Similar logic applies to every industry and every job. And it comes to the conclusion that we won’t have enough people for all the jobs that need to be done.

In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.

It's like chimpanzees seeing human society and saying "look how complex it is, imagine how many chimpanzees it needs to maintain it".

In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.

Let's be honest: we don't know. Maybe you're right, but for the moment it's more likely that you're not. And countries cannot bet on that vague intuition at the cost of destroying their research communities and world leadership (which takes decades if not a century to achieve).

But we're not building an AI society - to what end?

We're building tools to serve the human society.

A few people with way to much power are building AI to achieve their utopian, sci-fi dream. A few other powerful people are using that first group to achieve their post-democratic goals. It’s the most obvious top down attempt at imposing their vision onto the rest of the world. They aren’t building tools to serve anyone else than themselves

A few people with way to much power are building AI to achieve their utopian, sci-fi dream.

You mean dystopian. If it were a utopia everyone would be happy to welcome the new world order.

Math proofs are special because they’re verifiable, safe, and do not require physical experiments. You can perform exhaustive parallel search in simulation with RLVR.

Do you think this applies to say, surgery, as well? There are few useful problems that share these properties.

Not all of it would translate, but robotic surgery could be done on anesthetized animals in a mass RLVR way.

This is wrong, both morally and technically.

First off, I can’t imagine anything more torment nexus-y than throwing billions to automate and scale the torture of animals. If each token is a “cut”, how much suffering does 10 trillion training tokens (lower bound) corresponds to?

Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals. You cannot verify success so easily, either. Cancer cells, for example, could regrow over months. You would need to keep the animal alive and regularly test the animal, which would be difficult to scale. And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.

Do you think surgeons start performing surgery for the first time on people?

Sounds like you're saying that so long as the LLM with robot arms starts by operating on mice, cows and pigs you're fine with it

Here is a link to the original article:

https://poshenloh.com/posts/20260919-math-ai

The original posted link from OP is from Terry Tao’s website where the article was posted as a guest blog post.

It's astonishing how much more readable Tao's version of it is.

waiting for the xkcd for that

I always think of this one but I bet there's better

* https://m.xkcd.com/435/

Why do we need any job? Check "Bullshit Jobs" by David Graeber. Not saying being a mathematician is a bullshit-kinda job, but there have been so many made up positions way before this AI-era. In that sense AI isn't changing much.

There's one flaw in the evidence for the logic chain. The hugging face attack is used to demonstrate three things: the need for oversight today (fundamental to the article) and to demonstrate some kind of drift or unexpected capability gain, and finally to hint and some fundamental morality of the AI or at least the risk of drift from our morality.

I'd argue that what the hugging face attack illustrates is that large AI companies are motivated to have bombastic claims supported by bombastic demos. The model was clearly trained or encouraged to work as it did, as evidenced by the fact it keeps using this particular escape hatch.

And the fact that it aligns with prior and current calls for what very likely might be a regulatory capture / oversight capture move right before IPO. It aligns so well with this "barely constrained superweapon" narrative it might as well be PR.

How does your logic explain that OpenAI seems to want to hide the extent of the HuggingFace incident, and every week we learn from 3rd parties about new victims of the hack?

4D chess? They want others to find the hacked services, so the report of how dangerous the agents are seems more "legit"?

Pure math has been overvalued for a long time. Now that we don't need humans for it, it's really time for pure mathematicians to pack it in.

People keep talking about this like there's a finite amount of math to be done, and then the party's over. But that's never how math has worked, is it? Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post.

Math is meaningful because ... some people like to do it. The same as any other human pursuit. It doesn't need a reason beyond that. And AI won't change that. There will continue to be things to explore, things to find out, things that are maybe just at the edge of AI's reach and needs a human to decide whether it's worth continuing to explore or not. (Remember, AI isn't free).

So, IDK, I think for people who enjoy exploring math, there will always be interesting areas to explore. AI just gives us a better flashlight.

BTW I do agree that there's going to be an incident soon, whether intentional, accidental, or paperclip-factory, that leads governments around the world to shut all this down for some time, perhaps even shutting off access to GPUs entirely. It seems unavoidable. But that's just a temporary respite and skirts the core philosophical premise of the post.

It seems likely that there are an infinite number of math problems but only a finite number of interesting ones.

I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.

Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.

Why is x interesting? Just because it has a minimal description in P\I? That makes it interesting in strictly technical sense only.

I think that's a variation on the interesting numbers paradox joke. Statement: All numbers are interesting. Proof: Assume by contradiction that there's a non-empty set of uninteresting numbers. Then that set contains the smallest uninteresting number. That property makes it interesting.

An interesting problem must have a description that fits in a brain, at least for now. Your description-length argument assumes arbitrarily large storage.

the smallest problem that cannot fit in a brain would be pretty interesting

There probably is a finite amount of math to be done

The universe is bounded by rules, as far as we can tell, and not a lot of them

The universe is bounded by rules

That's physics. Not all math is physics.

