I really struggle to understand this discussion. Mathematics isn't a way to keep a specific kind of intellectual happy by providing fun, meaningless puzzles to solve in isolation. It's the bedrock of all other science. The goal is to find solutions and to advance the field - but we seem to be treating it like some sort of occupational therapy for bored geniuses.
As we solve lower level problems, we can unlock real world math solutions. Stellerator Design, Quantum error correction, Predictive superconductivity, Inverse protein and enzyme design - the list goes on. Many solutions AI can find can bring us closer to these - so let's fire up whatever datacenter we can find and get us to these outcomes.
Yet the creatives don't get into gamedev if they didn't have to spend any effort to make their art
The art will just change. The bar will be raised. What is considered as good art in 2021 is different in 2026.
In math, the speed and innovation will increase. The bar will be raised. Those mathematicians who do not want to be part of this new world will quit and go do something else, just like how some software devs have done so since AI agents. But not all will quit.
> It's the bedrock of all other science. The goal is to find solutions and to advance the field…
That may be true but that may not be the feature that attracts people to the field.
> As we solve lower level problems, we can unlock real world math solutions.
That is an argument often made for funding mathematics research. But again it may not be the thing that attracted people to the field.
Note I don’t share all the doom and gloom of some folks in the math community but it is a valid point that people who might otherwise have gone into the field may choose not to anymore.
Also, math research has typically focused a large amount of energy abstracting and generalizing results. Your suggestion about solving real problems usually moves in the other direction. If the powers that be with all the money decide AI can solve all the real problems that exist (rightly or wrongly) then mathematics research community will have a problem.
> Note I don’t share all the doom and gloom of some folks in the math community but it is a valid point that people who might otherwise have gone into the field may choose not to anymore.
Isn’t that only natural? I probably would have stayed in physics and not gone into software if AI had been around 25 years ago. People gravitate to areas where their innate abilities are appreciated and can make a difference.
I think "fun" and "meaningless" are important parts of math. There is applied math for sole utility. Other math is just random search that may or may not be useful in long term. But later people/AI will find important use cases from these short-term "meaningless" researches.
It's both. Some math has applications. Some math doesn't. Many mathematicians are interested in mathematical beauty, in solving problems, and playing with puzzles.
None of the results from October 6, as far as I know, have direct real-world applications today. We can't go build a better thing, or make a faster thing, etc. I might be missing something (I'm not an expert in any of the fields). I don't think most folks would have a problem with solving a bottleneck problem that let us flux a capacitor at 2x efficiency, or whatever.
I also think this is hard for a lot of folks who are in fields that heavily apply math to come to terms with. I do not think that programming and practicing mathematics are the same: many (most?) mathematicians are closer to a composer than a programmer: all three are making things, but there is much more emphasis on elegance and cleverness in mathematics than in programming.
But again, it's not binary, and a lot of the discussion does try to take a black-or-white view of it. It's tradeoffs all the way down.
For what it's worth, I think this quote from the article sums up my position best:
"To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof. Choosing worthwhile problems may end up mattering more than producing proofs."
> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof.
Yup. This is what Terence Tao has been saying about digestion of results for several months now. Mathematics has become a field where results themselves matter less, the social aspect of attracting others to understand your results and explore new problems has become more important.
In a sense it is a little
bit like software engineering, where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
> understands it thoroughly, and uses it to develop interesting further questions or directions
He phrases it like these criteria are requirements, but they really aren't for every problem. They make sense for those theoretical problems whose solutions' main values lie in their understandability and further development, but not every problem is like that. Some problems' solutions are just revolutionary on their own even if you contribute or understand absolutely nothing else. AI technology itself is an example of this!
To give a trivial hypothetical counterexample: if you came up with a solution for the traveling salesman problem that is both empirically sufficiently faster than the state of the art, that is an immense contribution and you absolutely deserve the credit for it regardless of whether you or anyone else understands how/why it works. It could literally even produce wrong results 50% of the time and still revolutionize humanity if it's efficient enough when it's correct. Denying them credit because they didn't understand the algorithm would be ludicrous - the results stand & speak for themselves.
Statisticians, traditional-AI researchers, etc. had to learn this the hard way when (basically black-box) neural nets became so good that their otherwise-elegant & explainable classical algorithms just stopped being able to compete in more and more applications, and got left behind. Are mathematicians looking to repeat that history?
I mean, of course, if you make the benefit of the result itself large enough than yes, by all means; I'm sure Terry would agree: if there was some algebraic geometry proof that could be used to build a life-saving drug, then I don't think folks would have a problem with using AI to prove it. But most math isn't like this. Most math has very little immediate effect on the real world.
There is a tradeoff that is implicit in this entire discussion: if I can go out and use a result to do a thing much much better/faster/easier than I could yesterday, like a real and useful thing in the real world, then the result itself has value. The cost incurred by swarm solving that individual problem is relatively low and isolated, and the value added is clear, so great, we should do it! But if proving a result only allows us to apply it to more mathematical questions, then there isn't really value added aside from our own value we place in the results, and that value comes from our understanding. So now, swarm solving random math problems is taking away our opportunity to solve these things ourselves.
And people who don't have math backgrounds will poo poo this whole "opportunity to solve problems ourselves" stance, but it is 100% the way mathers think about this stuff. When I was a math major in undergrad I actually had a friend who told me that he didn't want to give me hints on a difficult homework problem because he wanted me to have the opportunity to solve it myself. Like, his _present_ to me was to give me an opportunity to solve a hard but rewarding problem.
Breaking public key cryptosystems would be a better counterexample. Most NP-hard optimization problems can be solved in practice. While there are always some degenerate instances nobody knows how to handle, realistic instances tend to be easy enough to solve or at least approximate accurately.
> Breaking public key cryptosystems would be a better counterexample.
I deliberately avoided giving that example. Calling that a positive contribution to humanity and expecting credit for it feels a bit like calling the atomic bomb a positive contribution to humanity and expecting credit for it. Much of the world would rather stay away from that timeline.
> Most NP-hard optimization problems can be solved in practice.
Hence "a solution that is empirically sufficiently faster than the state of the art" and not merely "a solution".
Once you start solving problems in practice, you are probably no longer interested in the simple clean versions of the problem theoreticians think about. For example, instead of trying to find the shortest common superstring, you may be doing phased genome assembly. The problem is no longer unambiguously defined. The constraints and assumptions evolve over time, as the problem space becomes better understood and the nature of the data changes. While algorithmic ideas often transfer from one problem subtype to another, everything needs to be tailored for the specific problem.
Most day to day programmers prefer to keep the source code to their programs rather than the executable file, so I think it's not absurd at all, it just takes a brief moment of reflection.
It's not absurd if you think about the lifetime of software. It is created once and used immediately, but reread, adapted and tinkered with for decades after that if it is useful.
The number of reads is FAR greater than the number of writes for any given piece of code, and it needs to be comprehensible if it is to be extensible.
So in fact that instruction applies to all software, and I wish more people would take it seriously. It also applies in the AI era, because the AI has to understand the code too, and AI's spend more time reading code than writing it, indeed they base what they write more on what they read than humans do.
The fact that this wouldn't have been considered a contribution before, or that this is a novel idea, suggests that there was something wrong with mathematics culture all along. I'm glad they're figuring it out now, but it seems "too little too late" after decades of devaluing all expository and exploratory work other than writing difficult proofs.
These things definitely were considered valuable contributions pre-AI, but were perhaps second fiddle to proving theorems. For example, part of the reason Erdős is so acclaimed is that he posed many interesting open questions which have kept mathematicians busy for many years.
I think Tao's point is that there's a necessary rebalancing and more weight should now be given to the non-proof stuff.
I feel like the hyper-competitiveness of academia generally is the elephant in the room.
The disaster for a problem-solution-oriented mathematician isn't that once problem-solution is easy they'll just switch to theory-explanation and machinery-building. It is that they will have to compete (and won't be able to compete) with students and post-docs who've been doing theory-explanation and machinery-building for their entire career - and there are lots of these 'cause academic pipeline produces more people who can fill a job than jobs in every imaginable category and subcategory.
Academic competitiveness doesn't explain all the hysteria and why those at the top of the field (Field Medal and all) wrote that breakup letter over AI. There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence. Literally a math academic crying while being interviewed on AI impact, talking about not knowing what to do with himself.
Who knew 2-3 SD won't save you from being a pattern matcher that can't stand to see a better one.
it's not just math and CS. it's a human trait. we are seeing it in the arts and creative fields where technology and AI in particular are making their impact felt. there will be much, much more of this sort of reaction in the years ahead. we'll all have to learn how to adapt. and in the process rediscover that what makes us valuable as people is not our skills but our souls.
I agree that all people have value, but what does that mean in practice? Everyone needs a certain amount of resources just to survive. Should other people give you those resources because you have a soul?
A lot of people enjoy making art as a hobby; very few of them have combination of talent, discipline, and luck necessary to achieve commercial success. When I walk through a local art festival, most of what I see is so bad that the artists would have to pay me to take it. While taste in art is highly subjective, those artists don't seem to get much attention from other customers either. Going forward there will probably be even fewer people able to make a living in the arts. I'm not sure what that means for society?
Everyone seems to wrap up their criticisms on AI and labor saving technology in general as though it's a personal affront when in reality, as it's always been, it's an economic problem. Once we have UBI or something similar, you will see most people's opinions on AI doing math and art, for example, start to change to one of cooperation, not adversarial.
Everyone seems to wrap up their praise of AI by claiming that it will bring an era of abundance with UBI. Once we don't have UBI because why the fuck would we, you'll see most people's opinions of AI start to change from one of cooperation to adversarial.
The steam engine produced raw power but that still needed to be controlled and directed for a multitude of purposes, which required putting dirt farmers through education to turn them into factory workers. If machines now do both the thinking and the hard lifting, why exactly would their owners need us for? From where they're standing, they're now looking at a pragmatic solution to global warming: just reduce the "energy requirements" of the population.
the human touch. In cooking you would call it cooking with love. You know the reason your grandmas cooking will taste better to you than any michelin star restaurant.
Do male chicken have souls? Because female chicken have the skill to lay eggs, and that's what makes the latter valuable and the former bonemeal upon birth.
Its equally devastating when you're a renowned expert developer at a specific stack that falls out of favor and nobody cares about your skills any longer. This happens relatively often in tech.
Eh, this analogy really falls flat. The “stack” in this case has existed for hundreds of years, takes at least twenty years to master, and the best people started working with it when they were perhaps seven. It’s a bit different than Rails falling out of fashion.
The issue in my view isn't that a single tool has become obsolete, it's that there's nothing to retrain to. All the tools have become obsolete and the AI is eating away at the economic value of core human activities. If AI was really good at coding in one language or framework or something I wouldn't give a fuck. It's the difference between a tool and a replacement.
This is something that sci-fi and AI safety has been talking about for a while.
It isn't just about worrying if AI is going to wipe us out, it's trying to figure out how human kind survives itself as things so rapidly change.
The early 1900s also had very rapid change after the introduction of ICE vehicles and airplanes. We also had massive wars that killed millions and brought forth society ending weapons. What is this round of rapid change going to bring?
The slaughterbot drones are already pretty bad. In Gaza, Israel used the sounds of crying babies to lure out innocent people and shoot them with drones. They used drones to shoot doctors through a hospital window. A lovely preview for the rest of us.
In the Russia/Ukraine conflict, Ukraine is holding on basically because of drones. They don't have enough guys to populate the front, so the gaps are plugged with drones. So that means the drone technology race is getting supercharged and we'll all get to live with the results of flying killer robots.
In Émile Torres' book, he references how the superweapons of the 21st century were things like nuclear weapons, biological weapons, chemical weapons. In the 21st century, those risks haven't gone away, but to them we add the "GNR Bundle" (genetic technology, nanotechnology, and robotics technology) which are characterized by their autonomous self-replicating potential (with AI falling under robotics). Of course, in his book, he is also being somewhat critical and descriptive of the "futurological shift" that causes certain people to become obsessed with cataloguing every form of x-risk to the neglect of real people's problems in the present.
> There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence.
I don't see this as a toxic trait at all. Some people have the fortune and luxury to do what they love - I don't think that's a bad thing.
I think this quote from Gauss puts it very well: "It is not knowledge, but the act of learning, not possession, but the act of getting there, which grants the greatest enjoyment." So tons of mathematicians feel like AI is stealing this "act of learning which grants the greatest enjoyment", and I think it's fine that they are lamenting this.
This is inherently my whole problem with the "AI utopianists", who somehow think we'll all be happy once we're sitting around with no disease just eating bonbons all day or something.
This is inherently my whole problem with the "AI utopianists", who somehow think we'll all be happy once we're sitting around with no disease just eating bonbons all day or something.
I don't know about Gauss, but when I am given a more powerful tool, I use it to tackle harder problems, the kind that were out of reach before.
Call me when you run out of those sorts of problems. I'll propose some more.
Spoken like a true robot. For most of us, happiness (and I mean that broadly defined, e.g. "life satisfaction") is the only goal. I certainly thing happiness is a much more critical goal than "increasing shareholder value".
I know some real assholes who seem quite satisfied with their lives. A shot of heroin will make you happy … for a while. It's a meaningless goal. Regardless of your snarky comment about "shareholder value", we should all aim to make a positive contribution to society and achieve something of lasting value (not necessarily monetary). Happiness might come with that as a side effect but only a fool treats happiness as a goal.
It may be a bit unhealthy to only identify yourself only with your vocation - there's also family, friends and hobbies. But it's miserable to engage in "work" for forty hours a week when it's something you have no interest in whatsoever. And it is very unhealthy to think most people should do that. If "work is just work", how terrible is that?
If the sum-total of the activities you engage isn't "you", what is? The things you buy? "Your" sports team?
Plus a hypercompetitive academic world is a pretty good explanation for people identifying only with their vocation - how else to do you get to the top except putting your whole self into the challenge?
Seeing work as your purpose was supposed to be the holy grail of work that most people sadly couldn't achieve. Now we just immediately toss that away like a mouldy tomato and what's left? The new highest plane of human existence is working for capitalists to make them more profitable, in exchange for 10% of their profit?
I suggest reading some philosophy. It seems that we have spent a great deal of energy training people for a vocation, but in the process forgot to educate them.
I don't really understand this kind of triumphalism. Would you, personally, identify as a pattern matcher who's happy to see better ones? Or do you think you have some deeper skill more robust against disruption than the math and CS folks who don't know what to do with themselves?
Note that this may include things that you don't think of as being a "skill". I've seen some people brag about being a particularly hard worker, or about a deep instinct towards Getting Stuff Done, with the clear implication that these things can't be automated the way mere mathematical analysis can.
I don't read it as triumphalism. I read it as an accurate (to my experience, as a math PhD) reflection on the existentially toxic culture of math and other intellectual disciplines, one that AI is now forcing us to reckon with.
n.b. I think math culture is probably one of the least toxic cultures that have ever existed. Its equanimity and principles are a big part of what drew me to math in the first place. But humans are human.
I really don't see how to read "hysteria", "breakup letter", "literally a math academic crying" as anything other than an attempt to dunk on mathematicians.
I suppose, maybe they are dunking on mathematicians. I can't claim to speak for them. But I at least see truth in their observations. It would be inaccurate to downplay the sentimentality of the math community right now. I don't think that sentimentality has overly clouded the community's judgment (so "hysteria" is probably overdoing it) but there is a lot of heartache and unease.
The person who is brutally honest enjoys the brutality quite as much as the honesty. Possibly more
~Needham
>.. humans are humans
Since you've imputed intentions or inclinations-- once you see that "pattern matching", which, like a bastard "pigeonhole principle", has been seeing overuse in excusing bigotry -- you cant stop me from claiming to speak for _everyone _(TT)
Elsewhere here, people are imploring us to read philosophy, as if the LLMs haven't already mastered the one weird trick of enlightened stochastic parrots..
I'd recommend self-compassion/self-kindness for mathematicians (because contempt and pity [for mathematician, eg] are too cheap for far "right" and "left", respectively)
After you've felt those pangs of self-forgiveness you might be motivated to contribute to _distributed_ _crowdfunded_ LLMs that crater the value of datacenters instead...channel that brutality at uebermenschlichkeiten please!!
I will stick to my claim that math exists in the context of the hyper-competitive overall academic culture.
I should add that extreme competition combined with math being something people love, something with beauty unto itself, produces an especially weird world compared compared to a situation in, IDK, business school, where everything is about the money.
I mean, asking "who deserves to live the dream" really twists people's heads.
It's inaccurate because nothing about math or CS is "existentially" toxic. Or "distinctly" as the other commenter.
If you accept that as a premise then obviously you agree with their truth.
If you don't accept toxicity as somehow intrinsic to the subject but rather the structural conditions, say, of post-neoliberalized academia at large, and certain faculties being only a bit more susceptible (if even, or just different, say compare humanities vs STEM reactions in the future), then the whole comment reads as a conflation.
Instead the simpler answer is that academia at large values freedom of inquiry and other Enlightenment notions which are privileged compared to outside of academia. But that is not toxicity, it's just intellectual integrity and the academic autonomy in pursuit of one's interests avocationally. The toxicity comes from the intellectual profession having to function as a modern clerisy for socioeconomic reasons under post-neoliberal capitalism.
People may conflate the two attributions and that is a deep mistake, in my opinion. In fact the conflation due to internalized oppression easily creates and explains this kind of resentful, dismissive tone response that is a distortion of the true situation.
I only lay claim to criticize the mathematical culture I have myself experienced, Western and post-neoliberalized as it may be. I'm not sure it's even possible to discuss culture "intrinsic" to math, or any other subject for that matter.
> The toxicity comes from the intellectual profession having to function as a modern clerisy for socioeconomic reasons under post-neoliberal capitalism.
Sure. And I think (Western, post-neoliberalized) math has resisted that corruption possibly better than any of its sibling cultures. But what makes the toxicity "distinct" is the emphasis on enlightenment, as if being a top-class mathematician brings you closer to the Godhead than does, say, being a top-class chess player, or a top-class defense attorney. Mathematical theories are treated less like robust human creations percolating over decades and more like divine revelation channeled through chosen disciples in fits of inspiration (c.f. Cantor, Galois, Godel, Grothendieck). A lot of this is projected from the outside the culture into it, but it is not outright rejected by the culture. Instead we still openly endorse what amounts to Great Man theories of mathematical history, attributing to almost every noteworthy mathematician a moniker of "genius" and praising them as paragons of thought, not as the unhealthy subjects of cults of personality.
And the material conditions for this "great man" narrative is again the particular form of intellectual clerisy.
In other words, the move to experience and cultural relativist statements about "math culture" is exactly the cynical ideological move that an academic cleric does. It deflects from examination of where these narratives come from and attribute it to e.g. "math history" or "math culture" or just "something specific to math itself" etc, etc.
Neoliberal intellectuals love to attribute toxicity to the Enlightenment. They can't help it.
> Mathematics has become a field where results themselves matter less
"Results" in mathematics has always been the proof. (Preferably a beautiful and succinct one.)
Math is supposed to expose the workings of the epistemological foundations of the universe. The result itself (number or formula or whatever) is absolutely irrelevant to math.
(It may be important to physics, chemistry or engineering, but it isn't to math.)
fewer mathematicians means fewer problems to solve. All problems have human origin, right? Can AI create a problem on its own and what would be the point? Closer to can god create a stone he can't lift.
Posing problems is often easier thwn solving them, so yes? Fermats last theorem is posed in two lines. Took 300 years and massive advances in geometry to solve.
