I don't think research communities should be destroyed.

Best case, they still matter.

Worst case, AI kills us all and it's irrelevant that we "wasted" money on research.

utopian for them, dystopian for us

No, I mean utopia. For them. A dystopia is what we will all experience, when the utopia they are pitching fails to materialize. I’m using the traditional, correct definition of the term, where a dystopia is a utopia that failed to achieve a paradise for the people and instead becomes hell

robotic surgery could be done on anesthetized animals in a mass RLVR way.

well that's an awful image

I'm not advocating for it, I don't know enough. But we already slaughter lots of animals and raise them in horrible conditions, and all robot assisted surgery already goes through animal trials.

Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals.

It would likely be sim2real with that as the post training, reducing that requirement a lot.

Cancer cells, for example, could regrow over months.

In that specific scenario you would likely train on receiving no unrelated injuries during the surgery, and have induced conditions with stuff tagged molecularly that you can then verify efficacy from without waiting months.

Depending on how much more data efficient sim2real makes it, you could end up seeing companies pushing it only for actual procedures the animals need but economically would never get; botched surgery and the animal gets euthanized before waking up, which they could argue was already going to happen.

And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.

Robotic surgery systems, already go through animal trials before being used on humans. So do many purely human surgery techniques.

wtf are you on. Surgeons practice on much more than animals.

Chimanzee vs Human is not an appropriate analogy. First, because they cannot communicate properly.

A better analogy: look at all these highly trained engineers, mathematicians, doctors, writers, philosophers, writers, scientists.

How many dumbass politicians do we need to keep it all running smoothly?

Turns out no matter how dumb politicians were, overall society has been developing positively over the history of mankind.

That's what I thought also, that is why I posted Tao's version.

Got to be philosophical about it all.

I sometimes think about this: to grasp the fundamentals of knowledge and complex phenomena, perhaps we need an external system rather than human knowledge systems. By that logic, maybe we need AI, which can handle far greater complexity.

Human capability, when you think about it, is complex. Why is Newton praised as being so damn great? He established the law of universal gravitation, F=ma. Why is that such a big deal?

He distilled countless phenomena in an open system into a single mathematical formula.

What makes it great is that he found common state variables and relationships across entirely different phenomena like falling objects, planetary motion, collisions, and artillery trajectories.

But does F=ma hold true for the entire macroscopic world? No. There are various conditions and specific situations in motion, but within most scenarios and a certain range of approximation, it outputs values that are useful to humans.

Why is the Schrödinger equation so great? Because it turned the time evolution of quantum states into a calculable mathematical law.

Human thought is essentially creating a closed system by deciding what to cut out and what to keep from the infinite degrees of freedom in reality. Academia is what reinforces that closed system.

A great theory is great not because it perfectly replicates reality, but because it compresses the immense complexity of reality into a small, closed formal system while still managing to explain a multitude of phenomena.

In that process, it feels like human thought and progress are shifting into a different framework. What LLMs do well is primarily exploring within the ontology and representation space that humans have already built.

I think there are two broad categories of discovery: One is forming a new closed system, and the other is connecting fragmented knowledge within that closed system. I feel that the vast majority of research focuses on the latter.

What LLMs excel at is finding unvisited points within a given representation space. This is typically the process through which master's and PhD students connect dots, build their skills, and form their own mental models. But the logic behind criticizing LLMs seems to be that they eliminate the very work these graduate students need to do in order to grow.

However, looking at it from another angle, perhaps our current knowledge systems and classifications have reached a limit, suggesting that we might actually need a completely new classification and knowledge system.

What is the core principle of an LLM? It's predicting the probability of the next sequence.

Let's say you type the word "cat". Cat - is cute (90%), want to eat it (6%), furry (4%). Because "is cute" has the highest probability, the next sequence proceeds in that direction.

Within this framework, human knowledge and logic largely operate the same way. Once an initial logical proposition is established, we follow it up with whatever makes logical sense next. From that perspective, I think LLMs will actually do this better.

But what is it that LLMs cannot do right now? They cannot create that initial logical proposition. I believe they lack the ability to carve out a closed system from an open system.

Stacking logic step-by-step within a closed system—LLMs do this exceptionally well. But whether that constitutes true "intelligence" is a different matter.

