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> so advanced that it's just as readable to me as Greek

I used to feel this way about statistics.

The language and terms are hard to understand and many of the formulas are taught as "just memorize this" instead of building up from first principles.

But then I started using statistics to analyze something I cared a lot about (paintball) and I quickly realized it's like learning anything new:

- there is jargon

- and core concepts

- when you learn the above, it suddenly makes a lot more sense.

Not to diminish the comment, but most things are not as complex as they sound when phrased in everyday language or sound much more complex than they are when phrased in technical language.

Technical language is a tool that allows insiders to say less and refer to more, and to be specific, but it's just a tool. Most things can be described in accessible ways.

I think you'd be surprised at what you could understand and at just how few domains are truly complex enough that a layman couldn't understand with a little bit of patience and an accessible summary.

Thanks for posting this comment, it makes me proud of myself to be able to partially comprehend the comment :)

Very confused by this comment. The older (poorer) parts of the ML literature focus on models with convex and (gradient-)Lipschitz objectives, but that's not representative of reality, not even close. Modern objectives for AI models are famously nonconvex (catastrophically, from the point of view of classical optimisation theory), and that's where the interesting research is.

What do you mean by this? A neural network hypothesis space is not typically strictly convex or a lipschitz function.

AI is a threat to everyone. People who claim that AI will never be able to do X have consistently been proven wrong.

The only people who are safe are those whose jobs depend in some way on their humanity. e.g. yoga teachers, bouncers, etc

It's funny how you applied on your own argument several logical fallacies about why ai is only a threat to people who have been taught "recipies" versus who know what L1 cache miss is.

Actually it's sad there are people out there dumb enough to believe knowing L1 cache is any different than knowing recipies when it comes to the story which jobs AI will take. I'm convinced by now it will be the jobs of those people believing such crap.

Interesting, thanks. I don't know where "around here" is, but the signals I've seen in a lot of articles is that the demand for junior software people has taken a dive since a year or two back, with student programs etc getting cancelled. One googler said they were getting a junior to their team and that was kind of a big deal because it hadn't happened in that whole department for a long time.

In relation to that, I guess my question becomes: if the same thing will happen in math research, who will write the ten page math proof prompts in the future?

So... The AIs with no model of the world are replacing software developers that have no model of the world?

Unless you’re claiming that AIs will suddenly (and very soon) stop improving, they are obviously a threat to everyone’s job.

Calling notable conjectures that have been open for decades “low-hanging fruit” is an act of desperation. Most professional mathematicians couldn’t have proved those conjectures if their lives depended on it.

Indeed. Perhaps my article here will be of interest to some: https://blog.oberbrunner.com/blog/ai-math-as-humanities/

The author also used GPT-5.6 to write the prompt. This did involve giving GPT-5.6 access to his previous work and a back and forth process (so definitely still used the author's expertise to some degree), but the prompt itself is also largely AI generated.

Certainly. This feels similar, to me, to how building complex software with LLMs works today in practice. You need to know a lot to set up goals and guardrails and verify outputs. For me, making the bits change was always the fun part, not tangling with text in my editor, though that had its moments.

Yeah, back to the gold-in-gold out use of LLMs.

I've spent some time working both as a math researcher and as a software engineer, and I think this comment actually underrates the similarity between the two fields as they're actually practiced.

Some math research does involve grabbing a single, fully specified conjecture off the shelf and hunting for a proof of it, and it's true that if you manage to solve a long-standing open problem, other mathematicians will be interested no matter how you did it.

But this isn't all of what they do, probably not even most of what they do. Like in software engineering, it's not always obvious which question would be the most useful one to ask. A lot of mathematical work also goes into what we call "theory-building", where you could say that primary work goes into coming up with definitions rather than theorems. Mathematicians also care a great deal about how something is proved; a lot of them are some of the most aesthetically picky people I've ever met. Words like "ugly", "beautiful", "creative", and "boring" are used to describe both definitions and proofs all the time.

From the outside, it can look like all they're doing is pumping out proofs at any cost. But I promise you that when I talk to mathematicians who don't have any experience building software, they have a similarly narrow view of that field as well! Both fields, from the inside, look a lot more human than you might expect.

I think the difference is in math the problem is fully specified and easily verifiable and in programming it's not. I don't agree that we always know we can solve the problem.

> So I'll say it again, AI will win a fields medal for before managing a McDonald's simply because there are enough big problems within arms reach than their current capacity to plan over time

AI can manage a McDonald’s already. If manage means directing humans to do something to ensure the store is running. If manage means running robots, then yes maybe that is 5 years away but just directing humans to run a store, that is possible right now.