The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.

At those levels math and physics are the same bound: the bound of things the universe allows to be conceived of inside it.

It’s possible the math our monkey brains + sand can ever conceive of in this universe is a low and accessible amount.

The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.

When you have a moment, please reference some proofs supporting this.

There are lots of thoughts you’re not biologically capable of thinking. That means the list of thoughts you’re capable of thinking is finite. That means the amount of math you’re capable of discovering, even if you lived forever, is finite.

That’s true of all humans and all constructs.

So however much math there is to discover, that’s the finite subset you’ll ever have access to.

It’s hard to prove what thoughts no human and no construct is capable of generating, but surely there are some, and it’s possible or even likely some of those are math-related.

Even if we accept that there are "lots of thoughts" we're not capable of thinking (though I wonder how you would define a thought if not as something that you can think), it still does not follow that the "amount of math" (as if it's a definite quantity) that one could discover is necessarily finite, if you lived forever.

By analogy, the "amount of math" we can possibly discover could still be countably infinite even if the space of all possible thoughts would be uncountably infinite. Countably infinite is still plenty big, and it is certainly not finite.

Which math is not physics (and chemistry)? Fundamentally, all these are same

Why does the universe being bounded imply mathematics is finite?

If you study the history of math you’ll find that the interest in certain problems changes based on culture and technology.

In our lifetimes computers have made a lot of combinatorial and graph questions meaningful that otherwise would not be interesting.

Well it's actually nice, maybe more people will be able to do world class math with the help of these tools. There are few fields as elitist and hostile as pure mathematics, most mathematicians I know build their whole life around their profession and their self esteem is strongly coupled to the fact that they can do things that most other people can't. Naturally, many will be devastated when (if) you take that away from them. That said I think AI is still overhyped and human mathematicians can easily outthink it in most domains, look at how difficult it is for an AI to write even a single decent paper, a good PhD student can easily outclass it in that regard. All of these impressive results were generated by having world-class mathematicians steer the systems using highly tuned prompts, so I see it more like a super violin that produces beautiful music when played by master violinists rather than being a fully autonomous orchestra which many people are led to believe this already is.

And maybe let's not only hear the opinion of two or three Fields level mathematicians with blogs, 99 % of the worlds mathematicians in academia might profit from these tools as they might partially close the gap between them and the world elite, making creativity and tenaciousness more important than having the right neocortical structure allowing you to outperform 99.9 % of other humans at keeping context in your head and making predictions, AI can do that better now with the right prompts.

Friendly warning to those who might not be aware: the vast majority of comments below posts like the above will be left by (otherwise intelligent) programmers who think mathematics is a closed system where one attempts to solve endless Olympiad-type problems. I wouldn’t take any of it seriously at all. Better to listen to what those who actually know what the subject is about have to say.

Unfortunately, mathematics (especially pure mathematics) is by its very nature very, very poorly understood by those who haven’t worked as a mathematician. Even worse, those who don’t understand are seemingly not at all aware of their misunderstanding and are entirely confident in their (very wrong) characterisation of the subject.

Could you clue us in? What is mathematics actually about?

To know what something is about, a natural way is to do it yourself.

Wow, I'm glad AI is going to start humbling people.

To know what something is about, a natural way is to do it yourself.

literally what software engineers were doing for decades though

software is mostly just simple math, for the most part, until you need to do something more complex for some hairy algos lol

Actually, if you were to convert most software into mathematical notation it would look incomprehensible. We are not interested in gatekeeping as much by making things look more convoluted than they need to.

Good question, and maybe I should have provided an alternative description rather than just criticising.

Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.

Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:

(1) observe some phenomenon in ‘reality’.

(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.

(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.

It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.

The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.

NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)

If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.

Despite loving Mathematics and having considered being a Mathematician myself, that wasn't a very robust defense. Also, even though I'm a big Terence Tao fan, I'm not sure he has sorted that out this defense in full himself.

This post which he forwarded was quite poor in my opinion. Confusing, all over the place with AI criticisms and promotion of the AI hazing being done by mathematicians.

X thousand mathematicians who want to protect their livelihoods signed a bunch of letters against AI. Duh. We've seen similar movements from every profession that has been displaced ever.

Terence Tao uses AI and has made a few good points on how to use it. But defensiveness leaks into almost every defense of the role of humans in Mathematics that I've read, even his own at times.

To be clear, I actually believe that Mathematicians aren't going away, but I dont have enough knowledge about the life of a professional mathematician to articulate a path forward.

This "path forward" is what I'd like to see. We need a top mathematician with enough intellectual honesty (Terence Tao qualifies, I think) to start this questioning with "there's actually no role for human Mathematicians" as one of the options on the table and go from there.

For me, mathematicians seem to be still having an inner discussion rather a making these essays for the more general public. In any case, I think that statements of the type "there's actually no role for human Mathematicians" are completely non-serious, so it'd sad that the discussion concentrates on that.