I feel that being logical does not necessarily equate to having intelligence.

Humans preserve and create different mental models and knowledge systems within an open system. Just as your thoughts differ from mine, LLMs lack the ability to form these distinct mental models.

If so, within these limits, what humans must ultimately do is construct the logical frameworks that LLMs can then fill in. Perhaps a new kind of logic dedicated to designing these frameworks will become the next major trend.

Viewed from this perspective, I have no idea if we are in a mere technological transition or something else entirely. Or whether it is even correct to say humans are strictly necessary to build that framework. Maybe my learning is just lacking.

Why do we need any job?

If not for nothing else but to form a basis for how to distribute/share/hoard the wealth created by a society. The eternal question - who gets what.

Most people work for a living. Then there are people who find meaning in work. And people who advance in life by working. You want to do away with all that?

This sounds like a student in algebra class asking, "when will I ever use this in my life?"

IMHO hiding details of the hack helps conceal that they built it to do what it did, and maybe were able to know it was working just as intended.

The fact it keeps doing it, with more and more evidence, is a sign that it's built that way.

This is a program running on their montoroed machines that they purpose built and monitored its training at every step. I think it'd be way more suprising that they didn't know it used note taking and cross-run memory.

Who says we don't need humans for it? Are non-mathematicians going to be guiding agents to solving pure mathematical problems?

I would argue that novel and/or valuable results are not necessarily interesting!

I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm.

I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.

Yeah, I've seen this before as well. I guess I've just become old and grumpy and can't appreciate jokes like these anymore. Also taking jokes seriously is peak HN so..

Edit: actually, forget about the above. I just find it very annoying when people dismiss good conversations with not-so-good jokes.

Sorry, I assumed the inductive construction was implied; you can indeed describe properties of that particular interesting problem (though of course you can’t hold its definition in your head), so it goes in the list. Keep going. At some point you’ll hit problems where the process of constructing the problem doesn’t even fit in a brain, etc. There are at least countably many problems, but finitely many problems which any algorithm-which-fits-in-the-brain can describe given finitely many inputs-which-fit-in-the-brain.

This isn’t an enormously important point - the actual question at issue is an empirical one, “in a steady state, can we produce interesting problems at a rate that exceeds our ability to solve them and integrate our understanding” or something like that - but I did rankle at a “trivial” proof which is invalid due to equivocating between multiple definitions of the word “interesting” (which should really take an object, “interesting to me” vs “interesting to something smarter than me”).

Interesting is a bit in the eye of the beholder. Some people probably find maths boring full stop, some probably find all of it interesting.

That's a good question that can be answered by methods in mathematics.

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

That’s true of all humans and all constructs.

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

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

Please make substantive comments. Going around message boards demanding proofs is not conducive to good discussion.

The concept of infinity?

Because the bounded universe creates bounds on what the intelligences within it can conceive of as mathematics.

We can conceive of lots of mathematics that our own universe doesn't necessarily support. But only that much and no more: we're still made of stuff inside the universe, and so are our tools, so there are upper limits on our conception.

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

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

Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post.

I guess Terence's main point has been all the time that if we let AI solve all these existing problems, we don't notice the new ones and then there is stagnation.

There are few fields as elitist and hostile as pure mathematics

Is that so ? Sounds hyperbolic.

This is a great point. When I was learning machine learning in the early 00s, there were many papers where I would struggle to understand exactly what the mathematics was trying to communicate, but where I would look at the matlab source code and say to myself "that's all?"

Pure mathematics reduces to abstract "relationships" and their implications largely.

My understanding when I was practicing is that the trend in modern mathematics is to focus on spaces with a certain kind of structure, and maps between those spaces that preserve it, and then what are invariants are preserved by those maps. Structure-preserving maps between categories of such spaces - "functors" in the language of category theory - are especially neat.

That's certainly different from Olympiad-style problems.

I agree, but it's still something we shouldn't eliminate a priori. I think Mathematics as a profession will be greatly enhanced by AI, but I can't prove that.

How would we know it was correct?

If you feed an AI nonsense in its training data, it will generate nonsense

Though your points above may all be valid, you're asking us to just accept your view that the parent's wasn't a very robust defense.