I’d like to see four color conjecture and an elementary proof of FLT.

Seconded on the "not cheap" argument here. I've spent $25 worth of tokens completing a one-week task in an afternoon, or rather my company spent the money. I would never have personally felt OK with throwing this much money after some prompting back and forth for a few hours, one lazy Saturday afternoon. I ran the risk of not finding the solution before the token usage would be too high for me to want to carry on, if I was my own credit card linked to the account.

Of course money in this situation is a bit of a funny measurement, right, because if I was able to take the rest of the week off as soon as I had solved the one-week problem, then I would have no problem at all throwing even $100 worth of tokens at it, so I could enjoy a nice 4-day "mini-vacation".

How cheap "cheap" is, is indeed "in the eye of the beholder".

I mostly agree, these are things that couldn't or weren't solved with abundant capital at all, and now they are being solved

it went from not having a price, to having one, and we are trying to retroactively transpose economic viability or economic existence to it from some parallel and prior time.

"Lack" isn't the right word. "Lacking" is more like it.

If there was a deep fundamental inability, we wouldn't see things like newer generations of LLMs consistently improving on ARC-AGI series (heavy spatial reasoning loading) and SimpleBench (a lot of commonsense + spatial reasoning components).

In a way, it's a surprise that LLMs, notoriously lacking any sort of embodied experience, can even get this close to human baselines on tasks like this.

My takeaway is that text is a far richer modality than anyone has expected - and that high end LLMs are often sharp and flexible enough to recognize their weak points and substitute their strengths. I.e. all the LLMs implementing A* to optimally solve pathfinding in ARC-AGI-3 tasks, often unprompted.

There might still be unrealized gains there from true depth-unbounded recurrence, or maybe from finding better ways to integrate modalities in training. But clearly, a "fundamental limit" it ain't.

Is there any proof that they are not good at special reasoning? Arc agi 1 and 2 are saturated.

Llm's do have discipline and persistence, though.

It’s no diff to an educated human.

Stored potential.

But will that potential be converted that contributes to the economy..? That requires other traits.

Might be focus, might be discipline, might be the need to get revenge lmao.

This is what the llm-boosters miss. Progress is willed into existence.

AI / LLMs are not intelligence .. they are just a prediction engine that has been branded as 'intelligence' by marketing employees

At the end of the day it is still making a best guess at what the user wants based on data it has seen before.

It still requires someone smarter than the output to be able to evaluate if the result is any good, or just hand waving.

I generally agree, though direction, intuition, and domain knowledge are still relevant. Your breadth-first-search framing feels right, but you still need a sense of which paths are worth following, and you need to know when to trust the results.

I’ve been doing more math as a hobby in the past few weeks — working on lesser-known conjectures and exploring proofs of hard theorems — than I could have managed over the previous several years. It’s an exciting time.

If you really believe this, try to use GPT 5.6 to prove an open problem you know nothing about. You might get lucky, but if you don't, you will soon discover that 5.6 can make "progress" towards a theorem without actually getting anywhere pretty much indefinitely.

The infinite monkey theorem assumes random distribution of symbols*.

Given the tokenizers have a vocabulary in the 10k-100k range, "a million attempts" will generally still only get the first token of the answer correct.

Even really rubbish models, e.g. talkie, the "what if we only use pre-1930s data to train a model?"** model, had to be almost all the way to the right answer to reach the really low HumanEval pass@100 score of ~0.04 (I'm only eyeballing the relevant chart).

* Actual monkeys not being like this is, while amusing, irrelevant

** https://talkie-lm.com/introducing-talkie

>Ever heard of the infinite monkey theorem?

Even if every atom in the universe were a supercomputer generating a trillion trillion random characters every second since the Big Bang, the chance of producing Hamlet would still be essentially zero.

You've not seen how they react to noobs asking physics questions, I think.

Even when you've got an interesting idea, if you're an enthusiastic amateur who don't yet know enough to phrase the question right but does actually know the basics, they'll put you in the same category as the people who think healing crystals can power hyperspace telepathy with Anubis: "oh no not another one".

LLMs have infinite patience, but unfortunately come (came?) with too much sycophancy, giving even more people far too much confidence.

(within limits)

Do we employ mathematicians in customer service roles?

Uh.... Have you ever called customer service lately?

