Why do we need human mathematicians anymore? | Hacker News Reader

Why do we need human mathematicians anymore?

terrytao.wordpress.com

292 pointsauggierose383 comments

> We (humans) should help humanity flourish. I understand that not everyone agrees. I have been called a “speciesist” for being “too human-centric”.

this is insane. the "speciesist" people aside (the best way for all other species to flourish is for humans to eradicate themselves right now, which is completely mad), i have the opposite problem: the _axiom_ seems to be: we should help ME flourish. "Me" as in people who are raking in trillions for their own very special selves right now, at the cost of everyone else's future, while none of them can be trusted to hold my cell phone for a second.

> if an industry commits to the axiom of helping humanity flourish

where does he see such industries, outside of maybe nonprofits?

As I understand you referred to Elon Musk. But this is Poh Shen Loh talking. If you don’t know who he is, look up or at least read his articles.

weird bias by authority aside, do you really think that the daily decisions in this world today, and anytime soon, will be made based on the "let humanity flourish axiom"?

In daily decisions perhaps not, and certainly not when we were in the cave. But from time to time, we will face fundamental questions that need addressing clearly and decisively, and I believe it’s the case with AI. The more advances we make, the more frequent such questions will arise. That is also one of the main reasons why we don’t see any alien civilization. It’s getting harder and harder to keep the entropy low as we gain more knowledges and cognitive power.

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?

Because it will never change if you not try.

> Because it will never change if you not try.

That's why it's important to convince people to not try. It reduces the medium term risk to investor returns.

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.

> 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.

You picked three animals that will literally eat anything, I think they'll be fine.

But there will be fewer of them, because we generate their food.

That's only true for the rats. For the pigs and chickens (and rice), there will be fewer of them because they do not themselves have the capacity to exist at current population levels. The reason they're doing so well right now is purely that we want them to.

Even rats. They are supported by our food and garbage.

>> That's only true for the rats.

> Even rats.

...what do you think "even" means?

> doing so well right now

lol quantitatively well

We also actively cull them, for the rats. Recently watched an episode of survivorman where he was on this tropical island infested with rats. There's absolutely nothing on that island yet there's rats all over the place. Having no humans they may explode in population.

>> We (humans) should help humanity flourish. I understand that not everyone agrees. I have been called a “speciesist” for being “too human-centric”.

> this is insane. the "speciesist" people aside (the best way for all other species to flourish is for humans to eradicate themselves right now, which is completely mad), i have the opposite problem: the _axiom_ seems to be: we should help ME flourish. "Me" as in people who are raking in trillions for their own very special selves right now, at the cost of everyone else's future, while none of them can be trusted to hold my cell phone for a second.

I don't think there are actually any "speciesist people." I think most, if not all, are the selfish "me people," who have disingenuously created "speciesism" by mad-libbing racism and sexism in a ham-handed attempt deflect criticism/opposition to whatever they want to do. Most of them aren't even "raking in trillions," they just don't like thinking about others or being told "no."

there are a lot of people who put the lives of dogs above those of people]. I think you'd find that a lot of people will donate to save animals rather than humans

> there are a lot of people who put the lives of dogs above those of people]. I think you'd find that a lot of people will donate to save animals rather than humans

Ok, sure. But I was talking speciesist vs AI. Is there anyone who would genuinely put AI above on on part with humanity (beyond the rare nutjob weirdo)? I think the people who trot out "speciesism" to defend AI don't care about the consequences and just don't want to be told "no."

Umm… there are a lot of people out there, more than you can imagine… I'm sure there's quite a considerable community who want to upload their minds to skynet.

Every time I see something like the Panama or Epstein files I just get reminded how many people there are in the world doing weird shit.

I consider myself an applied mathematician though technically I’m an economist (macro models, mostly) and operations researcher and differential game theorist (odd combination, but whatever). For context I’m 45 and I’ve spent most of my working life in corporate life.

I’ve always been at a disadvantage academically because I’m rather clumsy with my manipulations, derivations, and I’m a disaster at mental arithmetic. I’m also dyslexic. But starting in the late 1990s when I was in High School I started to become fluent with CAS (Computer Algebra Systems): first Derive, then the Symbolics capabilities of Mathlab, and ultimately Mathematica.

The whole transition is turning out quite well for me: I’m now able to delegate exploring my intuitions to increasingly powerful tools, I no longer have to haul the pyramid blocks up the ramp myself but I can drop them in by helicopter (as if were) and what I bring to the party is intuition and understanding.

I view it a bit like astronomy: in ancient times, before telescopes, keen eyesight was a prerequisite to be an astronomer. Later, after telescopes, anybody with eyesight had enormously enhanced capabilities of observation, and now, with radioastronomy and other forms of remote sensing (neutrino observatories, gravitational interferometers) the whole field has opened up to people who by virtue of being blind would’ve literally been excluded only a few decades ago.

Go forth and multiply: everybody can become a mathematician now. There’s an infinite number of potential universes out there with an infinite number of facts to prove and disprove. And while we’re at it: there’s so much more to mathematics than conjectures, proofs and counterexamples. Solve, model, approximate, fiddle around: as long as you’re not doing trite numerics on arbitrary systems of equations you’ve cooked up for your own amusement you’re good in my books.

Enjoy the tools. Keep your wits about you. Work through the steps that are presented to you. Build your intuition. HAVE FUN.

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.

Absolutely true: there’ll be folks who understand no mathematics at all and who will try to elbow their way in. There will also be (extending the metaphor) an ecology of instrument-makers and lens-grinders who somehow become party to the debate. But to be Frank (though I am James) mathematicians will recognise their own.

> 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.

Also there is some instinctual understanding of the work moving a big heavy rock takes, and how to do it. No such corollary for algebra and hence some civilizations never discovered or understood it.

a scale?

Have you ever heard of concrete, aka “the stone for projects too large to be assembled in blocks”?

Maybe. As a parallel, I know C# and F#. They are programming languages. They are built on top of Microsoft products like the JIT and more basically Assembly language.

I have no idea how assembly works - in my years of experience I’ve never had an issue that required I dig that deep into it. Does that mean I can’t be effective with higher-level tooling because I do not know the absolute basics?

The entire point of a programming language design is to give you an abstracted environment with a complete coherent semantics so that you don't need to understand what's going on under the hood.

You can be effective at using C# and F# to solve other problems specifically because they are very well designed at the goal of abstracting over the lower level hardware. But if your goal is to build an intuition of how programming languages themselves are implemented and how computer hardware works, then, using those languages will be counter productive.

If you're trying to use math to solve arithmetical problems, then by all means using AI to help. But if you're trying to advance the math field itself, then actually knowing how math works is probably essential.

Put in more concrete terms: if someone wants to be a working mathematician without learning the fundamentals of math, then how do they even know what prompts to the AI are worth writing? In what way are they adding any value to the process at all?

I never said one should not know the fundamentals of mathematics, I was trying to express it’s a damned lot easier when once understood you can delegate the lesser aspects to machinery. Abacus, calculator, computer, CAS, now AI: the mathematician’s understanding and intuition are still fundamental, but the instruments they can rely upon are levelling the playing field (and indeed, it is a great field in which to play).

First of all, programming isn't necessarily a very skilled task as being a mathematician.

Secondly, I've encountered compiler bugs many times, so it is very helpful.

That's the difference between a senior and a junior: the junior goes to the senior and the senior figures it out. You can remain a junior at any age.

While true that doing things the hard way is critical to learn a discipline, the OP’s point is about what is possible with AI once you’ve built those skills and intuition. I think in the future it will be common for those passionate about a discipline to strategically avoid AI until they’ve “built the muscles.”

Exactly. I learnt to haul the blocks myself, but it has always been an uncertain and effortful exercise; now I can move them around as effortlessly as if I were playing Tetris.

This might be clumsy but I will say it anyway. If music is math, how do you explain people who have innate musical abilities without formal academic training in notation, music theory, circle of fifths, etc.

Well i can tell you what i think: music isn't math and whoever told you that is wrong.

To further underline the point: entire cultures' musical traditions don't follow the circle of fifths, or notation as you're likely familiar with it, or even notes that follow the kinds of ratios you might use to describe western musical scales of any kind. The mathematical representations of western music were applied in retrospect, math never played any role in the development of western music anyway.

So considering math was never a factor in the development of western music and the math used to describe western music doesn't even apply universally, i just can't really consider "music is math" to be a meaningful statement at all

Rhythm, the core of global music, is math.

2,000 year old Chinese music uses sanfen sunyi. You take a pipe length, cut it by a third, then extend the new one by a third, alternating. Hey look.....a petatonic scale.

Arabic music pitches even more. The maqam system uses intervals that fall between Western half steps, often called "quarter tones," Some used "commas" which divide the whole tone into 9 parts. This gives you octave with about 53 steps.

Math isn’t music, even in a metaphorical sense. The argument by analogy fails at the first hurdle.

Just to make sure I follow, do you mean the "music is math" idea fails, or that my question does? I'm not saying math has nothing to do with music. Frequency ratios and rhythmic divisions are clearly there. My point is that you can have deep musical ability without ever thinking in those terms, which suggests music is more than the math that describes it.

I’m saying that though argument by analogy is very persuasive, “math is music” is a metaphor that is fundamentally unsound as the grounds for reasoning from. I’m not talking about the mathematical aspects of music (which, as you correctly point out, can be found to exist) but that music is performative and aesthetic, whereas mathematics is all about internal consistency.

Sheet music is no more music than mathematical notation is mathematics. (Indeed both notations have roots in the same kind of two-dimensional seventeenth century calligraphic tradition.) However, as you point out, mathematics is indeed the study of patterns, structures, and relationships, and music is not. It is a structure and has relationships, but it is not a “metaphor of metaphors” as mathematics is. Case in point: modern visual art post-photography is much more affine to music than mathematics is, because after depiction became not quite trivial but literally photographic from the daguerreotype onwards, the role of the visual arts became the solicitation of emotion and the conveyance of narrative. Music has syntax, indeed it has very many, but it is not the study of syntaxes in the manner in which mathematics is.