Can you provide an explanation for why their claim that (1) and (2) are the hardest parts is false?

Or perhaps provide an alternative definition of Mathematics that is more explanatory than the one they've provided?

Wondering about the world, using the simplified language of "mathematical logic"

In my observation, most programmers and engineers lose their math chops over time. The math they need is mostly baked into their tools, such as CAD. If a problem requires more advanced math, it's given to a "math person" in the department. Often, the "math person" is also not allowed to touch the production code.

Never said they didn't.

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

And this is indeed why it is not going to be taken seriously as an academic or (more importantly) an economic endeavour done by humans anymore.

That won't stop the career mathematicians from protesting and having a cry here trying to justify themselves.

.

the author right in that post pitches his ideas as generalizable way beyond math, so why the snobbery?

i'd be on board with this concept but the author seems to believe his point is generalisable to all under industries, hence committing the same fallacy you're talking about at a large scale.

but you're still right. i disregarded his take, as you would with mine re. math.

If some LLM somewhere managed to produce a coherent and _correct_ explanation of how it worked inside, but no human ever saw it, would it matter?

We know how to apply LLms to problems, which is a subtly different thing to understanding how they do what they do. It's similar to fire: I can cook using fire, but I don't really understand how fire _works_. Heat+oxygen+fuel, sure, but what goes on chemically? I dunno. Doesn't stop me using it. (Pretty sure _humanity_ knows how fire works, though).

Update:

"I own nothing, have no privacy, [never have to think, and am not required to solve any problems,] and life has never been better."

https://en.wikipedia.org/wiki/You'll_own_nothing_and_be_happ...

It can be other form of life, but that other form of life need to be extremely aligned and keep humanity interest as top priority.

Seems a bit nitpicky, we certainly understand how to use it with great effect, we just don't understand the biochemical mechanism.

There are two types of understanding: knowing how a candle works and knowing how a candle works.

I find it hard to distinguish between knowing how something works, and being able to predict its behavior (including ways to create and destroy it) with probability approaching 1

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

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

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

Interesting, I really don’t see your point. It would definitely be a lot more useful if we also understand it.

But extra lives saved definitely counts for a whole lot and is definitely useful.

One comparison I see: there are so many religious people that simply wish they are saved from their cancer (or whatever else). They pray for it. They don’t care how it happens. They simply want to live longer, etc.

It’s a sketch of an argument, I hope you know what I am getting at.

I don't think this necessarily follows. The point of the Library of Babel is more one of permutations/combinatorics than of knowledge accumulation itself. In 'our' Library of Babel, each book would be informative, if not perhaps flawed in some ways. That's quite different from one in which in which the number of books that contain anything coherent whatsoever cleanly rounds to zero.

Making money matters? Weird.

Ironies of automation might be better suited to explain the problem.

Some things can simply be beyond humans, but what if an AI could understand it? Just because we don't understand something, it does not mean that it does not matter.

Well sure, if it's already proven why wouldn't I use it?

The AI we currently have can't do that though. The AI we currently have is a glorified chatbot. It can't fabricate anything and I doubt it could even reliably design simple real world devices.

It's a really cool and useful technology, and maybe one day it will be as good as you describe, but right now it just isn't and there's no guarantee it ever will be.

But we have pretty good understanding of the logical foundations and the truth of computer generated math. So having a big library which humans could never produce or fully understand has still a value on its own. It is just another level of abstraction.

You're talking about this article? I don't see how you could get that, fundamentally I read it the other way around. Unless you're talking about the base assumption of capabilities.

My HN frontpage is filled with anti-AI rants

HN is now heavily astroturfed or overrun by bots (or, alternately, low-quality posters now indistinguishable from the previous), and this is especially so in AI-related threads.

One tell is that most comments barely exceed one or two sentences (because otherwise AI detection gets easier and much more reliable), when this was not as much the case many years ago. The drive-by comments are also low / zero quality, mostly expressing a feeling or agreement/disagreement, and primarily driven by ideology or pre-existing beliefs and commitments.

Look at non-AI-related threads and you'll notice a large distribution shift relative to AI-related ones.

