“Math 2.0” will need to value mathematical progress more holistically

> You literally just have to ask it.

why doesnt it ask itself before proceeding?

OP didn’t suggest that.

The bar has been raised. Everyone has to meet it now. An inelegant solution squatted onto the internet doesn’t count as discovery per se, even if it’s impressive.

If a random person is given a 60 page proof to digest and not the author, those hidden insights that _aren't_ in the paper might be completely inaccessible. Maybe the AI will "just" be able to provide the insights. Maybe. But pedagogy is tricky work, and despite these AIs being able to do all this fancy math we can't get them to write good cover letters yet, so....

Ultimately we might be left with just a bunch of intellectually unsatisfying proofs. This means way less drive to simplify the proofs or rework them.

End result: we generate a layer of "less efficient" mathematics, that won't get built upon. We will not actually have any shoulders upon which to stand.

There is literally not a single shred of evidence to indicate either of your supposed eventualities. The core technology of an LLM is sampling from a distribution so there is literally no way to make it deterministically robust (only probabilistically).

Hold up, that's an even bigger assumption in the opposite direction, and I don't see anything to support it.

At least in terms LLMs getting all the "AI" hype these days, there is no structural/mathematical reason to believe they won't continue to have the same problem they've always had of generating plausible text over rational text, and I don't think anybody even has a clear idea how it could eventually be accomplished.

I've seen "then the magic singularity occurs and somehow it solves the problem for itself", but I would classify that more as mysticism than engineering.

It is an old saw at this point, but what an LLM does still cannot be divided into hallucination and non-hallucination. This is literally an anthropomorphism trap.

Layers and layers of application-specific verification can reduce the risks inherent to LLMs, to a really remarkable degree, but nothing about what these tools are suggests that this problem will go away; it will just bubble up again somewhere else.

Where does this "will probably not hold in the very near future" come from? People correctly warn about extrapolating current things onto the future, but then just throw some vague "probabilities" without providing any argument why their "probably" is somehow more grounded than others.

This means when writing documentation, tutorials or commit messages, their output is often a garbled jumble. Assuming shared context, using invented terminology without explaining, leaking conversational states due to improper epistemic boundaries and failing to model the reader. This all usually leads to their freely generated explanations being terrible. Getting good explanations requires chaining questions that force them to line things up properly, which is not easy the less you know. These failures as something LLMs naturally struggle with make sense, given the nature of attention and RL with weak signals from human data.

Math is not merely a collection of proofs, it's a way of understanding. A proof presented in a manner that cannot be incorporated remains useless. It does not make it's way to physics like Riemannian geometry and matrix math did. This is no less true when done by humans too.

Your hallucination conclusion, checking if a proof is one, is exactly the counterproductive cost.

Most of us cannot verify that the claims in the OpenAI lore dump are in fact all correct. It will take tons of work from experts to do this. It took subject expert mathematicians to identify the discrepancy and disconnect in the Navier Stokes proofs, for example. LLMs will struggle to make use of their own proofs or turn them into knowledge that accumulates over time.

The act of proving is often more valuable than the proof itself. Human constraints and limitations force us to invent tools and abstractions that a 100,000 x 1M context swarm can bypass. The tradeoff from that AI swarm advantage is work that doesn't usually lend itself to being built upon. It's like doing all the side quests and reading all the books of an RPG versus min maxing a straight path with a guide. We might try to identify new abstractions, but the fact that we don't get access to CoT and that much of it will be illegible means mining LLM traces for what human mathematicians produce naturally will be a tedious chore.

Here's a longer article which goes into a bit more of the details: https://terrytao.wordpress.com/2026/10/05/the-future-of-math...

You’re making the following assumptions:

1. the exercise of struggling to find proofs was not productive, but this is precisely how new techniques in math were produced. Brute forcing solutions doesn’t lend itself to the creation of much new mathematics (except maybe the exercise of developing verifiable proofs)

2. the point of doing mathematics is to be “productive” in the first place. This is silly. Many people get into mathematics because of the beauty of understanding, for example.

