I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.
I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.
Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.
And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.
As someone who mostly only applies math, that strikes me as a peculiarly academic take. Intuition is more important for me because it’s what enables me to know what methods are most applicable to whatever practical problem I’m trying to solve. The proof’s purpose is to verify my intuition. It’s just a means to an end. I only take the time to do my own when I can’t confirm what I need from a textbook or paper.
The explanation of proofs I got in school was terrible. I had no idea that you could actually reduce everything to one of the applications of the axioms until I found metamath's proof explorer.
Of course, it's too much work for most normal purposes, and in school they accepted whatever random breakdown people used inconsistently and never explained why.
Actually understanding that it wasn't about convincing anyone so much as having a chain of reasoning going all the way back to the axioms was something of a revelation for me.
The idea the proofs are the heart and soul of mathematics is an unfortunate unforced error, and will lead to the death of the professions now that machines are better at making proofs.
>You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself.
This isn't always the case. Our algebra (or analysis) course focused a lot on proofs for the exam. The result was that a lot of people learned the proofs by heart.
I think intuition is hard to test in a way that feels 'fair'.
You can do it - I doubt you could have got a first when I was at Oxford just by learning and understanding the material, but you should probably have been able to get an upper second. The final part of every question virtually always involved insight, but you'd obviously then have to prove what that insight helped you understand.
If you give people questions like those, there is the risk of complaints about the university not having been taught the material for the exams I guess, or you might find that nobody can answer those harder intuition parts. Certainly most students at Oxford couldn't answer that many of them - you needed to answer about three 'final' parts out of about ten questions say in each three hour exam to get a first and perhaps about 20 percent of students got firsts?
An aside, but tests should be that difficult. Otherwise you aren’t getting any signal at the high end. Even in the face of grade inflation that signal can be translated into grad school recommendations (in the uk case, getting all 10 might even garner a “really not bad.”)
Same here, but I didn't memorize the proofs, I tried to internalize their logic, so I could reconstruct them on demand by just thinking systematically. It did work for me pretty well on my real analysis final exam IIRC (27 years later).
Analogous to the Archimedean Property - there is no approach to teaching mathematics so intrinsically good that it cannot be done poorly enough to yield arbitrarily bad results.
Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and rightly so, certainly not in the way Poincaré was on about.
Maybe it's been done, but I'd like to see an LLM recreate Euclid from questioning without having seen it during training.
> Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that
Yes, I believe that, it's part of what I was implying (I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs)
I think we differ on what "mathematical intuition" is then. I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned.
The Euclid example also shows my bias towards spatial intuition of mathematical concepts (which is deeply unfashionable) but also exposes exactly where at least current LLMs break down; they do the symbol based pattern matching version, but they cannot leap outside of that, at least today.
This feels very related to the issues re: the presence or absence of world models in LLMs. Insofar as they have world models (or "intuitions"), these would seem to have to be primarily verbal-linguistic (or symbolic, when using math). LLM world models are not likely (currently) very spatial, in contrast to e.g. V-JEPA-2 models, which likely do have some basic spatial models (and perhaps "intuitions").
> I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned.
If you want to catch them, surely you can find proofs they aren't able to produce.
I used to be a game dev, and one of the interview questions someone came up with consisted of working out the surface area of a variant of Menger sponge to some given level of depth. The bifurcation for people that could do this vs those that couldn't was incredible, and did not follow obvious trends for academic achievement. (The same interview also included the gem "How wide is a pointer?" which also catches a frightening number of people).
I think it's actually the opposite. Intuition is the only thing LLMs can do, which is why they are prone to hallucinating when they can't validate their intuition against reality.
There is an idea that human intuition, expertise, and critical thinking are largely pattern recognition. When you encounter a situation, your brain gives you a plausible starting point, based on what it has experienced before. You then continue with explicit reasoning, which is slow and inefficient, and try to validate your ideas. The more relevant the patterns you have learned are to the situation, the more likely you reach a useful conclusion.