I think the big names behind the AI companies already have that problem solved. A lot of people probably won't like the solution very much though.

When machines are doing all the work - we no longer have to.

I can't stop wondering myself.... I'm writing some software with AI and wondering, why am I doing this? Will anyone need this? Will anyone have money to buy this?

Best I've come up with is we'll need to be adopted by technofeudlaist overlords to be our patrons like in the roman days

That view is incredibly reductionist. It really is an efficient encoding of how nature behaves. It might be a human construct, but given how best it allows to understand nature (through principles of physics), it is uncanny to be any different from the language of nature.

Reminds me of Wigner's Unreasonable effectiveness of mathematics in natural sciences [0].

[0]: https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

At a very high level mathematics is basically 100% text/symbolic rewriting. You start from some set of postulate assumed true and you do your thing to get a new different set of equivalent assertions in a form that is more useful.

I don’t know if LLMs will kill the working-mathematicians but at least seem like that it doesn’t seem absurd to imagine LLMs will be good at math…

We already drown in useful new things and especially knowledge. There's a limit to how fast we can adopt them.

just ignore these people

gig economy is a blessing to keep the LCD busy and doing something productive instead of being nihilistic losers

it’s better to focus on how to keep these “knowledge” workers busy for the AI age

they don’t like to work and complain even on a w2 job (where someone else had to figure out how to capitalize them) so in many ways they’re far more dangerous and hopeless.

Maybe we should also burn books and lobotomize ourselves to make science more fun and challenging. The ego here sickens me. I don't care about your ego. I want accelerated material science, medical science, energy, better outcomes for people across the world.

A rational reaction would be to identify the political coalition that would act in this manner and place yourself in opposition to it if you do not see yourself being able to change it.

I wonder which one that is.

wasn't that always the reaction historically whenever people felt like their livelihood was threatened by new technology?

Why? No one will be able to effectively ban AI. Pandora's box is open, and we can only witness the consequences.

unfortunately many won't get to experience it much because we'll be stressed and struggling to make a living

You ever play a video game with god mode cheats enabled, so you can unlock all the unlocks, get all the best gear, and be an unstoppable force with unlimited money?

Yeah, it's fun for 30 minutes.

If anything one wants to make is now possible to make, how is anyone supposed to make a living?

Can I make food with LLMs? Can I build a house and make clothes? This is stupid. No real wealth is being created for the general population here.

People like being needed and important to other people. That isn't some artifact of capitalism; it exists across all times and cultures.

I guess when cheap Chinese model was very close to SoTA models

The subtext of this whole post (or at least a subtext that some might read), is "we don't need mathematicians/programmers anymore" or "we will need much fewer mathematicians/programmers". So the fact that this result required a year of prior research and a 10 page prompt of specialized knowledge goes against that subtext. You still needed the human just as much to get to the result, and the LLM ended up being a tool to find the last bit.

> Saying “solve this problem” doesn’t get good results most of the time with humans either

Sure. That is not even remotely the point I was getting at. Already we see the thread filling up with comments about how human skills are irrelevant, using a mathematics PhD applying his expert skills in a way that the people who are saying that could never have done to justify their inane conclusion.

Calling it prompt engineer is doing it a disservice. With agents we’re well into process engineering, which is a ton more interesting.

The obvious baby’s first process is “plan -> execute” but as we learn about the strengths and weaknesses of LLMs you have to start unpacking that process into planning, prototyping, testing, validation, reviews, and tons of research. If you treat it like an extension of your brain that can automate some thought processes, it becomes a lot more powerful.

One of the key skills of a professor is asking the right questions. Figuring out something worth working on, and then framing it in an appropriate way and asking questions that allow someone with specific tools and skills to make progress in the topic. Usually the tools and skills are those available to a new student, but working with an LLM is similar.

That skill comes with experience. Most people don't have it immediately after PhD.

Ah, but who prompts the prompters?

I find it strange that people sometimes think of knowledge as 'public property for everyone.' The essence may be one, but the mental model of knowledge is individual. For an LLM's knowledge to become mine, I need to digest it to some extent.

And programming, as the programmer who created Eliza once said, is the act of becoming a legislator of your own universe. So even if there are black boxes, if you want to build a program that fits your own worldview, studying is essential.

Rather than prompt engineering, I think it should be called overall harness engineering. Anyway, that's how I feel these days

That doesn't make any sense; you can't have one LLM to read your mind to prompt another LLM.

