Mathematics has changed. Let's act like it
terrytao.wordpress.com
terrytao.wordpress.com
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.
Which is absurd on it's face for most day to day programmers, but a critical distinction for theoretical computer scientists.
The time saved using AI comes from reducing understanding.
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?
What benefit is a human soul to a feudal lord?
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.
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?
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!!
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.
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)
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.
AI-generated math textbooks
(but they probably already exist)
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.
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.
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.
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.
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.
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.
That analogy holds too.
(And even more if you can combine them.)
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.
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.
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.
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!'
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.
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...
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.
The wildcard is that we have no idea what the model after that will be capable of, or the model after that.
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.
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.
There's going to be some hobbyists doing math in their basements, but there's unlikely to be a future for professional mathematicians.
That's exactly what the Simons foundation was set up to reject, and it's one of most important funding streams for math
To address what you actually meant: it's possible but specifications are way easier to verify.
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 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.
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.
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.
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"
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.
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.
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.
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.
Define "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.
And it's becoming less clear over time that a breakthrough will be needed.
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.
I would like to be optimistic about the whole thing falling apart, but I don't know that it will.
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.
If you have a super-intelligent AI that can cure cancer, just sell the cure for cancer already?
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.
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
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.
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?
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.
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.
[1] 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-...
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.
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.
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.
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.
> AI is now undeniably capable of making groundbreaking contributions across many areas of mathematics.
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.
Some of these problems have had mathematicians dedicating their careers towards getting closer to a solution. Your comment reads like wishful thinking.
I won't even start on "AI is unconscious"...
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.
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
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.
They have been so far.
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?"
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".
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.
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?
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.
https://fortune.com/article/ai-godfather-radiologists-obsole...
Ironic that you resort to “think for yourself” as your silly name calling go to btw
I know, unrelated truths and coincidental timelines, part of me giggles nonetheless since the whole belief system behind the market relies on infinite coincidences
Sums up the reaction from a lot of developers too.
This guest author runs a seminar to lure people into AI: