems 12d ago
  > But for math: what is the point of a proof if no one will read it and no one uses its result? What is the point of writing software if nobody will run it? > So mathematicians should feel free to use AI as much as you want, but in the end, they should've gained some understanding on what happened. So they ask the AI to explain the proof.
- kenjackson 12d ago
  If the point of math is understanding then maybe our incentive structure is wrong. Maybe it should be teaching the concepts to as many people as possible rather than just continuing to write papers that 10 people in the world understand, which is the current state of a lot of math.

bonoboTP 12d ago

> We already interview faculty hires; we must now do the same for graduate admissions They hire PhD students without hearing them give a talk and then doing interviews? In Germany, the applicant gives a talk (30-40 min) to the research group they want to join, usually presenting their master's thesis, engage in discussion, often share lunch with the group, then do 1 on 1s with individual members of the group and a longer one with the PI. Obviously this can vary within Germany too, but I couldn't imagine hiring someone without something like this.

bobmarleybiceps 12d ago
1. US a lot of international PhD applicants, so traveling before even being accepted is difficult, and 2. lots of people don't do a masters. when I applied to PhD programs (not in math) it was basically CV + personal statement + recommendation letters + short chats with interested faculty :shrug: Maybe it was because my CV was "strong" but the chats were more see if interests were aligned, rather than actually interviewing me.
- bonoboTP 12d ago
  Remote talks are also a common alternative in such cases. Zoom etc. I also know that another major difference is that American universities tend to hire without a professors involvement, into a generic "program", then the PhD student seeks an advisor after being accepted, so applicants have to woo some unconnected committee pursuing various goals misaligned from the PIs instead of convincing the PI. In much of Europe a professor basically "owns" a chair and really is boss and decides hiring pretty much alone. Anyways, if they don't have a masters' yet, they could present their bachelor thesis. But I already think it's a bad model to combine the masters (courses) and the phd (research) into this American hybrid that's the direct-bachelor-to-PhD jump, but that is somewhat unrelated.
  - bobmarleybiceps 12d ago
    I agree the combined masters + phd is weird tbh. I think almost everyone treats it like "just focus on research and spend as little time on course work as possible." I would prefer it if courses were more flexible. I'm not a big fan of the US application setup. IDK how it is in Europe, but in the US, it feels like there's a lot of not-very-meritocratic "secret" stuff you need to know to up your chances.

youoy 12d ago

As with the rest of the domains, AI/LLMs will do syntax and search better than any human. In code, any developer whose differentiaton was clean code and knowledge of different technologies is now average. In math, any mathematitian whose differentiation was to manipulate formal systems and know tricks of different domains will be average. Fortunately, humans do more than syntax and search. The bad news for developers is that if you know what the output of your program should be (which happens most of the time), almost all of the job is syntax and search to build the code that reproduces the output. The good news for mathematitians is that for the majority of problems you never know the output, or you just know the output is either "True" or "False". There are some cases where you need something else, for example "a solution that blows up in finite time". For those cases AI will outperform you easily (see new Navier-Stokes solution) So if as a mathematitian you were doing more than syntax and search, then keep doing that and use AI just for what its best.

jplusequalt 12d ago

Another mathematician waxing poetically about a future that will not come to happen. Call me a pessimist if you'd like, but capital has no incentive to ensure mathematicians maintain their current status in society. If you're a mathematician you are in the same boat as the software engineer, and the Dodo. Better learn a trade buddy /s

jsrozner 12d ago

"A computer or monkey could easily start at the axioms of ZFC and iteratively apply deduction rules....simply conjecture all mathematical propositions in alphabetical order...The prospect of automating mathematics by enumerating all conjectures, and all proofs of ZFC, is probably not so disturbing to you." I thought we were going to get at least some brief comment on Godel here?

