However, it often makes conceptual errors that I can spot only because I have good knowledge of the topic I am discussing. For instance, in 3D Clifford algebras it repeatedly confuses exponential of bivectors and of pseudoscalars.
Good to know that ChatGPT 5.5 Pro can produce a publishable paper, but from what I have seen so far with Gemini, it seems to me that it is better to consider LLMs as very efficient students who can read papers and books in no time but still need a lot of mentoring.
Moreover, there's no reason to believe the progress of LLMs, which couldn't reliably solve high-school math problems just 3–4 years ago, will stop anytime soon.
You might want to track the progress of these models on the CritPt benchmark, which is built on *unpublished, research-level* physics problems:
Frontier models are still nowhere near solving it, but progress has been rapid.
* o3 (high) <1.5 years ago was at 1.4%
* GPT 5.4 (xhigh), 23.4%
* GPT-5.5 (xhigh), 27.1%
* GPT-5.5 Pro (xhigh) 30.6%.
Wrong. Every advancement has followed a s curve. Where we are on that curve is anyones guess. Or maybe "this time its different".
A scientific approach here is to look to falsify the statement. You start asking questions, running tests, experiments, etc. to prove the notion that it is done wrong. And at some point you run out of such tests and it's probably done for some useful notion of done-ness.
I've built some larger components and things with AI. It's never a one shot kind of deal. But the good news is that you can use more AI to do a lot of the evaluation work. And if you align your agents right, the process kind of runs itself, almost. Mostly I just nudge it along. "Did you think about X? What about Y? Let's test Z"
It’s also because it is so annoying to have to manage the memory of the LLM with custom prompts/instructions manually.
I have not yet played with the long term memory feature, but I fear it will be even less reliable than prompts, simply because in one year or two years so much will have changed again that this “memory” will have to be redone multiple times by then.
However, I think it's important to remember that LLMs are embedded in larger systems, and those larger systems do learn.
I think this is a bit pedantic. Obviously the parent you’re replying to is referring to the concept of “in-context learning”, which is the actual industry / academic term for this. So you feed it a paper, and then it can use that info, and it needs steering / “mentoring” to be guided into the right direction.
Heck the whole name of “machine learning” suggests these things can actually learn. “reasoning” suggests that these things can reason, instead of being fancy, directed autocomplete. Etc.
In other news: data hydration doesn’t actually make your data wet. People use / misuse words all the time, and that causes their meaning to evolve.
And that can be very hard to do given the ui we most interact with them in is a chat session.
what? training is learning, as long as weights are available continual learning is perfectly feasible: just keep training the LLM with the user corpus alternated with a frozen version to prevent catastrophic drift / collapse.
it's not because model providers don't want to provide user specific continual learning, that we don't know how to do it.
it would be a lot more expensive to host user-specific model weights, and would prevent amortizing the weights over many requests in batches...
There is a 50/50 chance that it turns out to be right or letting you jump of the cliff.
Only the trip stays the same beautiful 5 star plus travel.
Also, spotting an error and telling LLM makes it in most cases worse, because the LLM wants to please you and goes on to apologize and change course.
The moment I find myself in such a situation I save or cancel the session and start from scratch in most cases or pivot with drastic measures.
Gemini to me is the most unpredictable LLM while GPT works best overall for me.
Gemini lately gave me two different answers to the same question. This was an intentional test because I was bored and wanted to see what happens if you simply open a new chat and paste the same prompt everything else being the same.
Reasoning doesn’t help much in the Coding domain for me because it is very high level and formally right what the LLM comes up with as an explanation.
I google more due to LLMs than before, because essentially what I witnessed is someone producing something that I gotta control first before I hit the button that it comes with. However, you only find out shortly afterwards whether the polished button started working or gave you a warm welcome to hell.
In one case, it made a thoroughly convincing argument that an approach was justified. The second time it made exactly the opposite argument, which was equally compelling.
I now see LLMs as persuasion machines.
But noticed that the closer the domain they were talking about was to my area of competence the less convincing their arguments were. There were more holes, errors and wrong conclusions.
I recalibrated my bs meter thanks to that.
Since AI came I successfully used this strategy of being extremely cautious towards convincing arguments to not become mislead by AI.
However this year I'm working with AI more in the domain of software development. Where I can see the competence. And I see the competence. This had opposite effect on me. I tend to trust AI outside my domain of expertise much more after I saw what can it do in software.
One caveat though is that there are a lot of areas of human culture where there's very little actual knowledge, but a lot of opinions, like politics, economy, diet, business, health. I still don't trust AI in those domains. But then again, I don't trust humans there either.
For me basically AI achieved the threshold of useful reliability for any domain that humans are reliable at.
I don't really care about sycophancy. I might have a slight advantage that I don't talk to AI in my native language. So its responses don't have a direct line to my emotions.
For this sort of thing, using multiple LLMs is extremely helpful.
I was using Copilot and asked it a question about a PDF file (a concept search). It turned out the file was images of text. I was anticipating that and had the text ready to paste in.
Instead, it started writing an OCR program in python.
I stopped it after several minutes.
Often Copilot says it can't do something (sometimes it's even correct), that's preferential to the try-hard behaviour here.
This nails an important thing IMHO. I've absolutely noticed this, for better or worse. Gemini can produce surprisingly excellent things, but it's unpredictability make me go for GPT when I only want to ask it once.
If you had an infinite number of monkeys, each with a typewriter, one would eventually write Shakespeare. If you had an infinite number of college-educated interns, each with access to all the public records you can possibly get via FOIA, one would eventually get enough evidence to prove that a top politician is cheating on their partner, evidence which you could use to blackmail that politician.
You don't need that much intelligence to do that, you just need somebody who's willing to dedicate their life to knowing everything there is to know about that guy from Louisiana.