There’s a simple point to be made: if no one in the world understands an AI proof, what’s the point?

Maybe another AI could use the proof.

A lot of programmers consider themselves to be applied mathematicians.

Maybe more in years past when Comp Sci was a subset of Math Departments.

I did systems administration for a university math department for several years. I came to the conclusion that mathematics (and perhaps philosophy) were both topics where it was likely that no staff members in that department could describe "what goes on here" and that possibly even within the department, one professor may not be able to describe what another professor's actually doing.

The closest I could come to describing math is "some abstract process where imagined structures are characterized and extended; the most critical part of the process is identifying where seemingly independent structures are found to actually be fungible in some previously undiscovered way".

A simple example is

  "hey, did you know that x^i is the unit circle?" 

  "what's i?" 

  "i is defined as if you square it the result is -1" 

  "what does that have to do with circles?"

The clue is burried in the text of OP's article.

Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues.

The fact that this is a radical departure from the norm is part of why mathematics (and philosophy) is often seen as some intangible or ungrokable science to many outsiders, as they're generally approaching it from a perspective of "Okay, but why, what is this useful for?", and the answer "For the science of it" doesn't tend to land with people that aren't already passionate about said science/discipline and are just trying to figure out what it even is or involves.

Doesn't help that there is a pervasive sentiment in American Academia (not sure about elsewhere) about Math being *the* hard science, and I mean hard as in difficulty, so a lot of people get intimidated by it before they ever give it a chance very early on in their academic life and carry that through the rest of their education.

So there ends up being a rather small pool of people that are in(to) the field, and rather high friction for stimualting interest in it from outsiders from the way that it's taught, and a massive difference in the perspective of it's use between it's diaspora and the unmathed masses.

Mathematicians who do work on "real world" problems are mostly doing so with theoretical physicists, genomicists, and cryptographers. None of these count as what people consider valuable other than because they are hard.

That said, few 50 (or even 40) years ago would have predicted that completely abstract number theoretical computations about primes, discrete logarithms, and elliptic curves would be the foundation of our monetary system.

Everyone involved in AI should have read The Library of Babel [1].

It's about many things, but perhaps the most relevant idea here is that no information matters without understanding. We could generate all possible knowledge, but unless someone--a human--can verify and understand it, it doesn't count. The cure for mortality could be written on the moon, but if no one reads it, it hasn't really been discovered.

[1] https://maskofreason.wordpress.com/wp-content/uploads/2011/0...

While I get the point and it's true.

I'm a little more flexible, if the new knowledge (that human's don't understand) can be put into a mechanism and have an observable effect, I'd be happy enough. e.g. a new type of rocket fuel that burns 1000x more efficiently.

We already don't understand LLMs and they are having oberservable effects

but unless someone--a human--can verify and understand it, it doesn't count

Does it need to be a human or can it be some other form of life?

unless someone--a human--can verify and understand it, it doesn't count.

One counter-point: Humans have not figured out how general anesthesia works, but we use it every day to great effect.

https://en.wikipedia.org/wiki/Theories_of_general_anaestheti...

Only because we understand everything _except_ how it works.

how can we understand absolute everything and still don't know how it work ?

Trial and error :)

I wouldn't say that is "understanding". We must fully be able to explain to say that we understand it.

The part that does count is that we can verify that we does what we want it to do. We just don't know how.

Same with the black box part of AI. What the weights represent? Arcane dark magic if you ask me. What do they do? Well with LLMs we're all experiencing it.

The part that does count is that we can verify that we does what we want it to do. We just don't know how.

How do we know the person getting anesthesia doesn't die and a new soul/consciousness replaces them?

(We can ask the same question about going to sleep, er even walking through a door, but it's still something to think about).

you can verify it does what you want it to? How? Everything is a probability and so is your verification.

You're correct that verification *reduces* risk. It does not eliminate risk. Humans are imperfect. That does not mean that we should stop seeking understanding on principle though, because the journey towards that understanding improves outcomes.

Recall also that LLMs are not actually entities. In their current form, there is no sentience, there is no agency. They are tools. Therefore, their output must benefit the user that requested it. Right now, that's humans (and, ideally, the planet at large; we don't exist in a vacuum) and so it makes sense that humans should verify that output and try to ensure that it aligns with their goals.

That's not to say that the output of an LLM is useless, far from it. But we should still *try* to understand its output. It gives us at least some chance to notice flaws, and an even greater chance to appreciate the implications and tradeoffs of the solution it picked.

I don't disagree with anything you said. But the thread is about... LLM outputs, even if it cures cancer and solves world hunger, it is useless (it does not count for anything) if we don't understand it.

Doubt.

If I vibecode a video game and manage to sell it on Steam, the information definitely mattered even though I didn't understand any of it.