If the focus in placed on maximizing some easily measurable output on a narrow perspective, situation is unlikely going to match a sweet spot of holistic equilibrium which is maximizing harmony and happiness through humanity as a whole.

I was bitter about that back in the day as I hoped for more answers, more matches, more "truth" about chess being shown. Soon after that community project Leela Chess Zero was started and not only surpassed original AlphaZero but added few hundred ELO points over it. Then the combination of NN and classical engines happened with NNUE and current Stockfish is again a few hundred ELO points stronger.

Today we pretty much know the truth in chess for all practical purposes. Human analysts/preparation experts focus on finding interesting path and opponent profiling (what is the most unpleasant for the opponent to face). They don't look for truth anymore. The game is doing great, it's more popular than it ever was.

But, I do think you are right that there will be some level of moving on. The spotlight is currently on maths and that won't last. It will move to some other area where there is more impact to be had. So while they might shift gears and put less focus on math, it will always be there as part of the portfolio.

I definitely think that this is marketing, just "with good side effects". My doubt is when they will be able to move to "marketing with better side effects", that is, research with more concrete outcomes (health, materials etc.).

Problem is, that type of research is much harder. Some doubt that progress in such areas will be quick (https://www.noahpinion.blog/p/wheres-the-intelligence-explos...).

Being able to present useful novel ideas would likely generate a lot of press, for a while. I don’t know how this would look since I’m useless at math, but Im sure there are plenty of unknown problems with massive implications, that once formulated can be solved.

Pretty much the only enterprise that historically pays some mathematicians handsomely is quant finance, but those people are actually compensated not for proving theorems but rather for statistical modeling and programming skills. And even that industry is so technologically driven these days that pure research mathematicians no longer hold a clear edge over strong programmers with undergrad level probability and statistics at their fingertips.

Go was a specialized application. All the math results come as a side effect of reading the whole internet, and it will keep reading the whole internet. It will keep practicing thinking questions. Actually, math might be one of the best ways to keep them contemplating and measure their contemplation abilities, so math will always stay in the loop.

Also, math might not be useful just for humanity, but also for AI, so the system might actively benefit from new math results itself. (Not sure if any of the recent proofs qualify, but future work might.)

To take this to the next step, what happened after deep mind pretty much solved Go is that they started looking for the next set of things that hadn't been done yet. It does strike me as very likely that this will follow that same path.

Actually, the problem is, somehow the skill of building new theories in math is directly tied to slaving hard over a problem. It's the very experience of slaving away that actually somehow causes ideas to form. Pretty much all mathematicians understand this. Yes, senior mathematicians now can form some new theories, but what about junior ones who will have very little experience in working hard on a problem by hand?

Of course, they could work on the problem by hand anyway, but they won't because no one will pay them when a machine can do it.

It's unclear what more could be expected than releasing the presumably already verified results and write-ups for each problem. Should they run a mathematics school too?

Then the comment goes on to argue AI labs were not interested in actually advancing mathematics, and that investments into AI were manically excessive.

IMO none of this follows and demand is there to justify the investments.

The comment then goes further to argue that AI labs were putting too much effort into pretending there was exponential progress rather than actually making progress.

The factual basis for this claim seems to be that OpenAI released math results and write-ups, and it's not even clear what more they could do on that topic.

That's a very negative opinion.

So yes, that is exactly what they should do. Alternatively, if they are too lazy or incompetent to put in the effort themselves, do what AGMAI proposed and fund a third party to help out.

And if it's hard to understand (which seems to be the most common reaction) it's not exactly devoid of noise either

There was an opportunity for people to work with the AI to produce a proof, now it almost feels they're working against it.

I think a problem is that math seems like a deeply toxic, ego driven domain.

I think he argued that e.g because the navier stokes millennium problem ist considered solved now, you won't get any recognition for being the first human to solve.(How would you even proof you solved it yourself and not just regurgitated the ai proof?)

And since recognition is the main objective, noone would spend time on dissecting the proof, and perhaps finding some unique approach to solving the problem, that could be transferred to other open issues.