anyfoo 12d ago
Gödel effectively says ZFC must be incomplete, otherwise it would not be sound, but does that stop you from listing all mathematical propositions it can generate in some well-defined order?
- jsrozner 12d ago
  That's right. You could list all those propositions and search for proofs of them. In fact this is very similar to Hilbert's very program to which Godel's First Incompleteness Thm was a response (https://en.wikipedia.org/wiki/Hilbert%27s_program https://en.wikipedia.org/wiki/Hilbert%27s_program). Godel showed that there are true statements that cannot be proven, and also that among the unprovable statements from within the system is the consistency of the system itself. As I understand, mathematicians are still trying to figure out how much this matters. One of the best examples of its mattering is may be the Continuum Hypothesis: CH is consistent with ZFC, and ~CH (not CH) is also consistent with ZFC. In other words, you have enough flexibility in constructing your ZFC world such that in some ZFC-consistent worlds CH is true, and in others CH is false. Litt's statement is not wrong; it's just that what he wrote sounds so much like Hilbert's program, that I'm surprised we didn't get some even minor comment on what kinds of truths we could reach if we embarked on such an effort.
  - Chinjut 12d ago
    People always bring up the Continuum Hypothesis as though it has something to do with Gödel's incompleteness theorems, but it doesn't really. The Continuum Hypothesis being neither proven nor disproven by some particular axioms is a similar phenomenon as that the group axioms neither prove nor disprove commutativity, the ordered field axioms neither prove nor disprove the existence of a square root of 2, etc. There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is. Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent. But the inability of ZFC to consistently decide the Continuum Hypothesis is established in a completely different manner, with the Continuum Hypothesis not consistently decided by ZFC + Con(ZFC) either, or any such thing.

bonoboTP 12d ago

I think this goes in the right direction. You have to rethink the role of human work in math, can't put your head in the sand and cling to your comfy institutions and system just because you got to know it's ins and outs and just want it to be like that forever. But the bigger picture is: while I understand the author know his field best and wants to keep the post focused, the same issue will hit many more fields. We need to also have a broader discussion that involves more fields of knowledge work, largely academic scholarship but also regular office work, then it will come to engineering design, medicine, it's already coming for 3d modeling and vfx, software dev, it will come for a lot of middleman services. Not at the same rate, but we have to understand that it's not just that math will change. Change will be the default. Everything will change. It will be much easier to make math change because all things will change. You shouldn't worry and imagine that funding criteria will be like today or that journals or academia or politicians expectations will be like today. No, everything will adjust with some timing differences of course but it won't be a static world and then math changing and having to justify and fight to explain the change to other actors who are baffled. They won't be baffled they will themselves have to change. The world will transform as much as it did when society moved from feudal agrarian to urban capitalist industrial, or from the vast majority doing physical labor to a service economy with a huge amount of desk jobs. I can't tell how it will change exactly but it will be bigger than what we have seen in the last couple of generations or maybe more.

eliauelkouby 12d ago

[flagged]

asa123 12d ago

While I broadly agree with the premise of re-directing the “purpose” of math, I quite detest the idea that judgement might be primarily based upon some in person discussion, or oral presentation, and the claim that written mathematics that is not orally communicated might be less worthwhile in some sense (i know this isn’t the exact statement of the authors proposition). There are a good deal of people, whom, falter much more in oral discussions, whether this be for a psychological thing, stage fright, or difficulty explaining things on the spot. There are also certainly brilliant people, who can’t give an informative, discussion inviting talk to save their lives, but given enough time, can formalize their thoughts in writing at the highest levels of their field, and that writing is likewise very enlightening (sometimes). It’s not clear to me, that, AI as is, could not pose successfully in an oral discussion of a topic. I mention this because it seems that one implication of the article is that AI might write things that are logically correct, but devoid of understanding. I suggest rather that 1) it is not extremely improbably that AI is incapable of generating mathematics that furthers human understanding and if 2) it is indeed highly likely that they cannot generate mathematics that furthers human understanding in a textual format, then surely one could also differentiate between human and AI on a textual level, and judge the contribution of a human, without the need of oral discussion? I suppose another aside is, one might claim that the existence of AI means people have much much more text to filter for, and so, it becomes difficult to find one person’s good writing amidst a sea of, logically correct, yet understanding devoid textual content. But by and large much or mathematical academia certainly operates off of some reputation/vouching system presently anyways, that already serves as a “filter” in some sense. Perhaps the existence of such a system/culture is not a good thing, but oral discussions/seminars certainly aren’t immune from such predilections. Perhaps I’m babbling like an idiot, but the entire and sole purpose of this comment is just to say: for the love of god please don’t let the standard be judged by oral presentation

augment_me 12d ago

What is the incentive for a person to sit though seminars and evaluations? People already hate redundant meetings. What is the incentive to change the system from the existing one to one that rewards this verification somehow? Who benefits from this change?