The quote reads to me like “as long as you can analyze the solution after the machine solves it and make it suggest new ideas based on that, you’re almost the same as a discoverer yourself.” I’m deliberately avoiding “mathematician” here because I think Tao probably didn’t intend to take it that far. But, for many people that difference will be indistinguishable, especially because there will very likely be “math bros” just like “ai bros” and “crypto bros” before them who want to pretend they’re mathematicians just as we see people pretending to be engineers and financial experts based on how fast they got on the hype train.
I’m no mathematician, though I do have an undergrad degree in math, and unless I had a graduate degree or made substantial contributions at that level, I would never consider myself anything but a “student of mathematics,” and that too not just because I have a degree in the subject, but because I still have a great interest in certain branches of math and I keep learning in my spare time from books or from adjacent applied fields I continue to formally study. As a kid, I grew up hating math but I became interested in it later in life because of coding, computer science and formal logic, pushing myself a lot through my college years to get some foundational understanding of the subject at a deeper level than the general population, just to get my foot in the door, so to speak.
Even if I’m an outsider from the circle of mathematicians, this whole AI takeover of mathematics leaves such a bad taste in my mouth because I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general, and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject). It feels like a great loss for humanity to lose a brunt of our engagement with it to the machines, like a whole wing of a gigantic library starting to be closed off from us indefinitely because some people decided that we don’t _need_ to access it anymore, except on an even grander scale possibly than my imaginary library metaphor.
> I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general
Funny, I think it's a very widespread opinion to think of math as useless. In the US at least it's a major failure of the education system and STEM communicators.
> and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject).
There's plenty of controversy in mathematics. Controversy comes from the people, not the field of study.
How useless? Terence Tao made MRIs faster. That's a real thing that STEM and non-STEM people will maybe have to interact with at some point in their lives. Sure, we can blame science communicators. Should we put a bunch of banner ads on everything so that people know that math and science is involved in every thing?
> Funny, I think it's a very widespread opinion to think of math as useless. In the US at least it's a major failure of the education system and STEM communicators.
That is a US centric view, and it's curious how things came to be that way. European and Asian countries take mathematics more seriously.
How is that working out for them? I am by no means an anti-intellectual and I do value knowledge for it's own sake. But Europeans in particular seem to be unable to translate their mathematics educations into any sort of practical economic growth.
"What worth is education if the workers it produces are not making the rich richer."
Education is about cultivating true richness of culture and mind. The economy is in service of a higher purpose - of life, liberty, and the pursuit of happiness. Not the other way around.
I know there is some controversy, I would say that the field of study also “causes” controversy in a sense if the disagreement has to do with the subject itself, that is true for any field. For example, set theory “caused” significant controversy in the early years of its development, yes it was people in disagreement and insulting one another, but the fundamental questions were from the subject matter itself.
Regarding widespread perception of math as useless, I guess I was thinking mainly of academic circles rather than the general public in that statement. Even while I hated math, I knew it wasn’t useless because my late father (who was a civil engineer) was very good at it, and he was adamant on teaching it to us. It didn’t work on me back then though lol.
As I was getting more interested in math while getting my degree in computer science, I realized much of my own issue with math earlier in life was because of the same thing just about everyone else that struggles with it has: the lack of any practical relevance in my own life because of how it’s generally taught. And the fact that many math teachers themselves don’t know the foundational/philosophical aspects of the subject, which would go a long way I think towards explaining that relevance if introduced in some pieces earlier on, rather than treating it like a rote memorization subject and being so focused on continual practice to develop heuristics with exercises that some students couldn’t care less about. Of course, that works for some people clearly, but it didn’t for me at the time and I’m of the opinion high schools should focus more on discrete math like set theory, and formal logic rather than calculus, which is of little relevance at that stage unless you’re also invested in studying physics. Sets and logic on the other hand imo are general enough right at the beginning to be relevant to just about everything people of just about any age are concerned with.
Every theorem is already in the mind of God. No theorem prover has discovered it anymore than an astronomer discovers a star. We simply have better access to God's thoughts now. It makes us realize we are human and not divine.
What a complete nonsense. Math is no more 'the mind of god' than polio is. If anything math is a triumph of mind over religion because it gave us the tools to power a vast chunk of the scientific revolution. Without that we'd be cowering in the dark and hoping that the sun would rise tomorrow.
We need to rethink the idea of textbooks as static documents. LLMs make it possible to build something like "A Young Lady's Illustrated Primer" for real. Let's go.
Sure, but given the approximate tens of millions of dollars in API costs of solving these major math problems currently, we will also have to acknowledge that getting credit for major mathematical contributions is likely now to be far less about meritocracy and far more about access to capital to kickstart the original proof.
Less capable (but good enough) mathematicians with more capital will have a huge advantage over great mathematicians who are broke.
I'm not a mathematician, but that situation rubs everyone (except those with the most capital) the wrong way.
"On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above."
I will note that this includes the Quasi Riemann Hypothesis. The model that solved NS was significantly dumber and less efficient that OpenAIs more recent math-models and you shouldn't over index on "it's expensive now". AI has advanced so quickly that your <2 month old opinion is out of date by 5+ orders of magnitude!
Question: Have you actually used AI to do research-level math? At the current time, it is not "tackle any problem you wish and you can get headline results in ~3 hours." Their method is spraying-and-praying across the entire mathematical frontier, which is something only AI demonstrators are interested in. And as far as I know, they have never elaborated on what their "average of 3 hours" metric even means (e.g. including or excluding all unsuccessful leads?).
I suspect the AI frontier labs will also soon turn their tools loose on the entire corpus of historical math papers to check for errors. They'll likely find that some major areas of math are built on a very shaky foundation.
That sounds close to what human researchers do as well. Big part of getting grants/doing graduate work is finding a tractable problem. It requires constant refinement of what the scope and direction of focus.
I guess this is the part where you claim to have ever done graduate or higher level research before, that wasn’t just a PI just giving you pre defined, contained side project for a summer.
This is true in experimental sciences, where people who have grants and funding can obtain results using equipment unavailable to others. But their raw data can help much less funded theory groups to formulate new understanding that can lead to new experiments.
As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
> As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
My assumption (based on perceived human nature) is that the raw proofs will not be made available.
If all of the important credit goes to the person who attached understanding to them, I'd assume those with the money to take the first step will keep the AI proofs secret until they believe they have adequately done the follow-up that will give them the lion's share of any credit.
I never had a chance against Terence Tao. Neither did you. There are a few dozen people on the planet who had a giant advantage in math and could do things no one else could do.
It was not some great awesome fair thing that math was dominated by some outrageous outliers on the iq spectrum. It's not some great awesome fair thing now either.
> I never had a chance against Terence Tao. Neither did you.
I'm not a mathematician at all, so yeah I certainly didn't have a chance.
> It was not some great awesome fair thing that math was dominated by some outrageous outliers on the iq spectrum. It's not some great awesome fair thing now either.
I don't see these as equivalent circumstances. For most people if they get beat by someone smarter or someone stronger, it may be disappointing but they can still respect and admire that they were beat by talent/determination/etc.
Getting beat due to access to money just feels slimy, because at some level we all realize money is not distributed on a meritocratic curve at all, at least not if you are measuring values any well-adjusted person would consider positive.
You keep saying "meritocracy" and the like, as though it's just by nature a good thing. It is not. High IQ isn't distributed evenly or fairly or anything of that nature, and yet "meritocracy" seems fine with the idea that "some people are just smarter than others, deal with it".
Rawls had a concept of the "veil of ignorance" which is probably better for thinking about fairness rather than "meritocracy". No one should lose sleep over the fact that the tiny number of corporates with enough capital can now beat out the tiny number of 180 iqs at a task (mostly generating proofs) that is not really valuable.
> Rawls had a concept of the "veil of ignorance" which is probably better for thinking about fairness rather than "meritocracy". No one should lose sleep over the fact that the tiny number of corporates with enough capital can now beat out the tiny number of 180 iqs at a task (mostly generating proofs) that is not really valuable.
This is a weird framing. To begin with you didn't need a 180iq to work with math. What AI is doing though is making it so that an even smaller part of the population can work with intellectual, fun jobs and even more wealth gets concentrated to a small elite. That is absolutely something we should lose sleep over.
If you think the above comment reflects a change of heart from Terrence Tao, you never actually listened to him, only the strawmans of his opinion online.
For sure, he was arguably the first well known person to bring chatGPT to maths.
When his more "anti-ai side" (again, it's more nuanced but I'm calling it that) was showing, everyone was so quick to call him a Luddite. No one stopped to think why a well know ai in maths fan all of a sudden had a more negative opinion. Even reading his actual nuanced take was too much.
The best example was the following 2023 quote which has been repeatedly and wrongfully used as a criticism of Tao right up until he was proven exactly correct
>I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well. Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools? We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions. Our usual technology paradigms will not serve as an adequate guide for navigating these uncharted waters. Perhaps the greatest challenge will be transitioning to a new AI-assisted world as safely, wisely, and equitably as possible. https://unlocked.microsoft.com/ai-anthology/terence-tao/
The ever prescient Tao, over 3 years ago in 2023. You can search that quote and find lots of examples of "Lol he's so wrong" right up until very recently. Which is exactly the 2026 timeline Tao stated.
And his viewpoint really hasn't changed or needed to change. He was 100% correct.
To be fair, he’s had a bunch of guest posts on his blog that didn’t seem as reasonable to me (and some people seemed to not realize they were guest posts).
Edit: lol, this article is itself a guest post. My bad for not reading it first! But it does serve as an excellent demonstration of my point, since it seems like neither of us realized we weren’t talking about his words.
>> where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
This statement is a monstrosity, and the reason why real software engineers know vibe coding will fail. It is fundamental misunderstanding of thinking that that the intellectual work consists of knowing what you want, and coding is merely the mechanical labor of making it happen.
The coding is the intellectual work!
Its where the actual problems are discovered, where the difficult decisions are made, where vague ideas become precise, and where solutions are invented. There is not such thing as a fully formed solution waiting to be translated into code. :-)
Again just the modern variation of the idea guy who goes around saying I have a genius idea for a new startup...I just need some Developers to implement it ....
Yes, and the machines can do intellectual work now.
It's funny to see some commenters here cheer as the leopard eats the mathematicians' faces, when we can all see that the software developers are next in line.
1. That’s a legal grey area that could result in significant legal exposure for Anthropic.
2. Even if someone wants to do something that’s in a legal grey area, or more (hacking others, another item limited by Anthropic), is the idea that such people won’t have the ability to use AI to do those things? I don’t understand the criticism that just because one company might institute some guard rails that those who want to conduct legally grey or black activities will be unable to use other AIs to do so?
To a first approximation I agree that it’s similar to industrialization. However there’s a key factor which distinguishes this case, and which makes the analogy less than perfect: there isn’t a profit motive in mathematics, so there was no push for this development from inside the field. Instead, the industrialization was entirely exogenous, from companies which do in fact have that profit motive and view math as a means to achieve it.
There might be a “proof motive” in math which one could argue will drive the full automation of theorem proving. But we must recognize that profit is more universally valuable (i.e. it has value outside the industry in which it was created), whereas proofs are only valuable inside the math community (at least the pure math proofs the labs have been publishing thus far).
Do rich people like opera? Yes. Is that a profit motive? No... but it's a shibboleth for wanting to be rich. Do you get it?
It's ridiculous to say, no profit motive in mathematics. It's like saying, no profit motive in LISP. But cmon, you think the Collison brother who took Math 55 and liked LISP: I know that in a pedantic sense what you mean. But in terms of academic departments where people say they are no interested in money and it turns out they want to become billionaires... Math Department ranks pretty highly. Always has.
I agree that being technically gifted has been a great way to make boatloads of cash in the past few decades, and that this correlates with being good at math. But I don’t see why this implies that enjoying math means someone is more money-minded than others. If anything, the students in pure math departments are intentionally forgoing the potential cash conferred by their technical abilities in favor of the academic pursuit. In anticipation of a reply observing that math PhDs regularly go into quant, I counter that quant programs reach out to interview students in many other fields, and more to the point, spending half a decade studying algebraic number theory is surely not even close to an efficient way to build wealth.
In response to your second point: even if mathematicians desired wealth more than other professionals, how does this demonstrate the existence of some kind of profit motive in math?
> My recent take on AI is that people haven't adapted to the new huge potential of the tool, and don't dream big enough (yet).
I guess you can dream big enough only if you have access to the really interesting models. The 20$ subscription won't get you there, and the 100$ one is already out of reach for so many people.
Unfortunately, I cannot think of a single politician right now who is a thought leader on regulating technology, which is exactly the kind of person we should be looking for. People who understand that we don't need every website to give us an annoying popup saying that they are using tracking cookies but we do need the option to opt out of AI features for basic services.
What? Only websites that want to track you before opting in have cookie banners. HN doesn't have a cookie banner, in case you haven't noticed. I'd say your issue isn't with consumer protection, but with pushy marketing people (or worse). But they're good at marketing, hence the cookie banner nonsense complaints, which redirect the anger towards laws requiring that creeps announce their presence, rather than the creeps themselves.
Two possible walls for advancing math with AI: first, it could be that the kind of problems that AI solves are the ones that don't open new avenues to concepts and ideas, and second, problems complexity is usually not linear, and the new and interesting problems to solve could be at a level of complexity beyond what AI can currently solve. Perhaps mathematicians could work to find ways to help AI cope with those walls.
How you get somewhere is important and tools change how you get there.
You can drive 1000 miles and arrive at the destination on the highway quickly.
Or you can take the slow road and walk. You discover unrelated things, make new relationships, see the scenery, listen and smell what is around you and have time to contemplate things which may or not be related.
This particular tool economy robs us of all these things in the rush to the destination.
I'd sacrifice "progress" a little for some of the better bits of humanity.
Not OP, but actually I do because I have been forced to live in places where it’s dangerous to walk and where all the scenic landscape has been replaced with asphalt.
I get your point in general, but I think vehicles are a poor analogy because in many ways they have changed the places we live for the worse (and yes I’ll concede occasionally for the better). It turns out we need places to park all of these vehicles and they compete for space with humans.
That’s very narrow-minded. I love hiking and biking and exploring new places, meeting new people. I also appreciate that I can use the highway to get to the destination conveniently. Often so that I can enjoy the very experience I’d miss according to you.
The thing is that if you don't learn to drive some destinations are unachievable. And destination is more important than the journey unless you travel for pleasure or personal development.
The first sentence is actually a much better jumping off point for the other side of the argument than yours, because the mistakes involved in the statement should set off alarm bells for people who've walked with mathematics.
What do you mean by "place"? What do you mean by "more"?
Are there more places as marked by real numbers between 0 and 1, or places as marked by integers? Could a machine that can count integers 1000000 times faster than people "visit" all those real places between 0 and 1? What would it mean to visit all the places between 0 and 1? Once the machine visited all of them how would it help people explore that territory?
What do the general ratios and rules between paths and areas mean about how much path it takes to cover an area?
The real world is quantized in way that math isn't, of course, but that's one way the analogy can lead you astray.
And even just keeping to the real world rather than math, it's not all about the sheer velocity with which you can cover an arbitrary distance metric. The primary bottleneck is less that velocity than the amount of time you have to direct to place scaled to your capacities. The car's velocity can let you trade some of that time for fast travel between farther places, and with a road network can arguably even increase a certain variety of places, but that's a very different statement than more places. And it's also a wall against attention to everything in between.
Is this about driving to an island? If you need a plane or boat to the island and still call that trip "walking", then someone can take a plane or boat to the island and rent a car and call it "driving" or "driving plus walking".
Or if you're just saying that a very good backpacker that happens to be in greenland can get more places by walking because of a lack of roads, that might be true. But for a normal person in a typical household, their walking range is not that big. If you're in the US or Europe the roads reach almost everywhere.
They build cities around the car, they defund public transit, they close down rail lines, they build new housing a 5 km away from your closest grocery store, and 20 km away from any cafés and other cultural venues. Your job is located 50 km north of where you live, your partner’s job is 60 km in the other direction, and your child’s school is 10 km in the third direction.
They neglect to build sidewalks, and force you to walk 1-2 meters from cars racing at 70 km/h. With all the noise and high risk stressing you up to the level you just wanna scream.
stevenhuang: No one forces you to drive. You can still walk.
So drive then. You can also move. The point is there are options. Many cities have strong infrastructure and have embraced walk ability. If not driving is such a big deal to you, why not do something about it?
You're arguing like an edge lord here. What he is saying is that you're effectively forced to drive. Not so difficult to understand. There are plenty of people fighting for change but because our political and economic landscape is captured by big business, it's not so easy.
Not everybody in the world is effectively forced to drive. Sure, maybe if someone has the misfortune to be an US city they have no other option but there are other places that are not like that and cities with good public transport and relatively low car usage.
It's pretty well documented they deliberately made things impractically large distances apart and scrapped the idea of footpaths, to encourage car sales, paid for by car companies to your local city government several decades ago.
Could you please stop posting unsubstantive comments and flamebait? You've unfortunately been doing it repeatedly. It's not what this site is for, and destroys what it is for.
Yes, this. Unless there is a true singularity where no more questions are ever asked, there will be definition be infinitely many problems to solve, and most of them we have not even glimpsed yet.
Singularity? Probably not.
Tech bro dictatorship? Unfortunately a bit more likely.
None of the above? Then we have a small shot at a great future.
Having a machine that solves hard problems must be a good thing.
Why do nearly all the articles exploring the development of AI applied to math assume that capabilities won’t extend beyond theorem proving?
> other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world.
These all seem totally achievable by LLM agents, or at least, I don’t see an obvious reason why it wouldn’t be. 3blue1brown’s recent discussion on this subject was a bit better, but he spent the plurality of main body of the video assuming this scenario, devoting less time to the other possibilities. To me, it seems dangerous to build a math career on the foundation that AI won’t be able to explain math well in a few years, let alone a few months (which seems entirely plausible given the recent math progress).
Because the other option is that AI does what you say. And if you think about it (and you are a mathematitian and not an engineer) it means that AI is able to read our minds and know our desires. And at that point who cares about existence anymore?
"Choosing which terrain is worth exploring" is more clear if you have an objective function (if you are doing engineering). But mathematics is closer to art in that regard. What is a terrain worth exploring in art?
The joys of parenthood. The excitement of new love. The tragedy of aging. The rise and fall of a great family. Jealousy, vengeance, embarrassment, righteousness, venality. You obviously couldn’t create an all encompassing map of the terrain, but human art has a lot of recurring themes that are clearly “worth exploring” because they’ve consistently been explored in the past. I think the same kind of repetition probably also exists to some degree in other areas of human endeavor. An artificial intelligence could probably suss out a lot of those themes and produce valuable (if not maximally valuable) work without being a mind reader.
Im not saying it cant do those things. The mathematical equivalent of that are the hundred proofs that OpenAI dumped. And probably we still have a few decades of AI exploiting low hanging fruit in Maths and Art.
I was refering to going beyond and deciding what to explore. There are infinite paths in mathematics and a lot of ways you can describe and denote a particular theory. And the choice of notation and exposition is an art in some way.
You could probably say that "no, but this is done by minimizing memory usage in human brains so that its understandable by humans, maximizing compressability of previous ideas, and a link with some utility in the real world, etc"
But that just one example of loss function. There are many possible ones.
Do i think that there is a universal loss function that captures it all? Probably yes, in the limit, on the infinite energy regime. Are we in that case in practice? Probably not.
I tend to think that AI will fully replace us the same moment we are able to master prime factorization with our time and energy constraints.