With humans, the amount of money you'd need to pay such a person just isn't worth the reward. With LLMs, it may very well be.
you deserve opinions shaped by interactions with the best tools that are out there.
But regular reminder - All LLMs can be wrong all the time. I only work with LLMs in domains I'm expert in OR I have other sources to verify their output with utmost certainty.
When I'm cooking meatballs with sauce and the recipe calls for frying them, I'll have an LLM guestimate how long and which program to use in an air fryer to mimic the frying pan, based on a picture of balls in a Pyrex. So I can just move on with the sauce, instead of spending time browsing websites and stressing about getting it perfect.
I used to hate these non-deterministic instructions, now I treat it as their own game. When I will publish my first recipe, I'll have an LLM randomize the ingredient amounts, round them up to some imprecise units and also randomize the times. Psychologists say we artists need to participate and I WILL participate.
This. Should become a general rule for any non-trivial use of LLM in a professionel setting.
Claude has been utterly useless with most math problems in my experience because, much like less capable students, it tends to get overly bogged down in tedious details before it gets to the big picture. That's great for programming, not so much for frontier math. If you're giving it little lemmas, then sure it's great, but otherwise you're just burning tokens.
What I do to mitigate this is that I have fact checking agents configured to be extremely critical and non-biased on Opus, Gemini and GPT. Which are then handed the entire conversation to review it. Then it's handed off to a Opus agent which is setup to assume everything is wrong. After this, and if I'm convinced something is correct I'll hand the entire thing off to a sonnet agent, which is setup to go through the source material and give me a compiled list of exactly what I'll need to verify.
It's ridicilously effective, but I do wonder how it would work with someone who couldn't challenge to analytic agent on domain knowledge it gets wrong. Because despite knowing our architecture and needs, it'll often make conceptional errors in the "science" (I'm not sure what the English word for this is) of data architecture. Each iteration gets better though, and with the image generation tools, "drawing" the architecture for presentations from c-level to nerds is ridiclously easy.
Right now, we have a lot of smart people who have trained for decades to understand where these things go wrong and how to nudge them back, but the pool of people are going to slowly be replaced by less knowledgeable.
At some point, a rubicon will be crossed where these systems can't fallback to a human operator and will fail spectacularly.
It is troubling. It suggests a plateauing of human understanding.
Just in case if you don't want to disclose your name my email is northzen@gmail.com
Anthropomorphizing these systems is dangerous, whether coming from the bullish or bearish perspective. The output is statistically generated by a machine lacking the capability to be smug.
I have no idea what any of those words even mean. I'm sure LLMs make similar obvious-to-professors mistakes in all the domains. Not long ago, we didn't even have chatbots capable of basic conversation...
Bivectors and pseudoscalars (in a 3D context) are "just" signed areas and volumes. Easy!
Back around the GPT 3, 3.5, and 4.0 era I used to ask the bots to explain "counterfactual determinism", which is one of the most complex topics I personally understand.
Then I would lie to the bot about it, and see if it corrected me or not.
This test is useless now, the frontier models can't be fooled any longer on such "basic" concepts.
Conversely, LLMs are basically useless at anything that doesn't have enough (or no) public information for their training. Think: obscure proprietary product config files and the like, even if the concepts involved are trivial.
Similarly, Clifford Algebra is a relatively niche (even "alternative") area of mathematics and physics, with vastly less written material about it than the competing linear algebra. Hence, the AIs are bad at it.
Mine has been epically bad.
I put my stuff through several sota models and round robin them in adversarial collaboration and they are all useful even though, fundamentally, they don’t “understand” anything. But they are super useful delegates as long as deciding on the problem and approach and solution all sits safely in your head so you can challenge them and steer them.
So I know the article is about one particular new model acing something and each vendor wants these stories to position their model as now good enough to replace humans and all other models, but working somewhere where I am lucky enough to be able to use all the sota models all the time, I can say that all keep making obvious mistakes and using all adversarially is way better than trusting just one.
I look forward to the day one a small open model that we can run ourselves outperforms the sum of all today’s models. That’s when enough is enough and we can let things plateau.
That’s all they are. They don’t ‘know’ anything intrinsically and do know ‘know’ what reasoning even is.
Paying for Pro from any of my current academic budgets is completely ouf of the field of reality here -- all budgets tend to have restricted uses and software payments fit into very few categories. Effectively, I'd have to ask for a brand new grant and hope the grant rules allow for large software payments and I won't encounter an anti-AI reviewer; such a thing would take one year at least.
As a nail to the coffin, I was "denied" all Claude Opus recently as part of Microsoft's clampdown on individual (and academic) use of Copilot.
(Chagpt 5.5 Plus does not seem sufficient for any deeper investigations into new research topics, I've tried.)
Apologies for the rant.
An aside: It was a very nice gesture and completely unexpected by me, so even if it doesn't work out, it made my day. I personally believe that kind gestures have a lot of power.
Back on topic: There is a real danger of the gap between rich and poor universities significantly widening in all fields if the rich can afford Pro level models, or even hardware that can run their own comparable models, and this being fiscally inaccessible to the rest.
One can sweep this under the rug by blaming the educational funding but this just shoots down all discussion. Even if GDP of a country goes up by a lot -- such as Poland -- it takes time before any budget benefit trickles to the education budget, and with some governments it might never do.
I believe Microsoft et al do have the most power here to boost affordable access to AI for researchers on a large scale; the fact that they cut some too expensive models (Opus, 5.5) from their academic benefits package is a grim omen. I do realize they would like universities to pay them also, and ultimately the universities should do that -- but then we are back at the institutional level of the problem.
At present, the tools are available for whomever wants to buy them. Not OpenAI's fault that parent comment's government and/or institutions policies haven't been updated to allow for their purchase and use.