After looking at the title for a day, the idea occured that, without mathematicians doing mathematics, those people would end up in banking, so seeing the top post is most amusing. One of my local contacts is a semi retired bankster, who I am going to prank by pretending that theres something stuck in her fangs, looks like the still quivering flesh of a young accounts executive, which seems to be the thing that happens to mathematicians, money that is, triggers some sort of primitive predetory instinct better left dormant.

I’m so sorry but your metaphor totally sprained my brain.

Everyone involved in AI should have read The Library of Babel [1].

It's about many things, but perhaps the most relevant idea here is that no information matters without understanding. We could generate all possible knowledge, but unless someone--a human--can verify and understand it, it doesn't count. The cure for mortality could be written on the moon, but if no one reads it, it hasn't really been discovered.

[1] https://maskofreason.wordpress.com/wp-content/uploads/2011/0...

While I get the point and it's true.

I'm a little more flexible, if the new knowledge (that human's don't understand) can be put into a mechanism and have an observable effect, I'd be happy enough. e.g. a new type of rocket fuel that burns 1000x more efficiently.

We already don't understand LLMs and they are having oberservable effects

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...

> but unless someone--a human--can verify and understand it, it doesn't count

Does it need to be a human or can it be some other form of life?

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.

> unless someone--a human--can verify and understand it, it doesn't count.

One counter-point: Humans have not figured out how general anesthesia works, but we use it every day to great effect.

https://en.wikipedia.org/wiki/Theories_of_general_anaestheti...

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

Only because we understand everything _except_ how it works.

how can we understand absolute everything and still don't know how it work ?

Trial and error :)

I wouldn't say that is "understanding". We must fully be able to explain to say that we understand it.

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.

Same with the black box part of AI. What the weights represent? Arcane dark magic if you ask me. What do they do? Well with LLMs we're all experiencing it.

> 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).

This was my existential horror for most of my life as I almost never remembered dreaming or going to sleep - until I got a CPAP. Once I started using it, there was continuity to my life, I vividly dream and remember those dreams now. I never scored too badly on the apnea tests but just enough to qualify for one so I said what the hell and got it. Even though I don't use it regularly, something about my sleep has changed for the better.

Well generally we monitor their heart and lungs to see whether they died

you can verify it does what you want it to? How? Everything is a probability and so is your verification.

You're correct that verification *reduces* risk. It does not eliminate risk. Humans are imperfect. That does not mean that we should stop seeking understanding on principle though, because the journey towards that understanding improves outcomes.

Recall also that LLMs are not actually entities. In their current form, there is no sentience, there is no agency. They are tools. Therefore, their output must benefit the user that requested it. Right now, that's humans (and, ideally, the planet at large; we don't exist in a vacuum) and so it makes sense that humans should verify that output and try to ensure that it aligns with their goals.

That's not to say that the output of an LLM is useless, far from it. But we should still *try* to understand its output. It gives us at least some chance to notice flaws, and an even greater chance to appreciate the implications and tradeoffs of the solution it picked.

I don't disagree with anything you said. But the thread is about... LLM outputs, even if it cures cancer and solves world hunger, it is useless (it does not count for anything) if we don't understand it.

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.

We have just convinced ourselves it is statistically safe, much like most of medicine. Most drugs are not made from molecular simulation of an entire human body.

Of course we have also convinced ourselves before that cocaine in drinks, lead in petrol, asbestos in walls… were all safe…

This reminds me of the Feynman interview where the interviewer asks "how do magnets work" and he goes on this rant of how that's unanswerable and you have to decide on what is it you really want to ask. You can't expect to know the full chain of knowledge because at some point you will be asking about quarks and gluons and then hit a wall where "nobody knows". Similarly you cannot simply just give up any investigation at all because then you'll end up recommending people to put lead in their cars.

My favorite part of Feynman's magnetism rant about how the question is terrible and can't be answered is that he seemingly accidentally gives a very good explanation: that magnetic repulsion is a highly concentrated version of what repels his hand from the arm of his chair.

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.

Doubt.

If I vibecode a video game and manage to sell it on Steam, the information definitely mattered even though I didn't understand any of it.

There are drugs that nobody truly understands how they work and they are being used by professionals in actual treatments literally right now. We know purely statistical facts like "if drug X is used for condition Y it will help Z% of patients" and can only speculate as to their mechanism of action. They are used anyway and still benefit a lot of people.

Yeah that information mattered when someone wrote it down, not when the LLM regurgitated an approximation.

The regurgitated approximation can matter just as much depending on what results it accomplishes. If I somehow make money out of this, then it's mission accomplished. Arguably, they might even matter more than my artisanal hand crafted software projects which never paid me a living wage.

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.

> but unless someone--a human--can verify and understand it

Why though a human? Mathematical proofs are very ivory-towery, but if OpenAI would solve a subkind of Cancer, without mortal humans understanding, cancer is still be healed.

Not unless humans were able to produce the medicine or perform the surgery or whatever is necessary to actually heal someone. In order for that to happen it has to be verified and understood by humans, like the other commenter said.

Does it, though? What if future AI can do everything from designing the cure to building a self-contained universal surgical medcapsule? "Just lay down and I'll fix your cancer real good, trust me, bruh." If it was shown to work, would you use it?

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.

People keep talking about this like there's a finite amount of math to be done, and then the party's over. But that's never how math has worked, is it? 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.

Math is meaningful because ... some people like to do it. The same as any other human pursuit. It doesn't need a reason beyond that. And AI won't change that. There will continue to be things to explore, things to find out, things that are maybe just at the edge of AI's reach and needs a human to decide whether it's worth continuing to explore or not. (Remember, AI isn't free).

So, IDK, I think for people who enjoy exploring math, there will always be interesting areas to explore. AI just gives us a better flashlight.

BTW I do agree that there's going to be an incident soon, whether intentional, accidental, or paperclip-factory, that leads governments around the world to shut all this down for some time, perhaps even shutting off access to GPUs entirely. It seems unavoidable. But that's just a temporary respite and skirts the core philosophical premise of the post.

It seems likely that there are an infinite number of math problems but only a finite number of interesting ones.

I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.

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.

Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.

Why is x interesting? Just because it has a minimal description in P\I? That makes it interesting in strictly technical sense only.

I think that's a variation on the interesting numbers paradox joke. Statement: All numbers are interesting. Proof: Assume by contradiction that there's a non-empty set of uninteresting numbers. Then that set contains the smallest uninteresting number. That property makes it interesting.

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.

I mean, there's some truth to that joke in this case, so I wouldn't agree that it's dismissing. The point of it is that what matters is how we define "interesting", because by the joke's definition in particular, there can't be uninteresting problems. That seems to be a very subjective concept that can also change with time. Mathematics is so specialized that there can be 3 people in the world who find one specific problem interesting. If they solve it, they'll move on to something else.

I find it curious that you've protested against that joke but not against the statement that "it seems likely that there's a finite number of interesting math problems". It doesn't seem likely to me personally and I haven't seen proof of that, even in a joke form. There's a finite number of problems at any given time, obviously, because mathematicians are finite, but it would require a very good understanding of the whole of our mathematical knowledge to declare that if we keep expanding it we'll hit some kind of wall, of "interestingness" or whatever else.

Firstly, I'm very happy we're having this conversation. It's so pointless, yet pedantic it warms my heart in the best way possible.

Secondly, I stand by the statement that there is no merit to this joke. This is because the way it defines interesting is very hand-wavy. There are interesting and non-interesting problems, but by a sleigh of hand you can turn the non-interesting problems interesting, thus proving that basically everything in the universe is interesting. At least in the mathematically describable universe. When everything is interesting, nothing is interesting. So we can dismiss the proof as a silly joke.

What mathematicians find interesting is a different story. However, we can almost certainly say there is only a finite number of problems mathematicians as physical beings can solve. If we have 200 mathematical symbols at our disposal, and we consider all strings of these symbols of length 1000,000, we have captured all the descriptions of problems that fit to 1M symbols. But that's a finite number. Going beyond that starts to be difficult for a human to grasp (if 1M is not too much already), so all mathematical problems that are solvable by a physical mathematician are in that set of strings. And that's not even saying anything about whether or not they're interesting..

I still don't quite agree with the approach to mathematics as enumerating problems of a certain size, but to be honest I'm not prepared or motivated to keep this conversation going without turning to handwavy arguments based on my imperfect idea of what mathematics is and how it works. However, it's certainly given me something to think about, so it hasn't been completely pointless. Thank you for the discussion.

No worries, and thank you as well.

By the way, if I'm not completely mistaken, Gödel's argument to show the incompleteness of mathematics relies on encoding all mathematical statements as numbers. So I'm certainly not being very original here.

FWIW I think the concept of "interesting" is so subjective, time-dependent, and nebulous, that to say there's "only a finite number of interesting" mathematics problems feels absurd to me. What's interesting in ten years time will depend on what other interesting things have been discovered in the intervening time. But you're right, I shouldn't have responded with a cheap joke.

Godel's incompleteness theorem technically relates to individual axiomatic theories (i.e. the set of facts that logically derive from a given set of axioms). The numerical encoding you refer to applies to logical statements within that theory. Arguably, the kind of mathematics that humans do is not constrained to a single theory, but is a more general form of reasoning that is often flexible about which axioms may or may not be assumed.

An interesting problem must have a description that fits in a brain, at least for now. Your description-length argument assumes arbitrarily large storage.

the smallest problem that cannot fit in a brain would be pretty interesting

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”).

As a concept, sure. As a formal proof structure?

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 probably is a finite amount of math to be done

The universe is bounded by rules, as far as we can tell, and not a lot of them

>The universe is bounded by rules

That's physics. Not all math is physics.