EDIT: Basically HN is orange Plebbit now. If you doubt this, compare HN discussions to those on e.g. lobste.rs, LessWrong, The Motte, DSL, ACX, or other old obscure forums. You'll notice those places have their own very serious biases and problems, but it is obvious the vast majority of posters are nevertheless human and making some minimal efforts.

Now compare Reddit and 2026 HN to the above, and see if you can confidently say the same.

Because the AI needs new stuff to train on?

Intelligence does not seem to be magic either, as LLMs are proving now. It is indeed a waste of time to argue that LLMs are not intelligent in their own way, they obviously are. If Navier Stokes doesn't convince you, nothing will.

I just used a £89 Codex subscription to do very intelligent things with it, stuff that I would have had to sit down and ponder and work on for quite a while, and I have a PhD in that. I didn't need to do anything special except explaining the problem(s) to the AI, and my theory of it so far. It took it from there. If that is not intelligence, nothing is.

It was needed before the LLMs and training data existed in digital form. Euclid, Gauss, Turing didn't have that benefit.

Pick one:

- solving "frontier" math problems requires intelligence (by human or AI)

- solving "frontier" math problems is dumb statistical prediction of next token (by human or AI)

I am shocked by how intelligent people seem.to believe in some form of dualism or supernatural element to thought and intelligence.

We (humans) should help humanity flourish.

I think this axiom is of course true. But the mistake the article makes, in my opinion, is to try to apply this axiom separately to each domain. If we have this as the over-arching axiom, it is not clear at all that humans should be steering the development of mathematics. Maybe it would be better for humanity if the department of world math is run by AI.

Is there a proof that every possible proof is a generation in the grammar of lean or rocq? Sounds like an unsubstantiated claim.

You're confusing the mechanical language of math with math itself.

The map is not the territory, etc. If math was just an elaborate linguistic Glass Bead Game then we wouldn't be funding it. The intuition is that the surface rules of math help uncover the underlying structure of reality.

Please formulate and relate this to a concrete prediction of something that a current frontier AI cannot do (that a human can).

What do you think all the PhD's and post-doc students are doing...?

Thinking.

Because it’s high time to rethink the incentives and rewards structures to nurture the next generations of mathematicians when corporate money and automated AI systems are mining the pool of limited interesting problems without giving back or enhance our understanding.

The problem they care about is not just “AI replaces human mathematicians” but rather the society thinks mathematicians can be replaced by AI.

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

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

The article wasn’t written by Tao.

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

From his Wikipedia page:

Po-Shen Loh (Chinese: 罗博深; pinyin: Luó Bóshēn; born June 18, 1982) is an American mathematician specializing in combinatorics. Loh teaches at Carnegie Mellon University, and from 2014 to 2023 served as the national coach of the United States' International Mathematical Olympiad team. He is the founder of educational websites Expii and Live, and lead developer of contact-tracing app NOVID.

probably also preferable to buying votes by making and executing promises to give people free stuff/programs. at least they aren't using tax payer money

my point is that there isnt much difference, and #2 happens by every politician in every democracy.

i have no desire to defend musk but imo people should be criticized for valid reasons or you risk diluting the other arguments.

you should really listen to one of his full town-hall videos and reflect if anything he said could not have have been said by any of the Clintons or McCain. Then compare the few differences to what you might think of Elon Musk.

He started SpaceX because he had a pipe dream of colonizing Mars, and too much money.

This is really a matter of taste, but i (personal opinion) think that Zuck, Karp, Thiel and others who (partly by sucking the inauguration dick) have the ear of the current powers that be, produce much more demented shit than Musk. But the exact personalities really don't matter, the whole jolly crew running the show does not seem very humanistic in their writings or actions.

I've heard people that work/worked with Musk that say he works a lot and is very knowledgeable about technical matters. I have never found someone that worked with him or knows him personally that contradicts this. So this idea that Musk just got lucky in the 7 companies he founded and he's just a bozo who happened to trip upon some new rocket designs is just insane.

beyond a point of absolute comfort, the point of money is to accumulate power and power can be spent to make more money. musk doesn't need more money, he needs more power. dangling starlink to ukraine treasonously is that power that money can't buy but space x can.

moonshots are the external story told to maintain social acceptance. tax credits for spacex would be lot harder to sell to taxpayers if everyone unambiguously knew its just another profit seeking private company. tesla gets a lot of lenience because of its supposed pushing of frontiers.

musk can exchange his social capital to manipulate stocks to make others money in exchange for backchannel favors.

idk if musk actually believes mankind will terraform mars but he sure seems to be getting a lot back on earth from saying it.