I'd argue that the OpenAI dude/dudettes level of generosity is appropriate given the circumstances.
I probably will erase the contents in a few days.
Even if you just drop an email and it doesn't work out, I appreciate this gesture so much. Thank you.
Thank you.
There’s the example of a poor person and a rich person buying boots. The poor person’s boots wear out and have to be replaced while the rich persons boots last for many years due to higher quality craftsmanship. Over years, the poor person’s boots wear will pay may for boots.
Of course if you are really poor, then you have to take expensive shortcuts, but for most people that shouldn’t be the case. Learning to do more with less money isn’t as bad as many people think. It’s also good for the brain to be a bit more creative.
We are wading into philosophy here, but I believe this analogy doesn't track in this case -- my suspicion from this blog post and others is that already today, the Pro level thinking models are a positive multiplier to your research output similar to how the models one level lower are a multiplier to one's programming output.
Maybe one can someday use the cheaper models similar to how you can use cheaper models than Opus/5.5 and still be nearly as productive as a programmer -- but I am trying and failing doing exactly that for research questions.
But if you ask questions occasionally, (and don't resend, for example, your whole codebase with each request), then the API feels really cheap, even for the frontier models.
I'm not trying to shame here, just curious whether this is completely unattainable for most researchers in your area.
I am starting to see folks saying - ok, so LLMs can do this, what value have you added ? modulo llm is becoming the norm.
> It seems to me that training beginning PhD students to do research [...] has just got harder, since one obvious way to help somebody get started is to give them a problem that looks as though it might be a relatively gentle one. If LLMs are at the point where they can solve “gentle problems”, then that is no longer an option. The lower bound for contributing to mathematics will now be to prove something that LLMs can’t prove, rather than simply to prove something that nobody has proved up to now and that at least somebody finds interesting.
Training must start from the basics though. Of course everybody's training in math starts with summing small integers, which calculators have been doing without any mistake since a long time.
The point is perhaps confirmed by another comment further down in the post
> by solving hard problems you get an insight into the problem-solving process itself, at least in your area of expertise, in a way that you simply don’t if all you do is read other people’s solutions. One consequence of this is that people who have themselves solved difficult problems are likely to be significantly better at using solving problems with the help of AI, just as very good coders are better at vibe coding than not such good coders
People pay coders to build stuff that they will use to make money and I can happily use an AI to deliver faster and keep being hired. I'm not sure if there is a similar point with math. Again from the post
> suppose that a mathematician solved a major problem by having a long exchange with an LLM in which the mathematician played a useful guiding role but the LLM did all the technical work and had the main ideas. Would we regard that as a major achievement of the mathematician? I don’t think we would.
Yes but it's not just that if you solved a problem yourself, you're better at solving other problems; it's also that you actually understand the problem that you solved, much better than if you simply read a proof made by somebody (or something) else.
I see this happening in the enterprise. People delegate work to some LLM; work isn't always bad, sometimes it's even acceptable. But it's not their work, and as a result, the author doesn't know or understand it better than anyone else! They don't own it, they can't explain it. They literally have no value whatsoever; they're a passthrough; they're invisible.
According to the blog post linked in the OP, the LLM-generated results were read, understood, and confirmed by the mathematician whose work they built on.
I notice a dichotomy here between people who care about results and people who care about process. The former group wants to use LLMs insofar as they can contribute to getting results. The latter group is wary of LLMs because they're more interested in the process and less interested in the results themselves. Needless to say, I think the former group is right, and I'm happy to see that mathematicians (or some of them) agree.
Then middle management also have no value, since they're also a passthrough between upper management and ICs, yet they never went extinct.
I don't know or understand the binary executable.
I don't own the binary executable, I don't understand it, I can't explain it, it's not my work.
I'm a passthrough; I'm invisible.
I have literally no value whatsoever.
> Training must start from the basics though.
Sure, but the point is that at some point (e.g. when starting a PhD) one needs to do research, not learn the basics. And LLMs make that harder, because they solve the "easy research" part.
Take a young lion "fighting/playing" with another young lion as a way to learn how to fight, and later hunt. And suddenly they get TikTok and are not interested in playing anymore. Their first encounter with hunting will be a lot harder, won't it?
> People pay coders to build stuff that they will use to make money and I can happily use an AI to deliver faster and keep being hired.
Again, that's true but missing the point: if you never get to be a "good coder", you will always be a "bad vibe coder". Maybe you can make money out of it, but the point was about becoming good.
1. Does it matter, really? 2. Is it very different from previous computer-aided proofs, philosophically?
2. Yes, it is. Because pre-LLM era computer-aided proofs were about using the computer to either solve a large number of cases or to check that each step in a proof mechanically follows from the axioms.
Yeah, it's the same way with learning programming. LLMs can handle basic programming (and increasingly advanced programming) but I think it's necessary to write code by hand. As a beginner, of course, and arguably to maintain skill later too.
The alternative would be like, just asking ChatGPT to do your math homework and then "verifying" it by looking at it and saying "yeah, that looks okay." What are you going to learn?
We do stuff by hand for a reason.
The downside (not noted in the article, but noted by others here) is cost. It uses tokens at an insane rate, the tokens cost a lot, and using it with subagent flows that you can use to have it tackle large problems with high accuracy costs even more. It is also much "slower" for large scale problems because of context limitations -- it has to constantly rediscover context for each part of the problem, and in order to make it accurate you need to wipe its context before progressing to the next small part, or launch even more agents. For mathematical proofs like these, where the required context to understand the problem and proof besides stuff that's already available in its training set is small and the problems are considered "important" enough, this might not be a problem, but for many of the tasks I would like to use it for (ensuring correctness of code that affects large codebases, or validating subtle assumptions) it definitely is one.
