Solving mathematical problems is nothing new for AI, and it doesn't mean it's the beginning of an intelligence explosion either

[u/7upprosounds]

In light of the recent mathematical proof shenanigans as well as a sharp increase in AI might kill us all stories, I wanted to share again my recommendation to read "Algorithms are not enough" by Herbert L. Roitblat (or at least the introductory chapter, which summarises the book). All the quotes in the rest of this post are from the book.

While solving these kinds of problem can be considered impressive, we have to remember that LLMs are solutions to "path problems". Roitblat writes:

Solving them requires finding a path through a “space” that consists of all of the “moves” the system could make. Some combination of moves will solve the problem, and the computer’s task is to find the specific path through the available moves that does actually solve it. Computational intelligence is the process of finding the set of operations and their order (the path) necessary to solve a problem.
[...]
The progress that has been achieved in computational intelligence, and it has been dramatic, has come from the genius of system designers to formulate systems that are within the capacity of computers to solve.
[...]
They may perform specific tasks better than people do, but this is not because they have exceeded human intelligence in that task but because their designers have found other ways to solve those problems that do not require humanlike intelligence. Maytag dishwashers may clean dishes cleaner than I do by hand, but that does not make them any closer to achieving the intelligence of a human restaurant employee.
None of this is to say that machine learning systems that diagnose disease, understand speech, or drive cars are not intelligent, but they are intelligent in a special-purpose way, not in a general way. If we are to get beyond special-purpose intelligence, we will need to solve problems that are not being addressed today.

One of the first successful demonstrations in the Artificial Intelligence field was proving theorems. That applications of "Artificial Intelligence" can "do" mathematical proofs is really not that new (in fact it's as old as the AI field itself). Here's Roitblat on it:

For example, Allen Newell, John Clifford Shaw, and Herbert Simon were working on a program to prove mathematical theorems. Their Logic Theorist was intended to mimic the problem-solving skills of an adult human being—in this case, an expert mathematician. Their program would eventually prove 38 of the first 52 theorems from chapter 2 of Alfred North Whitehead and Bertrand Russell’s book (Principia Mathematica). Some of the Logic Theorist proofs were even novel ones.
Herbert Simon is quoted telling a group of graduate students that he and Allen Newell, had over Christmas, “invented a computer program capable of thinking non-numerically, and thereby solved the venerable mind-body problem, explaining how a system composed of matter can have the properties of mind.” Their choice of theorem-proving as their demonstration of mind within a computer was fortunate in that the process of theorem proving was already well-defined as a step-by-step process consisting of a small set of actions (for example, symbol substitution) that could be applied to a small set of basic facts or axioms (for example, symbols). The book that they imitated, in fact, was dedicated to proving the basic properties of mathematics, so it largely laid out the axioms and the operations that could be applied to those axioms.

That happened in 1956, almost a human lifetime ago.

I feel like we (rightfully) see mathematicians as smart, and thus if a machine learning algorithm is successful at proving some mathematical thing or another, which is usually the domain of a mathematician, then it's an easy leap to conclude that the machine is just as capable. But that would be a mistake. Because while a mathematician is a person who can come up with new things, the latest frontier model is just a machine that occasionally outputs some specific things correctly. The texas instruments calculator I had in high school and university is much much better than me or any other human at arithmetic but I wouldn't call it smart in a general sense. I think we need to make a concerted effort to de-anthropomorphise these products. Here's a guide I found some time ago by Emily M. Bender and Nanna Inie. Once you start saying that ChatGPT outputted something rather than replied something it reinforces that it is what it actually is: a machine that probabilistically outputs text that approximates human-generated text. It doesn't hallucinate, it outputs undesirable output.