And therefore the problem is now "poisoned". Since it's assumed to be solved noone will research it, and the potential revelations won't be found

But you are right, this is not OpenAI's "fault". The problem is - as others have said recently - that many people in mathematics want recognition for solving open questions more than they want the answers to the open questions. Everything about the economics and social environment of Mathematics will have to change.

I think that this is exactly the same split we see in software: there are those who mainly enjoy the craft aspect of building software, and are uninterested in the product or business they are supporting. Others are primarily interested in the production of useful software or building a platform or company.

I've always been in both camps myself. When it became obvious that AI was going to destroy the craft aspect - at least two years before it actually could do so - I became very discouraged, even depressed. But once it was actually good at building software, I became very excited about all the stuff I could now build. Sadly, I think a lot of people in our field have never had something they really wanted to build.

So effectively, stuff got proved, but people don't really understand how, so it's mostly fucking useless and done for OpenAI's marketing team, while also pissing off the maths world at large.

That community won't exist as many people simply won't even enter the field because it's reprehensible and contemptible, not to mention boring, just to read machine-generated proofs and verify them.

Collaborating on, or at least working on unsolved problems is what motivates most people.

AI is like a cheat code in a video game. You get to the end faster but fewer people want to play if the cheat code is always on. You can't turn it off either because the very challenge is to do something unique.

"When the architect completes a fine building, he removes the scaffolding." - Carl Friedrich Gauss

I’ve not heard of it either, but code is an abstraction of math, so I don’t see why this couldn’t theoretically happen.

Anecdotally, I’ve started spending time advancing my math skills beyond the early college level I stopped at and I’ve frequently found I already know concepts of more advanced math - I just didn’t know what they were called or how to apply them to an equation on paper, but I’ve been using them for years and intrinsically grasped the underlying academics.

Usually it's the opposite. "That's in prod? And it works? It shouldn't work and I thought it was doing something else. Why does it work?"

Every software design pattern came from elegant code. People wrote code, summarize code, learnt from code, and taught code. That's discovery

The S fell short in actual reality for the most part, as it was merely a hiring requirement. A hiring requirement that didn't even make sense, because the skillset of academic CS only marginally overlaps with the skillset one wants to hire for.

Material engineering at least for the most part has actual real-world applications where one can push humanity further. CS (as practiced, not necessarily the idea of real CS but the CS we got due to it being used as a hiring filter) for the most part is just self-referential spinning with mostly unclear results.

There is real impressive work being done in that field, of course, but I'd argue that the majority of it over the last decade or so at least was just performative nonsense.

Maybe by again allocating new resources to other fields, what hides under the label CS can become more pure actual CS again. I think that would also be a much less miserable experience for everyone involved.

The will to implement the stuff we learned, instead of letting dark patterns and churn for the sake of churn get the better of that :P

I don't think it's motivating to solve a black box by having AI generate another black box if what you want is to understand how the thing worked.

Not the mathematicians I know. They’d happily drop the academic admin stuff, but they’d absolutely keep doing mathematics in much the same way.

Quite a lot of people are not happy they aren't elite anymore, and many have spent years to decades to arrive here.

Simply put you invest years of your life to establish a kind of distinction over others, and that goes away. That hurts.

But its not something surprising. Most of these competitive programming problems were actually English languages puzzles, because you couldn't dial up the mathematical difficulty anymore making it a fields medal problem. And in most cases in simple language weren't even that hard to begin with, and you could look up solutions to these problems in an hour of Google searching.

Since the Greeks we've had the idea that "Being and thinking are one," or that Being (in the sense of all of existence as such) has some essential unity with thought, and therefore can be thought, and expressed or submitted to the logos or reason. Being is in some sense fundamentally intelligible, and mathematics is the most developed, exacting, and articulate expression of Being.

Logic was understood in this older sense up to roughly the the mid to late 19th century. This is why a work like Hegel's Science of Logic begins not with syllogisms or propositions but with Being and Nothing. But this was forgotten after logic was mathematized by the English around the time of Russell, and its connection to ontology was gradually overshadowed by a focus on epistemology (still, it should be remembered, originally as a means of getting back to ontology).