The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.

At those levels math and physics are the same bound: the bound of things the universe allows to be conceived of inside it.

It’s possible the math our monkey brains + sand can ever conceive of in this universe is a low and accessible amount.

> The universe imposes strict limits on the math that can exist within it, and certain broader limits on the math the creatures and well-organized sand within the universe can conceive of in the first place, whether or not it can maybe exist in other universes.

When you have a moment, please reference some proofs supporting this.

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.

Even if we accept that there are "lots of thoughts" we're not capable of thinking (though I wonder how you would define a thought if not as something that you can think), it still does not follow that the "amount of math" (as if it's a definite quantity) that one could discover is necessarily finite, if you lived forever.

By analogy, the "amount of math" we can possibly discover could still be countably infinite even if the space of all possible thoughts would be uncountably infinite. Countably infinite is still plenty big, and it is certainly not finite.

To put what I said another way, math may be infinite in principle, but the creatures in our universe can only conceive of or perceive so much of it: a finite amount.

We (or our constructs) can plausibly mine all there is and then there's no more that's physically possible for us or our constructs to mine within the universe in which we exist.

If an ant can't conceive of or perceive trigonometry that doesn't mean trigonometry doesn't exist. But if neither an ant nor a human nor any creature or construct or technology inside the current universe, now or ever, can conceive of or perceive trigonometry, then we might as well call it non-existent. It may exist: but we'd literally never know it and neither would our constructs or space aliens or inter-dimensional aliens, or their constructs, etc.

Say that there's a level of mathematics at which a blerg is a zorg, but our universe includes neither blergs nor zorgs and no intelligences in our universe can conceive of blergs/zorgs because that would require having evolved outside our universe... in that case we can consider our universe's mathematics solved without it needing to work down to the blerg and zorg level.

The natural numbers are countably infinite. For each n ∈ ℕ, there is a proof by reflexivity that n = n. Hence there are countably infinitely many such proofs, one for each natural number.

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

I didn’t demand anything. I requested some proofs since this thread is quite literally about a claim that requires proofs. As evidenced by the further comments in the thread. If you disapprove of the conversation, please feel free to use the tools and move on. No hard feelings!

Please don’t use pedantry.

Which math is not physics (and chemistry)? Fundamentally, all these are same

The concept of infinity?

That's why it is a concept. We lack sufficient ability to measure physical world, introduce imperfect abstractions, thus need concepts like infinity.

Why does the universe being bounded imply mathematics is finite?

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.

Bounds do not imply bounds. There are an infinite number of numbers between zero and one. Heck, with our finite minds in finite time we have created uncountable infinities!

Mathematics that our universe does not support is still mathematics. Also you're viewing this through a human lens. How do you know it doesn't support what you think it doesn't support? Mathematics after all is a very human pursuit. Some alien species may have started off with ternary or even continuous logic. We have no counterexamples to show that that is beyond our universe.

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.

The history of math was done on human scales. Same as the history of chess.

> There probably is a finite amount of math to be done

> The universe is bounded by rules, as far as we can tell, and not a lot of them

Respectfully, this is not a useful frame for the discussion. Nobody is expecting to reach the limit you have noted either with or without the assistance of LLMs. So there is always more math that could be done.

> 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.

> Similar logic applies to every industry and every job. And it comes to the conclusion that we won’t have enough people for all the jobs that need to be done.

In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.

It's like chimpanzees seeing human society and saying "look how complex it is, imagine how many chimpanzees it needs to maintain it".

> In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.

Let's be honest: we don't know. Maybe you're right, but for the moment it's more likely that you're not. And countries cannot bet on that vague intuition at the cost of destroying their research communities and world leadership (which takes decades if not a century to achieve).

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.

But we're not building an AI society - to what end?

We're building tools to serve the human society.

A few people with way to much power are building AI to achieve their utopian, sci-fi dream. A few other powerful people are using that first group to achieve their post-democratic goals. It’s the most obvious top down attempt at imposing their vision onto the rest of the world. They aren’t building tools to serve anyone else than themselves

>A few people with way to much power are building AI to achieve their utopian, sci-fi dream.

You mean dystopian. If it were a utopia everyone would be happy to welcome the new world order.

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

Math proofs are special because they’re verifiable, safe, and do not require physical experiments. You can perform exhaustive parallel search in simulation with RLVR.

Do you think this applies to say, surgery, as well? There are few useful problems that share these properties.

Not all of it would translate, but robotic surgery could be done on anesthetized animals in a mass RLVR way.

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

well that's an awful image

This is wrong, both morally and technically.

First off, I can’t imagine anything more torment nexus-y than throwing billions to automate and scale the torture of animals. If each token is a “cut”, how much suffering does 10 trillion training tokens (lower bound) corresponds to?

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. You cannot verify success so easily, either. Cancer cells, for example, could regrow over months. You would need to keep the animal alive and regularly test the animal, which would be difficult to scale. And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.

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.

Do you think surgeons start performing surgery for the first time on people?

Sounds like you're saying that so long as the LLM with robot arms starts by operating on mice, cows and pigs you're fine with it

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.

We need to pivot math towards intuition and accessibility.

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

I believe math has little to do with mathematical notation - intuition is much more important. One can have intuition but not be able to “read math” - much as many musicians don’t “read music”.

Reading math however does not guarantee ideas or intuition. And those are what math needs.

You would be surprised at the number of people (who study math formally) who just memorize proofs and having no clue what it all fundamentally is about.

Shut up and calculate.

No :). There are many who want to understand.

I am a mathematician, though I haven't been in academia for many years, and I would indeed be surprised if there were a lot of mathematicians who "just memorize proofs". Do you really mean that, and if so what's your evidence for it? (And: how do you suppose it is possible for someone who merely memorizes proofs without caring what they are about to prove new things in their field?)

I think LLMs are the single best things that were invented to make mathematics accessible.

I loved mathematics when I was young and wanted to be good at it, but I got burnt so many times by teachers who could not explain even if their life depended on it.

Asking an LLM and discovering how different theories belong together and what are genuinely open questions, along with different philosophical interpretations has been a wonderful gift.

I have long accepted that Mathematics is a language where you must go "all in" because for the most parts institutions are fundamentally unable to explain it in an intuitive way. There are people who "just get it" and are blind to the struggle of people who need a different approach in learning it. There's something about the field where a generational survivorship bias has created an environment where math is mostly taught to likeminded people, who then don't see why anybody could not see the difficulties learning it.

I know this is not true everywhere, but the only reason I could finish my studies was because there are some genuinely excellent online lectures on youtube, along with massive support from friends who have already been through the ordeal.

If I had had access to an LLM, I could have actually taught myself from scratch and be more playful in learning it. And no, I am not willing to learn lots of math "for fun", I can think of better "for fun" things that actually help my profession and cater to my actual interests and not just some academic ideal.

> If I had had access to an LLM, I could have actually taught myself from scratch and be more playful in learning it.

For every person with this mentality, how many people do you think there are that aren't learning from the LLM at all?

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.

You are not wrong, but I was just saying that the possibility is there. And especially for people who are despairing over bad math education I see this as a game changer. Seeing equations used to make me angry and powerless for over a decade because I felt absolutely hopeless during my studies.

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.

Yah there's certainly no fraud, bureaucracy, or gilded walls involved in the tech industry

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.

Sure. But it was a bitch to get access to a good maths textbook- either you forked out a lot of money, got a library loan (we had to renew ours every 7 days), etc. With an LLM there's good breadth of knowledge so if can quickly hop across topics rather than follow the progression proscribed by the author. There's also no element of judgement/ego etc etc.

They are definitely doing more than parroting. When learning mathematics, it's not uncommon to have some little misunderstanding that just isn't covered by a written explanation because it's "obvious" unless you happen to have misunderstood it that particular way. When that happens, you could look up the topic in 5 books on the subject and not make progress (never mind that you have to search through them to get to roughly the right point). LLMs are genuinely good at challenges like this - they make mistakes, but they can still identify and correct your misunderstanding with a bit of back and forth.

A good human on the subject will do better, and I'm not claiming the LLM has "understanding" in any philosophical sense. But it's more useful than something that just regurgitates material.

> it's not uncommon to have some little misunderstanding that just isn't covered by a written explanation because it's "obvious" unless you happen to have misunderstood it that particular way... LLMs are genuinely good at challenges like this

For undergrad math, sure, and again, I think it's often because on some random corner of the internet someone else had a very similar misunderstanding. On the other hand, when I ask for clarifications on niche topics that have few examples on the internet LLMs (even Opus 5) often confidently cite irrelevant papers / results or simply hallucinate.

I think you're desperate to think that LLMs cannot make any sort of deduction by themselves: they must have just remembered someone else doing it. But I've definitely had conversations where they've had insights that can't be explained by that. Yes, I've had plenty of hallucinations too, but it's worth it, for me at least, overall.

Having said that, I wouldn't attempt to learn a whole topic with an LLM, that would be frustrating.

I am not desperate; I just have enough expertise and work in a niche-enough field to recognize that LLMs just are bullshit machines.

See https://claude.ai/share/6431ccd9-f4f8-4052-8996-b046d1ea1f76 for an example that I ran into just now.

They don't parrot if you know what to ask for. You can name for examples, name your own thoughts and observations and these things will often quote something related or the fact that a famous mathematician has also made this observation some point in time which then lead to the study of something else.

I am not going to study a whole book just so I can find the answer to a couple things I might never need. I like to focus on a few areas, learn everything I need to understand it, know if there are any dependencies and see what is there to discover and then move on.

I know many people get joy from learning knowledge that is not directly applicable and might never be applicable. And I know that the academic spirit would be to learn these things. But my time is seriously finite and I am not using math in my life in any fashion whatsoever other than occasional curiosity.

