Posted by smilelamp 15 hours ago
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.
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.
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.
> 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.
no, its just a language. physics constrains 1+1. "one plus one equals two" is difficult to transform
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.
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 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".
Which is absurd on it's face for most day to day programmers, but a critical distinction for theoretical computer scientists.
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.
I think Tao's point is that there's a necessary rebalancing and more weight should now be given to the non-proof stuff.
The time saved using AI comes from reducing understanding.
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.
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.
Who knew 2-3 SD won't save you from being a pattern matcher that can't stand to see a better one.
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 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.
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.
What benefit is a human soul to a feudal lord?
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?
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.
This sounds horrible. Do you have an article where I can read about it?
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.
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".
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?
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.
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.
The person who is brutally honest enjoys the brutality quite as much as the honesty. Possibly more
~Needham
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.
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.
> 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.
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 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)
"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.)
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.
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.
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.
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.
https://news.ycombinator.com/item?id=50309981
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.
AI-generated math textbooks
(but they probably already exist)
https://www.penguinrandomhouse.com/books/172835/the-diamond-...
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!
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.
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.
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'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.
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.
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.
>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.
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 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.
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.
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.
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?
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 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.
There’s an immediate resistance towards change rather than embracing it and finding opportunities to improve and work with it.
Maybe some math departments could pay OpenAI for access to their models, but I doubt OpenAI would want to deal with most of them.
Maybe we should listen to people that care about negative externalities more often, and actually mitigate them.
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.
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.
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 analogy holds too.
(And even more if you can combine them.)
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.
I'm sure there's a coherent rebuttal but "That is plainly rubbish" isn't the reply I'd expect in good-faith discussion
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.
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 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.
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.
Or has he changed his mind again.
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.
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.
> 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).
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.
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.
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
Of course, general improvements may be sufficient to continue to drive math improvements even without specific attention - we’ll see.
Could also be that some things are not possible in some ways. I wonder if there is some general theory on "shortcuts".
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.
For useful materials science the numbers are still large but far lower than the numbers involved in 'a glass of water'.
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.
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.
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...
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.
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.
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 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.).
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.
I don't see it. What I see is more wealth and power concentration and with the worst group of people possible.
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.
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.
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.
It’s astounding that people expect the later to look like the former.
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.
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.
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...
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.