LLMs are largely the same, except that they cannot learn from experience in normal usage. And except that they experience the world only through symbolic data, while the human brain has access to plenty of sensory data.
My hunch (or intuition, hah!) is that intuition is an instinctive mental shortcut required to navigate large problem spaces that can’t entirely fit into our heads.
Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces.
> Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces.
My view is that is certainly true of smaller LLMs but becomes less true as they scale up.
To quote the parent bananaflag in a sub-comment:
> I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs
I think as the sort of spare space adjacent to pure language processing in LLMs grows the probability of the sort of reasoning bananaflag is getting at (or spatial reasoning, or anything else) emerging in that space grows enormously.
One of the questions for AI development over the coming months or years is going to be if deliberately cultivating the architecture of those sub models for specific reasoning types beats any emergent reasoning mechanisms or not.
Hmm we may be talking of two different interpretations of intuition here. I agree that LLM weights contain representations of abstract concepts, as a lot of prior research has shown. This surely includes Mathematical concepts.
But to me that is analogous to what human brains do, and a bit different from intuition. I think of intuition as “heuristics”, typically developed through experience, that may link seemingly unrelated concepts via vague, hard-to-define associations, but which let us make mental leaps (or shortcuts) while reasoning. (Maybe analogous to System 1 / 2 thinking.)
On the other hand, LLMs can do both: build “intuition” from patterns in data AND brute force a huge amount of potentially unrelated concepts. This gets fuzzier when we realize that even these “concepts” themselves are gleaned from patterns in data! But my point is we necessarily have to take shortcuts to scale, whereas machines can scale with hardware.
This is of course a layman theory! But it could explain why these models are progressing so fast.
Yes, in my case "intuition" comes a lot from visualizing things spatially, manipulating them, and being able to capture their properties in equations/proofs, and it's that which is (currently) conspicuously missing when dealing with LLMs. (And may yet appear with world models).
With the alternate view of intuition that many of you are describing it is clear LLMs are somewhat either there or heading there now.
This is an intriguing observation! LLMs were famously bad at spatial reasoning, until Astra which apparently has a huge improvement. I wonder if that has any bearing on the recent jump in Mathematical performance?
One thing that struck me from Dario's last podcast with Dwarkesh was that he said training LLMs on a diverse set of tasks does not make them better just at those tasks, but they get better at unrelated and other tasks overall. What you described could be a concrete example of how that dynamic works!
Your unstated major premise here is that their intent was to make a universal statement about how proofs work and not just talking to humans about how they teach humans.
Why? The entire subject of conversation is triggered by things which are not humans producing proofs.
If it's possible for a machine to produce a proof without intuition then clearly a human could also do it too. (And in fact I'd argue I've seen many people like that, simply very good at pattern matching over memorised items).
Because regardless of the point TFA is making, that interpretation makes less sense for the specific comment. It doesn’t fit with the immediate context, which was a response to a thoughtful comment about how humans do math. And it requires assuming a math professor doesn’t understand a very basic and obvious thing about their area of expertise.
That doesn’t really read as good faith engagement in the discussion. At best, it reads as being so AI pilled that you can’t even fathom that others might want to have a little side discussion about something other than AI.
That math prof was talking about his stance in discussion between mathematicians long before AI.
Plus, I studied math, I am from that environment. His description matches how math is done by people.
People who are good at pattern matching and memorize are, frankly, shit mathematicians. They are find in fun culture around math, but rarely in actual math. They cant really do it as science.
As a math professor, I care much more about the key idea, heuristics, and motivation than the proof. With the others in place the proof is clear, something an AI or a student can do.
Well, it's knowing when to push and when to not. You probably have an intuition for, I don't know, abstract algebra objects (I don't know your field of specialty :P), without needing to symbolically manipulate all of it, but you developed a deep intuition for them through many proofs and attempts at proofs with them.