So I think it will be a while before the impressive capabilities of these models really percolate into our lives as programmers, unless you're one of the lucky ones given unlimited access to 5.5 Pro.
I swear that people have said the same thing with effectively every new model that came out in the last six months.
The reality is likely that everyone is hitting similar barriers and the solutions are somewhat generalizable and get added to training new models.
Eventually people will reach the new limits and the cycle repeats.
That is definitely true, and at the same time, we can measure progress by who is making that claim. When Timothy Gowers, a Fields Medalist, says that models are now capable of "producing a piece of PhD-level research in an hour or so, with no serious mathematical input from me," we can be pretty confident that we are getting into seriously interesting territory.
Just as an fyi, the word you are looking for is jibes. Jive is something else entirely.
I don’t know about the rest of y’all but I find “rigidly guiding” LLM’s incredibly tedious and frustrating in the same way seeing an error code throw for the 40th time while troubleshooting something on my computer for two hours is frustrating. It also feels somewhat like micromanaging a direct report. I don’t find that process fun or enjoyable in the slightest and it teaches me little in the process. It’s just trading styles of work, and I guess the response to that is “some people prefer that of work.” I just don’t like being told by the world we all have to work that way now I guess.
The question that keep bothering me is can an LLM generate an idea that is truly novel? How would/could that actually happen? But then that leads to the question - what are we actually doing when we think?
Perhaps it's as simple as the ability to just make mistakes that matters, the same things that powers evolution. As long as the LLM can make mistakes, it's capable of generating something genuinely novel. And it can make more mistakes much faster than we can.
Some people like to parrot "next token prediction", "LLMs can only interpolate", and other nonsense, but it is obviously not true for many reasons, in particular since we introduced RL.
Humans do not have the monopoly on generating novel ideas, modern AI models using post training, RL etc can come to them in the same way we do, exploration.
See also verifier's law [0]: "The ease of training AI to solve a task is proportional to how verifiable the task is. All tasks that are possible to solve and easy to verify will be solved by AI."
This applied to chess, go, strategy games, and we can now see it applying to mathematics, algorithmic problems, etc.
It is incredibly humbling to see AI outperform humans at creative cognitive tasks, and realise that the bitter lesson [1] applies so generally, but here we are.
[0] https://www.jasonwei.net/blog/asymmetry-of-verification-and-...
[1] http://www.incompleteideas.net/IncIdeas/BitterLesson.html
Thank you for illustrating my point.
If it's "invented", then it requires ingenuity.
If it's "discovered", then it was always already there, just waiting for the right connections to be made for it to be uncovered and represented in a way we can understand.
Invention requires ingenuity, but discovery does not. So if LLMs can generate truly novel mathematics, for me that settles it that mathematics is indeed discovered, as LLMs are quite capable of discovery yet I don't consider them possible of invention.
Furthermore, the results of theorems aren’t an invention, they are a discovery of what the base assumptions (axioms) logically entail. Finding out which theorems are true and provable is a discovery process. For example, the results of Gödel’s incompleteness theorems were a discovery. They weren’t invented, in the sense that the results couldn’t have been otherwise. We merely could have failed to discover them.
This also holds for physical inventions. You discover a working way to build some functioning mechanism. It’s a process of discovery of what is possible in the physical world.
Whether you portray somethings as a discovery or as an invention is more a matter of degree, a matter of from which angle one is looking at it.
The possible states of an LLM are finitely enumerable. The same likely holds for the possible states and configurations of a human brain, in approximation. Therefore there is only a finite set of possible ideas, thoughts, and conceptualizations an LLM or a human can have, and in principle they could be exhaustively enumerated and thus “discovered”.
There is no ‘discovery’ here nor was it waiting to be found. The human has to sacrifice and pursue the path of exploring reality and thereby is inherently inventing.
Humans built up mathematics iteratively from smaller bases extending into large ones. Is this what LLM’s do? Of course not - They are fed with vast amounts of information from the off.
This works really well.
Now, it's clear that I have no idea how much of this is something we would consider new and original, and how much is a kind of systematic, but not novel, easy of thinking.
What I couldn't do so far is get an LLM to generate a truly new maths theory, with new abstract concepts and dimensions and points of view. The kind that is not just a combination of existing theories and logic.
To me, it's rearranging the information you had in a way that hasn't been applied or published before.
That's literally what LLMs are built for.
Limit the knowledge an llm to some point in time at which a discovery was made. And check to see if the llm could produce the discovery.
If you think OAI hasn’t already tried this then think again - they have every incentive to do so and announce it to the world.
You are worthy of doing this work because you are able to do it. Do the work because you love it and because you love the mystery. Enjoy every moment that you get to do it. Find joy in the great fortune you have to do this work while others toil away on tasks that bring them no satisfaction. Sometimes it's tedious, but sometimes it's incredibly rewarding in its own right.
Don't work for the possibility of eternal glory though, it just doesn't exist anymore.
Nobody looks at this species and goes hm, rational and reasonable :)
Any statement preceded by the word 'believe' is a coping mechanism.
> This notion of immortality was just a small intangible bonus I hoped for when I jumped into grad school
Any statement preceded by the word 'hope' is a coping mechanism.
> AI is making me feel less worthy
Worth comes from understanding, not achievement.
But I agree worth should be derived from understanding, not through achievement.
This is a cultural choice. It makes sense that in the mathematics culture we currently have, this is alien. But already, other fields, and many individuals, would disagree and say that the human did have a major achievement here. As long as human-AI collaborations are producing the best results, there is meaningful contribution by the humans, and people that are deeper experts and skilled LLM whisperers should be able to make outsized contributions. The real shoe drops when pure AI beats humans and human-AI collaboration.