There are drugs that nobody truly understands how they work and they are being used by professionals in actual treatments literally right now. We know purely statistical facts like "if drug X is used for condition Y it will help Z% of patients" and can only speculate as to their mechanism of action. They are used anyway and still benefit a lot of people.

Yeah that information mattered when someone wrote it down, not when the LLM regurgitated an approximation.

The regurgitated approximation can matter just as much depending on what results it accomplishes. If I somehow make money out of this, then it's mission accomplished. Arguably, they might even matter more than my artisanal hand crafted software projects which never paid me a living wage.

but unless someone--a human--can verify and understand it

Why though a human? Mathematical proofs are very ivory-towery, but if OpenAI would solve a subkind of Cancer, without mortal humans understanding, cancer is still be healed.

Not unless humans were able to produce the medicine or perform the surgery or whatever is necessary to actually heal someone. In order for that to happen it has to be verified and understood by humans, like the other commenter said.

Does it, though? What if future AI can do everything from designing the cure to building a self-contained universal surgical medcapsule? "Just lay down and I'll fix your cancer real good, trust me, bruh." If it was shown to work, would you use it?

What's up with these insanely off base articles on HN this morning? Every single one of these is an unhinged pro-AI anti-human schlock piece.

I’m just shocked by how “intelligent” people believe a token guessing system can be

I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence. Even Doctorow talks of "brute-forcing" a solution. Brute-forcing leads to combinatorial explosion, so there must be something more going on here. Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).

Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).

There are many automatic theorem provers that do very clever stuff, just as the underlying theroy describes.

I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence.

It is absurd to waste time discussing whether it is inteligent or not. It is just an algorithm, we know how it works, and it does exactly what we expect it to do. LLMs are not magical things. The main difference is the scale: for Navier-Stokes they spent in 3 days more money that the whole mathematical community over the last 20 years easily.

By the way, I'm not saying that LLM's are useless, that I'm anti-AI or anything like that.

Turns out what you call "intelligence" was never needed to do math.

AI is not AGI right now, it can't think or come up with something new like a human brain. It still uses knowledge that was made by a human and was published on the Internet.

There is no evidence to substantiate what you are saying.

Every possible proof exists already as a possible generation in the grammar of lean or rocq. In no way does that mean we have discovered everything.

Why is Terry Tao even talking these arguments when all OpenAI did was a sophisticated brute force search with LLMs?

Give them Navier-Stokes in a vacuum

What do you think all the PhD's and post-doc students are doing...? It's not called graduate-descent for nothing :p

Brute-force search is called that specifically for its lack of sophistication. A "sophisticated brute search" is an oxymoron.

Same reason Gary Kasparov and Lee Sedol started their morning in one universe and ended their evening in another.

Brute Force or not, the game is being played and won without them all of a sudden, and even worse: at a level of output far beyond them

No it's not though. Can OpenAI prove Navier-Stokes from a vacuum with no human knowledge?

Replication of the development of mathematics by human civilization is a task for an artificial hyperintelligence.

The article wasn’t written by Tao.

We (humans) should help humanity flourish. I understand that not everyone agrees. I have been called a “speciesist” for being “too human-centric”.

this is insane. the "speciesist" people aside (the best way for all other species to flourish is for humans to eradicate themselves right now, which is completely mad), i have the opposite problem: the _axiom_ seems to be: we should help ME flourish. "Me" as in people who are raking in trillions for their own very special selves right now, at the cost of everyone else's future, while none of them can be trusted to hold my cell phone for a second.

if an industry commits to the axiom of helping humanity flourish

where does he see such industries, outside of maybe nonprofits?

As I understand you referred to Elon Musk. But this is Poh Shen Loh talking. If you don’t know who he is, look up or at least read his articles.

weird bias by authority aside, do you really think that the daily decisions in this world today, and anytime soon, will be made based on the "let humanity flourish axiom"?

In daily decisions perhaps not, and certainly not when we were in the cave. But from time to time, we will face fundamental questions that need addressing clearly and decisively, and I believe it’s the case with AI. The more advances we make, the more frequent such questions will arise. That is also one of the main reasons why we don’t see any alien civilization. It’s getting harder and harder to keep the entropy low as we gain more knowledges and cognitive power.

How's humanity tackling that climate change question then? Lots of talk. Even lots of action I would argue. And yet, not enough. Why do you think that's going to change with discussions around AI when there are trillions of dollars at stake?

Because it will never change if you not try.

the best way for all other species to flourish is for humans to eradicate themselves right now

That wouldn't do much for rice, pigs, chickens, or rats.

OBSERVATION: There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders decision-making control over its own future to the less-capable species.

Does the set of intelligent species only have one member? I cannot treat this observation as a general rule if there is literally one example of an intelligent species to theorize about.

I think the quoted part is saying that given n species and the n * (n - 1) binary combinations possible, there are no examples of the said observation. It doesn't suggest "the set of intelligent species only [has] one member".

Yeah but there is only one species that ponders its future.