You don't think OpenAI has any interest in having agents identify new interesting research terrain? They just want to solve existing important millennium problems and then they're done?
Yes? I expect their motivation for devoting resources to solve these problems will dramatically drop as soon as they IPO. Maths is just a convenient vanity metric for OpenAI to demonstrate progress, but long term progress in mathematics has no commercial value to them.
Of course, general improvements may be sufficient to continue to drive math improvements even without specific attention - we’ll see.
Not a chance. They'll go for room temperature superconductors and other stuff that is currently in the quite possible but not yet proven and patent the hell out of it and charge rent to the rest of humanity.
AI systems will probably be able to, if not direct labs themselves, cut down a lot of the search space to only the most promising candidates. That’s basically what Isomorphic Labs is doing for candidate drug research. It doesn’t seem crazy that the same could be true for material sciences.
Simulation to the point that the accuracy is large enough to extrapolate to real materials. You can do a lot of stuff when you have access to a million GPUs that would otherwise be much harder.
A lot of materials science is already done through simulation, same with a large chunk of chemistry and biology. That fraction will only increase over time.
I am aware of advances, but I think a large chunk is a stretch in various ways. We'll just see what happens - simulating even a glass of water fully is tough.
That depends on the granularity of your simulation. For most - not all - purposes a glass of water can be simulated coarsely or by simulating a far smaller amount with greater fidelity and then to extrapolate.
For useful materials science the numbers are still large but far lower than the numbers involved in 'a glass of water'.
I really depends on what is needed. If you know that something quite "coarse" is enough to create/uncover room temperature superconductors with high critical current - fair enough. Again, we'll see what happens.
With the kind of compute they have at their disposal 'quite coarse' may actually be 'impressively detailed' and that is why I think they have an edge on this kind of stuff. It's like being able to borrow a trillion and then to go play research lab, it would be highly surprising if they weren't able to monetize such a head start.
> I expect their motivation for devoting resources to solve these problems will dramatically drop as soon as they IPO. Maths is just a convenient vanity metric for OpenAI to demonstrate progress, but long term progress in mathematics has no commercial value to them.
And why would IPO remove incentive that's present pre-IPO? Somehow math progress will influence price at IPO but won't influence stock price post-IPO? Or will they just stop caring about stock price once they are public. None of these options makes any sense.
No, post-IPO they will have to (at some point) demonstrate real commercial value. This just gets them to a stage to cash out insiders and (less importantly) get some additional capital to extend their runway.
That's the playbook, right - keep interest/investment flowing until you can figure out the actual commercial value/model. Until then, do whatever you need to keep the hype as high as possible and maximise your runway.
Because if you assume there is going to be an AI-god that can do everything better than us, there's nothing to really do. Like really, what do you want these people to write? "Well, math for humans is over as well as all intellectual activity, better just jerk off watching AI-generated hentai all day until I die"
I didn't say it was impossible, I said if it were true there would be nothing to do and thus no point in writing about that potentiality. It's sort of like writing about what mathematics should do if there was a thermonuclear war that wiped out all of civilisation.
There's a long collection of comments by mathematicians at [1].
A large number of workers in the field are rather discouraged. The main quality complaint is that the proofs are ugly, long, and incomprehensible. One mathematician comments that the first proofs of new major results are often like that. Elegance comes later.
The main social complaint is that the social foundations of mathematics just changed, in a big way. The main practical complaint is "what do I do now", especially from PhD students who just lost their thesis topic. They sound like the Uber drivers fighting Waymo.
One comment was "A common piece of wisdom is that mathematicians only have a few tricks in their bag – which they skillfully apply again and again."
Feynman made that point in his autobiography. For various reasons, he had
a different set of mathematical tools than others in his field, and that helped
him to solve problems that stumped others.
A big AI model has access to all known tools. AIs have more breadth than any human.
Some of these breakthrough proofs seem to come from applying techniques
from areas different from what people in the field tend to use. I'm not qualified to comment on math at that level, but the comments by mathematicians seem to hint in that direction.
That may indicate why throwing vast amounts of compute at these problems works - an AI can try everything.
Not only are the proofs long and unintelligent, but they don't cite previous work.
In fact, they completely disregard all standards of mathematical dissemination. They just post a bunch of 100+ page papers online and call it the day.
A real mathematician polishes and polishes for months until it is understandable by the community. Then they communicate the results to a journal where there are editors and peer reviewers who come with detailed feedback which is then implemented. And just then it is ready to publish.
"Polish this paper. Check all proofs with Lean. Try to find shorter proofs of lemmas where possible. Write for a undergraduate math student. Add brief explanatory material as needed. Cite previous work where relevant. Check all citations. Convert to LaTeX."
A. The capability is just not there. OpenAI and Anthropic can produce a brute-force Lean proof, but their models aren’t capable enough to digest the proof as human mathematicians do.
B. They haven’t seriously tried it yet. They prioritized grabbing the land first, and have not spent much effort to polish things up.
C. It’s fundamentally impossible. These theorems are provable only in a very ugly manner. This is the reason why human mathematicians haven’t been able to crack them before AI in the first place.
As a programmer, what I’m seeing is (C). Many recent cyber hacks are products of AI trying & combining every possible tricks, and I don’t think there is any elegant short path that human engineers can replicate.
The four-color theorem appears to be case C. That was the first really long computer-assisted proof. No one has since found a concise way to do that proof, and it's been half a century now.
D. They seek disruption as way to market dominance and worldly power. It's not unfortunate, in their minds, but the means to their ends.
Disruption has been a SV buzzword for a long time now. It's not a byproduct but a strategy for your new technology product. SV has long perfected disruption and the people behind the AI products are masters at it.
How can so many who have seen it (and used it) for so long be so taken in by this well-worn tactic. Over and over, they choose what is most disruptive. They could have dealt with the mathematical output in many ways, including working with mathematicians to polish and publish; you see what OpenAI chose.
Similarly was the recent events where AI software 'broke containment' and hacked others. Obviously they could have controlled it if they valued it; they got the result they wanted. Notice how Anthropic (or were they first and OpenAI followed?) urgently publicized that their product, too, 'broke containment'. 'We're disruptive too!'
> The main practical complaint is "what do I do now", especially from PhD students who just lost their thesis topic. They sound like the Uber drivers fighting Waymo.
I wish mathematicians would just focus on this. This really is the problem, at the end of the day.
Having models get better at writing elegant proofs and explaining their results will not solve this problem - it will only make it worse, if anything.
We are all sympathetic to this. I, personally, am much less sympathetic to the absurd notion that it is "bad" that this new technology can solve important open mathematical problems at all, that we should stop trying to solve them, that things like the Mathathon are dangerous and needs to be shut down, that ugly proofs are the real problem here, so much so that you wish the open question weren't resolved at all, and so on. So silly.
But being pissed off that people are going to be out of work is something I agree 100% on. Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.
Yeah I view it as AI being objectively good from a technological perspective - we get new medications, new technology is achievable faster overall.
But really bad from a social perspective - as people lose their jobs they also lose their tiny piece of power in the society that was the labor someone else needed. Now that power has concentrated to the frontier model owners, and everyone in the future will depend on them.
And on a personal level, sacrifices people have made to become good at their craft have been made worthless, as any rich PoS can now prompt their way to "good" art and call their end result as good as something made with skill that needed years of training. That I really hate.
> prompt their way to "good" art and call their end result as good as something made with skill that needed years of training.
Which is only possible because:
a) The models have been trained on work of millions of people,
b) They've done it for free, In other words, they've stolen entire people's lives. Millions of lives. They literally suck people dry and tell them to go look for another job.
I wonder what LLM's would achieve (if anything at all) if no mathematicians had used it to format their articles, etc (remember, no data is being used to train models! Of course).
It’s amazing for me to see math research going from craft to technology.
CS had this moment decades ago, sans some theoretical parts that are joining the bandwagon now.
I did a bit of math research in grad school collaborating with my math postdoc roommate. I found the aversion to any computation making its way into the craft to be odd decades ago.
Well, now the proof’s in the AI generated pudding.
Communicating results when zero other people are working on the same problem as you and your advisor aiming to submit to a peer reviewed journal that takes six months to respond; versus when competing teams of hundred engineers working with a few billion dollars of funding trying to get the results before the other teams has to be completely different.
It’s astounding that people expect the later to look like the former.
He did attach his name to it by posting it. I'd imagine if he disagreed strongly he wouldn't do that, or he would at least add a footnote.
What you could argue though is that he is muddying the waters with all these guest posts on a domain that is as personal as it gets. I appreciate he's trying to be the 'Carlos Santana of mathematics' by sharing the spotlight but it might be good to highlight the distance a bit more, for instance by adding his own views.
Well, you're wrong, given how many of the guest posts featured by Tao are inherently contradictory and so it would be strange for him to be personally endorsing each one! He also posts his own views elsewhere: see https://teorth.github.io/tao-web/ai-views.html.
The obvious goal is to highlight differing attitudes within the mathematics community so the mathematics community can have a dialogue and see what other people are thinking. Luckily, most mathematicians seem to be able to actually read and understand not all posts are from Tao. It is unfortunate it seems many people on this site cannot.
Posting something on a domain with your name attached to it is an endorsement of sorts. You can argue with that all you want but that's just how it works. Having a dialogue happens on neutral ground and Tao's blog is not neutral ground, even if that might be his intention.
If he thought it was fundamentally an unreasonable position he probably wouldn't do that. But he seems to be running his blog as something of an open(ish) forum for AI commentary. I suspect if he thinks it's a reasonable position that happens to be entirely incorrect he still would post it...
> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof. Choosing worthwhile problems may end up mattering more than producing proofs.
That is the correct take, but I can understand that given how current funding system (esp grants) work the incentive are stacked against doing that.
It's worth mentioning that many of these papers haven't been fully reviewed and confirmed as being correct. Several had to be retracted upon closer inspection. Too many people are assuming that passing lean checks for some of the papers means that all of them will hold up under scrutiny. It remains an open question if humans or any available system are capable of peer review for some of the more complicated proofs.
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
Of course if you solve any existing problem, new ones appear. That's not the point. The point is that AI will be able to advance on any problem better than any human.
But in a human way? I don't think so. The 700+ solutions that were put out by openAI are in no way accepted as solved until humans say so. It's kind of like saying SC2 was solved when Alpha beat the best players. The professional community just went on playing much the same. Math should do the same. Just with better tools
Applied mathematics is funded because of its practical use. Pure mathematics, like other theoretical fields, is funded to explore ideas practically oriented people ignore. Most of the time such exploration produces nothing of practical value. But occasionally theoreticians stumble upon ideas that turn out to be fundamental to understanding something practically oriented people discover much later.
If rich people want to use their money to fund math with the goal of NOT funding math with any known useful or practical purpose, that sounds like a hobby to me.
Mathematics as a field will not cease to exist, and not everyone will be interested in (or qualified to) managing the fields progress and its applications.
Plenty of these AI proofs have been accepted. On the most important result, on the Quasi-Riemann Hypothesis, there is no serious belief it's wrong. Mathematicians have already dissected and reformulated the claims.
The proof with the 11/12 constant, the simplest one, has already prompted what looks like chagrin from some. It's an unexpected combination of known results/techniques, a combination that apparently fell into a blind spot so it wasn't picked up before (even though it could have been, even 20 years ago).
I think these results are showing there was a lot of low hanging fruit that human mathematicians didn't have the breadth or mental bandwidth to find.
Isn't it also possible that the written proofs themselves are incorrect, say, for example, by mis-specification of constraints? Let's say I publish what I claim is a proof that 2+2=4, but in the Lean specification, I proved 1+1=2.
The main issue would rather that the formalization itself has a bug, eg. those proofs that you're trying to formalize are not "1 + 1 = 2" but theorems which take already multiple pages just to enunciate. And in these enunciations (that are made by humans) there can be errors.
In general, once the proof has been formalized, it's supposed to be the proof of the correct theorem (that's what the Lean proof-checker checks)
the impression i get is that people don’t per se assert that exploitable bugs exist in Lean, but that the volume of (code) proofs published represent a new magnitude of challenge to the integrity of Lean.
the obverse would be to assert that Lean (or any other theorem prover) is infallible under all circumstances, present and future - an even weightier claim in my opinion.
A single bug can usually allow you to prove anything you want. Saying that this is how the proofs were achieved is a totally different matter though, and probably incorrect.
Hmm, am no expert at Lean, but on one hand, since it's a programming language, guessing it necessarily means like all other programming languages, it has bugs.
But on the other hand, agree with you that the chances that one its bugs causes a solved proofs to be wrong is probably very, very low. Yeah, each version of Lean obviously must be run against hundreds if not thousands tests of existing proofs to ensure it's working correctly...
... Although, one thing that is a little worrying is that some mathematicians have said the Lean code generated in these OpenAI proofs is supposedly really complex and undecipherable. It might be pushing Lean to corner cases that are not well tested.
It can possibly have bugs. That said, the "core" of Lean is a somewhat "simple" core that does simple things to check the proof, so it's quite rare to find meaningful bugs
don't disbelieve you, but just looked at the reference language, it seems pretty full featured as a programming language, even with basic classes, polymorphism, scope, namespaces... But maybe most of the mathematicians who actual code with stick with the simpler language features? And also, didn't look to deeply, these could be very simple features that actually aren't very sophisticated.
I mean Lean has two things: a full programming language, and proof checker. The programming language is used to create either code, or proofs. But the "created proofs" are using a very simple subset (a bit like an "assembly language"), and only need somewhat simple rules to be checked
So as long as the "somewhat simple" core that checks to proof is correct, it doesn't matter if the "programming language" has bugs: either in the end the "proof assembly code" generated by the program is right and is validated by the core, or it's not, and you need to revise your proof, or to check for either a bug in the programming language (it didn't generate the proof you were expecting), or in very rare case detect a bug in the simple "core"
Ah, I see, that makes sense - and am guessing that most mathematicians (like most non-developers) who work with a programming language don't write "good" code and often ignore the organizational features like classes or namespaces. Yeah, could see how they really are just interested in the core feature-set of a programming language... although, have seen a bug in some high-usage low-level code before, which was greatly surprising. Still that was only one case in years of dev work.
I keep seeing people point to AI solved games as evidence that AI advancements are less impactful than we might think. Usually it's chess ("today there are more players than ever!"). Most of the time these are bad analogies. Games are played for entertainment, so productivity is meaningless.
I never advanced in math past high school, but my assumption would be that we as a society aren't paying mathematicians to have fun[0]. The discoveries mathematicians make must translate into real improvements in engineering and other life improving fields. So why would we pay mathematicians to work on math much the same way, if a new machine-led way makes these discoveries faster and cheaper?
[0] Being a little facetious. I know I don't know what I don't know.
Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
>Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
Obviously. In my ideal world even the janitors at the same institutions would have a blast, it's just not usually the end-goal reason humans are hired.
>With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
Right, but would it have made a difference for the application layer if that key bit of "pure math" had been discovered by a human (that had a great time) or a machine? That's my point about games being a bad analogy.
That depends. Many of these things can only be applied safely once they're understood.
For instance, almost all of cryptography is applied math. If your opponent understands the math better than you do you might end up with your secrets compromised.
Math is about as 'pure' a science as there is, the touchstone of applicability is missing the point entirely, it is first and foremost about understanding things at one of the most fundamental levels (even more fundamental than physics) that there is.
Nobody cares all that much about the discovery (except that we name things for the person that discovers it, it's sort of analogous to the discovery of a new piece of land). And given that the pay usually sucks this is a nice thing to be able to do. What mathematicians usually do really care about is their ability to understand the discovery and that this discovery has the proper foundations (other pieces of math that are also understood). It doesn't really matter what the answer is if you don't actually fully understand the question or the foundation the answer builds on.
I think it's likely that AI will create mathematical ideas that are so complicated and intricate that humans have no hope of ever understanding them. AI on the other hand will understand them, and understand them well enough to use them productively and safely. That is my prediction.
Many academic disciplines bring no everyday life improvements. They exist because they provide intrinsic value and long-term foundations, hence they aren't merely entertainment. Mathematics/physics happen to be fields that span the entirety of applied/pure spectrum. Those said, your question can be generalized: Should society continue funding human intellectual activities when AI can perform them substantially better?
LLM agents are perfectly capable of "advancing" my codebase down bizarre, esoteric rabbitholes. Meanwhile, the value remains undelivered.
Many crystals, likewise, are advancing their fantastic complexities in the deep, dark recesses beneath the crust of the earth, but does it make a sound? Who gives a shit.
Many paperclips, indeed, can be churned out. Even we can be converted into paperclips, remember.
I don’t think we know whether or not that’s true yet. Imagine a problem which requires a paradigm shift or new novel techniques to solve. There is no proof (yet) that an AI could create one.
We chose the problems to value. Literally every step of a proof is equally proved as the starting point. The choice of the starting point is what makes the problem, and it's a normative choice, which is to say it is a social choice.
We invested at least a couple trillion of speculative value into making ugly cartoon apes the future of art, finance and property. I'm not sure it's always possible to overwhelm reality with sufficient money.
I'm not saying it will "fall apart", I just think the burden of proof is on AI boosters for their extraordinary claims.
They are saying this thing will soon obsolete human thought; meanwhile I use these tools every day and can't get it to refactor a React app cleanly without making a mess.
It's telling that OpenAI's decided they're going to sell ads in ChatGPT. Instead of, you know, using their AI themselves to produce all these valuable and revolutionary breakthroughs they claim it's capable of making.
Medicine is a slow moving field. If they discovered the cure today how many years will it take to be certified as safe and effective by medical boards? More years than they have runway, in all likelihood.
It's not magical abilities - humans have proved for millenia that they can find novel solutions to problems - we have no idea how that comes to be though (which could be called magic I guess)
At issue with AI is that it's searching an existing corpus for fragments of solutions generated by humans - that's not necessarily novel, and we've not yet worked out how to make the machines actually invent.
But the question is whether the AI will itself be able to decide on which problem to advance, and do it in an ethical and aligned way with humanity. And that if it does, its significance will be understood by humans.
Otherwise we're just creating a black box, easily manipulated by their owners, who would be ruling over human lives. That's not what people want
The thing about much human progress on various problems is that human mathematicians formulate a human comprehensible machinery for the purposes of solving a given problem and then that machinery naturally suggests (to humans) further questions which then give rise to more machinery - And, in a sense, the value is ultimately the machinery, which has tended in the past to be relevant to physics, computer science, logistics, sociology and philosophy.
When AI writes proof no human can understand, it doesn't advance mathematics this way. Maybe AIs can understand their own proofs - meaning not just verify them but extract a machinery/broader-perspective (but I wouldn't count on it). But even machinery an AI isn't necessarily useful for humanity/society.
> The point is that AI will be able to advance on any problem better than any human.
If that was true it would of course be significant.
If you had no morals, 2 trillion dollars to spend, ability to commit intellectual property theft without consequences, and most of the smartest people in the world actively working to advance your cause... ...you really think you couldn't compete with the clankers?
Given the capabilities of OpenAI's unreleased model, I think the model helps mathematicians more than it hurts them. Some of the open problems that were solved were "It would be nice to know that", but many of the others represented ways in which we had been stuck for many years. There had been very little progress on analytic number theory, particularly the Riemann hypothesis or its variants. (One result, "no Landau-Siegal zeroes", even had a 10-page elementary proof that everyone missed for a century.) Complexity theory had seemingly more open questions than results. Progress could meaningfully accelerate now that we have a quick test of which of our ideas are good, and which are bad.
The wildcard is that we have no idea what the model after that will be capable of, or the model after that.