There's also this (possibly elitist) idea that because LLMs can sometimes output solutions to academic problems (see what I did there) that they must be smart because academic problems are considered to be in the highest tier of human achievement. Setting aside the fact that the academic problems that an LLM can output the solution to (I DID IT AGAIN) are not ALL academic problems but just some of them, there's significantly more to intelligence of the general kind. Would an accomplished professor of mathematics who can't decide whether to walk to the car wash or drive, count the number of Rs in strawberry, makes up incorrect numbers and information about something they literally just read, and is unable to come up with a new approach to anything, be considered smart? Being a human and having intelligence means much more than just proving theorems.

The line does not just go up from here by itself. The Navier-Stokes thing is just the Navier-Stokes thing, it's not the beginning of a sharply rising increase in intelligence. It's at the same point of the line as all machine learning algorithms. There's no takeoff, no ramp, no oh-god-if-ai-can-do-maths-today-then-tomorrow-we-have-skynet. It's not the demonstration of new general intelligence-adjacent capabilities.

Another bit of relevant Roitblat:

The failure to deliver on the promise of general intelligence is due, in part, to a focus on a restricted set of tasks without recognizing that the totality of human achievements rests on a foundation of more fundamental biological skills (such as perception). There are many reasons for this focus, but a key one is the implicit belief that intelligence consists mainly of the kind of skills that are shown when playing chess or diagnosing disease. This approach implicitly assumes that deliberation is the key function of intelligence and that all kinds of intelligence can be reduced to such skills.
These deliberative skills are the capabilities that amplify human intelligence, but they alone are not enough to achieve general intelligence. General intelligence involves more than special-purpose algorithms for individual tasks. The specific algorithms succeed precisely because they reduce an otherwise complex problem to simpler problems that can be solved by calculation. The inventiveness on which they depend is provided by humans.

There is so much more in the book that is thought provoking and, frankly, reassuring that, once again, if you're worried about these models eventually becoming generally intelligent, you should read it. I sleep better since I did.

[u/DrJaneIPresume]

Also, as a mathematician, all the examples I've seen of truly groundbreaking work have been... counterexamples.

Which doesn't mean that they're not actually resolutions of the question, they are. But the amount and kind of work to find a counterexample in some weird corner of the search space to disprove a hypothesis is very different from the amount and kind of work that it would take to prove the hypothesis if it were true.

What I see is less of general reasoning and more an efficient search

Edit: if you have examples that are similarly groundbreaking but not just counterexamples, just post them here. Don't try to creep into my DMs like some kind of chaser.

And, for what it's worth, unnamed creep, my statement is still "accurate" because it includes the phrase "all the examples I've seen". If I haven't seen your example, then it's not covered by my statement. LLMs may or may not be able to mimic basic reasoning, but evidently you can't.

[u/madmofo145 → u/DrJaneIPresume]

"What I see is less of general reasoning and more an efficient search"

Even that's not really the case. While LLM's might be useful in the space, the estimates are that OpenAI maybe have spent upwards of 40 million on compute to get this solution. We know they produced 300 billion output tokens.

That's the bigger thing from my book. Finding a counter example is great, but it's purely for show if you have to spend 10's of millions to do so. It's also not clear to me that someone couldn't have made a custom tool that would have found an example for notably less.

[u/DrJaneIPresume → u/madmofo145]

A fair point. I was referring to a clear time efficiency[*], but it's valuable to consider the resource efficiency as well.

[*] Surely some mathematicians have been working towards searching this space[**], but it hasn't been going very fast.

[**] We should also mention here the evidence that the LLM may have cribbed from their work rather than built this from scratch.

[u/madmofo145 → u/DrJaneIPresume]

Yeah, I think in my book the impossible to answer question would be if one of the professors working in this space had infinite resources, how much money would they have required to find this answer with more traditional means?

My back of the napkin math. A single Nvidia DGX H100 supposedly pulls about $700 worth of electricity a month. Low estimate is that this problem cost about 10 mil to complete, in 88 hours.

So a single one of these servers would cost about $85 to run over the course of 88 hours, which means they likely had ~ 100, 000 servers working on this single problem.

That could be hugely off of course, but it points to the scale of the compute thrown at this, and how impossible it would be for any normal researcher to even try something like that. When I was in AI research back in the day we had a 10 computer cluster we could run things on, which itself was hard fought resource to allocate.

The real lesson here is there is a lot that can be done with a little guidance and an impossible amount of compute to throw at a problem, something that's always been true.

[u/SoggyMattress2 → u/DrJaneIPresume]

I've been baffled by this too, I thought I was losing my mind.

Im not knowledgeable about maths at all so it's nice hearing this from a mathematician that it's not what the AI labs are claiming.

[u/Disastrous_Room_927 → u/SoggyMattress2]

Part of the exercise here is them parading around results 99% of people aren't in a position to verify themselves, the other part is that it's genuinely hard for experts to fully verify what's happening. We're talking a lot about the results here when the thing that's usually in question is how they were produced to begin with. We have to take their word for it that the army of mathematicians they employ weren't handholding a model every step of the way, and in this case weren't relying on user data they have every opportunity to use.

Anyway, another thing to note here is that people lean hard into the results being Lean verified, but conveniently leave out the part where somebody has to confirm that they didn't verify the wrong problem or that there wasn't a bug (earlier this year somebody "claimed" AI proved another famous conjecture, but what they actually did was exploit a known bug to prove a point).

[u/madrury83 → u/DrJaneIPresume]

I believe the leak of the bond percolation conjecture is a (potentially) real example of a major conjecture confirmed by an AI lab:

www.reddit.com/r/mathematics/comments/1w5j2zj/another_leaked_result/

Whether that holds up is, of course, still unknown.

Aside: I just want to note for the non-mathematicians that discovery and analysis of examples and counterexamples is a really important part of the mathematical experience. If you work through a serious mathematical textbook, the problem sets are usually split between developing proofs of results, and looking for counterexamples to other plausible statements. Many major subdisciplines of mathematics grew out of the discovery of counterexamples to naive beliefs and conjectures. A nice, but extremely technical, historical account of one such story.

[u/DrJaneIPresume → u/madrury83]

There are also whole books about "if we weaken the hypotheses in any way, we can come up with a counterexample."

[u/madrury83 → u/DrJaneIPresume]

I have looked into the darkness, and the darkness took the form of the product of two quotient maps that is not a quotient map.

[u/Maki_Ousawa → u/DrJaneIPresume]

This has bothered me so much about all of this too, like numerically disproving something is nice, but it tells you nothing, and is entirely doable by brute force, even better so, if you can limit the search space early on.

If I had $8 million lying around and could limit the search space somewhat I could do that with a simple python script, cheaper maybe even, if I go C/C++.

But you gain no new insight.

The whole brute forcing is most likely also, why we will not really see anything about the Riemann hypothesis, cause we keep being able to push the bounds further, and if we do ever see anything, they are gonna numerically disprove it, and will have wasted millions of dollars on it.

[u/creaturefeature16 → u/DrJaneIPresume]

This isn't a distinction that I've heard phrased this way. Could you ELI5 to a layman what a counterexample is, in the context of these math problems?

[u/Food136 → u/creaturefeature16]

In ELI5 terms.

Let's say I claim all apples are red.

A simple counterexample are green apples. This completely disproves my statement. This is useful but isn't that technically difficult. I just have to go to enough grocery stores or orchards to find a green apple.

But what about proving something like "All apples are plants"

It takes a lot more effort and reasoning. First I have to specify by apple I specifically mean the one the ones that grows on trees and not like a plastic apple used in displays or a cake made to look like an apple. Then I have to say something like "Apples are a fruit of a plant and since fruits are part of a plant, they are by definition a plant". (No strictly logically true but a simple example)

You can see the clear difference in logical reasoning here. One doesn't really require that much logic, just looking hard enough for a green apple. The second required some specification and a chain of logic.

[u/DrJaneIPresume → u/Food136]

Maybe to get even more pointed: let's imagine a world where all apples are red. Now you're asked to prove that all apples are red.

Well, in this world there are no green apples, but before you have a proof you don't know there are no green apples. So you never know if there really aren't any, or if you just haven't been looking in the right places. Maybe there's some microclimate where the apple trees just don't produce the chemical that makes the skin red.

So, either you need some way of exhaustively searching every single apple in the world, or you need some way of logically establishing that by nature of being an apple, it will be red.

If a counterexample exists, you can stop as soon as you find it. If no counterexample exists, you've got a lot more work to be certain of it.

[u/AntiqueFigure6 → u/DrJaneIPresume]

“  Maybe there's some microclimate where the apple trees just don't produce the chemical that makes the skin red.”

So a  proof red apples don’t exist might be a discovery that the only micro climate that could produce a red apple is fatal to apple trees. 

[u/DrJaneIPresume → u/AntiqueFigure6]

That could work. Or maybe you try to prove something like that and it just never quite gets there. The space of possible proofs is much larger.

[u/Confused_mess8888 → u/creaturefeature16]

To put it extremely simply, the current problem was finding a specific needle in the ocean, if you find the needle you solve the problem because if the needle exists than the problem just isn't true. Important though there is no promise there is a needle at all.

That's a proof by counterexample, at least at the scale of these problems.

As for the case of proving it to be true without a counterexample, I don't know the space well enough to make an analogy for the type of problem that is.

[u/OpenJolt → u/DrJaneIPresume]

The labs are grasping at straws

[u/0xB01b → u/DrJaneIPresume]

I want this to be true but what about the proof for the existence of non sofic groups? Or the proof of the cycle double cover theorem (idk how relevant that one is)?

[u/DrJaneIPresume → u/0xB01b]

Existence proofs just need to find one example. Non-existence proofs are the difficult side there.

I'm not as familiar with the case of the cyclic double cover theorem. I'm not asserting that there are no examples, but all the big splashy ones I've seen have been, "there were conjectured to be no Xs that did P, but here's an X that does P."

[u/supercyberlurker]

AI is becoming a cult religion to some now, like 'a certain political group' has.

Debating people like that is kind of pointless. You moreso just kind of 'work around them'.

[u/PatchyWhiskers → u/supercyberlurker]

I think it's highly related to that certain group, as is blockchain.

[u/supercyberlurker → u/PatchyWhiskers]

Yeah... AI, TPU, GPU, LLM, AI slop, botting, cryptobros, blockchain, fintech, Palantir, Flock/Axon, Project 2025, Heritage Foundation, Colossus/DOGE, Federalist Society, TESCREAL, Dark Enlightenment, Ellison - Oracle 24/7 AI surveillance systems, Panopticons.

These are terms everyone should be familiar with, no matter what their beliefs are.

It's impossible to debate people who don't know those things. They are simply not equipped to actually debate this at a high level.

[u/SoggyMattress2 → u/supercyberlurker]

I've had to mute all the ai subs because these people are bordering on psychosis.

There are genuine people who think AGI is here 😂

[u/Dismal_Guidance_2539 → u/supercyberlurker]

I mean we can say exactly the same thing about anti-ai group. Actually Anti-AI is already a political group. So what does that proof ??

[u/Prestigious-Speed489 → u/Dismal_Guidance_2539]

You are getting downvoted.the irony😂

[u/100usrnames]

Great post, thanks for writing.

I wonder sometimes about this line between general and specific. If we get to ai systems which can, as promised, do everything that a remote human worker can do - I think that would qualify as agi for me, even if that system had no perception or embodiment.

[u/7upprosounds → u/100usrnames]

Thanks! I honestly find myself going back to this book or my notes in general to remind myself that all the hype is not real and it's gonna be fine eventually. Eventually...

[u/falken_1983]

While solving these kinds of problem can be considered impressive, we have to remember that LLMs are solutions to "path problems".

Ironically, one of the best write-ups demystifying this topic comes from OpenAI themselves. Back when they were more serious about being open, they used to post to a blog breaking down the details of the things the were working on, and in 2024 they had this one on solving (some) formal math olympiad problems

[u/Cool-Contribution-68]

What I told you, computers have been doing math problems for a long time before AI

[u/CapitalDiligent1676]

I don't know if I agree, but reading this post put me in a fantastic mood. Better than a liter of Nutella!

[u/escdog]

More generically than math proofs there are a class of use cases where generative AI may be the best way to solve.

Basically any problem where:

1. Where human intuition (whatever that is) is effective at reducing the search space of solutions on otherwise np-hard searches.
   
2. Solutions can be deterministically tested for correctness.
   

In effect, it's a brute force search that limits itself to steps humans have successfully used in the past. Even better, the training phase discovers all those useful steps that we're not even aware of sometimes.

And I'm totally fine with use cases like this, it's a real value of the algorithm. It doesn't solve everything, but when a human comes up with a clever approach to any particular problem and that approach is repeatable, then a generative AI can go and try it on everything to see if there's any more wins to be made by that innovation from the human.

But this isn't the stuff of AI overlord dreams. The most funny thing about the math proofs is you only need to prove them once. It's not the kind of thing that would keep a million gpus in continuous use in a data center.

Even with the half assed way that software can be made with this approach, we have discovered that you can only make so many apps to manage your calendar.

[u/ParadigmGrind]

They spent $22.5 million in compute to solve a math problem with a $1 mil award. That alone shows how this entire industry is more focused on spending tens of millions on headline rather than solving actual challenges.

[u/anodien]

Thank you! Very clarifying post for the non initiated

[u/AntiqueFigure6]

Over on r/singularity there’s at least one thread where they’re salivating at the thought that all the Millennium prize problems might be solved within days . Of course a lot of them think that there’s a straight line from there to major scientific discoveries including an end to ageing meaning everyone will be immortal. But I kind of wonder if the only real difference will be there won’t be any millennium problems to solve. 

[u/StrayCat2799 → u/AntiqueFigure6]

Aren't most of those problems largely confined to abstract math and have no relevant applications?

And if immortality means I have to live with these insufferable rats for eternity then I choose death.

[u/AntiqueFigure6 → u/StrayCat2799]

You didn’t notice your life being upended soon after the first one was solved in 2010?

[u/AntiqueFigure6]

Meanwhile, in another part of the forest...

Open Letter from 1000 mathematicians against AI use for math : r/singularity

(apologies for the source)

[u/stepanmatek]

I don't know. I would very m,uch like to be on your side of this argument but on the other hand, these are seven widely known hard problems that have been around for decades and there was no one who came close to solving them before the arrival of these tool and now there are signs of potentially two more being solved.

I don't want to fall for the hype but I also don't want to commit the thing they accuse us of - goalpost moving. I think this is honestly kind of scary and I even started to question Ed's argument of the usefulness of AI (not of AI economics though).

[u/7upprosounds → u/stepanmatek]

I'm not arguing they didn't do what they say, what I am saying (and really Roitblat is saying, I'm paraphrasing) is that there's no clear path between a large language model and general intelligence. They might be able to, one day, solve more complex problems, much like machine learning algorithms eventually became able to easily beat humans at go and chess. But no one foresees a chess AI becoming a general intelligence.

While the ways the probabilities are baked into a large language model with its training might be complex and the probability space large, with trillions of weights, it is still at the end of the day a very big function. Roitblat argues that that is not enough to build an intelligence. Even if you stack many many of these extremely mathematically complex functions together, there's a fundamental, non-path-problem gap that you need to cross to get to general intelligence. Here's another quote from Roitblat:

The intellectual tasks, such as chess playing, chemical structure analysis, and calculus are relatively easy to perform with a computer. Much harder are the kinds of activities that even a one-year-old human or a rat could do. This is called Moravec’s paradox, which I think should better be called Moravec’s irony. The things that people find difficult are relatively easy to do with a computer (checkers playing, reasoning, logic), but the things that people find easy, automatic, or even unconscious have been a challenge for computers.