We praise car drivers despite most of the performance in their sport comes from the car. The driver makes the difference when two cars are close in performance. Brilliances or mistakes. Horse riders too.
In the case of math, the human can lead the LLM on the right track, point it to a problem or to another one. So it deserves some praise.
Then the team that built the car, cared about the horse, built the AI might deserve even more praise but we tend to care more about the single most visible human.
For some reason this reminds me of AI images and a domain like comedy.
If an image makes people laugh, the person who prompted it to make the image certainly doesn't get credit for the vast majority of the work in its creation, but perhaps they do get credit for the initial prompt idea and then the "taste" to select that particular one from whatever drafts they went through or otherwise guiding it.
So if a mathematician comes up with an amazing result that an LLM "did", I think they could still get a bit of credit for prompting it to do it and being its guide.
But whereas the first person could perhaps be called a comedian and not an artist, would the mathematician still be called a mathematician or something else?
We just call those ones “quant traders”.
This made me a little sad
The opening of the movie features the MIT campus full of students navigating its grounds and all the promise and status that higher education brings. [0]
Gave me the same sense of sadness realizing how much will fall to AI.
[0] - https://youtu.be/0lsUsWdkk0Y?si=TJl7f_b1RcWcDqF8&t=278
I don’t think I ever thought I was good enough to try and get (math) immortality by finding and naming some result that would live beyond me, but if I had, perhaps this bad news would have had a similar impact on me.
That said, I think I disagree with the premise at the margin, at least. I don’t care how many proof assistants or cluster compute is used - the team or person that proves the Riemann Hypothesis will be famous, or at least math famous.
Many mathematicians work because they love the breakthrough (a certain quote of Villani comes to mind). They love finding new results, uncovering new mysteries. From that point of view, having an AI that can build on your basic ideas and refine them into more powerful arguments is awesome, regardless of who gets the credit. There are those that treat it more like solving puzzles so the result is not of interest. From that point of view, I can see the dissatisfaction. But I have found those with that viewpoint don't tend to make it as far in academia as those with the other viewpoint.
We care about sports with humans.
> Where does the value of thinking and having deep ideas come from? We need to think about this now. If it comes primarily from their scarcity – the fact that having certain ideas is hard – then indeed this value may drop precipitously when the manufacture of ideas can be automated. But if the value comes from the utility of the ideas – the benefit that the idea brings – then the story changes: perhaps creating more good ideas is actually better, not worse. Here I’m using “utility” in a broad sense, not just in the sense of what people often call applied mathematics.
> In other words, mathematicians may need to adjust to a transformation from a scarcity economy to an abundance economy.
https://gowers.wordpress.com/2026/05/08/a-recent-experience-...
The first species is the pure problem solver. Tao is the poster child for this group. Their currency is interesting problems and solutions to those problems.
The second species is the pure theory builder. The poster child for this group is Conway. Their currency is theories and ideas rather than theorems, they are most interested in expanding the territory of mathematics and discovering new mathematical lands.
The third species is the applied mathematician. They see mathematics as a means to an end, they have some problem outside of mathematics and they want to use mathematics to solve it.
It seems like the first group (the problem solvers) are the most immediately threatened by AI, although so far AI is better at solving problems than finding new conjectures.
The second group (the theory builders) are more distantly threatened by AI, since thus far AI has shown limited ability to come up with novel and interesting mathematical ideas and nobody has any clue how to train an AI to do such a thing.
The third group stands to gain the most from AI. If an AI can answer your mathematical question then you can spend less time doing mathematics and more time on whatever it is outside of mathematics that you wanted to use mathematics to help solve.
Identifying suitable problems in this sense rather than solutions is an AI use-case you don't hear about much. We don't quite have the infrastructure for this yet but by combining language models / anomaly detection / knowledge-bases we might be in a position to give a Conway 25 interesting high-quality puzzles before breakfast. Funny that it's like a kids nightmare, chatgpt giving them homework instead of solving it, but if it had good taste, people would love it for research.
Anyway, for now, dreamers will probably find more inspiration by cross-pollination with colleagues from different disciplines, or just going for a walk.
Meanwhile Wiles and Perelman stayed offline and solved real problems.
Anyone spotting the issue here? What did that really cost?
I am not against compute being used for scientific or other important problems. We did that before LLMs. However, the major LLM gatekeepers want to make all industries and companies dependent on their models. And, at some point, they need to charge them the actual, unsubsidized costs for the compute. In the meantime, companies restructure in the hopes that the compute costs remain cheap.
Whatever the Joules... (convert to $ using your preferred benchmark price) it is a fraction to what it might take a human Ph. D. weeks to feed and sustain themselves when working on the same problem. The economics on LLMs is just unbeatable (sadly) when compared to us humans.
Do you understand the difference between an apple and an apple tree?
5.5pro is amazing but this implication might not be true & is the core argument of this piece.
AI will prove all sort of things - interesting, boring & incorrect.
To sort it will be the task of the PhD.
"Hey, Prove something a machine can't", sure I can't, "Hey, Say something worth proving & judge it well", ah, now I might have a few unique observation/ideas/curiosities/problems from my having being a human.
Imo, the feeling of intelligence or the process of originality(originativity) test for ai is subjective & is coming down to 4 paths: novel relative to a reference class, valuable within a domain, counterfactually sensitive to internal state and environment, and revisable through learning.
>> All I did was say things like, "Yes, it would be great if you could explore that idea and see whether you can get it to work," or "Could you rewrite that argument as a LaTeX file in the style of a standard mathematical preprint?"