Im glad you brought up abstract algebra—that was the one class in my math undergrad that I never developed an intuition for. I learned to do the proofs by pushing symbols around and putting bars on top of them but I never felt like I understood what was happening.
That's leaning into engineering, away from math. Heuristics aren't always accurate. Math history before proof is the history of delusion. Idea, heuristics, and motivation aren't nearly enough for correctness outside of a sandbox.
In my experience, proof is the gym reps that allows you to harness strong intuition elsewhere.
In practice as an engineer, intuition is far more useful, eg, being able to “feel” when something is off in our reasoning — but proofs are where I train those same sensibilities on “harder” problems, (eg) details about how to model identity, equality, and equivalence in a formal model.
There could be meta-faith, if you insert engineering into
Mathematics, physics, chemistry, astronomy, march in one front [lockstep]. Whichever lags behind is drawn after. Whichever hastens ahead helps on the others...
--Karl Schwarzschild
The problem or nonproblem before (elite) software engineers were pointed at rather bespoke conjectures, depending on one's specific denomination, was that the frontier mathematicians got too far ahead of the others to effectively drag them along (hence Tao's recent fundraising attempt using his one-off compressed-sensing work)
There is also the Experience<->Understanding "wave equation" if you will, codified by the popular engineers' joke about how mistakes/bugs mediate the two
Imho what academia+industry really need are GLM-wielding plumbers cheap but capable enough to find these abstraction leaks between silos. One taxes these plumbers so brutally that their clients can get by on basic tokens (morally speaking, so as not to drive demand in the farflung silos of billions bottles and babes)
There were interdisciplinarian buzzwords but these did not live outside the grant proposal, and probably won't survive better under the reign of Pangram
Similarly, programming is also a precise language of communication. Initially, we focused on direct machine behavior but every abstraction above the hardware (including assembly) has been to make that behavior legible to humans.
Developed notations and shared procedural abstractions have made thinking about computation more intentionally human and source control has established a protocol for conversing with other humans in the language of a program and changes to that program.
The moment just now feels like a neglecting of the idea of communication being central. If the program is a compile target but not sufficiently legible or if the conversation moves too quickly for us to keep up then we retain the effects of computation but loose its meaning as communication. We loose the understanding and the ability to develop and evolve further shared abstractions.
Open source programs could be more like motivated explanations of computation. For open source to survive, maybe we should start to make the distinction between free product distribution and programming as communication and community building.
This feels a bit too optimistic regarding programming. Or perhaps betrays an imprecise definition of programming. To me, programming is about expressing algorithms. But algorithms may be totally illegible and still work fine.
Math on the other hand is exclusively about being understood. It is ideas from math that made algorithms legible and thus made the act of programming an act of communication. If by programming you mean using notation and ideas that were borrowed from mathematics to specify algorithms, then for sure it’s communication, but only inasmuch as it was math first. If you mean only specifying algorithms, then no communication need take place; the executor of the algorithm will deterministically execute it irrespective of its ability to communicate.
Programming was never about communication. It was always about making the machine do the thing we want. Back in the day, a good game programmer knew which time intervals had writable video memory and which CPU cycles drew which scanlines, and spent more time rearranging the code to hit these timings than to write the actual algorithm. Later programmers (I hesitate to call them good) learned everything there is to learn about OS internals and wrote theoretically nonsensical and invalid code that still worked thanks to those internals, to save CPU cycles and especially memory use. And the next generation of programmers took the principles of late binding and abstraction to the logical extreme and created architectures that cannot be described in words anymore, only in diagrams - but are crazy good for code reuse, traceability and A/B testing.
> Open source programs could be more like motivated explanations of computation.
It is already that. Every time a method/function is created, a structure is defined, a variable is added, a file is created or renamed,… It’s all for the purpose of human communication. The computer only need binary in a single file.