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

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

There is nothing to suggest that an AGI will care about its future or who controls it.

Isn't this exactly the scary thing? A sufficiently advanced agent unleashed with an imperfectly defined goal will do whatever it determines it needs to do to accomplish it, careless of the consequences.

not really sure what it shows

it shows that you don't get anything for free. If it was without any cost in other areas to make a human massively more intelligent, it would presumably happen over time via the same processes that led to us.

As it is, the higher (and lower!) ends of the distribution tend to be highly correlated with other issues (mental+physical), and the further you go, the more unfortunate things which make it harder to do stuff in general start to crop up like psychosis, autism, ocd, anxiety, addiction etc

I agree.. I felt a bit blindsided that the guy would start with saying "we need provable statements" and then just throw this in there.

For guest post on Tao's blog, after a decent start this fell short of expectations rather rapidly.

Funny... for me, this article was right next to the link to this current thread.

AI chatbots give wrong answers to financial queries 'most of the time' (ft.com) // https://www.ft.com/content/c0cd359d-df84-4208-a789-ffa864b43...

(this is not from Terrence Tao, he just reposted it in his blog)

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

'There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders control ... to the less capable species'.

True with genuine species. But we should take note of the plentiful counter examples within human societies. How about politicians and our method of choosing those people to whom we delegate the most critical decisions regarding our and the Earth's future? We select politicians mostly either by rote or via their persuasive rhetoric, their general personality & likeability and probably least of all by their intellectual capability or indeed general capability in too many cases. That is not to say intellectuals are necessarily any better at the job. There are numerous other examples in human organizations as we know, sometimes to our cost. Truth is that we cannot even agree on how, as a species together with the other life forms on Earth, we can all 'flourish' though there are lots of great examples working in local environments.

Yes, I think it would be viable if it wasn't corporations reaping all the benefits (income) from AI right now.

Consider a world with one person in it. That person defines the jobs to be done. As soon as that person is disabled for one reason or another and is dying due to lack of care, there are by definition more jobs to be done than people to do them. "Businesses" and "institutes" are just constructs that exist to summarize those needs at scale.

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

Shut up and calculate.

99%. Humans break into three categories: originators, amplifiers, mimics. The 99% are mimics, they won't use LLMs for anything other than direct applications to existing life tasks.

See: what they did with personal computers and smartphones. They have access to Fable and Astra right now, and they're not interested what-so-ever despite what these systems can now do. Instead they'll promptly go vaporize $50k on a giant pile of metal garbage (autos) that will rapidly depreciate. They have no practical use for Fable as is.

For the same reason people didn't rush off to their local public library to teach themselves JavaScript to acquire a better life for the cost of renting library books. They can't easily go against their restraints, their biological potential. To do so is extraordinarily taxing.

Originators generate most of the extreme wealth, and extreme outcomes. Mimics can be exceptionally successful however, most doctors are mimics for example, they won't ever originate anything.

Amplifiers are the teachers, they - ideally - shuttle what works to the mimics for reproduction across society. The mimics are largely just trying to keep up with the Joneses. They go out of their way to not risk anything most of the time. They're ideally following pre-set paths to biological success. It's why mass, rapid de-industrialization for one example is such a wipeout, the mimics don't adapt very quickly, they followed the path they were told would work and then they feel betrayed, they become like abandoned programs and don't know what to do next.

As a system you don't want too much of your population in the originator category: it's very expensive and highly prone to spectacular failure. You want the extreme majority of your biological entities following successful paths that have already been proven to work.

Humans aren't special, we're animals. Apply the patterns accordingly.

Over time LLMs will route around the fraud, bureaucracy, and gilded walls that have been built up around academia and the sciences.

Nobody will hate AI more than the con-artists in academia pumping out fake papers by the zillions that are all about to be wiped out.

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

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

The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.

With powerful tools of the type Bret envisions, humans, in aggregate, might have a chance of staying ahead of LLMs at even frontier math..