Yeah, so all he did was take the horse to the water and make the horse drink. The collaboration with the other two mathematicians wasn't a trivial part of the problem solving either: every time Timothy Gowers figured ChatGPT had goe somewhere with its problem-solving, he stopped, asked it to render the answer in LaTex, and sent the answer off to be verified by the other two.
The reason for that is not to be underestimated: ChatGPT can produce answers to questions you ask it for as long as you ask it to do so but it has no capability to determine whether an answer is correct or not. That's why it needs a human with domain expertise to evaluate those answers. And of course to discard wrong answers in the process, because of course the process that's described here glosses over many false starts and back-and-forths and "you're absolutely rights, here's a new version of that"'s etc. that are common experience when using LLMs for problem-solving tasks.
The existential questions that the article poses about mathematics then are easily answered by taking all of the above into account. If LLMs are a useful tool for mathematicians, then nothing changes. Mathematicians of all levels can still do their job and perhaps do it faster or better with the new tool.
If you can sic ChatGPT on a mathematics problem and it can solve it without your input, that's a different matter but that's not what's happening.
thank you for the morning laugh
I mean that has happened so yeah ?
https://www.scientificamerican.com/article/amateur-armed-wit...
Actual GPT transcript. Zero such input https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...
And maybe the other guy wasn't the most polite about it but his point is very valid. Replace chatgpt with a human in both of these stories and nobody would say that timothy 'took the horse and made it drink'. The 'Horse' would be the first and likely only Author so this just sounds like denial.
That there are multiple of these stories in the last few months by the latest set of models (there are even more than these 2) should provoke this sort of consideration and discussion.
Then there's the kind of problem we're talking about. The "amateur" in the SA article solved one of Erdős problems and Gowers himself seems to think that, on its own, is not a cause for concern. He distinguishes his own result from that kind of earlier result at the start of his article:
>> The background is that, as has been widely reported, LLMs are now capable of solving research-level problems, and have managed to solve several of the Erdős problems listed on Thomas Bloom’s wonderful website. Initially it was possible to laugh this off: many of the “solutions” consisted in the LLM noticing that the problem had an answer sitting there in the literature already, or could be very easily deduced from known results.
So we have an "amateur" who "vibe-solved" an Erdős problem, on one hand, which may or may not already had a solutiuon lurking in the wings on the one hand; and an expert who solved a harder problem by interactive use rather than vibe-solving, on the other hand. There's no reason to believe that we can "Replace chatgpt with a human in both of these stories" as you say.
And btw there's scholarship that indicates vibe-solving is not yet ready to replace mathematicians like Timothy Gowers:
First Proof
To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.
https://arxiv.org/abs/2602.05192
See Appendix A for initial results.
Also if he did send me complete junk, I would still parse it for multiple days to see what is there.
I'm not criticising Gowers directly in this instance because he's exploring the possibilities, my disdain is towards the more general pattern I see emerging where people just send each other LLM outputs.
Graduate? Yes.
If we look where these models were 5-7 years ago...the existential threat of the Ph.D. was not even on the radar back then. The people finishing up their doctorate now are the first that can truly leverage these tools.
Now, if these to-be researcher students feel defeated (enough to quit), or completely lean on AI models the work for them, we're going to have a problem. Same with the funding of those Ph.D. positions. If we move away from "funding to produce researchers" to "funding to achieve results", will money that was usually spent to fund Ph.D. students start to flow towards compute?
If we look at it a bit cynically: Some researcher will be able to pump out a lot more papers by spending money on compute, than a couple of years of training students.
Interesting times. But also so much uncertainty. I feel terrible for the students that will have to decide now what they want to do, with all this knowledge.
Obviously this is already happening and will accelerate. Outside of grad work, you could already just buy a degree. Certainly in the softer disciplines, you can currently just buy a phd thesis and a good publication history. If you're in industry instead of academics, you can even buy a promotion. If your employer gives an AI budget to all workers then you quietly double that budget out of your own pocket for as long as it takes to get a promotion, then stop and just enjoy a bigger paycheck.
I see that they are able to do researches that they were not previously able to do. And although I see that using AI has certainly diminished their ability to code some stuff up, I see it the same way as someone using scikit-learn or Pytorch to code their ML models -- indeed the underlying details is abstracted away from you, and without AI, you won't be able to do much, but the research that you do is indeed happening because of you and wouldn't have happened with just the AI doing the research.
As an afterthought budget item, those funds aren’t exactly attractive targets to raid for pursuing an expensive, different process.
The implication is: we're ready to let everyone go wild with this very soon. Ok, go wild with what exactly? How do we know that influential VIP users who might make very friendly blog posts aren't getting allocated exclusive access to a billion dollars worth of hardware when they ask questions? I mean literally giving a certain group of people temporary privileged access to like 90% of all available compute would be a completely reasonable business decision for OpenAI.
Would a reveal like that change how we think about the result? What if half that amount of cash/compute could enable some completely non-AI approach of numerical brute forcing that settles the question even if it didn't write the paper?
My other question is always whether the latest is purely using giant models or if we're now deeply into harnesses that use MCTS and such. Understandable to keep that a trade secret I guess. But IMHO we should at least get the CoT trace as a proxy for true cost, or else maybe we're just getting played to do the hype for corporate.
Where's that? The stuff I've seen is from celebrities. Were those problems as hard as this one, or the ones that Tao posts about? Regardless.. what's the argument against more transparency here to just settle this kind of thing?
> which is too conspiratorial of an explanation for my liking regardless
OpenAI is not, in fact, open. Why do they deserve the benefit of the doubt?
Regardless.. special treatment for special customers isn't conspiracy, it's SOP literally everywhere and especially if you're helping to beta test. Anyone who's ever interacted with any technical account manager has seen waived quotas, free resource allocations, etc. The quid-pro-quo is obviously that your cheap early access means you get to give talks at a conference (or make a blog post that a lot of people read and talk about).