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

Sounds like someone was cruel to you at one point. The parent likely means the opposite of what you're suggesting.

Most people consider mathematical intuition something that only develops from deep study. What the parent is referring to is being able to anticipate how a system will behave before you actually do the calculation.

Most people who use mathematics in an industry job rely on their intuition rather than rigorous proofs. And most industry mathematicians will acknowledge that they're not as rigorous, nor as academically gifted, as their friends who stayed in academia.

Just like a programmer with 20 years of experience can look at a bug and anticipate the cause.

Brett Victor’s essay will resonate, if you haven’t read it yet. https://worrydream.com/KillMath/

(Attn: pls read more than the title before you downvote)

>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?

"explorable explanations" are big! google it, you are in for a treat.

> 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.

> 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

I really don't get the hate for mathematical notation here, do you have specific examples? Notation is a necessary tool to express complex ideas compactly so that others can understand them. It allows you to carry out technical proofs that confirm things that are "obvious". Then it uses previously introduced concepts to build new concepts on top of them. This keeps mathematics interconnected, there's value in explaining a concept in terms of already known things rather than just vibing it.

> makes itself intentionally hard

It doesn't make itself intentionally hard, it is just hard. You can simplify some expositions and make concepts easier to understand intuitively but it won't necessarily give you the skills to actually work with them in mathematical practice (which is the point of mathematics as a discipline).

My theory is that people run into the inevitable difficulty of math and rather than look inwards and come to the conclusion they aren't as smart, or special as they once thought, they blame "the system". It can't be that they haven't tried hard enough or aren't smart enough, it's the gated community who hides behind esoteric jargon that stops them from being successful in their mathematical studies.

They believe this despite the fact that there are a million different textbooks, all explaining the same concepts in different ways, or different branches that specify the same structures with unique methods. It's really strange when I read threads like this. I'm not the greatest at math either, but that's because it's a hard thing to be good at.

To learn math is an exercise in humility. You must struggle hard, do hundreds of problems, spend years refining your understanding, redoing proofs, finding counter examples, just to be at a good undergraduate level. Much harder than pretty much all CS classes, except maybe Theory of Computation, which is pretty much math.

Yes, that could be the case. I'm not a mathematician but I gave my fair share of struggle towards trying to understand the mathematics behind physics that I was studying. You don't need to know what a smooth manifold is to solve problems in classical mechanics, just as you don't need to know what a tensor product of vector spaces is to learn what tensors are in the context of neural networks. The former is always harder because the standard of rigor is higher, and that's because of different goals. So it could be that a person tries to look up what a tensor is, finds the mathematician's definition first, thinks "what the hell is this", then finds a simpler explanation in applied contexts and goes away with the idea that mathematicians unnecessarily complicate things, because the reason why the mathematical definition has to be so complicated escapes them.

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

Most musicians in history didn't know how to read music, yeah. Less so in the present day but just about every folk song has spread by the word of mouth, and even in pop music many genres have a strong oral tradition (jazz, blues, rock, you name it), they play by ear and not by sheet.

Well, that's not really what I mean. Obviously they had some concept of communicating music, somewhat standardized instruments, etc. Pythagoras didn't phrase his teachings using algebraic symbols either. I suppose what I mean is that

> Reading math however does not guarantee ideas or intuition.

is true, but the converse, having the mind for good mathematical ideas or intuition, almost surely means that you'll have very little trouble picking up mathematical notation (not just notation, but the reasoning and ability to write a proof, be it in natural language or some formal proof assistant language).

Like yeah, before the written language there were great storytellers who couldn't write. But now that humanity does have standardized languages, it's way more rare to come across such a person.

I've played music for twenty years and can't read sheet music to save my life. Pretty easy to learn the shapes and patterns on a keyboard or guitar without reading sheet music (easier on guitar) and i figure with drums like, while i might Google the chords of a song for guitar, I've never known any drummer that would learn a cover song via notation

I've never met someone who believed this who was actually able to solve simple math or physics questions given to them. Attitudes like this always feel like cope to me.

For the majority of math history, people wrote proofs using natural language. Formalized notation is relatively recent. I have read both, and the notation is (on the whole) so much easier to read. But anybody is free to learn math beyond an undergraduate level with natural language texts. If you want to be traditional, start with Euclid's "Elements."

Plus, in the good articles I've read, the structure is usually: dissection of subproblem, intuitive description of solution, converting that solution into notation, proceed to next subproblem. If you see an unfamiliar symbol in an article, look it up. If you don't understand its definition, you don't yet have an intuitive grasp of the concept. Keep going down that rabbit hole until you have the intuitive building blocks to understand the article.

Frankly, I love how much math we can teach before a kid graduates high school. Thousands of years of mathematical ideas, up to calculus invented less than 400 years ago. All before they're 18 years old and probably don't care much about math.

This reads well enough as a platitude, but what do you actually mean?

How is mathematics gated and inaccessible? How do you specify mathematics without notation or some language? What intuition do you mean, as many mathematical ideas are non-intuitive?

A village builds a clock tower so everyone can glance up at it and know what time it is. The tower helps as people meet more easily, work schedules align, and meals are shared. Over time, fewer people watch the sun and the clock becomes authoritative. When it stops working, nobody knows what time it is. Discord ensues and arguments follow as meetings are missed and people wait impatiently for repairs. An older woman in the village kept baking at dawn and never stopped watching the sun. She never relied on the clock. Villagers gathers at her door and call her wise, for her practice ends up inadvertently helping to re-calibrate the clock once it is repaired. She replies that she is not wise or special and was just going about her usual baking routine.

I assume this is satire, as obviously they could recalibrate by the sun itself, and the baker is superfluous here.

This assumes the people fixing the clock know how to tell time based on the movement of the sun.

In the story, the baker was the only one left with that ability, as others left the skill behind when the clock became the authority.

This problem has been seen when missionaries, or whoever else, try and bring tribes into more modern civilization. They have trouble adapting to modern civilization, but the skill loss during that process makes it almost impossible to go back to the jungle.

When skills are lost due to modern technology, it isn’t so trivial to get them back. Look at how much time people have spent trying to figure out how stone was moved in ancient Egypt, for example.

I am not implying that skill loss does not happen in other situations, but I don't understand how someone would be able to calibrate a clock by the time a baker starts her work day, but not by the time the sun comes up. What am I missing?

Maybe the baker got up and started baking every day at 4am, year-round.

Sunrise changes throughout the year. Sunrise in December and July are not the same thing.

To set the clock, they’d need a reference for the solar time to clock time for the current day.

I ran into this in Rome when I visited last year. A church there has a giant solar clock where the sun shows up on a line on the floor at noon (or so I was told). At noon, nothing happened. After thinking for a second, I realized I would need to find solar noon, and went to Wolfram Alpha to find that solar noon, in Rome, on that day, was 1:19pm.

The other option would be for the town to arbitrarily set the clock to solar noon one day, and just go with it, letting the drift be the drift. But the time would not be the same as it was before the reset. It could be an hour or more off.

Relevant talk about preventing the collapse of civilization https://www.youtube.com/watch?v=ZSRHeXYDLko

Fun story. The actual reality, as it happened: the older woman became a meme, a topic of gossip, possibly moralizing stories, eventually even brought up by commenters of tech-related social media boards.

Meanwhile, as people got used to the clock, the society became used to staying in synch at hour and sub-hour intervals. As better clocks were invented, synchronization became counted in minutes. This simplified things, allowing people to do more in the day and coordinate less, as they learned to rely on agreements to independently do things at specific time. Shops started posting opening hours, school started posting lesson plans, professionals started billing by minutes.

Clocks became widespread, and synchronized remotely - first from the church clock, then over the wire, over the radio, finally over the Internet. The loss of shared time reference at minute-granularity became rarer and rarer, until it became unheard of (except in excuses of those who failed to keep their commitments).

Losing a clock now would indeed throw society into disarray. If all clocks magically stopped tomorrow, many would starve and die before society figured out some way to restore time-keeping by proxy.

Clocks are a perfect example of the ratched effect of technology. A new technology at first helps you, or gives you advantage. As it spreads out more, it becomes relied on. Eventually, reaching ubiquity, it becomes a dependency, and everyone is assumed to be using it.

In a modern society, you cannot live without a properly synced clock anymore.

The clock is repaired, but a banker doesn't use it. Inspired by the baker, she follows the sun. In the summer she opens while everyone else is asleep. In the winter the windows stay closed until nearly 1000. Locals become increasingly confused by the erratic hours. No one can agree on when the bank will open on a cloudy morning. When confronted she mutters something about the baker. At the local banking conference she presents on her newfound wisdom and receives a standing ovation.

> 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.

Does the set of intelligent species only have one member? I cannot treat this observation as a general rule if there is literally one example of an intelligent species to theorize about.

I think the quoted part is saying that given n species and the n * (n - 1) binary combinations possible, there are no examples of the said observation. It doesn't suggest "the set of intelligent species only [has] one member".

Yeah but there is only one species that ponders its future.

I stopped reading at this point because I reject this initial assertion, and I assumed everything that follows is based on it.

There is nothing to suggest that an AGI will care about its future or who controls it. (whatever that means). that just more anthropomorphism.

We care a lot about our future so an ai pretending to be like us will also pretend to care. But we have a lot of biology driving us. Our emotions (Fear, Hate, Love, Pride etc) are baked into us in a way just wont be the same for ai.

> 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.

The statement that the set has 1 member is my criticism of the text, not what the text is trying to say.

> given n species and the n * (n - 1) binary combinations possible, there are no examples of the said observation

I would expect that in the vast majority of combinations, those n species are not intelligent enough to make decisions for another species, making the statement nearly useless.

The absence of such relationships given here are so many species is exactly the point being made by the author: there is no data to show that it would be safe, let along good, for us to surrender decision making to some more intelligent entity.

> I would expect that in the vast majority of combinations, those n species are not intelligent enough to make decisions for another species...