I stopped reading at this point because I reject this initial assertion, and I assumed everything that follows is based on it.

There is nothing to suggest that an AGI will care about its future or who controls it. (whatever that means). that just more anthropomorphism.

We care a lot about our future so an ai pretending to be like us will also pretend to care. But we have a lot of biology driving us. Our emotions (Fear, Hate, Love, Pride etc) are baked into us in a way just wont be the same for ai.

My problem with this is that yes, looking at the species level from our sample of ~1 the most intelligent species wins out (although there seems to be a lot of debate as to if Neanderthals were more or less intelligent than humans).

But within the sample of the human species it is more common for less intelligent people to control more intelligent people.

This happens within political systems (political leaders are usually above average intelligence, but hardly the most intelligent) and within companies and other economic systems.

I don't see this fairly obvious point discussed, and not really sure what it shows.

I find Tao's writing on this very helpful, particularly because a lot of the ideas generalise to most knowledge work. It is darkly fortunate that some of the smartest and most articulate people on the planet are affected by the AI race - it gives me a little hope that alignment and control can be solved.

Humans still excel at posing the right questions and finding truly novel, insightful proofs. AI just brute-forces.

AI does NOT “brute-force”, else we could have had those math breakthroughs decades ago. Seeing this dishonest goalpost-moving every few months is so tiring.

we won’t have enough people for all the jobs that need to be done

Who defines jobs that need to be done? Businesses and institudes do.

There is no magical entity that observes the need for jobs and creates them at ideal rate.

I would say aggregate human desires determines what jobs need to be done businesses come about to fulfill the gap.

We need human mathematicians for humans.

But companies are our gods (don't believe? E.g. companies cannot die from natural causes). They don't need puny humans. They need AI.

Companies are nothing other than the tool of rich people used to control normal people. There's some sort of mass delusion that "companies" are evil. Guess what, a company has no own agency to destroy the environment, exploit people, profit from suffering, etc. It's just following the whims of its shareholders, who are humans with accountability.

I didn't say companies are evil. But there are companies and banks in our world, which are much older than any human could ever be. There are people who think they are in charge of these companies, however sooner or later they will be replaced and the company will still exist without them. In that sense companies are eternal entities, which gives them a godlike status.

Alright sorry if I was unclear. I wasn't saying that you specifically claimed companies are evil. But there is a general sentiment / understanding that companies are (being used as a tool for) doing evil things in general. I suppose whether you think that companies are (being used for) doing more good than bad is a different question, so let's not get bogged down in that.

My point however is that companies have no agency of their own. Their age doesn't really matter for that. There are guns that are older than all living people too. If somebody used a flintlock pistol to go around robbing people that wouldn't make the pistol a "godlike eternal entity". Or for that matter if somebody used a 200 year old shovel to create a really nice garden (in the case you believe companies are a net positive).

So you are saying a company is basically a tool, like a shovel? But AIs are also tools like shovels, they have no agency on their own. Adding an AI to a company makes sense. It is an additional tool.

We need to pivot math towards intuition and accessibility.

Math has built a gated, inaccessible institution which - by design or not - served as a moat.

I believe math has little to do with mathematical notation - intuition is much more important. One can have intuition but not be able to “read math” - much as many musicians don’t “read music”.

Reading math however does not guarantee ideas or intuition. And those are what math needs.

You would be surprised at the number of people (who study math formally) who just memorize proofs and having no clue what it all fundamentally is about.

I think LLMs are the single best things that were invented to make mathematics accessible.

I loved mathematics when I was young and wanted to be good at it, but I got burnt so many times by teachers who could not explain even if their life depended on it.

Asking an LLM and discovering how different theories belong together and what are genuinely open questions, along with different philosophical interpretations has been a wonderful gift.

I have long accepted that Mathematics is a language where you must go "all in" because for the most parts institutions are fundamentally unable to explain it in an intuitive way. There are people who "just get it" and are blind to the struggle of people who need a different approach in learning it. There's something about the field where a generational survivorship bias has created an environment where math is mostly taught to likeminded people, who then don't see why anybody could not see the difficulties learning it.

I know this is not true everywhere, but the only reason I could finish my studies was because there are some genuinely excellent online lectures on youtube, along with massive support from friends who have already been through the ordeal.

If I had had access to an LLM, I could have actually taught myself from scratch and be more playful in learning it. And no, I am not willing to learn lots of math "for fun", I can think of better "for fun" things that actually help my profession and cater to my actual interests and not just some academic ideal.

If I had had access to an LLM, I could have actually taught myself from scratch and be more playful in learning it.

For every person with this mentality, how many people do you think there are that aren't learning from the LLM at all?

LLMs are decent at explaining up to undergrad mathematics but that is because they are parroting / plagiarizing the hundreds of human-written books that have been written for the express purpose of teaching undergrads. LLMs are actually fairly terrible at explaining higher mathematics; humans are much better.