Mathematicians have engaged, vigorously, on this very philosophical question for centuries - is math discovered truth, or is it more akin to building an edifice where you first define the materials, then the structure, and see where it leads?
There are lots of strong feelings on both sides. For instance: “God created the integers, the rest is the creation of man” — Kronecker, 19th century sums up one particular perspective.
To me, it’s probably a mix of both - some fantastic results in imaginary numbers show up as describing key electromagnetic effects many decades after they were first ‘discovered’ by theoretical mathematicians.
NB: My original comment led with a pejorative, which was rightly flagged.
Please don't respond to a bad comment by breaking the site guidelines yourself. That only makes things worse.
We're trying for curious conversation here, and you've clearly got something interesting to say, but when you put it this aggressively, curiosity gets fried (https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...)
The "non-trivial" is for human abilities. The weights lifted by a crane are also "non-trivial". People keep getting amazed at machine's abilities. Just like a radio telescope can see things humans can't, microscope can see the detail humans can't, we need not be amazed. The sensory perception of patterns is at different level for AI. It's a machine.
It usually takes dissolving that, often through difficult experiences, before they can see it as a machine, something that could be separated from them.
OAI and Anthropic both require KYC for models of similar intelligence. They both do account bans if the classifiers fire wrong. You simply hear about it less with OAI because Codex has fewer prosumers.
This is as AGI as it needs to be to get my vote. And it's scary.
It still sounds to me like remarkable automation rather than something that's expanding the frontier of human knowledge, for now at least.
jagged AGI
Creativity is connecting ideas from different domains and see if something from one field applies to another. I do think AI is overhyped generally; but a major benefit from AI could be that after ingesting all the existing human knowledge (something no single human can ever hope to achieve) it would "mix and connect" it and come up with novel insights.
Most published research sits ignored and unread; AI can uncover and use everything.
That's true. The question is whether the produced pattern has any value. LLMs are incapable of determining this, and will still often hallucinate, and make random baseless claims that can convince anyone except human domain experts. And that's still a difficult challenge: a domain expert is still needed to verify the output, which in some fields is very labor intensive, especially if the subject is at the edge of human knowledge.
The second related issue is the lack of reproducibility. The same LLM given the same prompt and context can produce different results. This probability increases with more input and output tokens, and with more obscure subjects.
The tools are certainly improving, but these two issues are still a major hurdle that don't get nearly as much attention as "agents", "skills", and whatever adjacent trend influencers are pushing today.
And can we please stop calling pattern matching and generation "intelligence"? This farce has gone on long enough.
thats literally what an IQ test tests - abstract pattern matching. but I guess you dont like IQ tests either
Maybe if you find AI to be doing stuff you find impressive, the stuff you were doing wasn't that impressive? Worth ruminating on your priors at least.
For those that don't know, this is Timothy Gowers. He is one of the most accomplished mathematicians in the world. Like Terence Tao, he is considered one of the world leaders in mathematics and tends to have good judgement in where the field is going.
Even without that knowledge, no, this article is certainly not AI generated. It has none of the tells.
At the time I thought the key missing tool was a natural language search that acted like mathoverflow, where you could explain your problem or ideas as you understood them and get references to relevant literature (possibly outside your experience or vocabulary).
Being a gifted mathematician does not make you right. In fact, mathematicians have a lot of bizarre theories.
And certainly not to send it to a fellow colleague to ask its opinion first.
LLMs are certainly becoming capable to code, find vulnerabilities, solve mathematical problems, but we need to avoid putting their works in production, or in front of other humans, without assessing it by any possible mean.
Otherwise tech leads, maintainers, experts get overwhelmed and this is how the « AI slop » fatigue begins.
To be clear I’m talking about this step:
> That preprint would have been hard for me to read, as that would have meant carefully reading Rajagopal’s paper first, but I sent it to Nathanson, who forwarded it to Rajagopal, who said he thought it looked correct.
I think this is good advice in general, maybe with an emphasis on public vs. private, friendly contact. Having 0 thought AI slop thrown at you out of the blue is rude. "could have been a prompt" indeed. But having a friend/colleague ask for a quick glance at something they know you handle well is another story for me.
If I've worked on a subject for a few years, and know the particulars in and out, I'd have no trouble skimming something that a friend or a colleague sent me. I am sparing those 5-10 minutes for the friend, not for what they sent. And for an expert in a particular domain, often 5 minutes is all it takes for a "lgtm" or "lol no".
https://github.com/vjeranc/fixed-rtrt
M3 module was formalized fully purely from experimental data and from a nudge by earlier versions of codex in 15-30 minutes in a simple write/compile/fix-first-error loop. I was a bit surprised how fast it picked up the pattern but given there was a paper from '70s it became clear why later.
https://www.renaissancephilanthropy.org/
The "brighter future" of course is that everyone is redundant and all capital is further concentrated.
It is always Gowers, Tao and Lichtman (math.ínc startup) who are pushing these technologies.
In your mind does this mean that they are lying, or driven by motivated reasoning and cognitive bias, or whatever you'd like to say?
Because I feel like people bring up these facts as a way to discount everything that these people are saying, but whether or not they've chosen to align themselves with AI aligned venture capital funding or not. The question is really, did what they say is happening happen or not? Are these capabilities real or not?
To my mind, mathematics is pretty definitely, externally, objectively verifiable, so it would be easy to catch them in a lie. In the case of the Erdös problem that was recently solved in a novel and productive way, it wasn't even initiated by them and the chat GPT transcript is public for all to see. And the proof could easily be verified by other people, for instance.