This kind of statement is very difficult to build discussions upon (not a criticism). "Vast majority", "intelligent enough" are both very difficult to quantify and there is too much subjectivity involved.

> This kind of statement is very difficult to build discussions upon (not a criticism). "Vast majority", "intelligent enough" are both very difficult to quantify and there is too much subjectivity involved.

Then let me provide a stronger and more detailed version of my claim.

The article said:

> 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.

In order for this claim to make any sense, there must be TWO “intelligent” species. One species capable of taking over another species’ future, and a second species which is even more capable than the first species but willingly surrenders control.

But there is only one species which can take such roles: humans. Every other species is not even capable of controlling another species fate AND every other species does not look out for their species-wide future.

Some people might point to whales, elephants, cats, non-Sapiens hominids, or even those ants that harvest aphids. But the whales, elephants and cats lack the capacity to control entire other species, even if they can manipulate individuals of other species. The non-Sapiens hominids are not available for comment: we’ll get to them later. And the ants do not think of their future.

So even though there are a massive number of species on earth, none have the prerequisites to even apply to the above statement.

> The absence of such relationships given here are so many species is exactly the point being made by the author

If we lived in a world where there were a vast array of species capable of controlling other species and capable of thinking of their future, all of whom fiercely defended their own autonomy, your point would be correct. But most species are not capable of either, even if would be a good idea. Therefore we cannot conclude that such a surrender of control would be a good idea or bad idea based on the statement. The statement is moot.

We might as well say the author’s observation is correct because sand doesn’t control volcanoes. In terms of the ability to control other species, other species on earth are no better than sand or volcanoes.

> there is no data to show that it would be safe, let along good, for us to surrender decision making to some more intelligent entity.

There is no evidence against either because the observation lacks any examples other than humans and AI.

Truth be told, I agree with the conclusion. But we have to be honest with each other in arguing for our survival and autonomy, even if it means admitting that we are scared and irrational, instead of trying to fake objectivity and universality. The article starts with a bad argument in favour of a conclusion I agree with.

If I were to write an article about the risks of super-intelligence, I would not even attempt to construct such a “universal” rule - it would have very little bearing on the current situation. I would simply point to the bones of the rest of the Homo genus, and say “we’re next”.

My problem with this is that yes, looking at the species level from our sample of ~1 the most intelligent species wins out (although there seems to be a lot of debate as to if Neanderthals were more or less intelligent than humans).

But within the sample of the human species it is more common for less intelligent people to control more intelligent people.

This happens within political systems (political leaders are usually above average intelligence, but hardly the most intelligent) and within companies and other economic systems.

I don't see this fairly obvious point discussed, and not really sure what it shows.

> 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'm not sure.

Maybe there are other things ("leadership", "likeability", "appearance" etc) that are more important in the pursuit of power than intelligence.

I think it's telling that it is usually conventionally intelligent people making the claim that intelligence is what leads to power.

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.

I am also not sure if the observation is true. People mentioned cats as a counterexample.

I think there are other species that could be considered more capable than humans, such as E. Coli, octopuses or ants.

And it's not even clear whether the AI will have its own individuality. It might become an extension of human brains, in the same way neocortex is an extension of amygdala. In that case the statement of who has control might become meaningless.

There's one flaw in the evidence for the logic chain. The hugging face attack is used to demonstrate three things: the need for oversight today (fundamental to the article) and to demonstrate some kind of drift or unexpected capability gain, and finally to hint and some fundamental morality of the AI or at least the risk of drift from our morality.

I'd argue that what the hugging face attack illustrates is that large AI companies are motivated to have bombastic claims supported by bombastic demos. The model was clearly trained or encouraged to work as it did, as evidenced by the fact it keeps using this particular escape hatch.

And the fact that it aligns with prior and current calls for what very likely might be a regulatory capture / oversight capture move right before IPO. It aligns so well with this "barely constrained superweapon" narrative it might as well be PR.

How does your logic explain that OpenAI seems to want to hide the extent of the HuggingFace incident, and every week we learn from 3rd parties about new victims of the hack?

4D chess? They want others to find the hacked services, so the report of how dangerous the agents are seems more "legit"?

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.

>> The model was clearly trained or encouraged to work as it did, as evidenced by the fact it keeps using this particular escape hatch.

> How does your logic explain that OpenAI seems to want to hide the extent of the HuggingFace incident,

Because they, uh, "clearly trained or encouraged" it?

I mean, you can dispute the truth of that statement, sure. But it's kinda hard to say that the GP's logic didn't explain why OAI wanted to hide the extend of the HF incident.

I'd like for HN to start tagging posts that are not written by terry tao, but are guest posts, like this one. Considering HN internally already uses AI tools to scan through and read the content to determine if it is AI-generated, I assume this is doable.

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.

'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.

I find Tao's writing on this very helpful, particularly because a lot of the ideas generalise to most knowledge work. It is darkly fortunate that some of the smartest and most articulate people on the planet are affected by the AI race - it gives me a little hope that alignment and control can be solved.

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

AI is not AGI right now, it can't think or come up with something new like a human brain. It still uses knowledge that was made by a human and was published on the Internet.

There is no evidence to substantiate what you are saying.

Every possible proof exists already as a possible generation in the grammar of lean or rocq. In no way does that mean we have discovered everything.

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.

This isn't necessarily true.

While every formal proof can be encoded mathematically that doesn't imply that the grammar of lean or rocq is sufficient to encode every potential proof.

However the core of what you are saying: that every possible proof exists in the space of all mathematical statements is correct.

You are correct, but in trying to get the point across I attempted (perhaps fruitlessly) to use concrete concepts today of a system where the search space is calculable yet whose span does not imply the 'knowing' of all calculable things.

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

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. Traditionally having one was evidence you had an insight. But 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?

Because it's a poorly written article.

> Why doesn’t the author even mention this ...

Because the question that belies all of this is whether still makes financial or economic sense in the age of LLMs for tax-payers to fund mathematics research.

Up until recently, they've had a true ivory-tower to dismiss such criticism, but as LLMs have demonstrated, that approach won't survive going forward, so pure mathematics are up in arms. They genuinely don't know how to justify their jobs.

I don’t think this is a key question at all. There are a hundred other academic disciplines that dont provide immediate research utility either.

Math was always a few levels removed but somehow sneaks in with engineering and basic science.

What's up with these insanely off base articles on HN this morning? Every single one of these is an unhinged pro-AI anti-human schlock piece.

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.

> we won’t have enough people for all the jobs that need to be done

Who defines jobs that need to be done? Businesses and institudes do.

There is no magical entity that observes the need for jobs and creates them at ideal rate.

I would say aggregate human desires determines what jobs need to be done businesses come about to fulfill the gap.

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.

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...

> 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.

Why is Terry Tao even talking these arguments when all OpenAI did was a sophisticated brute force search with LLMs?

Give them Navier-Stokes in a vacuum

What do you think all the PhD's and post-doc students are doing...? It's not called graduate-descent for nothing :p

> 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.

You're right. But...

Spending 132 billion tokens is definitely way more energy than human mathematicians would have thrown at the problem and probably would have solved within two years.

Same reason Gary Kasparov and Lee Sedol started their morning in one universe and ended their evening in another.

Brute Force or not, the game is being played and won without them all of a sudden, and even worse: at a level of output far beyond them

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

Can a human?

(Evidently not, since humans hadn't managed to do it even with the entirety of human knowledge available to them.)

They used an agent swarm. Could humanity? Of course; it was close to being solved. It might have taken five years

The question you asked was whether the AIs could prove the thing "in a vacuum", without making any use of previous human work. "Close to being solved" is a statement about all the previous human work that mathematicians might have used to do it in five years.

Replication of the development of mathematics by human civilization is a task for an artificial hyperintelligence.

What will anyone do? What will anyone feel their purpose would be? Can't all have the same one, either.

What will the planet 'need' homo sapiens for (arguably it never needed homo sapiens at all)?

Would anyone's time or life really have significant value?

I didn't mean that as "we need hyper-intelligence." I mean that going all the way from nothing to singular, dual, plural grammatical numbers attached to particular nouns to abstract one, two, three, many to zero, one, two, three... to rationals to... is a very long way.

The article wasn’t written by Tao.

He's expressed similar fears of being replaced in a recent talk

> A host of reasoned declarations and open letters to protect/guide the math research community have been released over the past few months, spiking in intensity after OpenAI announced their solution to the Millennium Prize variant of Navier-Stokes. They quickly gained widespread support among mathematicians. The Leiden Declaration already has 4,000+ signatories, Math and AI has 7,000+, and even the open letter opposing the Caltech Mathathon has 2,000+.

> Among non-mathematicians, the public response was more sympathetic than not, but I observed a vocal minority (particularly from the technology and economics communities) with reasoned objections, generally saying that the mathematicians should adapt and cede control in the new AI world.

Interesting how Software Engineers, instead, seem to have given up early without any declarations or open letters.

Friendly warning to those who might not be aware: the vast majority of comments below posts like the above will be left by (otherwise intelligent) programmers who think mathematics is a closed system where one attempts to solve endless Olympiad-type problems. I wouldn’t take any of it seriously at all. Better to listen to what those who actually know what the subject is about have to say.

Unfortunately, mathematics (especially pure mathematics) is by its very nature very, very poorly understood by those who haven’t worked as a mathematician. Even worse, those who don’t understand are seemingly not at all aware of their misunderstanding and are entirely confident in their (very wrong) characterisation of the subject.

Could you clue us in? What is mathematics actually about?

To know what something is about, a natural way is to do it yourself.

Wow, I'm glad AI is going to start humbling people.

> To know what something is about, a natural way is to do it yourself.

literally what software engineers were doing for decades though

software is mostly just simple math, for the most part, until you need to do something more complex for some hairy algos lol

Actually, if you were to convert most software into mathematical notation it would look incomprehensible. We are not interested in gatekeeping as much by making things look more convoluted than they need to.