But people feels like they should be able to jumpninto curl code without any understanding of networking, or linux code with no knowlede of computer architecture. Few code are meant for total beginners.
> There’s a wonderful book, How to Prove It by Daniel Velleman
the name sounds familiar but i don't think i have read that one, i did enjoy "introduction to mathematical reasoning" by eccles.
personally my relationship with mathematical proofs has been complicated. it took some work to understand basic proofs (dedekind cuts, ideas vs. instructions with mathematical notation), but all of the theory of computation proofs, which supposedly are difficult for many, were completely intuitively easy for me.
i think mathematicians are facing a similar confusion as computer programmers. the medium used to require precise thinking and the simple act of reading, writing and composing it was a mechanism for thinking and learning. in the llm era, the question is: should there be a new mechanism and if so, what should it look like?
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
> usually by people who are good mathematicians but know close to nothing about teaching.
I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens.
And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
I think the problem is partly circular. Most people do not like maths. This includes most primary school teachers - in places I know primary school teachers are not subject specialists so just reflect the population of those with the required level of education in terms of their attitude to maths.
My daughter hated maths when I took her out of school at the age of nine. A few years later she was very good at it and enjoying maths and STEM subjects. When she went to a sixth form college[1] she liked it well enough to pick it as one of her A levels[2].
Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
This is tricky, because, in fact, hard math having an intelligence floor is one of the nastier realities of the human condition. Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling (and this term is a common one thrown around in people studying mathematics, because intelligence denial is so obviously false when you do hit your abstraction ceiling).
Most people are correct that they lack the intelligence / mind for a lot of hard math (even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are, and these are just basic undergrad calc).
And yeah, sure, people have different kinds of intelligence and such, but there is still a g-factor, and people of low intelligence almost universally can't do hard math, whereas most people who can do e.g. advanced undergrad math can generally do almost all other advanced undergrad fields reasonably well. The world isn't fair here.
> Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling
Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it. I simply ran out of motivation to pursue higher math, especially when following through on learning and research became more and more labor. (It was always labor; but it was a labor I used to love.) I am far from a prodigy.
> even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are
Disagree. It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind. What your average mind (and student) lacks is exposure to formalism, abstraction, and how these things tie in with what they are familiar with, which is symbolic manipulation. With the exception of geometric proofs (another educational bugbear), they have little context for what formalism is or why it matters.
> Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it.
This sounds a lot like you may have in fact succumbed to your abstraction ceiling, because in practice, the ceiling manifests as not as it being impossible for you to learn something, but that it would take you years and inordinate effort to master what you notice others mastering easily in just a fraction of the time. You may have not heard the exact term (comes from Douglas Hofstadter), and you may be talking about just the academic busywork, but I find it hard to believe you never encountered discussions about this kind of stuff. I would also politely suggest that unless you are Terry Tao posting under some kind of alt, you most certainly do have an abstraction ceiling (or your own mathematical limits) too.
> It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind
The latter statement is obviously false, but regardless, intelligence explains some of the difficulty, and much other difficulties far more parsimoniously than "everyone could just learn any math if they just tried hard enough and had good enough teachers". E-d is merely an obvious and generally familiar example, and nothing I said really relies on this very specific aspect of maths, obviously. We also shouldn't pretend your (almost certainly false) view of math and intelligence isn't also often harmful to struggling students in its own way.
> Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle - while for myself I felt clumsy. even oafish, wandering painfully up a arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn ( so I was assured), things I felt incapable of understanding the essentials or following through to the end.
(Alexander Grothendieck, Recoltes et Semailles)
Amazing that he managed to keep going after hitting his abstract ceiling in graduate school.
You clearly don't understand the meaning of the term. Grothendieck was almost certainly wrong about his gifts here, and even if not, your mathematical ability and output isn't fully explained by your ability ceiling.
Honestly, the pushback on this post is utterly baffling. Clearly the human mind has limits on what it can comprehend and the rate at which it can learn difficult things. Clearly these limits differ among individuals and are related to intelligence broadly.