There may be truth in art, but it's always haunted by its own historicity or contingency, which is to say untruth. Mathematics seems on the contrary the only really timeless, absolute thing we have. Part of what makes it captivating is stumbling on a construction or concept or proposition or theorem that simply must be, independent of us.

The AIs are certainly now more than automatic theorem provers, mechanically traversing some space of true propositions. They are able to push things forward and connect seemingly disparate domains to get to a proof, but to my mind it remains to be seen how well they will be able to form new concepts and definitions.

Imagine the controversy surrounding Cantor, for example, but put an AI in the place of Cantor. If an AI proposed something like the (infinite) hierarchy of infinity, would we have accepted it? What would the intuitionism debates have looked like? Would they even have taken place? And aside from that, has it actually been shown conclusively that an AI could propose such a thing?

There are lots of attempts right now to recover a humanism for mathematics, or restore man's pride of place with respect to it, but maybe we don't need to worry about that. Tao's attempts to preserve the mathematical community, while allowing for practices to change through the crisis may look like a kind of rearguard action, but seems reasonable to me and not really dependent on any kind of humanism. It's a way to avoid the question for now while things play out (and not conservative/reactionary like Scholze and others), which may be exactly what we need, because after all, perhaps we still don't understand why we do mathematics, what it's really for, and what our relation is to it. Whether it's enough to preserve funding is another issue.

Its different when your own job is on the line.

When human manual arts were being automated away it was supposed to be not only acceptable but any complain and you were told you were a progress blocking luddite.

Now that mental labor is getting automated, the response to automation is very different.

I feel like we’re circling back to that 2010s energy of “everyone can be an entrepreneur.” Now it’s “everyone can build software”

Most people would be happier with La Marzocco coffe machines which costs thousands of dollars, but if you get an OK shot with a 100 USD DeLongi, then the choice is clear for majority of the population.

Like given the choice, the vast majority of people would prefer one quality game like Minecraft, LoL, or Fortnite, vs. thousands of one-shot generated games, and looking at user playtime this is exactly what we see. If anything AI is just going to entrench these pre-AI franchises even more.

Everyone should have a portfolio of prompts that are indecipherable by other humans but when fed to a frontier model, produces shocking one paragraph english version of a 10000 line lean proof

Obfuscated prompt grant contest

I'd end up with the buckets

a) Fully human

b) Hybrid human-AI

c) Fully AI

You say marathons are just for fun, but prior to the wheel it was the only way to get around. (Other than horses in some places)

So it’s not that running is immune to automation, it’s that we’re already post-automation and that only people doing it for fun are left.

There's a reason there are maybe a few hundred professional marathon runners in the world vs tens of thousands of professional mathematicians. Bucket 1 is basically an "amusement for the upper classes" type of deal. Any field that goes in that direction would have to shrink down massively.

It also devalues the field in my opinion from something really profound with actual impact in the world to a somewhat vain leisure activity. (basically going back to gentleman scientists) But I know other people would see it exactly the opposite way.

I didn't follow through to the PhD, but I spent few years building up to understanding of Fluid Dynamics and Functional Analysis to come close to NS. It's intriguing that it's "solved", but what interests me is then "what do we learn from it" and what lies beyond in non-linearity.

In the 10 years I've spent away from academia, I still cherish what Math taught me best: looking at equivalences and I still feel the kick that I surely wouldn't want an Agent to do on my behalf. NS was never the point. And who can't see that I can only feel that they missed out.

It's fascinating how people like to push any statement to its limits, because it circles around to absurdity and they believe they've made a point. Moderation seems to be chasing you, but you clearly are faster!

You may say "hey, before AI people payed for math salaries even though they didnt understand the math or the economic outcome". But the issue is there are now "mathematitians" trying to convince not to fund.

In this new reality, you will get "mathematitians" trying to convince that only AI maths matter. And on the other side someone speaking about "understanding", "taste", "community". And the people deciding to fund dont have the skills to differentiate. So they will fund the AI boosters with a higher probability.