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.

Good question, and maybe I should have provided an alternative description rather than just criticising.

Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.

Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:

(1) observe some phenomenon in ‘reality’.

(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.

(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.

It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.

The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.

NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)

If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.

Despite loving Mathematics and having considered being a Mathematician myself, that wasn't a very robust defense. Also, even though I'm a big Terence Tao fan, I'm not sure he has sorted that out this defense in full himself.

This post which he forwarded was quite poor in my opinion. Confusing, all over the place with AI criticisms and promotion of the AI hazing being done by mathematicians.

X thousand mathematicians who want to protect their livelihoods signed a bunch of letters against AI. Duh. We've seen similar movements from every profession that has been displaced ever.

Terence Tao uses AI and has made a few good points on how to use it. But defensiveness leaks into almost every defense of the role of humans in Mathematics that I've read, even his own at times.

To be clear, I actually believe that Mathematicians aren't going away, but I dont have enough knowledge about the life of a professional mathematician to articulate a path forward.

This "path forward" is what I'd like to see. We need a top mathematician with enough intellectual honesty (Terence Tao qualifies, I think) to start this questioning with "there's actually no role for human Mathematicians" as one of the options on the table and go from there.

For me, mathematicians seem to be still having an inner discussion rather a making these essays for the more general public. In any case, I think that statements of the type "there's actually no role for human Mathematicians" are completely non-serious, so it'd sad that the discussion concentrates on that.

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.

There’s a simple point to be made: if no one in the world understands an AI proof, what’s the point?

Maybe another AI could use the proof.

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"

A lot of programmers consider themselves to be applied mathematicians.

Maybe more in years past when Comp Sci was a subset of Math Departments.

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.

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.

The closest I could come to describing math is "some abstract process where imagined structures are characterized and extended; the most critical part of the process is identifying where seemingly independent structures are found to actually be fungible in some previously undiscovered way".

A simple example is

  "hey, did you know that x^i is the unit circle?" 

  "what's i?" 

  "i is defined as if you square it the result is -1" 

  "what does that have to do with circles?"

The clue is burried in the text of OP's article.

> Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues.

The fact that this is a radical departure from the norm is part of why mathematics (and philosophy) is often seen as some intangible or ungrokable science to many outsiders, as they're generally approaching it from a perspective of "Okay, but why, what is this useful for?", and the answer "For the science of it" doesn't tend to land with people that aren't already passionate about said science/discipline and are just trying to figure out what it even is or involves.

Doesn't help that there is a pervasive sentiment in American Academia (not sure about elsewhere) about Math being *the* hard science, and I mean hard as in difficulty, so a lot of people get intimidated by it before they ever give it a chance very early on in their academic life and carry that through the rest of their education.

So there ends up being a rather small pool of people that are in(to) the field, and rather high friction for stimualting interest in it from outsiders from the way that it's taught, and a massive difference in the perspective of it's use between it's diaspora and the unmathed masses.

Mathematicians who do work on "real world" problems are mostly doing so with theoretical physicists, genomicists, and cryptographers. None of these count as what people consider valuable other than because they are hard.

That said, few 50 (or even 40) years ago would have predicted that completely abstract number theoretical computations about primes, discrete logarithms, and elliptic curves would be the foundation of our monetary system.

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.

Humans still excel at posing the right questions and finding truly novel, insightful proofs. AI just brute-forces.

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.

> AI does NOT “brute-force”, else we could have had those math breakthroughs decades ago.

Brute-force as an approach, or partial approach, might only become feasible with more computing power. Decades ago, we had less computing power.

> 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).

Here is a link to the original article:

https://poshenloh.com/posts/20260919-math-ai

The original posted link from OP is from Terry Tao’s website where the article was posted as a guest blog post.

It's astonishing how much more readable Tao's version of it is.

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

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

Well it's actually nice, maybe more people will be able to do world class math with the help of these tools. There are few fields as elitist and hostile as pure mathematics, most mathematicians I know build their whole life around their profession and their self esteem is strongly coupled to the fact that they can do things that most other people can't. Naturally, many will be devastated when (if) you take that away from them. That said I think AI is still overhyped and human mathematicians can easily outthink it in most domains, look at how difficult it is for an AI to write even a single decent paper, a good PhD student can easily outclass it in that regard. All of these impressive results were generated by having world-class mathematicians steer the systems using highly tuned prompts, so I see it more like a super violin that produces beautiful music when played by master violinists rather than being a fully autonomous orchestra which many people are led to believe this already is.

And maybe let's not only hear the opinion of two or three Fields level mathematicians with blogs, 99 % of the worlds mathematicians in academia might profit from these tools as they might partially close the gap between them and the world elite, making creativity and tenaciousness more important than having the right neocortical structure allowing you to outperform 99.9 % of other humans at keeping context in your head and making predictions, AI can do that better now with the right prompts.

> There are few fields as elitist and hostile as pure mathematics

Is that so ? Sounds hyperbolic.

I haven't sampled all mathematicians on the planet and I don't think there's a study on this but I would argue a lot of people will agree with mathematics with being elitist, and from my experience mathematicians aren't very keen of interacting with math "enthusiasts" or the wider community in general, they interact with a very select number of other specialist through a small set of conferences.

Mathematics is by far the most accessible "science" in academia. Anyone can, in theory, produce novel results with nothing more than a pencil and paper. And while the existing literature may not always be readily "accessible" (in the easy-to-understand sense) it is pretty widely available (often online and certainly via any decent academic library).

But "frontier" mathematics is still a highly advanced, highly specialized field. It can take years of study to be prepared to understand the established theory and results for a given subtopic.

These two observations, taken together, lead to the predictable outcome of a lot of math "enthusiasts" with an incomplete understanding of the field loudly asserting that they have discovered a radical new result. Often they lack the foundation to even understand what they are doing wrong.

The reluctance of mathematicians to engage with amateurs that come off as cranks is a symptom of how accessible the field is.

This is true of professionals in any specialized and difficult to acquire skill to be world class.

I haven't had any interaction with mathematicians that I would describe as hostile.

The premise of the article assumes that the work (number of true statements to prove) is finite.

But Godel's Incompleteness Theorem and Tarski’s Undefinability theorem ensure an infinite space of provable true statements.

Neither LLM's nor humans can exhaust it. So yes, both mathematicians and LLMs are needed.

Both can contribute and there will still be work leftover.

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.

Mathematicians insist that in many cases the route toward solving a problem is more valuable than solving the problem itself. This may have been true, and may still be true for now, but this seems to implicitly assume that future generations of AI cannot be creative in similar ways that human mathematicians have been. It assumes that the absence of human comprehension during problem exploration loses a valuable aspect of mathematics that cannot be recovered by aiming future AI at solved problems in a naive environment.

If we get to a future where all frontier mathematics contributions are by AI. A future where AI displays creativity in ways that expand mathematic exploration similarly to the ways humans have in the past. A future where AI explains frontier mathematics to curious humans. What will have been lost? Perhaps just "The pleasure of finding things out".

> What will have been lost? Perhaps just "The pleasure of finding things out".

I'll kindly disagree on this front, because when I'm walking a path toward solving a solution, I mark a lot of steps for possible diversions to other paths hence solving adjacent or different problems with the method I have at hand.

Currently, AI takes us from A to B, and is improving on that front. However, the paths in science are not lines, but a trees. Methods are cross-pollinated from each other.

Human intuition enables this cross-pollination. AI works with a laser focus. Human intuition and resulting wide perspective sow the seeds for solutions in many areas at once.

I recognize that if a document were to be published tomorrow containing solutions to all formulated open problems in mathematics we will have gained very little in mathematical understanding.

But surely, if we know that the dead ends of exploring a problem are valuable, we should be able to explore them even if a solution is already known. It just requires that the mathematics community reshapes itself. And it must. Two years from now people might be able to run the computation that solved NS on their Iphone.

I'm sure that whatever has been discarded during the NS exploration as a dead end you would be able to rediscover using purpose built tooling in the near future. The purpose of human mathematicians in the medium term might be to explore dead ends, and to provide human insights as context to attack other problems. But whether this type of work will remain necessary in the long term im not sure.

> But whether this type of work will remain necessary in the long term im not sure.

Any and every capability AI can demonstrate today is the result of us, humans doing it in the first place for a very long time. The transfer method of these abilities is a subject of another comment, but as Microsoft and NVIDIA puts it, it's a theft of unprecedented scale [0].

AI labs dream of recursive self improvement as an escape from that, but until we arrive there, humans have to do something so AI can do it as well.

And, as of fully autonomous self driving which should have arrived 5 years ago, RSI and AGI is just around the corner, a corner with a radius so large that it never ends.

We dream of building utopias with these tools, but it's a path to dystopia paved with stones made from utopic dreams.

[0]: https://news.ycombinator.com/item?id=49752056

I dont disagree. Clearly AI leans on the cumulative cognitive labor of humanity without porpper attribution. This is in part due to the design of these sytems. Information is compressed into the weightspace in ways we dont understand, and a concept becomes unlinked from its origin.

I hope you're right about AGI. What we need is time, and we may not have it. https://www.dwarkesh.com/p/noam-brown

>this seems to implicitly assume that future generations of AI cannot be creative in similar ways that human mathematicians have been

It's because they can't.

It's a mathematical theorem that no algorithm that "solves mathematics" can exist.

>In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an input statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.

https://en.wikipedia.org/wiki/Entscheidungsproblem

The halting problem constrains human and machine alike, right? Or are you suggesting there is something unqiue about the human mind that allows us to do things machines cannot do in principle? Most people in the field believe in substrate-independence.

https://www.edge.org/response-detail/27126

The post above this is titled: AI chatbots give wrong answers to financial queries 'most of the time' (ft.com)

I think the headline alone makes a reasonable argument based on the AI tools we have today.