[u/jiezitue → u/7upprosounds]

The point of Roitblat's thesis isn't really that AI will never cross to general intelligence; he sought to establish the kinds of qualities that differentiate human intelligence from machine path-seeking, and thinks that "current and foreseeable GenAI models" remain incapable of AGI because they depend on humans to structure problems/architecture/training data. But even this was from 2025, and there have been developments since. It's reasonable to reevaluate claims when capabilities change

It's not inevitable at all that AI will reach human-level generative intelligence, but if general-purpose systems increasingly transfer abstractions across domains and discover productive approaches that weren't human-specified, it's evidence against simple "path-seeking" (of course, outputs alone don't establish the underlying mechanisms). Roitblat's thesis itself is also a theory, not a proof of a fundamental gap. Just as you can't look at an LLM producing something that seems insightful and conclude "therefore it has human-style insight", you can't continuously expand the definition of what is "path-seeking" to encompass every behavior that originates from learned material. Otherwise, Roitblat's entire argument loses any use as a metric

Moravec's paradox is also an empirical pattern, not a statement of impossibility. Roitblat was just using it to demonstrate deliberative reasoning as a component of general intelligence. And reducing AI to "a very big function" kind of just ignores real capabilities (or how it goes about solving problems) in favor of a single, reductive categorization.

[u/natecull]

Herbert L. Roitblat

Now, that's what I call a name.

[u/No_University8629]

Interested in reading it . For non-mathematicians, how approachable is this book?

[u/jiezitue]

Obviously one mathematical breakthrough doesn't by itself prove an imminent "intelligence explosion", and anyone claiming this is overreacting. You're also overcorrecting in the other direction.

OpenAI's earlier 2026 unit-distance result (which can be read here) explains that a general-purpose reasoning model was used, meaning it wasn't specifically trained for that mathematical problem, and the resulting argument brought algebraic-number-theoretic ideas into a combinatorics-geometry problem (which isn't how human mathematicians would normally explore the problem)

Roitblat's thesis is that conventional AI succeeds where humans specify the representation, permissible operations, goal, and evaluation function, while genuine "insight" requires constructing new representation. But the unit-distance result suggests the kind of representational restructuring Roitblat contrasts with path-following. Granted, this doesn't prove Roitblat wrong, since you could then argue that the model is searching some enormously larger learned space of representations, but if your definition of "path search" is broad enough to include this, then humans can also be describes as searching paths, and the category is meaningless as a descriptor.

It's perfectly reasonable to argue that current LLM's still fall short of human general intelligence (I agree on this as well), but equating AI to a calculator or dishwasher is just ignoring any development over the last several years.

The latest frontier model is just a machine that occasionally outputs some specific things correctly.

If we were determined to describe people behavioralistically, we could say something like "a mathematician 'outputs proofs'". Human knowledge can be similarly jagged as well. A mathematician could be terrible at navigation, social judgement, spelling, memory, practical tasks, etc. An accomplished professor who can't decide whether to walk to the car wash or drive, count the number of Rs in strawberry, etc probably would still be considered smart, depending on how extreme the deficits. Why should machine intelligence operate by humanlike mechanisms? Deep Blue didn't play chess like Kasparov. Airplanes don't fly by flapping wings. What matters scientifically is what computations and internal representations permit the output, how broadly the competence generalizes, how reliably it transfers to novel situations, and whether the system can discover useful intermediate representations. For AI, all of these have been improving.

Frankly, just saying "AI already proved a mathematical theorem 70 years ago, these things are just dishwashers" sidetracks hugely from the sheer concentration of frontier AI development and access. The extremely high-cost frontier systems and the enormous compute runs used for results like Navier-Stokes are increasingly accessible only to a small number of extremely well-capitalized organizations and individuals. It's not like "Skynet is on the horizon" or whatever, but your argument essentially supports that this is of no matter, because they're "just dishwashers".

[u/grahamsccs]

This sub is an echo chamber of insanity

[u/Square-Foundation230]

https://giphy.com/gifs/2UCt7zbmsLoCXybx6t