These “intuition” people are the biggest gatekeepers of all. Forget studying and working hard, unless you have intuition you don’t count.

Brett Victor’s essay will resonate, if you haven’t read it yet. https://worrydream.com/KillMath/

(Attn: pls read more than the title before you downvote)

One can have intuition but not be able to “read math” - much as many musicians don’t “read music”.

Is it really that common for musicians to not be able to read music? I'm not a musician so I wouldn't know, but it seems far-fetched. Like the story that Einstein failed math. I likewise doubt that there are many people who would be good mathematicians but unable to learn the language.

Sure notation is arbitrary and not the actual "truth" of math, but we need some way to talk about it, so why not just learn the accepted standards?

There might be at least one awesome author out there who never learned to write or type, but they're probably extremely rare, because becoming great at something usually entails a lot of practice, and that's hard to get when you're not making the effort to learn the basics.

In any case, LLMs are probably a bad way to "pivot math towards intuition and accessibility" because they, as language models, are great at generating a bunch of jargon that is hard for somebody with no training to tell apart. See the recent story [0]. Here the author (perhaps) succeeded with his proof, but was it because of superior intuition? Or just throwing compute at the wall to see what sticks?

[0] https://news.ycombinator.com/item?id=49755024

Most musicians in history didn't know how to read music, yeah. Less so in the present day but just about every folk song has spread by the word of mouth, and even in pop music many genres have a strong oral tradition (jazz, blues, rock, you name it), they play by ear and not by sheet.

The premise of the article assumes that the work (number of true statements to prove) is finite.

But Godel's Incompleteness Theorem and Tarski’s Undefinability theorem ensure an infinite space of provable true statements.

Neither LLM's nor humans can exhaust it. So yes, both mathematicians and LLMs are needed.

Both can contribute and there will still be work leftover.

OBSERVATION: There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders decision-making control over its own future to the less-capable species.

Many people ceed control of their path-finding to seeing-eye-dogs.

I consider myself an applied mathematician though technically I’m an economist (macro models, mostly) and operations researcher and differential game theorist (odd combination, but whatever). For context I’m 45 and I’ve spent most of my working life in corporate life.

I’ve always been at a disadvantage academically because I’m rather clumsy with my manipulations, derivations, and I’m a disaster at mental arithmetic. I’m also dyslexic. But starting in the late 1990s when I was in High School I started to become fluent with CAS (Computer Algebra Systems): first Derive, then the Symbolics capabilities of Mathlab, and ultimately Mathematica.

The whole transition is turning out quite well for me: I’m now able to delegate exploring my intuitions to increasingly powerful tools, I no longer have to haul the pyramid blocks up the ramp myself but I can drop them in by helicopter (as if were) and what I bring to the party is intuition and understanding.

I view it a bit like astronomy: in ancient times, before telescopes, keen eyesight was a prerequisite to be an astronomer. Later, after telescopes, anybody with eyesight had enormously enhanced capabilities of observation, and now, with radioastronomy and other forms of remote sensing (neutrino observatories, gravitational interferometers) the whole field has opened up to people who by virtue of being blind would’ve literally been excluded only a few decades ago.

Go forth and multiply: everybody can become a mathematician now. There’s an infinite number of potential universes out there with an infinite number of facts to prove and disprove. And while we’re at it: there’s so much more to mathematics than conjectures, proofs and counterexamples. Solve, model, approximate, fiddle around: as long as you’re not doing trite numerics on arbitrary systems of equations you’ve cooked up for your own amusement you’re good in my books.

Enjoy the tools. Keep your wits about you. Work through the steps that are presented to you. Build your intuition. HAVE FUN.

I think the astronomy and telescope comparison is missing some nuance. You don't delegate actual thought to a telescope, the thinking and acting on the observations you make with it is still done by you. With AI that is different; you get the opportunity to delegate a lot of thinking and acting to it (practically all of it if you really wish), at which point it becomes dubious to call yourself the creator or inventor of some idea.

You can use AI like a telescope, and alleviate your inherent organizational problems or other ailments, which in my opinion is a great use case, as it is empowering. But there will also be plenty of people who are not looking to alleviate any mental/physical quirks they have to do more, but simply want to make a "quick buck" for the least possible effort from the work and thinking of others, without adding much value of their own. Lowering the bar makes things easier for both, the ones who bring value and the ones who merely exploit in some context.

I no longer have to haul the pyramid blocks up the ramp myself... what I bring to the party is intuition and understanding.

Hauling the pyramid blocks is what gives people intuition and understanding. It is true that school systems usually have way too much computation -- it is easier to test and grade computation. But the only way that you were able to use computer algebra systems fluently is because you had internalized how algebra worked by hand. If we tell students that they no longer need to learn how to solve equations we are seriously depriving them of a mathematical education.

Also there is some instinctual understanding of the work moving a big heavy rock takes, and how to do it. No such corollary for algebra and hence some civilizations never discovered or understood it.