> I think the model helps mathematicians more than it hurts them
I would agree with you if these models were available to mathematicians, and OpenAI wasn't effectively competing with them by spending millions of dollars in compute to rush out solutions.
> We have much more control over how we hire mathematicians, publish papers, and train students than over the behavior of trillion-dollar companies, and that is where our efforts would be better spent.
We should also make sure that there are no trillion-dollar companies. The existence of large and powerful companies is a problem, whether or not the field of mathematics is in crisis. All these companies should be shattered so that they cannot do things until after society as a whole is sure they're okay with those things.
I wonder if this happened to math first because the goal is so clear (proving a statement, etc.). Math is just going through it first, and I think other fields will get their turn soon too. If so, I'm not sure if things like a PhD are even needed. Someone in a bachelor's program who just learns the basics and how to use AI tools might be better.
Better at investigating the idea that one could conceive of elements in the different commensurability classes of compact arithmetic hyperbolic three-manifolds indexed by non-split quaternion algebras {D/F} as forming some type of geometric Kolyvagin system (“Heegner points”) associated to {K_3(F)}, which one could yoke together by some unknown integral arithmetic version of the Jacquet–Langlands correspondence?
It has changed from elegance and human understanding to brute force. But that trend was started a while ago, before AI became really useful.
From TFA:
"I’m ready to board the plane, because I want to see what’s there. At the same time, I would add that people who previously have been walking around on earth with their eyes closed are not going to recognize anything however high they fly; I still believe that many of the skills that mathematicians have now will not only be transferable but vital to the new world order."
IMO, this captures well the essense of AI: it's an alien swarm intelligence that solves problems with brute force and pattern matching. For comparison, we prefer simple and elegant solutions, partly because our time and attention scope is very limited, but mainly because it rhymes with our nature.
I mean he's basically admitting that math academia is outdated.
If we don't interview candidates based on results, what is it? Politics. It already was. I had a JEMS paper in grad school and got zero postdoc offers. Many other cases.
The real shift is that math will become a hobbyists game. Make money other ways and then spend your money on humans and tokens to solve your problems.
Terry's stance has shifted gradually. This reminds me of Garry Kasporov's response and stance a few decades ago agains deep blue (and deeper blue). I think in a closed system/game (or "field"), this kind of AI superiority is inevitable. However, I do agree with Terry that we need the right form of transition if we want to continue holding the rein or at least playing a part in the subject.
> gulf between ... novel proofs and ... understanding
Machine proofs are useful machine understanding. You don't read the proofs, that's holding it wrong. You point another agent at the proof and prompt "use these ideas to prove this lemma that's useful to me". And it works, agents are faster given access to the proofs than without.
“Note there is also a difference between knowing mathematics deeply and explaining it fluently; they are related but different skills. Ideally I want both, but it is the first that I ultimately value more highly.”
Once again perpetuating the cardinal sin of lionizing mathematics heroes. Which to be fair is just THE culture of higher mathematics.
What good is a minuscule elite Cabal of savants? If these deep insights are inaccessible to 99.999% of the population, then they are no better than the manuscripts written by AI. We already have those.
The whole situation in mathematics reminds me of two years ago when models started coding well and all the developers were really sad. Now we use it as a tool and, as a whole, are able to make much more.
> someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution
among all the discourse, it seems mathematicians are vaguely converging on this higher layer of abstraction being where the human value in math is now. It seems across domains (such as in software too), people speak of “judgement”, “taste” and other fuzzy concepts around defining problems rather than solving them. The lowest level of abstraction where humans win is asking the questions, rather than merely answering them.
As math is so verifiable, the “transition” seems most poignant and quick there. Watching these big name mathematicians discuss this unfolding revolution in real time is fascinating.
I theorize that number of problems AI is possible of solving is limited. My theory rests on an assumption that all human knowledge is a result of lived experience. AI as a statistical machine doesn't know lived experience, therefore it can't solve problems that a living person interacting with reality could solve. Therefore human brain > AI, or so long AI is unconscious anyway.
Sure it can solve math problems by just following the implication arrows. But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
These fall under the class of problems that have solutions which can be inferred more easily from applying massive computation, which is a subset of problems that human minds are trying to solve.
It's like being a CIA analyst, maybe. You have an overwhelming number of information sources of varying quality and non-obvious relationships. Applying massive computation will definitely allow you to draw new insights and identify trends that would have been difficult, or nearly impossible for unassisted human reasoning.
From what I understand, this is the sort of thing that the LLMs have been particularly effective at in mathematics, applying massive computation to the existing field and with a combination of impressive vector space search and runtime evaluation, deriving plausible solutions to certain narrowly defined problems.
Here's a harder one: you're in a bar, sitting having drinks with friends. There's an attractive girl sitting with some of her friends that has glanced at you a couple of times. What should you do?
Taking into account, of course, the entire human experience from marriage status, age, rejection likelihood, your friends opinions, her friends opinions, your family, your hormones, everyone's intoxication level, how well you slept, the noise level, how long her glance lingered, and the billions of other things that go into it - entire lifetimes of experience, genetic behavior predisposition, etc...
This is just a small example of a brief problem that a normal human has to solve hundreds or thousands of times per day that don't fit into the class of solutions that LLMs would be particularly adept at. It's easy to come up with a lot more.
So yes, I'd argue that those neatly formulated, clean problems with clearly provable solutions are low hanging fruit.
> But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
Some of these problems have had mathematicians dedicating their careers towards getting closer to a solution. Your comment reads like wishful thinking.
Low hanging fruits for the agents that solved them. It's very possible that there are whole classes of problems that AI won't be able to solve or will require another few orders of magnitude of compute to solve. A human might even be required to first make a breakthrough on some field before AI can re-apply it exhaustively and find new discoveries on top of that.
It's amazing to still see posts like this in late 2026. I suggest you recalibrate your ideas on what both AI and human intelligence is and can do. My take: both are trained systems. Both are as "statistical" as the other. Any idea that "because statistics" means that humans are different or better is bogus. Both can do things based on their training data. You are totally correct that a human has access to training data that AI doesn't (the smell of new mown hay, for example, or the fear of being crushed by the garbage compactor room in Star Wars). But there's no "humans have more training data therefore better" argument to be made about fields like mathematics. In fact the opposite argument works quite well: AI can have all mathematics ever invented in its training data, something that's pretty hard to achieve as a human.
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
This analogy is pants on head stupid. They're talking as though the only thing AI will ever solve is the Langlands program, not as though we have discovered a generic tool.
It's like saying "they drove the car one mile -- and every mile driven leads to more roads that we can trot along, why are you so worried about the future of horses?"
People are allowed to process their anxieties. And processing takes time.
That said publicly doing it on the great modern internet attention/noise factory complicates the story.
Everyone is surprised that a person that some consider to be one of the greatest modern mathematicians can have such obtuse views of technology and its own field.
I don't think it was a particularly well-reasoned essay (I thought the classification of finite simple groups is almost evidence for the opposite of what he says), but at the same time I don't think you understood his point.
If we have a theorem proving machine, then we can use it to prove theorems. If we have a machine to solve the Langlands program in one day, then we'll think of more things to prove after that. There's a million unsolved questions in number theory unrelated to Langlands, and if we knew Langlands (and why it was true) we could make new conjectures that our theorem proving machine could solve, rather than having to speculate for 50 or 100 years like now.
The possibility he's not considering is that the machine might be better even at conjectures, and then there really is no role for humans once the machine already knows things our primitive meat brains are too small to understand. But what he's saying isn't "pants on head stupid".
Sure, or you can think for yourself for five minutes on the implications of a general proof machine, and whether it's likely to continue producing proofs.
If you've forgotten how to think for yourself, you can also ask an LLM.
The host of the blog, Tao, was an early booster. He didn't think through the implications, and seems to have expected it to remain unable to do interesting work. Today he's on a "war footing".
I don't think mathematics are uniquely skilled at predicting the future, and the impacts of AI on their profession.
It is weird. From what I can tell, you and I have quite similar expectations on what AI will be able to do in the future, but for some reason you are much more dismissive of Taos opinion.
I perfectly understand his point and it is not about being selfishly frustrated by a machine being better than him. Is that really your interpretation of the discourse?
Logic is logic, you don't need a specialist's opinion on every aspect of a specialist field. I've had eye opening insights delivered to me on problems I'm working on by people who barely know how to use a computer.
What is the current thought on AI creating new emergent forms of mathematics? Even the most stunning results like the proof of the quasi-RH are a clever combination of known mathematics. Humans have created (discovered?) stunning new ideas (e.g. complex numbers) that have opened wide new horizons -- do we think AI will be capable of creting / discovering novel new things like this?
Are humans going to be able to recognize the things AI discovers?
It took us a while to make the planets line up neatly in a heliocentric system. Just last month I read that a whale mother "might" grief while carrying her stillborn child.
The limiting factor is human imagination and comprehension. No offense, but see your comment.
I think AI has already invented new math, but the guys haven't looked for it because they didn't think it possible. And even under their noses, they wouldn't understand what they see.
> It's not what you look at that matters, it's what you see. —Thoreau
LEAN and other tools verify from what do know -- at least stepwise. Anything that deviates from these things should stick out. I suppose some million line LEAN program may have embedded within it something that is novel and not readily apparent, but I imagine truly novel approaches should stick out like a sore thumb.
Could AI create a novel set of axioms (a new math) from which things like RH become a by product? I think that would be hard to miss.
Dozens of agents probably already invented it, but they were given ephemeral containers where noone ever looked or they only wrote it in direct CPU instructions.
I would really like to see an AI invent a new set of (consistent) axioms from which a whole slew of new problems and solutions arise. Something akin to Euclidean geometry, but for a whole new landscape of theorems.
That analogy makes no sense to me. Wouldn't OpenAI have the incentive to release results that are incomprehensible, to showcase how beyond humans their models are?
There’s an elegance to how the Feds just got out of funding this stuff and within a year all proofs were solved by the private sector
I know, unrelated truths and coincidental timelines, part of me giggles nonetheless since the whole belief system behind the market relies on infinite coincidences
Mathematicians can just say no to this technology.
1. Make mathematics a closed community, only people who join this community can read and work on its stuff. I imagine a VPN that contains all maths knowledge that only members can view, similar to how big tech handle their codebases. It needs to be hard to copy and paste that stuff into the open internet and doing so should involve serious punishments like jail time, similar to posting say proprietary Google code on the internet.
2. Anyone caught using AI is excommunicated from the society, his access to the VPN permanently revoked, and his work is never interacted with again.
3. There will be a bit of a disruptive period as maths moves from open to closed model. During that time AI will keep solving more problems, but as long as mathematicians stop contributing more open problems, at some point AI will stop (either because it has solved all open problems, or it can’t make progress anymore because scaling is no longer economical).
I think Association for Human Mathematics should do this, I think they will be surprised by how much funding the public is willing to allocate to this (public hates AI), because fundamentally humans like thinking about and doing maths, and do not mind dedicating a bit of the surplus produced from society to let others do this, similar to arts. Just because I can prompt a better melody from Suno, doesn’t mean humans will stop wanting to make music.
Of course other mathematical associations can and should form, which fully embrace AI. But if the VPNs secure, these associations will not be competing as they won’t even have shared consensus of what problems to solve. They both will still need to compete to get public funding, which is fine, I think AI math will have a harder time justifying their funding since anyone can prompt a robot.
Three years ago people were laughing at LLM arithmetic mistakes. Today the Simons Foundation is starting up a fund to keep professional mathematicians from suiciding:
> Under a partnership, the Simons Foundation, Anthropic and OpenAI will commit $15 million in combined funding and resources. The foundation will manage and award funding, including through an open call for proposals, to support workshops and initiatives that help researchers evaluate recent AI-assisted results, learn and use new tools, share perspectives and shape collective standards for AI use.
I really struggle to understand this discussion. Mathematics isn't a way to keep a specific kind of intellectual happy by providing fun, meaningless puzzles to solve in isolation. It's the bedrock of all other science. The goal is to find solutions and to advance the field - but we seem to be treating it like some sort of occupational therapy for bored geniuses.
As we solve lower level problems, we can unlock real world math solutions. Stellerator Design, Quantum error correction, Predictive superconductivity, Inverse protein and enzyme design - the list goes on. Many solutions AI can find can bring us closer to these - so let's fire up whatever datacenter we can find and get us to these outcomes.
Second order effects, that rational types fail to grasp
Gamedevs can make better games with AI! Yet the creatives don't get into gamedev if they didn't have to spend any effort to make their art
Mathematicians can advance mathematics with AI! Yet mathematicians don't get into mathematics because they want to "advance" mathematics
Teachers can teach better with AI, writers can write better with AI.. Can they?
Can't you people understand romance and the beauty of the human endeavor
In math, the speed and innovation will increase. The bar will be raised. Those mathematicians who do not want to be part of this new world will quit and go do something else, just like how some software devs have done so since AI agents. But not all will quit.
> It's the bedrock of all other science. The goal is to find solutions and to advance the field…
That may be true but that may not be the feature that attracts people to the field.
> As we solve lower level problems, we can unlock real world math solutions.
That is an argument often made for funding mathematics research. But again it may not be the thing that attracted people to the field.
Note I don’t share all the doom and gloom of some folks in the math community but it is a valid point that people who might otherwise have gone into the field may choose not to anymore.
Also, math research has typically focused a large amount of energy abstracting and generalizing results. Your suggestion about solving real problems usually moves in the other direction. If the powers that be with all the money decide AI can solve all the real problems that exist (rightly or wrongly) then mathematics research community will have a problem.
> Note I don’t share all the doom and gloom of some folks in the math community but it is a valid point that people who might otherwise have gone into the field may choose not to anymore.
Isn’t that only natural? I probably would have stayed in physics and not gone into software if AI had been around 25 years ago. People gravitate to areas where their innate abilities are appreciated and can make a difference.
Let me help you by breaking down your point.
> The goal is to find solutions
> and to advance the field
Nobody seems to be disputing LLMs help with the former, but some are concerned that the latter is jeopardized by having aliens do our homework.
It’s too early to tell but math could be an AI only field with automatic research and advancement.
You still need humans to tell the AI what is important.
I think "fun" and "meaningless" are important parts of math. There is applied math for sole utility. Other math is just random search that may or may not be useful in long term. But later people/AI will find important use cases from these short-term "meaningless" researches.
> It's the bedrock of all other science.
no, its just a language. physics constrains 1+1. "one plus one equals two" is difficult to transform
It's both. Some math has applications. Some math doesn't. Many mathematicians are interested in mathematical beauty, in solving problems, and playing with puzzles.
None of the results from October 6, as far as I know, have direct real-world applications today. We can't go build a better thing, or make a faster thing, etc. I might be missing something (I'm not an expert in any of the fields). I don't think most folks would have a problem with solving a bottleneck problem that let us flux a capacitor at 2x efficiency, or whatever.
I also think this is hard for a lot of folks who are in fields that heavily apply math to come to terms with. I do not think that programming and practicing mathematics are the same: many (most?) mathematicians are closer to a composer than a programmer: all three are making things, but there is much more emphasis on elegance and cleverness in mathematics than in programming.
But again, it's not binary, and a lot of the discussion does try to take a black-or-white view of it. It's tradeoffs all the way down.
For what it's worth, I think this quote from the article sums up my position best:
"To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof. Choosing worthwhile problems may end up mattering more than producing proofs."
> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof.
Yup. This is what Terence Tao has been saying about digestion of results for several months now. Mathematics has become a field where results themselves matter less, the social aspect of attracting others to understand your results and explore new problems has become more important.
In a sense it is a little bit like software engineering, where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
> understands it thoroughly, and uses it to develop interesting further questions or directions
He phrases it like these criteria are requirements, but they really aren't for every problem. They make sense for those theoretical problems whose solutions' main values lie in their understandability and further development, but not every problem is like that. Some problems' solutions are just revolutionary on their own even if you contribute or understand absolutely nothing else. AI technology itself is an example of this!
To give a trivial hypothetical counterexample: if you came up with a solution for the traveling salesman problem that is both empirically sufficiently faster than the state of the art, that is an immense contribution and you absolutely deserve the credit for it regardless of whether you or anyone else understands how/why it works. It could literally even produce wrong results 50% of the time and still revolutionize humanity if it's efficient enough when it's correct. Denying them credit because they didn't understand the algorithm would be ludicrous - the results stand & speak for themselves.
Statisticians, traditional-AI researchers, etc. had to learn this the hard way when (basically black-box) neural nets became so good that their otherwise-elegant & explainable classical algorithms just stopped being able to compete in more and more applications, and got left behind. Are mathematicians looking to repeat that history?
I mean, of course, if you make the benefit of the result itself large enough than yes, by all means; I'm sure Terry would agree: if there was some algebraic geometry proof that could be used to build a life-saving drug, then I don't think folks would have a problem with using AI to prove it. But most math isn't like this. Most math has very little immediate effect on the real world.
There is a tradeoff that is implicit in this entire discussion: if I can go out and use a result to do a thing much much better/faster/easier than I could yesterday, like a real and useful thing in the real world, then the result itself has value. The cost incurred by swarm solving that individual problem is relatively low and isolated, and the value added is clear, so great, we should do it! But if proving a result only allows us to apply it to more mathematical questions, then there isn't really value added aside from our own value we place in the results, and that value comes from our understanding. So now, swarm solving random math problems is taking away our opportunity to solve these things ourselves.
And people who don't have math backgrounds will poo poo this whole "opportunity to solve problems ourselves" stance, but it is 100% the way mathers think about this stuff. When I was a math major in undergrad I actually had a friend who told me that he didn't want to give me hints on a difficult homework problem because he wanted me to have the opportunity to solve it myself. Like, his _present_ to me was to give me an opportunity to solve a hard but rewarding problem.
Breaking public key cryptosystems would be a better counterexample. Most NP-hard optimization problems can be solved in practice. While there are always some degenerate instances nobody knows how to handle, realistic instances tend to be easy enough to solve or at least approximate accurately.
> Breaking public key cryptosystems would be a better counterexample.
I deliberately avoided giving that example. Calling that a positive contribution to humanity and expecting credit for it feels a bit like calling the atomic bomb a positive contribution to humanity and expecting credit for it. Much of the world would rather stay away from that timeline.
> Most NP-hard optimization problems can be solved in practice.
Hence "a solution that is empirically sufficiently faster than the state of the art" and not merely "a solution".
Once you start solving problems in practice, you are probably no longer interested in the simple clean versions of the problem theoreticians think about. For example, instead of trying to find the shortest common superstring, you may be doing phased genome assembly. The problem is no longer unambiguously defined. The constraints and assumptions evolve over time, as the problem space becomes better understood and the nature of the data changes. While algorithmic ideas often transfer from one problem subtype to another, everything needs to be tailored for the specific problem.
This echoes SICP's admonition that "Programs must be written for people to read, and only incidentally for machines to execute."
Which is absurd on it's face for most day to day programmers, but a critical distinction for theoretical computer scientists.
Most day to day programmers prefer to keep the source code to their programs rather than the executable file, so I think it's not absurd at all, it just takes a brief moment of reflection.
It's not absurd if you think about the lifetime of software. It is created once and used immediately, but reread, adapted and tinkered with for decades after that if it is useful.
The number of reads is FAR greater than the number of writes for any given piece of code, and it needs to be comprehensible if it is to be extensible.
So in fact that instruction applies to all software, and I wish more people would take it seriously. It also applies in the AI era, because the AI has to understand the code too, and AI's spend more time reading code than writing it, indeed they base what they write more on what they read than humans do.