And yet in this case a human prompted the LLM for this result, not another LLM

So these days I've been writing down my thoughts on my personal homepage. Things I've learned, my background knowledge, and so on.

I've been realizing that there are more books tied to my background knowledge than I expected, but I'm not sure what will happen as AI advances further.

These days, I'm living for the fun of building my own personal wiki on my homepage

If knowledge is a Swiss cheese, LLMs can help fill the holes, but not make the cheese bigger.

this is a fairly bleak outlook even when you're trying to make it sound the opposite. Only the cream of the crop talent will have value going on?

Most of us aren't Terence Tao

so it seems like The New Big Question In Math is

How's It Hanging, Brother?

The author explains he's an expert in the domain and that he had worked sporadically on the problem for about a year, also with the help of previous LLMs. So whatever he means by "I wouldn't really say that this result is using or creating some fundamentally new techniques" it doesn't mean that the result was trivial. Also, says it might not make sense to work on low or even medium hanging fruits in the future- and I bet that's by far the largest share of work for most mathematicians.

Sure, it's not a breakthrough that opens new roads in mathematics- is this where the goalpost has moved now?

A lot of people who thought they were special and “better” than mere blue-collar workers are realizing that in fact;l they are the same just working with a different medium.

It's such a weird train of thoughts lol. You're using the fact that

- Claude isn't doing that

as evidence to support the assumption that

- it's a marketing trick

Which is obviously non sequitur, as if it were a marketing trick, Anthropic could do it too. Anthropic isn't known for not spending on marketing.

Honestly, nowadays I question human's reasoning ability more than I question AI's.

kind of a hilarious conspiracy theory

You are correct that LLMs are trained on existing proofs but hiring researchers to solve unsolved problems is just unrealistic, both in terms of how none of the mathematicians simply came out and took credit for their own discovery or exposed this, and how training sets are not easily memorized (rather, the meta techniques are learned).

OpenAI just has better training methods and techniques for pure math over Anthropic, it’s one of their biggest strengths

Terrence Tao getting paid by openAI is, to you, the most probable conclusion... much more so then the LLM actually being able to come up with math proofs?

> Why is it only GPT doing this, why not Claude?

Because Claude can't do it. Anyone who tells you that Fable is better than GPT 5.6 at pure math is lying to you.

Terence Tao also uses Anthropic's models in his work. Oh, you didn't know that? Well, now you can pivot to saying that he's getting paid off by both companies. This is actually one of the hallmarks of both conspiratorial nonsense and military-grade cope. Any fact, regardless of how mundane or extraordinary, gets re-imagined as evidence of the same mad-hatter conclusion.

I hope people are screenshotting this stuff. This really needs to be documented. It's remarkable how wild it's getting.

I won't touch creativity, but if this and other results like it do not demonstrate intelligence, what does? How was it able to solve problems that specialist mathematicians have tried and failed to solve for years?

Comical human arrogance...

With such high standards, most HN commenters also do not have intelligence nor creativity. I don’t think we can set the bar that high.

I have absolutely no problem with people disliking or fearing AI. It's energy consumption, effects on education and potential for displacing good jobs are all quite disturbing. But "stochastic parrot" means that "all it does is randomly repeat things that it has seen before without understanding them." It's infuriating to see this written about an instance of an AI solving an open math probably. Do you think the models are just randomly repeating facts until they accidentally emit a proof? If so, then how do they synthesize that knowledge into something logically coherent?

Alternatively, if you think that even Maxwell was a stochastic parrot, then presumably almost every human who has ever lived was also a stochastic parrot except a few rare examples like Einstein. Not sure what definition you are using but it seems too broad to be useful.

I think it's quite clear the proof here shows that it is not a parrot. It objectively isn't.... that's the only rational conclusion. Yet many people claim that it is, so the main conclusion is fear/hatred is causing people to rationalize their logic to fit the narrative they prefer.

Is it not objective reality that the feat performed by the LLM here is much more then parroting or summarizing something?

It's doing math proofs. At this point, it's fully clear that objective reality is that the LLM is not parroting anything here.

Mizar is an early theorem prover. It still exists, see the 2025 issue of Formalized Mathematics journal [1] that publishes math articles formally verified by Mizar (since 1990).

[1] https://reference-global.com/issue/FORMA/33/1

The demographics of HN have changed drastically over the past 10-15 years. I don't want to be the "back in my day", but back in my day, there were a lot more technical people here, and politics was much smaller. Now there's a ton more people, ton more politics, and a lot less "hey here's something really cool I built, it's like rsync but nice"