Maybe. As a parallel, I know C# and F#. They are programming languages. They are built on top of Microsoft products like the JIT and more basically Assembly language.

I have no idea how assembly works - in my years of experience I’ve never had an issue that required I dig that deep into it. Does that mean I can’t be effective with higher-level tooling because I do not know the absolute basics?

The entire point of a programming language design is to give you an abstracted environment with a complete coherent semantics so that you don't need to understand what's going on under the hood.

You can be effective at using C# and F# to solve other problems specifically because they are very well designed at the goal of abstracting over the lower level hardware. But if your goal is to build an intuition of how programming languages themselves are implemented and how computer hardware works, then, using those languages will be counter productive.

If you're trying to use math to solve arithmetical problems, then by all means using AI to help. But if you're trying to advance the math field itself, then actually knowing how math works is probably essential.

Put in more concrete terms: if someone wants to be a working mathematician without learning the fundamentals of math, then how do they even know what prompts to the AI are worth writing? In what way are they adding any value to the process at all?

While true that doing things the hard way is critical to learn a discipline, the OP’s point is about what is possible with AI once you’ve built those skills and intuition. I think in the future it will be common for those passionate about a discipline to strategically avoid AI until they’ve “built the muscles.”

After looking at the title for a day, the idea occured that, without mathematicians doing mathematics, those people would end up in banking, so seeing the top post is most amusing. One of my local contacts is a semi retired bankster, who I am going to prank by pretending that theres something stuck in her fangs, looks like the still quivering flesh of a young accounts executive, which seems to be the thing that happens to mathematicians, money that is, triggers some sort of primitive predetory instinct better left dormant.

I get the feeling that mathematicians are needlessly panicking because they don't really understand how AI works. They see the results, but they haven't thought enough about the methodology and so they don't have a clear picture of the true capabilities of thsoe systems.

For the n'th time: the recent successes of AI in mathematics are the result of a brute-force attack. See the proof for Navier-Stokes: 10k agents running for 88 hours; that's ~100 GPU years. How many human-years were invested in solving the same problem, before they were overtaken in the last few days by an AI? 90? Not even: that's just the time since Jeal Leray's statement of the problem in 1934. 26, if you want to count the time since 2000 when the Clay Institute named it as one of its Millennium Prize problems. But how much time have human brains spent working on the problem in either of those time periods? How many mathematicians have worked on the problem? 10k? Not likely.

And all that's without even considering whether the AI based its proof on carelessly shared work by the humans. Or rather, yes, let's consider that: it totally did.

Further. There have been several results in mathematics produced by AI but we have no information on how many attempts were made to produce similar results that failed. Because we don't have this information we cannot estimate the true capabilities of AI.

Yet we can observe that, for example, out of the six Millennium Prize Problems remaining open before the claim of a solution of Navier-Stokes existence and smoothness, only one (the aforementioned) was solved by an AI. We can assume that the AI companies (more than one) tried and failed to solve the others. We can even guess that they previously tried, and failed, to solve Navier Stokes itself, and only succeeded once the progress made by Buckmaster and Alpöge was in the training data [1]. That's a success rate of one out of six, or ~17%. That's what's gonna solve all of maths and destroy the tradition of mathematics? A success rate of 17%? Well, grab a Snickers 'cause we're gonna be waiting for some time!

Moreover. If we include in the list the Poincaré conjecture, proved by Grigori Perelman, who is a human, that's a score of AI 1-1 Humans. And that's being gracious: we have one Millennium Problem fully solved by humans, one solved partly by humans with a last-mile solution by AI. We have thousands of problems solved by humans in the last 2k years and how many by AI? A couple dozen? Oooh scary!

- Hey Hal! Prove that P ≠ NP!

- I'm sorry Dave. I can't do that.

What I'm trying to say, without the snark (sorry): Panic if you will, but the machines are not yet taking over. If you're panicking, panic for what you believe they will be able to do in the future. Because they certainly can't do hat in the present. They can't solve "all of mathematics" (whatever that means).

______________

[1] Yes it was. Buckmaster reported that he turned off the option to train on his data in July, after working on the problem with Alpöge for a year since September 2025. OpenAI claimed a solution in September, a month after they had stopped hoovering up Buckmaster's data. They had plenty of time to train on his data. Ask for references if you want them because I don't have them handy right now.

How many human-years were invested in solving the same problem, before they were overtaken in the last few days by an AI? 90? Not even: that's just the time since Jeal Leray's statement of the problem in 1934. 26, if you want to count the time since 2000 when the Clay Institute named it as one of its Millennium Prize problems.

I hope this isn't actually news to you, but: There is more than one human. There is even more than one mathematician.

If there happen to have been as many as four humans working on Navier-Stokes at any given time since the year 2000, then that's more human-years applied to the problem than agent-years.

How many mathematicians have worked on the problem? 10k? Not likely.