This can still be true, but it will be in a language divorced from registers and stacks.
The fact that this wouldn't have been considered a contribution before, or that this is a novel idea, suggests that there was something wrong with mathematics culture all along. I'm glad they're figuring it out now, but it seems "too little too late" after decades of devaluing all expository and exploratory work other than writing difficult proofs.
These things definitely were considered valuable contributions pre-AI, but were perhaps second fiddle to proving theorems. For example, part of the reason Erdős is so acclaimed is that he posed many interesting open questions which have kept mathematicians busy for many years.
I think Tao's point is that there's a necessary rebalancing and more weight should now be given to the non-proof stuff.
Understanding is what takes the time. Understanding what someone else wrote is harder than understanding something you wrote.
The time saved using AI comes from reducing understanding.
> Mathematics has become a field where results themselves matter less
Results are the only things that truly matter. Everything else is a position advocating for someone’s quality of life.
> writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
It’s still early days and the LLM can already ask the customer what they want directly.
I feel like the hyper-competitiveness of academia generally is the elephant in the room.
The disaster for a problem-solution-oriented mathematician isn't that once problem-solution is easy they'll just switch to theory-explanation and machinery-building. It is that they will have to compete (and won't be able to compete) with students and post-docs who've been doing theory-explanation and machinery-building for their entire career - and there are lots of these 'cause academic pipeline produces more people who can fill a job than jobs in every imaginable category and subcategory.
Academic competitiveness doesn't explain all the hysteria and why those at the top of the field (Field Medal and all) wrote that breakup letter over AI. There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence. Literally a math academic crying while being interviewed on AI impact, talking about not knowing what to do with himself.
Who knew 2-3 SD won't save you from being a pattern matcher that can't stand to see a better one.
it's not just math and CS. it's a human trait. we are seeing it in the arts and creative fields where technology and AI in particular are making their impact felt. there will be much, much more of this sort of reaction in the years ahead. we'll all have to learn how to adapt. and in the process rediscover that what makes us valuable as people is not our skills but our souls.
I agree that all people have value, but what does that mean in practice? Everyone needs a certain amount of resources just to survive. Should other people give you those resources because you have a soul?
A lot of people enjoy making art as a hobby; very few of them have combination of talent, discipline, and luck necessary to achieve commercial success. When I walk through a local art festival, most of what I see is so bad that the artists would have to pay me to take it. While taste in art is highly subjective, those artists don't seem to get much attention from other customers either. Going forward there will probably be even fewer people able to make a living in the arts. I'm not sure what that means for society?
Everyone seems to wrap up their criticisms on AI and labor saving technology in general as though it's a personal affront when in reality, as it's always been, it's an economic problem. Once we have UBI or something similar, you will see most people's opinions on AI doing math and art, for example, start to change to one of cooperation, not adversarial.
You can flip this backwards:
Everyone seems to wrap up their praise of AI by claiming that it will bring an era of abundance with UBI. Once we don't have UBI because why the fuck would we, you'll see most people's opinions of AI start to change from one of cooperation to adversarial.
The steam engine produced raw power but that still needed to be controlled and directed for a multitude of purposes, which required putting dirt farmers through education to turn them into factory workers. If machines now do both the thinking and the hard lifting, why exactly would their owners need us for? From where they're standing, they're now looking at a pragmatic solution to global warming: just reduce the "energy requirements" of the population.
>Should other people give you those resources because you have a soul?
For what it's worth, yes, I believe this, I think people should give each other the resources needed to survive because we have souls.
Not when others are busy producing useful things for society while you are busy doing nothing or doing shitty art while calling it modern.
What do you mean by "souls" here?
the human touch. In cooking you would call it cooking with love. You know the reason your grandmas cooking will taste better to you than any michelin star restaurant.
I am waiting for his answer.
Do male chicken have souls? Because female chicken have the skill to lay eggs, and that's what makes the latter valuable and the former bonemeal upon birth.
What benefit is a human soul to a feudal lord?
Won't be surprised if he believes in literal souls but I am waiting to see what he says.
What benefit is a feudal lord to a human soul?
None, but nobody asked them.
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Its equally devastating when you're a renowned expert developer at a specific stack that falls out of favor and nobody cares about your skills any longer. This happens relatively often in tech.
Eh, this analogy really falls flat. The “stack” in this case has existed for hundreds of years, takes at least twenty years to master, and the best people started working with it when they were perhaps seven. It’s a bit different than Rails falling out of fashion.
The issue in my view isn't that a single tool has become obsolete, it's that there's nothing to retrain to. All the tools have become obsolete and the AI is eating away at the economic value of core human activities. If AI was really good at coding in one language or framework or something I wouldn't give a fuck. It's the difference between a tool and a replacement.
This is something that sci-fi and AI safety has been talking about for a while.
It isn't just about worrying if AI is going to wipe us out, it's trying to figure out how human kind survives itself as things so rapidly change.
The early 1900s also had very rapid change after the introduction of ICE vehicles and airplanes. We also had massive wars that killed millions and brought forth society ending weapons. What is this round of rapid change going to bring?
Massive wars that will kill millions and new, society ending weapons.
The slaughterbot drones are already pretty bad. In Gaza, Israel used the sounds of crying babies to lure out innocent people and shoot them with drones. They used drones to shoot doctors through a hospital window. A lovely preview for the rest of us.
In the Russia/Ukraine conflict, Ukraine is holding on basically because of drones. They don't have enough guys to populate the front, so the gaps are plugged with drones. So that means the drone technology race is getting supercharged and we'll all get to live with the results of flying killer robots.
In Émile Torres' book, he references how the superweapons of the 21st century were things like nuclear weapons, biological weapons, chemical weapons. In the 21st century, those risks haven't gone away, but to them we add the "GNR Bundle" (genetic technology, nanotechnology, and robotics technology) which are characterized by their autonomous self-replicating potential (with AI falling under robotics). Of course, in his book, he is also being somewhat critical and descriptive of the "futurological shift" that causes certain people to become obsessed with cataloguing every form of x-risk to the neglect of real people's problems in the present.
> Israel used the sounds of crying babies to lure out innocent people and shoot them with drones.
This sounds horrible. Do you have an article where I can read about it?
https://skeptics.stackexchange.com/questions/56904/do-israel...
> There seems to be this distinctly toxic trait common among math and CS folks where you don't see work as work, like the median worker, but as intrinsically linked to your self-worth and reason for existence.
I don't see this as a toxic trait at all. Some people have the fortune and luxury to do what they love - I don't think that's a bad thing.
I think this quote from Gauss puts it very well: "It is not knowledge, but the act of learning, not possession, but the act of getting there, which grants the greatest enjoyment." So tons of mathematicians feel like AI is stealing this "act of learning which grants the greatest enjoyment", and I think it's fine that they are lamenting this.
This is inherently my whole problem with the "AI utopianists", who somehow think we'll all be happy once we're sitting around with no disease just eating bonbons all day or something.
Did Gauss say that learning was only the greatest enjoyment when no one else was known to already know the thing you are learning?
This is inherently my whole problem with the "AI utopianists", who somehow think we'll all be happy once we're sitting around with no disease just eating bonbons all day or something.
I don't know about Gauss, but when I am given a more powerful tool, I use it to tackle harder problems, the kind that were out of reach before.
Call me when you run out of those sorts of problems. I'll propose some more.
Why would he call you? AI will propose those new problems instead, and tackle them without any need for your (or any human’s) intervention
Awesome. Next stop Kardashev Type III, I guess!
Happiness is a pointless goal anyway. It's not a metric that can be used to measure the success or failure of any particular policy.
> Happiness is a pointless goal anyway.
Spoken like a true robot. For most of us, happiness (and I mean that broadly defined, e.g. "life satisfaction") is the only goal. I certainly thing happiness is a much more critical goal than "increasing shareholder value".
I know some real assholes who seem quite satisfied with their lives. A shot of heroin will make you happy … for a while. It's a meaningless goal. Regardless of your snarky comment about "shareholder value", we should all aim to make a positive contribution to society and achieve something of lasting value (not necessarily monetary). Happiness might come with that as a side effect but only a fool treats happiness as a goal.
It may be a bit unhealthy to only identify yourself only with your vocation - there's also family, friends and hobbies. But it's miserable to engage in "work" for forty hours a week when it's something you have no interest in whatsoever. And it is very unhealthy to think most people should do that. If "work is just work", how terrible is that?
If the sum-total of the activities you engage isn't "you", what is? The things you buy? "Your" sports team?
Plus a hypercompetitive academic world is a pretty good explanation for people identifying only with their vocation - how else to do you get to the top except putting your whole self into the challenge?
Given the pay of a typical career in math it would seem logical that the typical mathematician's self worth is associated with the work.
Seeing work as your purpose was supposed to be the holy grail of work that most people sadly couldn't achieve. Now we just immediately toss that away like a mouldy tomato and what's left? The new highest plane of human existence is working for capitalists to make them more profitable, in exchange for 10% of their profit?
I suggest reading some philosophy. It seems that we have spent a great deal of energy training people for a vocation, but in the process forgot to educate them.
I don't really understand this kind of triumphalism. Would you, personally, identify as a pattern matcher who's happy to see better ones? Or do you think you have some deeper skill more robust against disruption than the math and CS folks who don't know what to do with themselves?
Note that this may include things that you don't think of as being a "skill". I've seen some people brag about being a particularly hard worker, or about a deep instinct towards Getting Stuff Done, with the clear implication that these things can't be automated the way mere mathematical analysis can.
I don't read it as triumphalism. I read it as an accurate (to my experience, as a math PhD) reflection on the existentially toxic culture of math and other intellectual disciplines, one that AI is now forcing us to reckon with.
n.b. I think math culture is probably one of the least toxic cultures that have ever existed. Its equanimity and principles are a big part of what drew me to math in the first place. But humans are human.
I really don't see how to read "hysteria", "breakup letter", "literally a math academic crying" as anything other than an attempt to dunk on mathematicians.
I suppose, maybe they are dunking on mathematicians. I can't claim to speak for them. But I at least see truth in their observations. It would be inaccurate to downplay the sentimentality of the math community right now. I don't think that sentimentality has overly clouded the community's judgment (so "hysteria" is probably overdoing it) but there is a lot of heartache and unease.
>..at least I see truth
>.. humans are humansSince you've imputed intentions or inclinations-- once you see that "pattern matching", which, like a bastard "pigeonhole principle", has been seeing overuse in excusing bigotry -- you cant stop me from claiming to speak for _everyone _(TT)
Elsewhere here, people are imploring us to read philosophy, as if the LLMs haven't already mastered the one weird trick of enlightened stochastic parrots..
I'd recommend self-compassion/self-kindness for mathematicians (because contempt and pity [for mathematician, eg] are too cheap for far "right" and "left", respectively)
After you've felt those pangs of self-forgiveness you might be motivated to contribute to _distributed_ _crowdfunded_ LLMs that crater the value of datacenters instead...channel that brutality at uebermenschlichkeiten please!!
Math is toxic yet the least toxic?
I will stick to my claim that math exists in the context of the hyper-competitive overall academic culture.
I should add that extreme competition combined with math being something people love, something with beauty unto itself, produces an especially weird world compared compared to a situation in, IDK, business school, where everything is about the money.
I mean, asking "who deserves to live the dream" really twists people's heads.
It's inaccurate because nothing about math or CS is "existentially" toxic. Or "distinctly" as the other commenter.
If you accept that as a premise then obviously you agree with their truth.
If you don't accept toxicity as somehow intrinsic to the subject but rather the structural conditions, say, of post-neoliberalized academia at large, and certain faculties being only a bit more susceptible (if even, or just different, say compare humanities vs STEM reactions in the future), then the whole comment reads as a conflation.
Instead the simpler answer is that academia at large values freedom of inquiry and other Enlightenment notions which are privileged compared to outside of academia. But that is not toxicity, it's just intellectual integrity and the academic autonomy in pursuit of one's interests avocationally. The toxicity comes from the intellectual profession having to function as a modern clerisy for socioeconomic reasons under post-neoliberal capitalism.
People may conflate the two attributions and that is a deep mistake, in my opinion. In fact the conflation due to internalized oppression easily creates and explains this kind of resentful, dismissive tone response that is a distortion of the true situation.
I only lay claim to criticize the mathematical culture I have myself experienced, Western and post-neoliberalized as it may be. I'm not sure it's even possible to discuss culture "intrinsic" to math, or any other subject for that matter.
> The toxicity comes from the intellectual profession having to function as a modern clerisy for socioeconomic reasons under post-neoliberal capitalism.
Sure. And I think (Western, post-neoliberalized) math has resisted that corruption possibly better than any of its sibling cultures. But what makes the toxicity "distinct" is the emphasis on enlightenment, as if being a top-class mathematician brings you closer to the Godhead than does, say, being a top-class chess player, or a top-class defense attorney. Mathematical theories are treated less like robust human creations percolating over decades and more like divine revelation channeled through chosen disciples in fits of inspiration (c.f. Cantor, Galois, Godel, Grothendieck). A lot of this is projected from the outside the culture into it, but it is not outright rejected by the culture. Instead we still openly endorse what amounts to Great Man theories of mathematical history, attributing to almost every noteworthy mathematician a moniker of "genius" and praising them as paragons of thought, not as the unhealthy subjects of cults of personality.
And the material conditions for this "great man" narrative is again the particular form of intellectual clerisy.
In other words, the move to experience and cultural relativist statements about "math culture" is exactly the cynical ideological move that an academic cleric does. It deflects from examination of where these narratives come from and attribute it to e.g. "math history" or "math culture" or just "something specific to math itself" etc, etc.
Neoliberal intellectuals love to attribute toxicity to the Enlightenment. They can't help it.
>simpler answer
Simpler conflation: need and ability
Eg
1. "from each according to his ability (the tyranny of academy)" with "from each according to his need (the freedom afforded to clerisy)"
2. "to each according to his ability to get rich" (meritocracy) with "to each according to his need to get rich" (post-liberal capitalism)
One of the best comments I read this month and in October
> Mathematics has become a field where results themselves matter less
"Results" in mathematics has always been the proof. (Preferably a beautiful and succinct one.)
Math is supposed to expose the workings of the epistemological foundations of the universe. The result itself (number or formula or whatever) is absolutely irrelevant to math.
(It may be important to physics, chemistry or engineering, but it isn't to math.)
fewer mathematicians means fewer problems to solve. All problems have human origin, right? Can AI create a problem on its own and what would be the point? Closer to can god create a stone he can't lift.
Of course AI can create problems.
but philosophically could it create a problem it couldn't solve? And what would be the value in writing that out?
Posing problems is often easier thwn solving them, so yes? Fermats last theorem is posed in two lines. Took 300 years and massive advances in geometry to solve.
The quote reads to me like “as long as you can analyze the solution after the machine solves it and make it suggest new ideas based on that, you’re almost the same as a discoverer yourself.” I’m deliberately avoiding “mathematician” here because I think Tao probably didn’t intend to take it that far. But, for many people that difference will be indistinguishable, especially because there will very likely be “math bros” just like “ai bros” and “crypto bros” before them who want to pretend they’re mathematicians just as we see people pretending to be engineers and financial experts based on how fast they got on the hype train.
I’m no mathematician, though I do have an undergrad degree in math, and unless I had a graduate degree or made substantial contributions at that level, I would never consider myself anything but a “student of mathematics,” and that too not just because I have a degree in the subject, but because I still have a great interest in certain branches of math and I keep learning in my spare time from books or from adjacent applied fields I continue to formally study. As a kid, I grew up hating math but I became interested in it later in life because of coding, computer science and formal logic, pushing myself a lot through my college years to get some foundational understanding of the subject at a deeper level than the general population, just to get my foot in the door, so to speak.
Even if I’m an outsider from the circle of mathematicians, this whole AI takeover of mathematics leaves such a bad taste in my mouth because I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general, and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject). It feels like a great loss for humanity to lose a brunt of our engagement with it to the machines, like a whole wing of a gigantic library starting to be closed off from us indefinitely because some people decided that we don’t _need_ to access it anymore, except on an even grander scale possibly than my imaginary library metaphor.
> I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general
Funny, I think it's a very widespread opinion to think of math as useless. In the US at least it's a major failure of the education system and STEM communicators.
> and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject).
There's plenty of controversy in mathematics. Controversy comes from the people, not the field of study.
How useless? Terence Tao made MRIs faster. That's a real thing that STEM and non-STEM people will maybe have to interact with at some point in their lives. Sure, we can blame science communicators. Should we put a bunch of banner ads on everything so that people know that math and science is involved in every thing?
> Funny, I think it's a very widespread opinion to think of math as useless. In the US at least it's a major failure of the education system and STEM communicators.
That is a US centric view, and it's curious how things came to be that way. European and Asian countries take mathematics more seriously.
How is that working out for them? I am by no means an anti-intellectual and I do value knowledge for it's own sake. But Europeans in particular seem to be unable to translate their mathematics educations into any sort of practical economic growth.
Poland is growing a lot. The Soviet block countries were big on math.
"What worth is education if the workers it produces are not making the rich richer."
Education is about cultivating true richness of culture and mind. The economy is in service of a higher purpose - of life, liberty, and the pursuit of happiness. Not the other way around.
Can you stop with the flamebait posts? Especially the country specific ones, we're an international community.
https://news.ycombinator.com/item?id=50309981
https://news.ycombinator.com/item?id=50295724
https://news.ycombinator.com/item?id=50295076
Nope, those have nothing to do with any specific country, but rather a specific mindset that is ruining the world.
I know there is some controversy, I would say that the field of study also “causes” controversy in a sense if the disagreement has to do with the subject itself, that is true for any field. For example, set theory “caused” significant controversy in the early years of its development, yes it was people in disagreement and insulting one another, but the fundamental questions were from the subject matter itself.
Regarding widespread perception of math as useless, I guess I was thinking mainly of academic circles rather than the general public in that statement. Even while I hated math, I knew it wasn’t useless because my late father (who was a civil engineer) was very good at it, and he was adamant on teaching it to us. It didn’t work on me back then though lol.
As I was getting more interested in math while getting my degree in computer science, I realized much of my own issue with math earlier in life was because of the same thing just about everyone else that struggles with it has: the lack of any practical relevance in my own life because of how it’s generally taught. And the fact that many math teachers themselves don’t know the foundational/philosophical aspects of the subject, which would go a long way I think towards explaining that relevance if introduced in some pieces earlier on, rather than treating it like a rote memorization subject and being so focused on continual practice to develop heuristics with exercises that some students couldn’t care less about. Of course, that works for some people clearly, but it didn’t for me at the time and I’m of the opinion high schools should focus more on discrete math like set theory, and formal logic rather than calculus, which is of little relevance at that stage unless you’re also invested in studying physics. Sets and logic on the other hand imo are general enough right at the beginning to be relevant to just about everything people of just about any age are concerned with.
Every theorem is already in the mind of God. No theorem prover has discovered it anymore than an astronomer discovers a star. We simply have better access to God's thoughts now. It makes us realize we are human and not divine.
What a complete nonsense. Math is no more 'the mind of god' than polio is. If anything math is a triumph of mind over religion because it gave us the tools to power a vast chunk of the scientific revolution. Without that we'd be cowering in the dark and hoping that the sun would rise tomorrow.
You completely misunderstood the comment by inserting your own alternat version of God
Assuming you are right, can you explain?
You just made me think of something but I'm almost afraid of saying out loud because I don't want to speak it into existence...