In addition, I think it's unlikely that they're not explaining things as they honestly see them and also doing their due diligence to make sure that they are seeing them as close to correctly as possible. Because their positions with these organizations not to mention their entire reputation and life's work and passion depends on their reputation in academic mathematics. If they were to give that up by falsifying these claims or not verifying them sufficiently, they would lose everything.
I think it's also worth pointing out that it is totally possible for someone to align themselves with such organizations after the fact because they agree with them instead of being bought out by such organizations. Otherwise, it would be possible to dismiss the opinion of anyone working at any NGO dedicated to being against AI and denying AI's capabilities or whatever, as well by the same logic of their salary being paid by an organization dedicated to pushing those ideas.
> This reminds me of Antirez's "Don't fall into the anti-AI hype". In a sentence: These foundation models are really good at optimizing these extremely high level, extremely well defined problem spaces (ie multiply matrices faster). In Antirez's case, it's "make Redis faster".
> Conversely, for problems where one’s initial reaction is to be impressed that an LLM has come up with a clever argument, it often turns out on closer inspection that there are precedents for those arguments, so it is still just about possible to comfort oneself that LLMs are merely putting together existing knowledge rather than having truly original ideas. How much of a comfort that is I will not discuss here, other than to note that quite a lot of perfectly good human mathematics consists in putting together existing knowledge and proof techniques.
This is exactly what leads me to believe that the real impact of LLMs in human history is yet to come. My work as a researcher was mostly spent on two classes of workloads: reading papers that were recently published to gather ideas and keep up with the state of the art, and work on a selection of ideas gathered from said papers to build my research upon. It turns out that LLMs excel at the most critical component of both workloads: parsing existing content and use it when prompting the model to generate additional content based on specific goals and constraints. I mean, papers are already a way to store and distribute context.
This comment about time is very interesting to me. I know it's "just" doing mathematical proofs but the possibilities of speeding up planning, proposals and decision making in the physical world should excite people.
However I think it’s very important to approach such questions objectively, or at least self uninterested, and not as one who’s worried about one’s job or sense of self worth threatened by LLM technology.
The value is in the development of one’s own mental faculties. In math classes they tell you you have to work the problems. Even if LLMs become capable of solving entire classes of problems that that set expands over time, the value in developing one’s ability never goes out of style.
Somewhat ironic then, to not make this more explicit in an article about solving a combinatorial problem.
I also do not think this makes people less capable of solving hard problems. The bar just moves up. More people can now work on harder problems with better tools.
If the goal is credit or proving real skill, then focus on harder problems, like ones AI can't reach
Does the author know about CAISc 2026 [0]?
And then the answer isn’t even right.
The Bourbaki group was one of the first who attempted a mechanized approach (using pen and paper still of course) to set theory and were literally accused of wanting to end all mathematics. The approach was largely ignored in practice.
Gowers and a handful of others who work on computerized approaches also seem to want to end human mathematics and have sharecropper mathematics for a monthly tithe. So far they are largely ignored in practice.
Interesting question, I guess a starting point is “moltbook”, but perhaps a better one is something like GitHub, where Lean proofs and preprints can go, and trending items can get boosted.
I also think that posting this stuff on x or bluesky has merit, but again the existing paradigm doesn’t quite work; perhaps you can create a completely separate identity for your agent (à la Moltbook) but I think you want some sort of reputational association with the human piloting the agent, at least for now. (Maybe eventually there are enough agents critically engaging with content so that “interesting” results get agent likes, and so we’ll-piloted agents stand on their own merit.)
This is of enormous importance but still is being actively ignored by many professionals or dismissed as as a minor issue.
Our emotional human brains are very enthusiastic about these new kind of "intelligent" products ("partners") and we want to believe so hard that they are finally "there" that we tend to ignore how big of a problem it is that LLMs carry a fundamental design problem with them that will make them produce errors even when we use a grotesque amount of resources to build "bigger" versions of them. The potential for errors will never go away with the current AI architecture.
This is a fundamental paradigm shift in computing. Instead of putting a lot of energy into building an architecture that will produce reliable results, we are now maximizing on a system / idea that will never give us 100% reliable results.
Basically it is just a marketing stunt. Probably the computer science guy building it knew very well that he would still need some fundamental break troughs to get to a real product, but the marketing guy saw that there is still potential to make a lot of money by selling a product that will produce correct results only 80% of the time.
The marketing guy was right and marketing is now dominating science, but humanity will pay a big price for that.
Putting enormous amounts of money into a fundamentally flawed system that we can not optimize to produce reliably error free results is just stupid.
The big achievement of "classical" computing is that the results are reliably error free. We have still some known issues eg. with floating point math and bad blocks on disk / bit flipping etc. but these are observable and we can handle / avoid them. Generally "non-ai-computing" was made so reliable, that we can depend on it for many very important things. This came not by accident but was created by a lot of people who put a lot of resources into research to achieve that result.
LLMs introduce a level of uncertainty and unreliability into computing that makes them practically useless.
Because if you have enough knowledge to verify the result and AI is only quicker in producing the result, what is the point then putting so much resources in it (besides making money by re-centralizing computing, of course). Verifying a lot of results that have been produced quicker is still slow, so the people who are now just AI verifiers should just produce the results themselves, makes the whole process quicker.
AI is only of value if it can produce results about things that you or your organization does not know anything about. But these results you can not verify and therefore potentially wrong results can be fatal for you, your organization and all the people that are affected by actions generated based on these wrong results.
Many people have already been killed because decision makers are not able to follow that very simple logic.
So we can still create "interesting and enjoyable results", but finally it is a gigantic miss-allocation of resources of historic idiocy. It fits, of course, very well in a timeline where grifters are on top of societies around the world.
It is a fundamentally wrong path that should not be followed and scientists around the world should articulate exactly that instead of producing marketing blog posts for a system with such fatal inherent issues.