I’m just shocked by how “intelligent” people believe a token guessing system can be

I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence. Even Doctorow talks of "brute-forcing" a solution. Brute-forcing leads to combinatorial explosion, so there must be something more going on here. Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).

> Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).

There are many automatic theorem provers that do very clever stuff, just as the underlying theroy describes.

> I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence.

It is absurd to waste time discussing whether it is inteligent or not. It is just an algorithm, we know how it works, and it does exactly what we expect it to do. LLMs are not magical things. The main difference is the scale: for Navier-Stokes they spent in 3 days more money that the whole mathematical community over the last 20 years easily.

By the way, I'm not saying that LLM's are useless, that I'm anti-AI or anything like that.

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 is just an algorithm, we know how it works,

We know what calculations it does. We have some hazy idea of some bits of how those calculations lead to something that at least somewhat resembles intelligent behaviour. But that's a far cry from actually knowing how it works.

For instance, suppose you give one of today's frontier models some of those chain-of-cubes rotation puzzles (the sort that infamously men are about 1sd better at than women, statistically speaking). How well will it do? I have absolutely no idea and I'm quite sure that a more detailed understanding of the transformer architecture would not make my guesses any better. (Actually, I do kinda have some guesses but they're based on a vague notion about how the models might be partitioned between vision-y bits and language-y bits, and it's very possible that that notion is out of date.)

> it does exactly what we expect it to do

Were you, let's say 6 months ago, expecting it to resolve one of the Millennium Prize problems?

(I do agree that it is more productive to ask "what can and can't they do?" than "should we classify that as intelligent or not?".)

> for Navier-Stokes they spent in 3 days more money than the whole mathematical community over the last 20 years easily.

Are you sure?

(The numbers I've heard, which I admittedly have no very strong reason to trust, don't seem that way to me.)

> Were you, let's say 6 months ago, expecting it to resolve one of the Millennium Prize problems?

I didn't expect them to throw millions of dollars at each famous math problem. But one year ago we already had LLMs that solved IMO problems, no?

> Are you sure? (The numbers I've heard, which I admittedly have no very strong reason to trust, don't seem that way to me.)

Math has very little founding compared to other science domains. Also, if you filter mathematicians by specialization in PDE and that have worked on Navier-Stokes, then you end up with a very niche community.

> For instance, suppose you give one of today's frontier models some of those chain-of-cubes rotation puzzles. How well will it do?

I feel like this is not the correct way of thinking about it. We can also ask, for instance, how well a state-of-the-art algorithm for the salesman problem works on a particular graph topology. People do PhD thesis on topics like that, so the answer is not obvious at all. For LLMs we still don't have a curated theory that explains what they're good/bad at, and that you don't see how to extract an answer from the definitions is no surprise since this is obviously not an easy problem. But all this is normal because this is a rather new topic (models of this scale appeared when? 3 years ago? That's nothing for science).

Anthropomorphizing LLMs has added so much noise to this discussion.

Yes, one year ago we had LLM-based AI systems solving some IMO problems. My impression is that most observers at that time didn't expect them to be solving Millennium Prize problems within a year.

> Math has very little funding compared to other science domains.

True. But to whatever extent the numbers I've seen are correct, for the whole mathematical community to have spent less on Navier-Stokes than OpenAI did -- even if we value the tokens they spent at something like market rate rather than at what the compute actually costs them (which might be right since any capacity they use internally can't be sold to customers) -- the average number of mathematicians working on Navier-Stokes since 2000 would need to be somewhere around four (depending of course on how well paid they are), and that seems too low to me.

> I feel like this is not the correct way of thinking about it.

It seems to me that if you say "It is absurd to waste time discussing whether it is intelligent or not. It is just an algorithm, we know how it works, and it does exactly what we expect it to do." then this only makes any sense if your "knowing how it works" and "what we expect it to do" enable you to predict what it can and can't do.

(I repeat that I agree that what matters is what it can do, not whether we choose to apply the term "intelligent" to it. But unless I misunderstood you were saying somewhat more than that.)

> Anthropomorphizing LLMs has added so much noise to this discussion.

I think sometimes it helps, sometimes it hurts, and sometimes it's indifferent, because LLMs are like us in some ways and unlike us in some ways. (The same goes for many other things, but LLMs are much more like us in some important ways than any other human-made artefacts.)

Turns out what you call "intelligence" was never needed to do math.

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 need human mathematicians for humans.

But companies are our gods (don't believe? E.g. companies cannot die from natural causes). They don't need puny humans. They need AI.

Companies are nothing other than the tool of rich people used to control normal people. There's some sort of mass delusion that "companies" are evil. Guess what, a company has no own agency to destroy the environment, exploit people, profit from suffering, etc. It's just following the whims of its shareholders, who are humans with accountability.

I didn't say companies are evil. But there are companies and banks in our world, which are much older than any human could ever be. There are people who think they are in charge of these companies, however sooner or later they will be replaced and the company will still exist without them. In that sense companies are eternal entities, which gives them a godlike status.

Alright sorry if I was unclear. I wasn't saying that you specifically claimed companies are evil. But there is a general sentiment / understanding that companies are (being used as a tool for) doing evil things in general. I suppose whether you think that companies are (being used for) doing more good than bad is a different question, so let's not get bogged down in that.

My point however is that companies have no agency of their own. Their age doesn't really matter for that. There are guns that are older than all living people too. If somebody used a flintlock pistol to go around robbing people that wouldn't make the pistol a "godlike eternal entity". Or for that matter if somebody used a 200 year old shovel to create a really nice garden (in the case you believe companies are a net positive).

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.

I wouldn't know anything about that. My point is that this is a completely ridiculous statement:

> But shovels are our gods (don't believe? E.g. shovels cannot die from natural causes). They don't need puny humans. They need AI.

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

We need human mathematicians so that someone is creating the data that the models are trained with...

Because the AI needs new stuff to train on?

Lots of deans and college presidents asking themselves the same question.

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).

______________

[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.

> 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.

I hope this isn't actually news to you, but: There is more than one human. There is even more than one mathematician.

If there happen to have been as many as four humans working on Navier-Stokes at any given time since the year 2000, then that's more human-years applied to the problem than agent-years.

> How many mathematicians have worked on the problem? 10k? Not likely.

You don't get to count the factor of 10k once when working out how many agent-years OpenAI gave to the problem and again when demanding that for parity there would need to have been 10k mathematicians on it.

> 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.

Let's suppose that indeed what Buckmaster and Alpöge had done was in the model's training data. Well, it didn't enable Buckmaster and Alpöge to solve the problem for Navier-Stokes (they could only do Euler), and it did enable OpenAI's model to do that.

Also: we don't actually know that what they'd done was in the training data; the latest bits of what they'd done that could plausibly have been in the training data were from before when Buckmaster said they progressed from preliminaries ("We worked through the literature and upgraded various preliminary results") to actually making substantial progress on the problem ("This was until about a month ago, when we had real progress"); and from what Buckmaster wrote it sure seems like a lot of the Buckmaster/Alpöge progress was in fact done by LLMs. (E.g., Buckmaster says that he and Alpöge have been working frantically to try to understand the proof for their Euler solution. That sounds to me much more like "an LLM did this thing" than "we figured out all the hard bits and the LLM did nothing more than filling in a few details".)

Buckmaster's own account of things is that all the really clever ideas were those of Córdoba and Martínez-Zoroa. (Which are already out there in the open literature, and there is nothing remotely improper about making use of them.) And my understanding (but, note, I am not an expert on fluid dynamics or PDEs and I could be wrong) is that actually the OpenAI model's construction is quite different from that of C&MZ. On what basis are you confident that "the AI based its proof on" what B&A did?

(For the avoidance of doubt: I am not arguing that what OpenAI did was OK. Even if they actually didn't train at all on any of the Buckmaster/Alpöge chats, it's very much not good professional ethics to hear that someone else is working on something and rush to try to scoop them, and there is absolutely no question that they did that. The question here is how impressed we should be by the model's mathematical prowess.)

> A success rate of 17%?

A success rate of 17% on problems of this difficulty and significance is something that for any human being would be a career-defining triumph.

> We have thousands of problems solved by humans in the last 2k years and how many by AI? A couple dozen? Oooh scary!

That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.

>> If there happen to have been as many as four humans working on Navier-Stokes at any given time since the year 2000, then that's more human-years applied to the problem than agent-years.

My bad for not showing my work and inadvertently leading you down the garden path, but the "~100 agent-years" calculation goes like this:

10,000 agents * 88 hours = 880,000 agent-hours

88,000 agent-hours / 24 hours = 36,666.7 agent-days

36,666.7 agent-days / 365 days = 100.5 agent-years.

That's what you get for working 24 hours a day, 7 days a week, 365 days a year. Realistically speaking, that's not a work schedule any human can follow.

It's hard to make a realistic estimate because normally even a very dedicated mathematician will not be working exclusively on one problem all their waking time, or even all their working time. But, let's ignore this and assume a pretty standard work schedule of 8 working hours, five working days a week, and 52 working weeks a year.

Now, that's:

8 hours * 5 days = 40 working hours a week

40 hours * 52 weeks a year = 2080 hours a year

880,000 agent-hours / 4 humans = 220,000 hours per human

220,000 hours per human / 2080 hours a year = ~105.8 years

To clarify, that's how I estimate the number of years it would take a mathematician to do a quarter of the work of the 10k OpenAI agents if that mathematician worked only on solving Navier-Stokes and did nothing else in their entire career.

That's just not a realistic work schedule for any human. You can adjust the working hours if you want but I don't believe you'll get any realistic estimate. Don't forget that most academics' careers last around 30 years from PhD to Professor Emeritus. If you want a more realistic estimate of how much time it would take how many humans to do the work of the 10k OpenAI agents, you can start from that assumption and work your way up from that.