Thats how the job of professional mathematitian dissapears. By being replaced by something that on the surface looks similar, but its just an ugly copy.

In my view, over the last century, math has turned into an intellectual analogue of extreme bodybuilding competitions. A navel gazing runaway optimization in making useless stuff just to demonstrate cleverness. That's fine, why not. But society has no obligation to fund that, just as it doesn't fund other extreme hobbies. Ideally if we ever get something like UBI, math can be still their hobby.

In my experience, every new model release allows me to go further, although every time i see every time the limitations, and i identify where i add value. And this value gets bigger every time.

But on the other hand, every model release reduces the amount of people that are able to value this "added value", because it requires more skills.

So we have the paradox that the added value i can bring on top gets bigger and bigger, but the perception of the economic value for the majority of the population gets smaller.

Yes, but that's a social issue, external to technology but exacerbated by every new technology, AI or not:

UBI should be a thing: let people work on what they find fulfilling, instead of having to work to survive.

AI could help design a system for UBI that everyone agrees with, since it's so good at maths and shit now.

This problem HAS to be tackled. Removing/slowing AI will only kick it further down the road, not eliminate it.

Their applications?

For example I'm not a mathematician but I love thinking about weird "useless" shit like how math might be like for aliens? Are numbers as fundamental as we assume? i.e. humans developed math for "arithmetic" first, then latched geometry etc on top of that. We took ages to admit zero and negative numbers.. what if an alien species develops math for "navigation" first, and starts out with complex numbers right away!?

> What happens when humanity becomes obsolete?

There's an infinity out there to explore.

> And what happens when the machines that cause this obsolescence are controlled by a tiny amount of people, who suddenly don't need the rest of us?

That's a social problem we needed to tackle more than 100 years before AI or even computers appeared.

Apparently the minutes hand was added to clock to keep time in factories, for the benefit of the factory owners, not the workers — something I learned from this 1991 show from the BBC with Terry Jones: "So This Is Progress" https://www.youtube.com/watch?v=-Em96NVxO9Q

The less said about Gary Marcus the better.

This is a technology not like prior technologies. Are we okay if the technology discourages a whole generation of Mathematicians? If the technology leads to 10x fewer mathematicians -- what impact does that have on the field? These are the questions Tao is asking. And I don't think he himself claims to have all the answers, he just doesn't wanna see the math _community_ die.

So as measured by utility, I absolutely believe we don’t need humans doing science into the future. I’m sure people will continue doing it, but not for utility, for enjoyment - as a hobby. Probably we’ll all end up as dedicated hobbyists.

However, with AI it actually may become so cheap that the scattergun random approach becomes more viable rather than less. It’s when human time and resources are scarce that you need to optimise. The hobbyist approach may therefore ironically continue, but without the hobbyists.

Note, I think the debate is mainly over what research should be funded, not whether any research should be funded.

Thats great! So in other words it should be perfectly fine for AI to solve all of the supposedly useless math problems so that it might possibly be useful later. No need to worry about the mathematicians hobbies here.

> things that I do and that my colleagues do, this kind of like curiosity-driven, you know, applied math, computational physics type research has always been justified by, I would argue, intentionally blurring the line between what I would call, you know, science as product versus science as process

> science as product is very kind of clear-cut. It’s, you know, things like, you know, cure cancer, solve nuclear fusion, generate, you know, clean energy.

> And then there’s science as process, which is kind of the curiosity-driven stuff about, you know, like, “I want to understand protein folding,” or, “I want to understand, you know, turbulence,” or, “I want to understand quantum gravity,” or something. And broadly speaking, we have tended to justify the latter by kind of laundering it through the former

And the examples he gives are actually the more defensible ones, he talks about a friend of his working on some abstract algebra under the false guise of cryptography research later.

And it’s not just him saying it, this is simply true. He should be lauded for admitting it publicly, this is the only way any progress is made. At least, it used to be. Now it’ll be AI instead.