AI-generated math textbooks
(but they probably already exist)
We need to rethink the idea of textbooks as static documents. LLMs make it possible to build something like "A Young Lady's Illustrated Primer" for real. Let's go.
https://www.penguinrandomhouse.com/books/172835/the-diamond-...
Sure, but given the approximate tens of millions of dollars in API costs of solving these major math problems currently, we will also have to acknowledge that getting credit for major mathematical contributions is likely now to be far less about meritocracy and far more about access to capital to kickstart the original proof.
Less capable (but good enough) mathematicians with more capital will have a huge advantage over great mathematicians who are broke.
I'm not a mathematician, but that situation rubs everyone (except those with the most capital) the wrong way.
Tens of millions? You're thinking specifically of the proof of Navier-Stokes. According to https://github.com/openai/math/blob/main/README.md:
"On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above."
I will note that this includes the Quasi Riemann Hypothesis. The model that solved NS was significantly dumber and less efficient that OpenAIs more recent math-models and you shouldn't over index on "it's expensive now". AI has advanced so quickly that your <2 month old opinion is out of date by 5+ orders of magnitude!
Question: Have you actually used AI to do research-level math? At the current time, it is not "tackle any problem you wish and you can get headline results in ~3 hours." Their method is spraying-and-praying across the entire mathematical frontier, which is something only AI demonstrators are interested in. And as far as I know, they have never elaborated on what their "average of 3 hours" metric even means (e.g. including or excluding all unsuccessful leads?).
I would say no. I am just finishing my MS in applied computational math at Johns Hopkins, but am not specialized to a PhD level.
You shouldn't be so cofindent then, specially since your opinion is based on OpenAI's PR.
I suspect the AI frontier labs will also soon turn their tools loose on the entire corpus of historical math papers to check for errors. They'll likely find that some major areas of math are built on a very shaky foundation.
>Have you actually used AI to do research-level math
Being that OpenAI has done most of this on models that are not released, I don't think you have either...
The other thing you're not looking at is mathematics is becoming a means of actual training a model versus just something to try out.
That sounds close to what human researchers do as well. Big part of getting grants/doing graduate work is finding a tractable problem. It requires constant refinement of what the scope and direction of focus.
That's not even remotely how human research works.
I guess this is the part where you claim to have ever done graduate or higher level research before, that wasn’t just a PI just giving you pre defined, contained side project for a summer.
This is true in experimental sciences, where people who have grants and funding can obtain results using equipment unavailable to others. But their raw data can help much less funded theory groups to formulate new understanding that can lead to new experiments.
As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
> As long as the raw proofs, potentially ai generated, are available a lot of less funded mathematicians can still use them to generate understanding and created new programs to be resolved with new ai aided mathematical experiments.
My assumption (based on perceived human nature) is that the raw proofs will not be made available.
If all of the important credit goes to the person who attached understanding to them, I'd assume those with the money to take the first step will keep the AI proofs secret until they believe they have adequately done the follow-up that will give them the lion's share of any credit.
You're assuming the people with the money care about credit.
I never had a chance against Terence Tao. Neither did you. There are a few dozen people on the planet who had a giant advantage in math and could do things no one else could do.
It was not some great awesome fair thing that math was dominated by some outrageous outliers on the iq spectrum. It's not some great awesome fair thing now either.
> I never had a chance against Terence Tao. Neither did you.
I'm not a mathematician at all, so yeah I certainly didn't have a chance.
> It was not some great awesome fair thing that math was dominated by some outrageous outliers on the iq spectrum. It's not some great awesome fair thing now either.
I don't see these as equivalent circumstances. For most people if they get beat by someone smarter or someone stronger, it may be disappointing but they can still respect and admire that they were beat by talent/determination/etc.
Getting beat due to access to money just feels slimy, because at some level we all realize money is not distributed on a meritocratic curve at all, at least not if you are measuring values any well-adjusted person would consider positive.
You keep saying "meritocracy" and the like, as though it's just by nature a good thing. It is not. High IQ isn't distributed evenly or fairly or anything of that nature, and yet "meritocracy" seems fine with the idea that "some people are just smarter than others, deal with it".
Rawls had a concept of the "veil of ignorance" which is probably better for thinking about fairness rather than "meritocracy". No one should lose sleep over the fact that the tiny number of corporates with enough capital can now beat out the tiny number of 180 iqs at a task (mostly generating proofs) that is not really valuable.
> Rawls had a concept of the "veil of ignorance" which is probably better for thinking about fairness rather than "meritocracy". No one should lose sleep over the fact that the tiny number of corporates with enough capital can now beat out the tiny number of 180 iqs at a task (mostly generating proofs) that is not really valuable.
This is a weird framing. To begin with you didn't need a 180iq to work with math. What AI is doing though is making it so that an even smaller part of the population can work with intellectual, fun jobs and even more wealth gets concentrated to a small elite. That is absolutely something we should lose sleep over.
This has unfortunately been the case in experimental academia forever.
For now. It once took Deep Blue hardware to beat Kasparov. Now most laptops can do it.
It's still confusing to call that a "change". Apart from the way the media covers the topic, I've never known mathematics to be about proofs.
[dead]
I'm so glad that sanity ulitmately won in his mind.
If you think the above comment reflects a change of heart from Terrence Tao, you never actually listened to him, only the strawmans of his opinion online.
yeah, Tao has been extremely reasonable throughout this, while the coverage has been... maybe 'motivated' is the right word?
For sure, he was arguably the first well known person to bring chatGPT to maths.
When his more "anti-ai side" (again, it's more nuanced but I'm calling it that) was showing, everyone was so quick to call him a Luddite. No one stopped to think why a well know ai in maths fan all of a sudden had a more negative opinion. Even reading his actual nuanced take was too much.
The best example was the following 2023 quote which has been repeatedly and wrongfully used as a criticism of Tao right up until he was proven exactly correct
>I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well. Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools? We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions. Our usual technology paradigms will not serve as an adequate guide for navigating these uncharted waters. Perhaps the greatest challenge will be transitioning to a new AI-assisted world as safely, wisely, and equitably as possible. https://unlocked.microsoft.com/ai-anthology/terence-tao/
The ever prescient Tao, over 3 years ago in 2023. You can search that quote and find lots of examples of "Lol he's so wrong" right up until very recently. Which is exactly the 2026 timeline Tao stated.
And his viewpoint really hasn't changed or needed to change. He was 100% correct.
To be fair, he’s had a bunch of guest posts on his blog that didn’t seem as reasonable to me (and some people seemed to not realize they were guest posts).
Edit: lol, this article is itself a guest post. My bad for not reading it first! But it does serve as an excellent demonstration of my point, since it seems like neither of us realized we weren’t talking about his words.
"This is a guest post by Frank Calegari."
He's not out of the woods yet.
>> where writing code can sometimes be delegated to AI, and is arguably less important than understanding what the customer really wants from this software.
This statement is a monstrosity, and the reason why real software engineers know vibe coding will fail. It is fundamental misunderstanding of thinking that that the intellectual work consists of knowing what you want, and coding is merely the mechanical labor of making it happen.
The coding is the intellectual work!
Its where the actual problems are discovered, where the difficult decisions are made, where vague ideas become precise, and where solutions are invented. There is not such thing as a fully formed solution waiting to be translated into code. :-)
Again just the modern variation of the idea guy who goes around saying I have a genius idea for a new startup...I just need some Developers to implement it ....
> The coding is the intellectual work!
Yes, and the machines can do intellectual work now.
It's funny to see some commenters here cheer as the leopard eats the mathematicians' faces, when we can all see that the software developers are next in line.
Isn't it the passage of mathematics from the age of craftsmanship to the age of industrial making ?
My recent take on AI is that people haven't adapted to the new huge potential of the tool, and don't dream big enough (yet).
It's like seeing a car engine for the first time, lamenting about the future of horses, and not realizing it'll also help us fly.
>and don't dream big enough
People have been dreaming running PS5 games on PC for a long time. Just when AI starting to become capable of doing that, Anthropic shut it down.
Yeah you can dream, but only what we allow.
1. That’s a legal grey area that could result in significant legal exposure for Anthropic.
2. Even if someone wants to do something that’s in a legal grey area, or more (hacking others, another item limited by Anthropic), is the idea that such people won’t have the ability to use AI to do those things? I don’t understand the criticism that just because one company might institute some guard rails that those who want to conduct legally grey or black activities will be unable to use other AIs to do so?
To a first approximation I agree that it’s similar to industrialization. However there’s a key factor which distinguishes this case, and which makes the analogy less than perfect: there isn’t a profit motive in mathematics, so there was no push for this development from inside the field. Instead, the industrialization was entirely exogenous, from companies which do in fact have that profit motive and view math as a means to achieve it.
There might be a “proof motive” in math which one could argue will drive the full automation of theorem proving. But we must recognize that profit is more universally valuable (i.e. it has value outside the industry in which it was created), whereas proofs are only valuable inside the math community (at least the pure math proofs the labs have been publishing thus far).
It's more like opera. Math has always been opera.
Do rich people like opera? Yes. Is that a profit motive? No... but it's a shibboleth for wanting to be rich. Do you get it?
It's ridiculous to say, no profit motive in mathematics. It's like saying, no profit motive in LISP. But cmon, you think the Collison brother who took Math 55 and liked LISP: I know that in a pedantic sense what you mean. But in terms of academic departments where people say they are no interested in money and it turns out they want to become billionaires... Math Department ranks pretty highly. Always has.
I agree that being technically gifted has been a great way to make boatloads of cash in the past few decades, and that this correlates with being good at math. But I don’t see why this implies that enjoying math means someone is more money-minded than others. If anything, the students in pure math departments are intentionally forgoing the potential cash conferred by their technical abilities in favor of the academic pursuit. In anticipation of a reply observing that math PhDs regularly go into quant, I counter that quant programs reach out to interview students in many other fields, and more to the point, spending half a decade studying algebraic number theory is surely not even close to an efficient way to build wealth. In response to your second point: even if mathematicians desired wealth more than other professionals, how does this demonstrate the existence of some kind of profit motive in math?
This is really well put, I totally agree with you.
There’s an immediate resistance towards change rather than embracing it and finding opportunities to improve and work with it.
How do you expect mathematicians to "embrace" OpenAI having unreleased models that are much better at solving big-ticket problems than they are?
Maybe some math departments could pay OpenAI for access to their models, but I doubt OpenAI would want to deal with most of them.
> My recent take on AI is that people haven't adapted to the new huge potential of the tool, and don't dream big enough (yet).
I guess you can dream big enough only if you have access to the really interesting models. The 20$ subscription won't get you there, and the 100$ one is already out of reach for so many people.
That is a great example, car engines (and derivatives) did create tremendous economical value but are in the process of making our planet unhabitable.
Maybe we should listen to people that care about negative externalities more often, and actually mitigate them.
Unfortunately, I cannot think of a single politician right now who is a thought leader on regulating technology, which is exactly the kind of person we should be looking for. People who understand that we don't need every website to give us an annoying popup saying that they are using tracking cookies but we do need the option to opt out of AI features for basic services.
> every website?
What? Only websites that want to track you before opting in have cookie banners. HN doesn't have a cookie banner, in case you haven't noticed. I'd say your issue isn't with consumer protection, but with pushy marketing people (or worse). But they're good at marketing, hence the cookie banner nonsense complaints, which redirect the anger towards laws requiring that creeps announce their presence, rather than the creeps themselves.
Two possible walls for advancing math with AI: first, it could be that the kind of problems that AI solves are the ones that don't open new avenues to concepts and ideas, and second, problems complexity is usually not linear, and the new and interesting problems to solve could be at a level of complexity beyond what AI can currently solve. Perhaps mathematicians could work to find ways to help AI cope with those walls.
How you get somewhere is important and tools change how you get there.
You can drive 1000 miles and arrive at the destination on the highway quickly.
Or you can take the slow road and walk. You discover unrelated things, make new relationships, see the scenery, listen and smell what is around you and have time to contemplate things which may or not be related.
This particular tool economy robs us of all these things in the rush to the destination.
I'd sacrifice "progress" a little for some of the better bits of humanity.
Do you feel robbed of your scenic expeditions because vehicles and highways exist?
Not OP, but actually I do because I have been forced to live in places where it’s dangerous to walk and where all the scenic landscape has been replaced with asphalt.
I get your point in general, but I think vehicles are a poor analogy because in many ways they have changed the places we live for the worse (and yes I’ll concede occasionally for the better). It turns out we need places to park all of these vehicles and they compete for space with humans.
Not really. I feel sad for the people who miss them.
That’s very narrow-minded. I love hiking and biking and exploring new places, meeting new people. I also appreciate that I can use the highway to get to the destination conveniently. Often so that I can enjoy the very experience I’d miss according to you.
The thing is that if you don't learn to drive some destinations are unachievable. And destination is more important than the journey unless you travel for pleasure or personal development.
There are many more places you can't drive. I've walked to quite a few of them.
That analogy holds too.
To keep with the analogy, there are far more places you can feasibly drive to than walk to.
(And even more if you can combine them.)
Your best argument is the 2nd sentence.
The first sentence is actually a much better jumping off point for the other side of the argument than yours, because the mistakes involved in the statement should set off alarm bells for people who've walked with mathematics.
What do you mean by "place"? What do you mean by "more"?
Are there more places as marked by real numbers between 0 and 1, or places as marked by integers? Could a machine that can count integers 1000000 times faster than people "visit" all those real places between 0 and 1? What would it mean to visit all the places between 0 and 1? Once the machine visited all of them how would it help people explore that territory?
What do the general ratios and rules between paths and areas mean about how much path it takes to cover an area?
The real world is quantized in way that math isn't, of course, but that's one way the analogy can lead you astray.
And even just keeping to the real world rather than math, it's not all about the sheer velocity with which you can cover an arbitrary distance metric. The primary bottleneck is less that velocity than the amount of time you have to direct to place scaled to your capacities. The car's velocity can let you trade some of that time for fast travel between farther places, and with a road network can arguably even increase a certain variety of places, but that's a very different statement than more places. And it's also a wall against attention to everything in between.
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It's a pretty straightforward geometric argument, from your own analogy
I'm sure there's a coherent rebuttal but "That is plainly rubbish" isn't the reply I'd expect in good-faith discussion
It's well in the realm of obvious, but possibly outside the realm of personal experience on your part.
I think pointing at Greenland for a start and asking you to think about the problem for a bit will lead to the same coherent view on the matter.
And how will you be getting to the island?
Is this about driving to an island? If you need a plane or boat to the island and still call that trip "walking", then someone can take a plane or boat to the island and rent a car and call it "driving" or "driving plus walking".
Or if you're just saying that a very good backpacker that happens to be in greenland can get more places by walking because of a lack of roads, that might be true. But for a normal person in a typical household, their walking range is not that big. If you're in the US or Europe the roads reach almost everywhere.
No one forces you to drive. You can still walk.
But it now takes 3 times as long because you constantly have to cross roads.
They build cities around the car, they defund public transit, they close down rail lines, they build new housing a 5 km away from your closest grocery store, and 20 km away from any cafés and other cultural venues. Your job is located 50 km north of where you live, your partner’s job is 60 km in the other direction, and your child’s school is 10 km in the third direction.
They neglect to build sidewalks, and force you to walk 1-2 meters from cars racing at 70 km/h. With all the noise and high risk stressing you up to the level you just wanna scream.
stevenhuang: No one forces you to drive. You can still walk.
So drive then. You can also move. The point is there are options. Many cities have strong infrastructure and have embraced walk ability. If not driving is such a big deal to you, why not do something about it?
You're arguing like an edge lord here. What he is saying is that you're effectively forced to drive. Not so difficult to understand. There are plenty of people fighting for change but because our political and economic landscape is captured by big business, it's not so easy.
This is extremely location dependant.
Not everybody in the world is effectively forced to drive. Sure, maybe if someone has the misfortune to be an US city they have no other option but there are other places that are not like that and cities with good public transport and relatively low car usage.
It's pretty well documented they deliberately made things impractically large distances apart and scrapped the idea of footpaths, to encourage car sales, paid for by car companies to your local city government several decades ago.
But Terry T said I have to drive or die.
Or has he changed his mind again.
Could you please stop posting unsubstantive comments and flamebait? You've unfortunately been doing it repeatedly. It's not what this site is for, and destroys what it is for.
If you wouldn't mind reviewing https://news.ycombinator.com/newsguidelines.html and taking the intended spirit of the site more to heart, we'd be grateful.
Yes, this. Unless there is a true singularity where no more questions are ever asked, there will be definition be infinitely many problems to solve, and most of them we have not even glimpsed yet.
Singularity? Probably not.
Tech bro dictatorship? Unfortunately a bit more likely.
None of the above? Then we have a small shot at a great future.
Having a machine that solves hard problems must be a good thing.
Solving the hardest problems doesn't always result in the best outcomes unless the answers are shared equitably and implemented compassionately.
Absolutely. This is an aspect of the tech bro dictatorship.
Why do nearly all the articles exploring the development of AI applied to math assume that capabilities won’t extend beyond theorem proving?
> other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world.
These all seem totally achievable by LLM agents, or at least, I don’t see an obvious reason why it wouldn’t be. 3blue1brown’s recent discussion on this subject was a bit better, but he spent the plurality of main body of the video assuming this scenario, devoting less time to the other possibilities. To me, it seems dangerous to build a math career on the foundation that AI won’t be able to explain math well in a few years, let alone a few months (which seems entirely plausible given the recent math progress).
Because the other option is that AI does what you say. And if you think about it (and you are a mathematitian and not an engineer) it means that AI is able to read our minds and know our desires. And at that point who cares about existence anymore?
Sorry, why would it need to read our minds exactly?
"Choosing which terrain is worth exploring" is more clear if you have an objective function (if you are doing engineering). But mathematics is closer to art in that regard. What is a terrain worth exploring in art?
Sometimes I wonder if "art" is also just some form of pattern matching on a very meta level.
In some sense, any art can’t be random in order to be distinguishable from noise
>What is a terrain worth exploring in art?
The joys of parenthood. The excitement of new love. The tragedy of aging. The rise and fall of a great family. Jealousy, vengeance, embarrassment, righteousness, venality. You obviously couldn’t create an all encompassing map of the terrain, but human art has a lot of recurring themes that are clearly “worth exploring” because they’ve consistently been explored in the past. I think the same kind of repetition probably also exists to some degree in other areas of human endeavor. An artificial intelligence could probably suss out a lot of those themes and produce valuable (if not maximally valuable) work without being a mind reader.
Im not saying it cant do those things. The mathematical equivalent of that are the hundred proofs that OpenAI dumped. And probably we still have a few decades of AI exploiting low hanging fruit in Maths and Art.
I was refering to going beyond and deciding what to explore. There are infinite paths in mathematics and a lot of ways you can describe and denote a particular theory. And the choice of notation and exposition is an art in some way.
You could probably say that "no, but this is done by minimizing memory usage in human brains so that its understandable by humans, maximizing compressability of previous ideas, and a link with some utility in the real world, etc"
But that just one example of loss function. There are many possible ones.
Do i think that there is a universal loss function that captures it all? Probably yes, in the limit, on the infinite energy regime. Are we in that case in practice? Probably not.
I tend to think that AI will fully replace us the same moment we are able to master prime factorization with our time and energy constraints.
Ive noticed that too, I don’t know why people talk like we need to figure this out right now
Like we can even know what the future of math is this early
It’s not like you can just declare that up front, it’s something that happens naturally when people spend time using the tools
Right now they have to figure out some kind of story to tell the grad students. Even if it will be subject to change they have to have something.
Just because they could, doesn't mean anyone would bother having the AI agents do so.
You don't think OpenAI has any interest in having agents identify new interesting research terrain? They just want to solve existing important millennium problems and then they're done?
Yes? I expect their motivation for devoting resources to solve these problems will dramatically drop as soon as they IPO. Maths is just a convenient vanity metric for OpenAI to demonstrate progress, but long term progress in mathematics has no commercial value to them.
Of course, general improvements may be sufficient to continue to drive math improvements even without specific attention - we’ll see.
Not a chance. They'll go for room temperature superconductors and other stuff that is currently in the quite possible but not yet proven and patent the hell out of it and charge rent to the rest of humanity.
Why would that necessarily be accessible by thinking only? Or do you see them running or directing large laboratories of various types?
AI systems will probably be able to, if not direct labs themselves, cut down a lot of the search space to only the most promising candidates. That’s basically what Isomorphic Labs is doing for candidate drug research. It doesn’t seem crazy that the same could be true for material sciences.
Maybe. Are we seeing any indication of increased success rates in stage 3 trials already for "AI drug candidates"?
Could also be that some things are not possible in some ways. I wonder if there is some general theory on "shortcuts".
Stage 1, yes, stage 2, it's a wash (for now at least), which does not promise much for stage 3, but it is still a bit early for that.
Simulation to the point that the accuracy is large enough to extrapolate to real materials. You can do a lot of stuff when you have access to a million GPUs that would otherwise be much harder.
I have my doubts there, but we will see.
A lot of materials science is already done through simulation, same with a large chunk of chemistry and biology. That fraction will only increase over time.
https://en.wikipedia.org/wiki/Computational_chemistry
https://en.wikipedia.org/wiki/Computational_biology
https://en.wikipedia.org/wiki/Computational_materials_scienc...
These fields are all going through a tremendous acceleration right now.
I am aware of advances, but I think a large chunk is a stretch in various ways. We'll just see what happens - simulating even a glass of water fully is tough.
That depends on the granularity of your simulation. For most - not all - purposes a glass of water can be simulated coarsely or by simulating a far smaller amount with greater fidelity and then to extrapolate.
For useful materials science the numbers are still large but far lower than the numbers involved in 'a glass of water'.
I really depends on what is needed. If you know that something quite "coarse" is enough to create/uncover room temperature superconductors with high critical current - fair enough. Again, we'll see what happens.
With the kind of compute they have at their disposal 'quite coarse' may actually be 'impressively detailed' and that is why I think they have an edge on this kind of stuff. It's like being able to borrow a trillion and then to go play research lab, it would be highly surprising if they weren't able to monetize such a head start.
If they did that and were able to find someone to pay them, that'd be a pretty good contribution, no?
> I expect their motivation for devoting resources to solve these problems will dramatically drop as soon as they IPO. Maths is just a convenient vanity metric for OpenAI to demonstrate progress, but long term progress in mathematics has no commercial value to them.
And why would IPO remove incentive that's present pre-IPO? Somehow math progress will influence price at IPO but won't influence stock price post-IPO? Or will they just stop caring about stock price once they are public. None of these options makes any sense.
No, post-IPO they will have to (at some point) demonstrate real commercial value. This just gets them to a stage to cash out insiders and (less importantly) get some additional capital to extend their runway.
That's the playbook, right - keep interest/investment flowing until you can figure out the actual commercial value/model. Until then, do whatever you need to keep the hype as high as possible and maximise your runway.
Because if you assume there is going to be an AI-god that can do everything better than us, there's nothing to really do. Like really, what do you want these people to write? "Well, math for humans is over as well as all intellectual activity, better just jerk off watching AI-generated hentai all day until I die"
Just because the truth is inconvenient doesn't make it false.
I didn't say it was impossible, I said if it were true there would be nothing to do and thus no point in writing about that potentiality. It's sort of like writing about what mathematics should do if there was a thermonuclear war that wiped out all of civilisation.
There's a long collection of comments by mathematicians at [1]. A large number of workers in the field are rather discouraged. The main quality complaint is that the proofs are ugly, long, and incomprehensible. One mathematician comments that the first proofs of new major results are often like that. Elegance comes later. The main social complaint is that the social foundations of mathematics just changed, in a big way. The main practical complaint is "what do I do now", especially from PhD students who just lost their thesis topic. They sound like the Uber drivers fighting Waymo.
One comment was "A common piece of wisdom is that mathematicians only have a few tricks in their bag – which they skillfully apply again and again." Feynman made that point in his autobiography. For various reasons, he had a different set of mathematical tools than others in his field, and that helped him to solve problems that stumped others.
A big AI model has access to all known tools. AIs have more breadth than any human. Some of these breakthrough proofs seem to come from applying techniques from areas different from what people in the field tend to use. I'm not qualified to comment on math at that level, but the comments by mathematicians seem to hint in that direction. That may indicate why throwing vast amounts of compute at these problems works - an AI can try everything.
[1] https://proofsandprompts.com/2026/10/08/100-reactions-to-100...
Not only are the proofs long and unintelligent, but they don't cite previous work.
In fact, they completely disregard all standards of mathematical dissemination. They just post a bunch of 100+ page papers online and call it the day.
A real mathematician polishes and polishes for months until it is understandable by the community. Then they communicate the results to a journal where there are editors and peer reviewers who come with detailed feedback which is then implemented. And just then it is ready to publish.
"Polish this paper. Check all proofs with Lean. Try to find shorter proofs of lemmas where possible. Write for a undergraduate math student. Add brief explanatory material as needed. Cite previous work where relevant. Check all citations. Convert to LaTeX."
I’m wondering why frontier labs haven’t done that.
A few possibilities I can think of are:
A. The capability is just not there. OpenAI and Anthropic can produce a brute-force Lean proof, but their models aren’t capable enough to digest the proof as human mathematicians do.
B. They haven’t seriously tried it yet. They prioritized grabbing the land first, and have not spent much effort to polish things up.
C. It’s fundamentally impossible. These theorems are provable only in a very ugly manner. This is the reason why human mathematicians haven’t been able to crack them before AI in the first place.
As a programmer, what I’m seeing is (C). Many recent cyber hacks are products of AI trying & combining every possible tricks, and I don’t think there is any elegant short path that human engineers can replicate.
The four-color theorem appears to be case C. That was the first really long computer-assisted proof. No one has since found a concise way to do that proof, and it's been half a century now.
D. They seek disruption as way to market dominance and worldly power. It's not unfortunate, in their minds, but the means to their ends.
Disruption has been a SV buzzword for a long time now. It's not a byproduct but a strategy for your new technology product. SV has long perfected disruption and the people behind the AI products are masters at it.
How can so many who have seen it (and used it) for so long be so taken in by this well-worn tactic. Over and over, they choose what is most disruptive. They could have dealt with the mathematical output in many ways, including working with mathematicians to polish and publish; you see what OpenAI chose.
Similarly was the recent events where AI software 'broke containment' and hacked others. Obviously they could have controlled it if they valued it; they got the result they wanted. Notice how Anthropic (or were they first and OpenAI followed?) urgently publicized that their product, too, 'broke containment'. 'We're disruptive too!'
Doctorow talks about this dynamic in his book about reverse centaurism. It is a “growth story” for investors.
Funny.
The problem is that this is not something that LLMs (currently) can do.
Lean has nothing to do with understanding and intuition.
Brevity of proofs are also not the target.
The target isn't undergrads, either, but all of this is pointless because LLMs aren't up to this task (p.t.).
> The main practical complaint is "what do I do now", especially from PhD students who just lost their thesis topic. They sound like the Uber drivers fighting Waymo.
I wish mathematicians would just focus on this. This really is the problem, at the end of the day.
Having models get better at writing elegant proofs and explaining their results will not solve this problem - it will only make it worse, if anything.
We are all sympathetic to this. I, personally, am much less sympathetic to the absurd notion that it is "bad" that this new technology can solve important open mathematical problems at all, that we should stop trying to solve them, that things like the Mathathon are dangerous and needs to be shut down, that ugly proofs are the real problem here, so much so that you wish the open question weren't resolved at all, and so on. So silly.
But being pissed off that people are going to be out of work is something I agree 100% on. Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.
> Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.
I don't see it. What I see is more wealth and power concentration and with the worst group of people possible.
Yeah I view it as AI being objectively good from a technological perspective - we get new medications, new technology is achievable faster overall.
But really bad from a social perspective - as people lose their jobs they also lose their tiny piece of power in the society that was the labor someone else needed. Now that power has concentrated to the frontier model owners, and everyone in the future will depend on them.
And on a personal level, sacrifices people have made to become good at their craft have been made worthless, as any rich PoS can now prompt their way to "good" art and call their end result as good as something made with skill that needed years of training. That I really hate.
> prompt their way to "good" art and call their end result as good as something made with skill that needed years of training.
Which is only possible because:
a) The models have been trained on work of millions of people,
b) They've done it for free, In other words, they've stolen entire people's lives. Millions of lives. They literally suck people dry and tell them to go look for another job.
I wonder what LLM's would achieve (if anything at all) if no mathematicians had used it to format their articles, etc (remember, no data is being used to train models! Of course).
It’s amazing for me to see math research going from craft to technology.
CS had this moment decades ago, sans some theoretical parts that are joining the bandwagon now.
I did a bit of math research in grad school collaborating with my math postdoc roommate. I found the aversion to any computation making its way into the craft to be odd decades ago.
Well, now the proof’s in the AI generated pudding.
Technology has changed and extended the craft, but everything about developing applications and communicating results remains.
Communicating results when zero other people are working on the same problem as you and your advisor aiming to submit to a peer reviewed journal that takes six months to respond; versus when competing teams of hundred engineers working with a few billion dollars of funding trying to get the results before the other teams has to be completely different.
It’s astounding that people expect the later to look like the former.
It’s strange to me seeing multiple talk about how Terry’s viewpoint has evolved or not as a response to this post. He didn’t write this.
He did attach his name to it by posting it. I'd imagine if he disagreed strongly he wouldn't do that, or he would at least add a footnote.
What you could argue though is that he is muddying the waters with all these guest posts on a domain that is as personal as it gets. I appreciate he's trying to be the 'Carlos Santana of mathematics' by sharing the spotlight but it might be good to highlight the distance a bit more, for instance by adding his own views.
Well, you're wrong, given how many of the guest posts featured by Tao are inherently contradictory and so it would be strange for him to be personally endorsing each one! He also posts his own views elsewhere: see https://teorth.github.io/tao-web/ai-views.html.
The obvious goal is to highlight differing attitudes within the mathematics community so the mathematics community can have a dialogue and see what other people are thinking. Luckily, most mathematicians seem to be able to actually read and understand not all posts are from Tao. It is unfortunate it seems many people on this site cannot.
Posting something on a domain with your name attached to it is an endorsement of sorts. You can argue with that all you want but that's just how it works. Having a dialogue happens on neutral ground and Tao's blog is not neutral ground, even if that might be his intention.
If he thought it was fundamentally an unreasonable position he probably wouldn't do that. But he seems to be running his blog as something of an open(ish) forum for AI commentary. I suspect if he thinks it's a reasonable position that happens to be entirely incorrect he still would post it...
I believe you, but without a stated policy that is speculative. To me it is indistinguishable from curated guest posts.
I don't have such a stated policy... I do have that he both
1. Signed an open letter https://terrytao.wordpress.com/2026/09/11/a-severe-misalignm...
2. Posted a guest post about why they didn't sign said open letter https://terrytao.wordpress.com/2026/09/17/why-i-didnt-sign-t...
Good example, thank you. Position adjusted.
> To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof. Choosing worthwhile problems may end up mattering more than producing proofs.
That is the correct take, but I can understand that given how current funding system (esp grants) work the incentive are stacked against doing that.
It's worth mentioning that many of these papers haven't been fully reviewed and confirmed as being correct. Several had to be retracted upon closer inspection. Too many people are assuming that passing lean checks for some of the papers means that all of them will hold up under scrutiny. It remains an open question if humans or any available system are capable of peer review for some of the more complicated proofs.
None of the OpenAI papers published with a Lean formalization have been retracted.
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
Of course if you solve any existing problem, new ones appear. That's not the point. The point is that AI will be able to advance on any problem better than any human.
But in a human way? I don't think so. The 700+ solutions that were put out by openAI are in no way accepted as solved until humans say so. It's kind of like saying SC2 was solved when Alpha beat the best players. The professional community just went on playing much the same. Math should do the same. Just with better tools
Math is too boring as a spectator sport. It's funded because of its practical use.
There's going to be some hobbyists doing math in their basements, but there's unlikely to be a future for professional mathematicians.
Applied mathematics is funded because of its practical use. Pure mathematics, like other theoretical fields, is funded to explore ideas practically oriented people ignore. Most of the time such exploration produces nothing of practical value. But occasionally theoreticians stumble upon ideas that turn out to be fundamental to understanding something practically oriented people discover much later.
> It's funded because of its practical use.
That's exactly what the Simons foundation was set up to reject, and it's one of most important funding streams for math
If rich people want to use their money to fund math with the goal of NOT funding math with any known useful or practical purpose, that sounds like a hobby to me.
The rich people in this case, yes, wanted to fund math to protect it from people like you
Mathematics as a field will not cease to exist, and not everyone will be interested in (or qualified to) managing the fields progress and its applications.
Mathematics as field won't. It is the future of mathematicians that is uncertain.
They’re going to become sysadmins imho
Haha, the breaks from reality there are two, one professionals do not spend most of their time inventing proofs, and two, practical use? :-)
The proofs may not yet be understood, but to claim they should not yet be accepted as solved is to claim there is a bug in Lean, which is a big claim.
Plenty of these AI proofs have been accepted. On the most important result, on the Quasi-Riemann Hypothesis, there is no serious belief it's wrong. Mathematicians have already dissected and reformulated the claims.
The proof with the 11/12 constant, the simplest one, has already prompted what looks like chagrin from some. It's an unexpected combination of known results/techniques, a combination that apparently fell into a blind spot so it wasn't picked up before (even though it could have been, even 20 years ago).
I think these results are showing there was a lot of low hanging fruit that human mathematicians didn't have the breadth or mental bandwidth to find.
Isn't it also possible that the written proofs themselves are incorrect, say, for example, by mis-specification of constraints? Let's say I publish what I claim is a proof that 2+2=4, but in the Lean specification, I proved 1+1=2.
To be pedantic, the proofs would be correct, they would just be proving the wrong thing.
To address what you actually meant: it's possible but specifications are way easier to verify.
The main issue would rather that the formalization itself has a bug, eg. those proofs that you're trying to formalize are not "1 + 1 = 2" but theorems which take already multiple pages just to enunciate. And in these enunciations (that are made by humans) there can be errors.
In general, once the proof has been formalized, it's supposed to be the proof of the correct theorem (that's what the Lean proof-checker checks)
the impression i get is that people don’t per se assert that exploitable bugs exist in Lean, but that the volume of (code) proofs published represent a new magnitude of challenge to the integrity of Lean.
the obverse would be to assert that Lean (or any other theorem prover) is infallible under all circumstances, present and future - an even weightier claim in my opinion.
No. Lean being bug-free is a far bigger claim.
A bug which has allowed 700 proofs from diverse fields of math to appear as proven, and noone has figured it out yet?
A single bug can usually allow you to prove anything you want. Saying that this is how the proofs were achieved is a totally different matter though, and probably incorrect.
Hmm, am no expert at Lean, but on one hand, since it's a programming language, guessing it necessarily means like all other programming languages, it has bugs.
But on the other hand, agree with you that the chances that one its bugs causes a solved proofs to be wrong is probably very, very low. Yeah, each version of Lean obviously must be run against hundreds if not thousands tests of existing proofs to ensure it's working correctly...
... Although, one thing that is a little worrying is that some mathematicians have said the Lean code generated in these OpenAI proofs is supposedly really complex and undecipherable. It might be pushing Lean to corner cases that are not well tested.
It can possibly have bugs. That said, the "core" of Lean is a somewhat "simple" core that does simple things to check the proof, so it's quite rare to find meaningful bugs
don't disbelieve you, but just looked at the reference language, it seems pretty full featured as a programming language, even with basic classes, polymorphism, scope, namespaces... But maybe most of the mathematicians who actual code with stick with the simpler language features? And also, didn't look to deeply, these could be very simple features that actually aren't very sophisticated.
I mean Lean has two things: a full programming language, and proof checker. The programming language is used to create either code, or proofs. But the "created proofs" are using a very simple subset (a bit like an "assembly language"), and only need somewhat simple rules to be checked
So as long as the "somewhat simple" core that checks to proof is correct, it doesn't matter if the "programming language" has bugs: either in the end the "proof assembly code" generated by the program is right and is validated by the core, or it's not, and you need to revise your proof, or to check for either a bug in the programming language (it didn't generate the proof you were expecting), or in very rare case detect a bug in the simple "core"
Ah, I see, that makes sense - and am guessing that most mathematicians (like most non-developers) who work with a programming language don't write "good" code and often ignore the organizational features like classes or namespaces. Yeah, could see how they really are just interested in the core feature-set of a programming language... although, have seen a bug in some high-usage low-level code before, which was greatly surprising. Still that was only one case in years of dev work.
There's been many bugs found in Lean already, and there are probably still a few. So that's not the main issue.
I keep seeing people point to AI solved games as evidence that AI advancements are less impactful than we might think. Usually it's chess ("today there are more players than ever!"). Most of the time these are bad analogies. Games are played for entertainment, so productivity is meaningless.
I never advanced in math past high school, but my assumption would be that we as a society aren't paying mathematicians to have fun[0]. The discoveries mathematicians make must translate into real improvements in engineering and other life improving fields. So why would we pay mathematicians to work on math much the same way, if a new machine-led way makes these discoveries faster and cheaper?
[0] Being a little facetious. I know I don't know what I don't know.
Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
>Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?
Obviously. In my ideal world even the janitors at the same institutions would have a blast, it's just not usually the end-goal reason humans are hired.
>With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
Right, but would it have made a difference for the application layer if that key bit of "pure math" had been discovered by a human (that had a great time) or a machine? That's my point about games being a bad analogy.
That depends. Many of these things can only be applied safely once they're understood.
For instance, almost all of cryptography is applied math. If your opponent understands the math better than you do you might end up with your secrets compromised.
Math is about as 'pure' a science as there is, the touchstone of applicability is missing the point entirely, it is first and foremost about understanding things at one of the most fundamental levels (even more fundamental than physics) that there is.
Nobody cares all that much about the discovery (except that we name things for the person that discovers it, it's sort of analogous to the discovery of a new piece of land). And given that the pay usually sucks this is a nice thing to be able to do. What mathematicians usually do really care about is their ability to understand the discovery and that this discovery has the proper foundations (other pieces of math that are also understood). It doesn't really matter what the answer is if you don't actually fully understand the question or the foundation the answer builds on.
I think it's likely that AI will create mathematical ideas that are so complicated and intricate that humans have no hope of ever understanding them. AI on the other hand will understand them, and understand them well enough to use them productively and safely. That is my prediction.
Many academic disciplines bring no everyday life improvements. They exist because they provide intrinsic value and long-term foundations, hence they aren't merely entertainment. Mathematics/physics happen to be fields that span the entirety of applied/pure spectrum. Those said, your question can be generalized: Should society continue funding human intellectual activities when AI can perform them substantially better?
> The point is that AI will be able to advance on any problem better than any human.
Define "better".
Define "advance"!
LLM agents are perfectly capable of "advancing" my codebase down bizarre, esoteric rabbitholes. Meanwhile, the value remains undelivered.