>> That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.

Sure. But the thing is agents can run 24/7, 365/365 in parallel and as you see above they can cover 2000 years of human work in much less time. I'm not going to estimate how much because the only bottleneck is the amount of compute and money that an AI company wishes to spend, and that depends on their motivation to solve a particular problem. However, with sufficient motivation 2k years of human research (keeping mind that's not 2k years of continuous work) can be covered in a few ... months? Probably.

> My bad for not showing my work and inadvertently leading you down the garden path

The problem isn't that you didn't show your work, it's that your work was wrong.

I entirely agree with your calculation that 10k agents for 88 hours is about 100 agent-years if we assume 24/7/365 operation. That's not what I was disagreeing with.

But then you said "How many human-years ...?" followed by estimating not the number of human-years that have gone into the problem but merely the number of years.

You can compare elapsed years for humans (26) and elapsed years for AI systems (about 0.01). You can compare agent-years (about 100) and human-years (26 times the average number of humans working on Navier-Stokes at any given time). Either of those is defensible.

But it makes absolutely no sense at all to compare agent-years for the AIs and elapsed years for the humans. Which is what you did.

If a typical human mathematician works 2000 hours a year (actual human mathematicians generally find that they can't do 8 hours a day of focused hard intellectual work, but I think we should count some of their "percolation time" too) then that's about 6 human-years per mathematician. So to get the same amount of mathematician-work as agent-work the average number of mathematicians you need to have been on the job is about 100/6, or about 16.

So when you wrote

> How many mathematicians have worked on the problem? 10k? Not likely.

the 10k figure was a total irrelevance. The number it would actually have to have been is about 16.

(My earlier "as many as four" ignored the fact that humans don't work 24/7/365, as you point out. But my point is that however you slice it the relevant number is more like four than it is like 10,000.)

My guess, for what it's worth is that that is roughly the order of magnitude of the number of human mathematicians working primarily on things that could be classified as "trying to make progress toward resolving the Navier-Stokes problem" during that time. I wouldn't be surprised if the actual figure were 3x bigger or 3x smaller. It probably depends on how broadly you interpret "trying to make progress toward resolving the Navier-Stokes problem", and one important difference is that all those human mathematicians leave behind them a trail of papers proving things that, whether or not they end up on the path to Navier-Stokes, may turn out to be useful later, whereas if OpenAI's agent swarm proved a lot of useful theorems along the way most of them never got published.

I don't, of course, disagree that it's possible for an AI company to put a lot of AI agents to work on a problem, but I'm not sure how that makes what they can do less impressive. The fact that you can do that has always been a major part of why AI could be such a big deal. "A country of geniuses in a datacentre" is the kind of thing people have said; we aren't quite there yet, but the "country" part is as important as the "geniuses" part.

I'm sorry but I'm not sure I understand your argument. I think you're saying I'm comparing apples to oranges. I'm not: I'm comparing apples to apples and oranges to oranges. These are two different questions:

>> 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?

So neither 10k humans worked on Navier-Stokes, nor has any human spent a century of non-stop work on it.

But I could have made the point more clear maybe.

>> I don't, of course, disagree that it's possible for an AI company to put a lot of AI agents to work on a problem, but I'm not sure how that makes what they can do less impressive. The fact that you can do that has always been a major part of why AI could be such a big deal. "A country of geniuses in a datacentre" is the kind of thing people have said; we aren't quite there yet, but the "country" part is as important as the "geniuses" part.

Yes, I see your point, but those are not geniuses. Grigori Perelman proved the Poincaré conjecture alone, though as he has emphasised his work was based on advances made by others, particularly Richard S. Hamilton. That we can call a genius: a single man who solves one of the most interesting problems in all of mathematics building on the work of his predecessors. 10k agents that search blindly and find a result by luck (or by stealing it), I don't agree we can call "genius". That's what I call "brute force". Anyone who wants to call OpenAI's agents "a country of geniuses" has first to deal with the fact that they look a lot like monkeys on typewriters.

(My apologies for the very slow response; I hadn't noticed your reply until just now. If YGQ doesn't reply to this, future readers should note that the long delay between their last comment and my reply is a sufficient explanation.)

You wrote, earlier:

> For the n'th time: the recent successes of AI in mathematics are the result of a brute-force attack. [...] ~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?

I don't see how this makes sense except as a comparison between total AI-agent effort and total human effort. And the result of that comparison is not that the AI agents have put dramatically more effort into attacking the problem than human mathematicians have. Any one of those 10k AI agents has spent a lot less time on Navier-Stokes than a human who's been working hard on it has. All of them collectively have spent less time on Navier-Stokes than the mathematical community at large.

The AIs put an amount of agent-time into the problem that's in something like the same ballpark as the amount of human-time the mathematical community has put in, and (I think) in fact clearly quite a bit smaller. The AIs got to the solution, which the humans hadn't. I don't claim that this makes the AIs more impressive than the humans -- it's plausible that humans were already substantially more than half-way there. But it means that the AIs are at least comparable to the humans when it comes to attacking this sort of problem.

I agree that it doesn't seem like OpenAI's models are as good at mathematics as Grigori Perelman. But a year ago they weren't as good at mathematics as me[1], and a year before that they weren't as good at mathematics as a typical university undergraduate, and a year before that they weren't as good at mathematics as a typical secondary school student. (Don't take any of those timings/levels too seriously; I haven't checked exactly how the progress went. But it's something along those lines.) How do you expect things to stand a year or two from now?

[1] I was an IMO silver medallist[2] and have a mathematics PhD from a good university, but I'm in my fifties and necessarily a bit rusty at this sort of thing, and while I spent a couple of years trying to be a pure mathematics researcher I wasn't terribly good at it and escaped to the Real World.

[2] For those who don't know how it works, this doesn't mean "second place"; they give out a lot of medals of each type. Roughly speaking the proportions nothing:bronze:silver:gold are 6:3:2:1, IIRC. I was a fairly mediocre silver medallist, so maybe somewhere around the 20th percentile of IMO team members worldwide.)

> they look at lot like monkeys on typewriters.

It is hard to take your arguments seriously when you say this sort of thing. 10k monkeys on typewriters, typing one letter per second per monkey, would take something like 300,000 (elapsed) years to get as far as typing "NAVIERSTOKES". (I am assuming typewriters with only letters, case-insensitive.)

The AIs made an amount of progress that was at least broadly comparable to the amount of progress humans had made, with a number of agent-hours of work that was not dramatically greater than the amount humans had taken, to complete the solution of the Navier-Stokes thing. So the AIs were, individually, in something like the same class of ability as the humans.

You may be as impressed or unimpressed by that as you please, of course. You may think it unwise to extrapolate superhuman performance in the future, perhaps e.g. on the grounds that the smarter we try to make them the less relevant training data there is from what humans have done before. You may absolutely have moral or legal or political objections to what the AI companies are doing or what you expect them to do in future. But whatever they are, these systems are a long long long way from being monkeys on typewriters now.

> I get the feeling that mathematicians are needlessly panicking because they don't really understand how AI works.

I sunno if mathematicians would be having problems understanding how matrix multiplication, backprop, sigmoid functions, attention, embedding distances, probabilities, etc work.

As a group, they are probably more likely to understand it than everyone else.

That's a bit like saying that a physicist is more like to understand how a car works than anyone else because they understand all the principles of an internal combustion engine. And yet, curiously, when we take our car to the garage the person fixing it does not tend to have a physics degree.

Wanna guess why? I'm too tired now to expand the argument properly but basically understanding the components of a complex system doesn't mean you understand the principles of the system. A mathematician who is not an expert in AI has no reason to be particularly capable of understanding how AI works, i.e. how all the maths that go into creating an AI system come together to create. An AI system.

> the result of a brute-force attack

I would say "the result of easily scalable intelligent attack."

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.

Lol, this really struck a nerve. He has been posting non-stop for about a week now, wrote that lame open letter, gave a talk on it, etc.

What a disgrace, but I'm happy he's showing his true self to the public.

You realize this is a guest post, right? Terrence Tao didn't write it. He's been using his blog to facilitate discussions about math's future.

waiting for the xkcd for that

I always think of this one but I bet there's better

* https://m.xkcd.com/435/

Got to be philosophical about it all.

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

I thought "oh, I'll just click through to the original post and read it there" and wow, what a mistake. Just the most obnoxious slop-filled over-animated endlessly distracting claudesite I've ever seen in my life.

Why do we need any job? Check "Bullshit Jobs" by David Graeber. Not saying being a mathematician is a bullshit-kinda job, but there have been so many made up positions way before this AI-era. In that sense AI isn't changing much.

> 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 is a loss of control from incumbents in a scientific field

Lol, the ultimate delusion, laughing at people in the flood zone and not seeing the tsunami... What goes around comes around

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.

We do not.

Yet another extremely long and winding "defense" of the mathematics status quo.

What's the tldr;?

What are the mathematicians arguing for exactly? And against? Totally unclear to me even after chomping through that massive word salad.

This all feels very John Henry to me. Why is it not a good thing that we have new tools that can accelerate discovery? Because it disrupts the current order?

If you can't make the argument in 100 words, you probably don't have one.

Pure math has been overvalued for a long time. Now that we don't need humans for it, it's really time for pure mathematicians to pack it in.

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

Many people ceed control of their path-finding to seeing-eye-dogs.

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?

Airplanes are not a species, and it’s facetious to compare boarding a flight (which requires thousands of human hours to make sure you arrive safely) to humans who walk where their dog guides them.

The statement I quoted said “zero examples of intelligent species” doing something. It took me 2 seconds to think of a counterexample. At the very least it was poorly phrased, if not altogether wrong. Moreover, whether or not people are doing it voluntarily is orthogonal to the statement.

I agree with the gist of the article, but having such a *BOLD:* _False Statement_ does not do it any favors.

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