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".
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.
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.
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?
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.
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 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?
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?
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 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.
"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.
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.
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
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.