Many crystals, likewise, are advancing their fantastic complexities in the deep, dark recesses beneath the crust of the earth, but does it make a sound? Who gives a shit.
Many paperclips, indeed, can be churned out. Even we can be converted into paperclips, remember.
All but a negligible amount of novel mathematical results going forward will have had substantial LLM involvement
I don’t think we know whether or not that’s true yet. Imagine a problem which requires a paradigm shift or new novel techniques to solve. There is no proof (yet) that an AI could create one.
Why do you believe human brains are magic? We may need some breakthroughs, but trillions of investment tends to produce those.
And it's becoming less clear over time that a breakthrough will be needed.
> Why do you believe human brains are magic?
We chose the problems to value. Literally every step of a proof is equally proved as the starting point. The choice of the starting point is what makes the problem, and it's a normative choice, which is to say it is a social choice.
Why do you believe that LLMs are good at the same things humans are?
I believe that we're investing trillions into making it so.
We invested at least a couple trillion of speculative value into making ugly cartoon apes the future of art, finance and property. I'm not sure it's always possible to overwhelm reality with sufficient money.
We didn't. Investment into it peaked at a few fractions of a percent of what's going into AI.
I would like to be optimistic about the whole thing falling apart, but I don't know that it will.
I'm not saying it will "fall apart", I just think the burden of proof is on AI boosters for their extraordinary claims.
They are saying this thing will soon obsolete human thought; meanwhile I use these tools every day and can't get it to refactor a React app cleanly without making a mess.
It's telling that OpenAI's decided they're going to sell ads in ChatGPT. Instead of, you know, using their AI themselves to produce all these valuable and revolutionary breakthroughs they claim it's capable of making.
Like how Anthropic has started to run their own biology labs?
Probably wildly irresponsible given they think it's likely to kill everyone with a virus, but yes.
If you have a super-intelligent AI that can cure cancer, just sell the cure for cancer already?
Medicine is a slow moving field. If they discovered the cure today how many years will it take to be certified as safe and effective by medical boards? More years than they have runway, in all likelihood.
That's another way of saying the same thing: even if AI is great, there's all kinds of reasons why it won't be a new industrial revolution.
Intelligence is very rarely the limiting reagent in progress.
It's not magical abilities - humans have proved for millenia that they can find novel solutions to problems - we have no idea how that comes to be though (which could be called magic I guess)
At issue with AI is that it's searching an existing corpus for fragments of solutions generated by humans - that's not necessarily novel, and we've not yet worked out how to make the machines actually invent.
We have to come to terms with the reality, and we must find a way of existing with this new super tool
Did any ai made any new interesting question or conjecture?
No.
ANY problem?
But the question is whether the AI will itself be able to decide on which problem to advance, and do it in an ethical and aligned way with humanity. And that if it does, its significance will be understood by humans.
Otherwise we're just creating a black box, easily manipulated by their owners, who would be ruling over human lives. That's not what people want
The thing about much human progress on various problems is that human mathematicians formulate a human comprehensible machinery for the purposes of solving a given problem and then that machinery naturally suggests (to humans) further questions which then give rise to more machinery - And, in a sense, the value is ultimately the machinery, which has tended in the past to be relevant to physics, computer science, logistics, sociology and philosophy.
When AI writes proof no human can understand, it doesn't advance mathematics this way. Maybe AIs can understand their own proofs - meaning not just verify them but extract a machinery/broader-perspective (but I wouldn't count on it). But even machinery an AI isn't necessarily useful for humanity/society.
But how do you know which problems are worth making progress on? Traditionally, you'd find those by working on solving previous problems.
How do you build a society where everyone wants to live in it and has a means to?
How do you reach the conclusion that society is something that admits rational design?
How can you type that into a device and then go continue social charades and be like yeah this isn't rationally designed
What?
> The point is that AI will be able to advance on any problem better than any human.
If that was true it would of course be significant.
If you had no morals, 2 trillion dollars to spend, ability to commit intellectual property theft without consequences, and most of the smartest people in the world actively working to advance your cause... ...you really think you couldn't compete with the clankers?
Given the capabilities of OpenAI's unreleased model, I think the model helps mathematicians more than it hurts them. Some of the open problems that were solved were "It would be nice to know that", but many of the others represented ways in which we had been stuck for many years. There had been very little progress on analytic number theory, particularly the Riemann hypothesis or its variants. (One result, "no Landau-Siegal zeroes", even had a 10-page elementary proof that everyone missed for a century.) Complexity theory had seemingly more open questions than results. Progress could meaningfully accelerate now that we have a quick test of which of our ideas are good, and which are bad.
The wildcard is that we have no idea what the model after that will be capable of, or the model after that.
> I think the model helps mathematicians more than it hurts them
I would agree with you if these models were available to mathematicians, and OpenAI wasn't effectively competing with them by spending millions of dollars in compute to rush out solutions.
This is just a side point, but:
> We have much more control over how we hire mathematicians, publish papers, and train students than over the behavior of trillion-dollar companies, and that is where our efforts would be better spent.
We should also make sure that there are no trillion-dollar companies. The existence of large and powerful companies is a problem, whether or not the field of mathematics is in crisis. All these companies should be shattered so that they cannot do things until after society as a whole is sure they're okay with those things.
I wonder if this happened to math first because the goal is so clear (proving a statement, etc.). Math is just going through it first, and I think other fields will get their turn soon too. If so, I'm not sure if things like a PhD are even needed. Someone in a bachelor's program who just learns the basics and how to use AI tools might be better.
Better at investigating the idea that one could conceive of elements in the different commensurability classes of compact arithmetic hyperbolic three-manifolds indexed by non-split quaternion algebras {D/F} as forming some type of geometric Kolyvagin system (“Heegner points”) associated to {K_3(F)}, which one could yoke together by some unknown integral arithmetic version of the Jacquet–Langlands correspondence?
Please can we add the name of the guest authors in the titles of these posts?
It has changed from elegance and human understanding to brute force. But that trend was started a while ago, before AI became really useful.
From TFA:
"I’m ready to board the plane, because I want to see what’s there. At the same time, I would add that people who previously have been walking around on earth with their eyes closed are not going to recognize anything however high they fly; I still believe that many of the skills that mathematicians have now will not only be transferable but vital to the new world order."
That sounds a bit disconnected and rude.
IMO, this captures well the essense of AI: it's an alien swarm intelligence that solves problems with brute force and pattern matching. For comparison, we prefer simple and elegant solutions, partly because our time and attention scope is very limited, but mainly because it rhymes with our nature.
It sounds like Maths Co. will be only hiring forward deployed mathematicians from now on.
I mean he's basically admitting that math academia is outdated.
If we don't interview candidates based on results, what is it? Politics. It already was. I had a JEMS paper in grad school and got zero postdoc offers. Many other cases.
The real shift is that math will become a hobbyists game. Make money other ways and then spend your money on humans and tokens to solve your problems.
Academia is almost entirely a patronage network. It was never about results.
Terry's stance has shifted gradually. This reminds me of Garry Kasporov's response and stance a few decades ago agains deep blue (and deeper blue). I think in a closed system/game (or "field"), this kind of AI superiority is inevitable. However, I do agree with Terry that we need the right form of transition if we want to continue holding the rein or at least playing a part in the subject.
Useful reference to "Colossus - the Forbin Project".[1] Watch starting at 00:30:00, and see this scenario predicted in detail, half a century ago.
[1] https://archive.org/details/colossus-the-forbin-project-1970...
https://archive.org/details/colossus-the-forbin-project-1970...
here is the direct link of the video which starts at 1800 seconds.
Offtopic: Fun fact but aside from archive.org there is also a way to have a link which links to any direct video at any time.
Someone made a good HN article about it that I vaguely remembered[0]
[0]: https://www.autodidacts.io/how-to-deeplink-to-specific-part-...
> gulf between ... novel proofs and ... understanding
Machine proofs are useful machine understanding. You don't read the proofs, that's holding it wrong. You point another agent at the proof and prompt "use these ideas to prove this lemma that's useful to me". And it works, agents are faster given access to the proofs than without.
“Note there is also a difference between knowing mathematics deeply and explaining it fluently; they are related but different skills. Ideally I want both, but it is the first that I ultimately value more highly.”
Once again perpetuating the cardinal sin of lionizing mathematics heroes. Which to be fair is just THE culture of higher mathematics.
What good is a minuscule elite Cabal of savants? If these deep insights are inaccessible to 99.999% of the population, then they are no better than the manuscripts written by AI. We already have those.
The whole situation in mathematics reminds me of two years ago when models started coding well and all the developers were really sad. Now we use it as a tool and, as a whole, are able to make much more.
This is another 15 min phase like “prompt engineering”. Soon it’s going to be just a CEO and an LLM.
> someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution
among all the discourse, it seems mathematicians are vaguely converging on this higher layer of abstraction being where the human value in math is now. It seems across domains (such as in software too), people speak of “judgement”, “taste” and other fuzzy concepts around defining problems rather than solving them. The lowest level of abstraction where humans win is asking the questions, rather than merely answering them.
As math is so verifiable, the “transition” seems most poignant and quick there. Watching these big name mathematicians discuss this unfolding revolution in real time is fascinating.
I theorize that number of problems AI is possible of solving is limited. My theory rests on an assumption that all human knowledge is a result of lived experience. AI as a statistical machine doesn't know lived experience, therefore it can't solve problems that a living person interacting with reality could solve. Therefore human brain > AI, or so long AI is unconscious anyway.
Sure it can solve math problems by just following the implication arrows. But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
They solved a millennium prize problem… they disproved the exponential time hypothesis. Are those low hanging fruits?
GP missed the most important part of TFM's introduction:
> AI is now undeniably capable of making groundbreaking contributions across many areas of mathematics.
In the "theory of mind" sense, yes, probably.
These fall under the class of problems that have solutions which can be inferred more easily from applying massive computation, which is a subset of problems that human minds are trying to solve.
It's like being a CIA analyst, maybe. You have an overwhelming number of information sources of varying quality and non-obvious relationships. Applying massive computation will definitely allow you to draw new insights and identify trends that would have been difficult, or nearly impossible for unassisted human reasoning.
From what I understand, this is the sort of thing that the LLMs have been particularly effective at in mathematics, applying massive computation to the existing field and with a combination of impressive vector space search and runtime evaluation, deriving plausible solutions to certain narrowly defined problems.
Here's a harder one: you're in a bar, sitting having drinks with friends. There's an attractive girl sitting with some of her friends that has glanced at you a couple of times. What should you do?
Taking into account, of course, the entire human experience from marriage status, age, rejection likelihood, your friends opinions, her friends opinions, your family, your hormones, everyone's intoxication level, how well you slept, the noise level, how long her glance lingered, and the billions of other things that go into it - entire lifetimes of experience, genetic behavior predisposition, etc...
This is just a small example of a brief problem that a normal human has to solve hundreds or thousands of times per day that don't fit into the class of solutions that LLMs would be particularly adept at. It's easy to come up with a lot more.
So yes, I'd argue that those neatly formulated, clean problems with clearly provable solutions are low hanging fruit.
Wait until you find about RizzGPT.
> But those are just low hanging fruits, i.e problems that humans were too lazy to deal with.
Some of these problems have had mathematicians dedicating their careers towards getting closer to a solution. Your comment reads like wishful thinking.
Low hanging fruits for the agents that solved them. It's very possible that there are whole classes of problems that AI won't be able to solve or will require another few orders of magnitude of compute to solve. A human might even be required to first make a breakthrough on some field before AI can re-apply it exhaustively and find new discoveries on top of that.
It's amazing to still see posts like this in late 2026. I suggest you recalibrate your ideas on what both AI and human intelligence is and can do. My take: both are trained systems. Both are as "statistical" as the other. Any idea that "because statistics" means that humans are different or better is bogus. Both can do things based on their training data. You are totally correct that a human has access to training data that AI doesn't (the smell of new mown hay, for example, or the fear of being crushed by the garbage compactor room in Star Wars). But there's no "humans have more training data therefore better" argument to be made about fields like mathematics. In fact the opposite argument works quite well: AI can have all mathematics ever invented in its training data, something that's pretty hard to achieve as a human.
I won't even start on "AI is unconscious"...
This hangup on sensory data is weird. Sensory data is just like any other signal and can be fed.
I think Sutton agrees [1].
[1] https://youtu.be/ThFq87Rp21s?is=xvgU-cKCpK0KwG_M
> But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry.
This analogy is pants on head stupid. They're talking as though the only thing AI will ever solve is the Langlands program, not as though we have discovered a generic tool.
It's like saying "they drove the car one mile -- and every mile driven leads to more roads that we can trot along, why are you so worried about the future of horses?"
People are allowed to process their anxieties. And processing takes time. That said publicly doing it on the great modern internet attention/noise factory complicates the story.
Everyone is surprised that a person that some consider to be one of the greatest modern mathematicians can have such obtuse views of technology and its own field.
Note, this is a guest post.
I don't think it was a particularly well-reasoned essay (I thought the classification of finite simple groups is almost evidence for the opposite of what he says), but at the same time I don't think you understood his point.
If we have a theorem proving machine, then we can use it to prove theorems. If we have a machine to solve the Langlands program in one day, then we'll think of more things to prove after that. There's a million unsolved questions in number theory unrelated to Langlands, and if we knew Langlands (and why it was true) we could make new conjectures that our theorem proving machine could solve, rather than having to speculate for 50 or 100 years like now.
The possibility he's not considering is that the machine might be better even at conjectures, and then there really is no role for humans once the machine already knows things our primitive meat brains are too small to understand. But what he's saying isn't "pants on head stupid".
I think I’ll take the professional mathematician perception of how AI affects his field over random HN commenter saying it’s “pants on head stupid”
Sure, or you can think for yourself for five minutes on the implications of a general proof machine, and whether it's likely to continue producing proofs.
If you've forgotten how to think for yourself, you can also ask an LLM.
The host of the blog, Tao, was an early booster. He didn't think through the implications, and seems to have expected it to remain unable to do interesting work. Today he's on a "war footing".
I don't think mathematics are uniquely skilled at predicting the future, and the impacts of AI on their profession.
It is weird. From what I can tell, you and I have quite similar expectations on what AI will be able to do in the future, but for some reason you are much more dismissive of Taos opinion.
I perfectly understand his point and it is not about being selfishly frustrated by a machine being better than him. Is that really your interpretation of the discourse?
Tao didn't write this article.
Either way, either there's no future for AI in math, or there's no future for people in math. I think this is true for many professions.
Neither are mindless AI boosters
https://fortune.com/article/ai-godfather-radiologists-obsole...
Ironic that you resort to “think for yourself” as your silly name calling go to btw
Logic is logic, you don't need a specialist's opinion on every aspect of a specialist field. I've had eye opening insights delivered to me on problems I'm working on by people who barely know how to use a computer.
You are welcome to opine on anything but there’s a reason we’re reading his blog and you’re down here in the comments
Speak for yourself. I'm reading these comments and not the blog.
On the flip side, the professional mathematician has all the incentive to pearl clutch and rent seek.
What is the current thought on AI creating new emergent forms of mathematics? Even the most stunning results like the proof of the quasi-RH are a clever combination of known mathematics. Humans have created (discovered?) stunning new ideas (e.g. complex numbers) that have opened wide new horizons -- do we think AI will be capable of creting / discovering novel new things like this?
Are humans going to be able to recognize the things AI discovers?
It took us a while to make the planets line up neatly in a heliocentric system. Just last month I read that a whale mother "might" grief while carrying her stillborn child.
The limiting factor is human imagination and comprehension. No offense, but see your comment.
I think AI has already invented new math, but the guys haven't looked for it because they didn't think it possible. And even under their noses, they wouldn't understand what they see.
> It's not what you look at that matters, it's what you see. —Thoreau
LEAN and other tools verify from what do know -- at least stepwise. Anything that deviates from these things should stick out. I suppose some million line LEAN program may have embedded within it something that is novel and not readily apparent, but I imagine truly novel approaches should stick out like a sore thumb.
Could AI create a novel set of axioms (a new math) from which things like RH become a by product? I think that would be hard to miss.
Dozens of agents probably already invented it, but they were given ephemeral containers where noone ever looked or they only wrote it in direct CPU instructions.
I would really like to see an AI invent a new set of (consistent) axioms from which a whole slew of new problems and solutions arise. Something akin to Euclidean geometry, but for a whole new landscape of theorems.
> Are humans going to be able to recognize the things AI discovers?
They have been so far.
That's like saying that all serial killers make a mistake that gets them caught, which is true only of those that were caught (and studied.)
That analogy makes no sense to me. Wouldn't OpenAI have the incentive to release results that are incomprehensible, to showcase how beyond humans their models are?
There’s an elegance to how the Feds just got out of funding this stuff and within a year all proofs were solved by the private sector
I know, unrelated truths and coincidental timelines, part of me giggles nonetheless since the whole belief system behind the market relies on infinite coincidences
> Much of the reaction seems to reflect a feeling that the subject that they love is under attack, and is being “hollowed out”.
Sums up the reaction from a lot of developers too.
Mathematicians can just say no to this technology.
1. Make mathematics a closed community, only people who join this community can read and work on its stuff. I imagine a VPN that contains all maths knowledge that only members can view, similar to how big tech handle their codebases. It needs to be hard to copy and paste that stuff into the open internet and doing so should involve serious punishments like jail time, similar to posting say proprietary Google code on the internet.
2. Anyone caught using AI is excommunicated from the society, his access to the VPN permanently revoked, and his work is never interacted with again.
3. There will be a bit of a disruptive period as maths moves from open to closed model. During that time AI will keep solving more problems, but as long as mathematicians stop contributing more open problems, at some point AI will stop (either because it has solved all open problems, or it can’t make progress anymore because scaling is no longer economical).
I think Association for Human Mathematics should do this, I think they will be surprised by how much funding the public is willing to allocate to this (public hates AI), because fundamentally humans like thinking about and doing maths, and do not mind dedicating a bit of the surplus produced from society to let others do this, similar to arts. Just because I can prompt a better melody from Suno, doesn’t mean humans will stop wanting to make music.
Of course other mathematical associations can and should form, which fully embrace AI. But if the VPNs secure, these associations will not be competing as they won’t even have shared consensus of what problems to solve. They both will still need to compete to get public funding, which is fine, I think AI math will have a harder time justifying their funding since anyone can prompt a robot.
This is the dumbest, most troglodyte “solution” to this frontier pushing development. WHO does this benefit besides a handful of mathematicians?
Wake me up when AI solves Collatz.
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Can we stop with this ridiculous narrative? That didn't happen.
Can you elaborate or provide a good source to explain why this is not the case?
The original OpenAI blog post.
Three years ago people were laughing at LLM arithmetic mistakes. Today the Simons Foundation is starting up a fund to keep professional mathematicians from suiciding:
https://www.simonsfoundation.org/news/simons-foundation-laun...
Anyone still pretending this is normal technology is completely delusional. We're in the beginning of a singularity takeoff.
> Under a partnership, the Simons Foundation, Anthropic and OpenAI will commit $15 million in combined funding and resources. The foundation will manage and award funding, including through an open call for proposals, to support workshops and initiatives that help researchers evaluate recent AI-assisted results, learn and use new tools, share perspectives and shape collective standards for AI use.
"Normal technology"?
Like a toaster that just toasts bread.
Have the pro-AI people finally found a new mathematician? Otherwise it is always the same 10 people on Tao's blog citing each other.
This guest author runs a seminar to lure people into AI:
https://math.uchicago.edu/~fcale/AI.html
Lure people? It isn't a cult.