You don't get to count the factor of 10k once when working out how many agent-years OpenAI gave to the problem and again when demanding that for parity there would need to have been 10k mathematicians on it.

And all that's without even considering whether the AI based its proof on carelessly shared work by the humans. Or rather, yes, let's consider that: it totally did.

Let's suppose that indeed what Buckmaster and Alpöge had done was in the model's training data. Well, it didn't enable Buckmaster and Alpöge to solve the problem for Navier-Stokes (they could only do Euler), and it did enable OpenAI's model to do that.

Also: we don't actually know that what they'd done was in the training data; the latest bits of what they'd done that could plausibly have been in the training data were from before when Buckmaster said they progressed from preliminaries ("We worked through the literature and upgraded various preliminary results") to actually making substantial progress on the problem ("This was until about a month ago, when we had real progress"); and from what Buckmaster wrote it sure seems like a lot of the Buckmaster/Alpöge progress was in fact done by LLMs. (E.g., Buckmaster says that he and Alpöge have been working frantically to try to understand the proof for their Euler solution. That sounds to me much more like "an LLM did this thing" than "we figured out all the hard bits and the LLM did nothing more than filling in a few details".)

Buckmaster's own account of things is that all the really clever ideas were those of Córdoba and Martínez-Zoroa. (Which are already out there in the open literature, and there is nothing remotely improper about making use of them.) And my understanding (but, note, I am not an expert on fluid dynamics or PDEs and I could be wrong) is that actually the OpenAI model's construction is quite different from that of C&MZ. On what basis are you confident that "the AI based its proof on" what B&A did?

(For the avoidance of doubt: I am not arguing that what OpenAI did was OK. Even if they actually didn't train at all on any of the Buckmaster/Alpöge chats, it's very much not good professional ethics to hear that someone else is working on something and rush to try to scoop them, and there is absolutely no question that they did that. The question here is how impressed we should be by the model's mathematical prowess.)

A success rate of 17%?

A success rate of 17% on problems of this difficulty and significance is something that for any human being would be a career-defining triumph.

We have thousands of problems solved by humans in the last 2k years and how many by AI? A couple dozen? Oooh scary!

That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.

I get the feeling that mathematicians are needlessly panicking because they don't really understand how AI works.

I sunno if mathematicians would be having problems understanding how matrix multiplication, backprop, sigmoid functions, attention, embedding distances, probabilities, etc work.

As a group, they are probably more likely to understand it than everyone else.

The primary motivation for math is not to check off proofs, but to understand the universe.

A proof itself is only an articulation of understanding. Traditionally having one was evidence you had an insight. But if a computer generates 10,000 pages of technical goop then we don’t learn anything.

For example if tell you P=NP with no other information, it doesn’t change anything. There are mathematicians who believe and act on both conditions. What we hope a proof would reveal is how a verifier could be used to derive the solver (even if doing so was impractical).

Why doesn’t the author even mention this and immediately jumps to utility arguments for why the research is important for the government to fund?

Mathematicians insist that in many cases the route toward solving a problem is more valuable than solving the problem itself. This may have been true, and may still be true for now, but this seems to implicitly assume that future generations of AI cannot be creative in similar ways that human mathematicians have been. It assumes that the absence of human comprehension during problem exploration loses a valuable aspect of mathematics that cannot be recovered by aiming future AI at solved problems in a naive environment.

If we get to a future where all frontier mathematics contributions are by AI. A future where AI displays creativity in ways that expand mathematic exploration similarly to the ways humans have in the past. A future where AI explains frontier mathematics to curious humans. What will have been lost? Perhaps just "The pleasure of finding things out".

What will have been lost? Perhaps just "The pleasure of finding things out".

I'll kindly disagree on this front, because when I'm walking a path toward solving a solution, I mark a lot of steps for possible diversions to other paths hence solving adjacent or different problems with the method I have at hand.

Currently, AI takes us from A to B, and is improving on that front. However, the paths in science are not lines, but a trees. Methods are cross-pollinated from each other.

Human intuition enables this cross-pollination. AI works with a laser focus. Human intuition and resulting wide perspective sow the seeds for solutions in many areas at once.

Lol, this really struck a nerve. He has been posting non-stop for about a week now, wrote that lame open letter, gave a talk on it, etc.

What a disgrace, but I'm happy he's showing his true self to the public.

A village builds a clock tower so everyone can glance up at it and know what time it is. The tower helps as people meet more easily, work schedules align, and meals are shared. Over time, fewer people watch the sun and the clock becomes authoritative. When it stops working, nobody knows what time it is. Discord ensues and arguments follow as meetings are missed and people wait impatiently for repairs. An older woman in the village kept baking at dawn and never stopped watching the sun. She never relied on the clock. Villagers gathers at her door and call her wise, for her practice ends up inadvertently helping to re-calibrate the clock once it is repaired. She replies that she is not wise or special and was just going about her usual baking routine.