sabujp 12d ago

[flagged]

soundworlds 12d ago

I was involved in this research a few years ago: https://metamiditoolkit.com/research-paper/ https://metamiditoolkit.com/research-paper/ One of the things that struck me was that most people have an area of their workflow that they would be happy to hand off to an AI, so that they can focus on the thing they love. However, that area is different for each person. One person's grind-work is another person's love-work. This is why it's so hard to build consensus around where the "red lines" are in this space.

gulugawa 12d ago

Misanthropic and Open Artificial Ignorance are misleading the public. They are leeching off the work of mathematicians and pretending that new math was invented.

Jblx2 12d ago

Does anyone have a rough estimate of how many research mathematicians there are in the U.S. or world?

sashank_1509 12d ago

I’m not a mathematician, but if I were I would imagine the main attraction would be the act of thinking hard and solving problems. Maths felt like the ultimate profession for someone who loved puzzle solving. If you delegate the thinking, any percent of it to an LLM, I don’t think it’s maths anymore. Might as well join any other job and earn better money now. A completely ludicrous proposition, that maths will no longer involve mathematical problem solving.

vouaobrasil 11d ago
Modern incentives, unfortunately, reward producing solutions rather than going after the joy of solving problems. And it's the latter that inspires others, not the former.

GMoromisato 12d ago

My worry is that the frontier of math is too far away for most humans to reach. AI is only going to make that worse. If today it takes twenty years of math study to reach the frontier (in a narrow field), what's it going to be like when it takes forty years? Or four-hundred years? Will the fields just get narrower and narrower to accommodate finite human intelligence?

bell-cot 11d ago
> If today it takes twenty years of math study to reach the frontier ... Are you counting from when a kid first learn 1 + 1 = 2? That seems a misleading metric. The rate at which kids are learning math in the first half to 2/3 of their schooling is very low. Or are you asserting that Ph.D.'s don't reach the frontier until age 40+? Actually, the frontier of math is too far away for most humans to reach because >99% of humans don't have the drive and talent to study and learn enough advanced math. Similar barriers exclude >99% of people from Olympic-level athletics, or being famed violinists, or being important architects, or being US Senators, or being successful ancient historians, or many other things.

MarceliusK 11d ago

The key shift seems to be from "can you produce mathematics?" to "do you understand mathematics?"... Those used to be correlated strongly enough that a thesis or paper could serve as evidence of both. If AI breaks that correlation, evaluating people through defenses, discussion and live problem solving suddenly makes a lot more sense.

Kvarnek 11d ago

Always fascinated by how foundational math concepts built up. Euclid's Elements still blows my mind for its logical rigor.

Tbarlow 11d ago

Pretty neat how these abstract origins still shape our world, especially in CS. Kinda makes you appreciate the shoulders we stand on.

dropshade77 11d ago

As a recent CompSci graduate and software engineer, I can sympathise with the author’s position. At the very least, these times are highly unsettling.

samayashar 11d ago

> AI does not care if you are anti-AI. This >>>

econ 11d ago

Imagine, eventually, a large ever growing library full of unread books. One could roam around forever and discover new things. Might be possible to generate a 3d world using some taxonomy estimate the amount of titles and generate the books as the area is explored. Like the manuscripts in Sakya Monastery in Tibet and like the millions(?) of cuneiform clay tablets. Vast amounts of data that may contain information.

darkstarsys 11d ago

This is an excellent essay, and especially relevant when read together with the MIT report on the future of education in an AI era: https://aiandeducation.mit.edu/report/ https://aiandeducation.mit.edu/report/ (and my own https://oberbrunner.com/blog/ai-future-of-mathematics https://oberbrunner.com/blog/ai-future-of-mathematics). I'm heartened to see this kind of thoughtful, reasoned discussion around long-needed reforms in the educational community, with AI as the catalyst. We certainly have interesting times ahead.