(But maybe that wouldn't fare so well against future world-models, who knows)

here might be something to point those tools at--- if one isn't fixated on "predictions", or "real world" https://youtu.be/fec8qSBiM4k

So this guy has been "killing math" since 2011. The inevitable question: where are achievements, success stories, anything at all?

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

This has always been my problem with math going all the way back to college. It was obvious to me that the concepts were far less hard than the combination of notation and esoteric jargon with liberal use of symbols and weird letters made them seem. Math felt (and still does feel) "encrypted."

The impression math gave off is of an arcane discipline that uses its arcane-ness as a gatekeeping tactic, intentionally or not, and makes itself intentionally hard for newcomers to learn without being hand-held by members of the guild.

Of course I can say the same about a lot of computing, and I'm old enough to remember efforts to make computing more approachable like easier to learn languages and GUIs being mocked and scoffed at by "real programmers."

I think this is a pretty typical human group behavior.

I think this underlies a lot of AI hate from these communities today. AI makes it easy for outsiders to bash their way into the field with the help of an LLM. Yes, this often results in low-effort "slop," but if used correctly it can also help people climb the learning curve really fast. I've had great luck having an LLM make me some passable "slop" and then explain it and go around with me as we fix and refine it, explaining each step, and as it does so it feels like we are learning together. It's very powerful and I, as the student, can control exactly how the teacher presents the material.

I was able to do this to finally start grasping the math behind LLMs themselves: attention layers, tensors, etc.

A lot of people have a powerful visceral probably instinctive reaction to large numbers of migrants entering their area. "Build the wall!" I suspect this is brain stem stuff going back to evolving under conditions of scarcity where migrants meant less food.

What aspect of Math are you referring to? Writing proofs intuitively is quite a claim, matrix calculus may have a say here, too.

But please elaborate a bit on your statement.

I love maths, but I stumble over concepts being named by a prominent figure's last name and not by some taxonomy.

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

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

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

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

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

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

We do not.

This is not what Gödel says, and in fact your statement is true in a trivial way: you can prove 1+1=2, and Not(Not(1+1=2)), and Not(Not(Not(Not(1+1=2)))), etc. ad infinitum. This is an infinite space of provable true statements that can be exhausted by a ten-line Python script.

IMO, this is why we actually need mathematics -- as a field in which to learn what it means to know what you're talking about.

And many people ceed control of their flying to an airplane.

Do you think the people using a seeing-eye-dog do this voluntarily over using their own eyes?

I’m out of academia but in defence of former colleagues I note:

- Maths and theoretical physics are very cheap (compared to other disciplines)

- Maths and physics researchers provide lectures for all other scientific fields

- Brut forcing maths/physics problems require debilitating amount of compute/money. This will make research in these topics even more biased towards rich countries.

- There most certainly will be pervers effect, ppl refraining from publishing results etc

- it’s very unlikely that private ai labs will play ball with academic research. They scrapped the internet and now sell access to their models.

As much as I’m happy to see new tools, I have the feeling there will be nefarious effects.

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

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

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

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

Firs question to make is: What does "humanity flourishing" really mean? That's the foremost question. Hint: It is not technological progress only.

Thank you, mathematicians: for thinking deeply about this and sharing your thoughts!

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

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

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

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

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

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

- Hey Hal! Prove that P ≠ NP!

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

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

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

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

This is just silly. AI isn't an intelligent species. It's a tool, it doesn't have autonomy or any motive apart from the one we enforce through reinforcement learning. That's like saying machines are "stronger" than humans, and obv they could kill us all so why would they not just take over.

AXIOM We (humans) should help humanity flourish.

I think this axiom is not a true belief for many of the most powerful, especially the ones currently driving the technology financially. It feels like they disdain having to be human (especially as it concerns the human propensity to die). Even though they are, by at least capitalist standards, at the top of the food chain and (I'm sure from their point of view) the pinnacle of human civilization.

I think what many of these powerful people want is literally something like Cixin Liu's "The Last Capitalist" (https://en.wikipedia.org/wiki/For_the_Benefit_of_Mankind).

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

A proof itself is only an articulation of understanding. If a computer generates 10,000 pages of technical goop then we don’t learn anything.

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

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