Huge proportions of the population struggle to ever even grasp simple fractions, and not for a lack of effort from them or society. Fourth-year undergraduate mathematics is another beast entirely. Pretending the world is otherwise is pure fantasy and also plainly harmful, to the world and people that are unfairly pushed beyond their capabilities.
> Grothendieck was almost certainly wrong about his gifts here, and even if not, your mathematical ability and output isn't fully explained by your ability ceiling
lol, see, it's unfalsifiable. No true abstraction ceiling.
Find me the magical teaching method that can make anyone learn any kind of mathematics at nearly the same rate, and you have falsified the idea that anyone has mathematical limits.
Given we as a society can't even figure out how to do this for educating stuff involving simple fractions, my theory is far superior than whatever exactly it is you think.
That's far different than telling a math PhD whom you've never met that they hit their "abstraction ceiling" based on a 2/3 paragraph comment on hacker news. I'm sure you'd have said a similar thing to a young Grothendieck if he were describing his early struggles in graduate school. You're getting pushback because you were being rude and presumptuous.
Work on your reading comprehension, I made it clear that increased effort is what an abstraction ceiling feels like, but also made it clear that GP could have been talking about the effort of academic busywork.
Let's also not pretend that "you could have learned epsilon delta proofs, you just didn't try hard enough or your teachers weren't competent" or "you just didn't have enough time" and etc. is also not rude and presumptuous. Denying the existence of such limits is equally offensive.
You’re in a kind of compulsive ideology here. This same vein of thought and why it is harmful is described by David Bessis in the book Mathematica: A Secret World of Intuition and Curiosity.
“To stop thinking in terms of “gifts” and “talents,” one has to find an alternate explanation. My way of looking at things, which has served me well throughout my career, was to imagine that creative mathematicians were hackers who had found ways to unlock “hidden modes” of our cognition. Most of the time, they’d done so unwittingly, and were entirely incapable of explaining how.”
It is a phenomenon like child-like mental yoga of attention.
Uh huh. A bunch of quasi-mystical bullshit to justify utterly inept garden variety intelligence denialism (it also just shifts terminology: if we accept your metaphysics, it would still be strange to propose that everyone has exactly equal "hacking" ability).
The hubris and willful ignorance required to imagine that everyone is just equally and infinitely unbounded in their cognitive ability is simply mind-boggling in 2026.
But everyone is as a kid when they learn language and everything else. Some people retain that level of watching, listening, and babbling (hacking) when they don't know what to do. So the task is just to get people back into that mindset. Innate intelligence isn't necessarily only symbolic manipulation (analytical), it also is experiential/creative and practical per Sternberg.
> But everyone is as a kid when they learn language and everything else
Also clearly false by almost all current research.
> So the task is just to get people back into that mindset.
Again, you have no evidence, and this is clearly wrong in cases of mental retardation or brain damage. Modern genetic studies also seem to suggest intelligence is related to a lucky absence of errors / genetic problems that are otherwise inconsequential (or even advantageous) in other domains, so really, your "everyone starts perfectly equal in intellectual ability" is just empirically disconnected and ignorant fantasy.
I haven't talked to a child educational psychologist lately (I imagine that would be quite renewing to spend some time in a kindergarten even if only virtually, don't you?), but I don't think biological claims about elite abstraction is the only way when it comes to explaining more and more with less and less for mathematical education. For example, non-symbolic distinction and indication operations are far more fundamental than symbolic manipulation, and using that is a bottoms-up foundation more in keeping with constructivist understanding exhibited by math exemplars.
Cognitive disparity is because of compounding investment of attention and metacognition in development, preferably in a self-referential non-symbolic universal way because intuition is partly based on sensual metaphors and embodied cognition. Everyone has issues distinguishing ungrounded concepts if they don't have a map of them from their attention previously..
Mental rigidity (aka "fragile perfects") is a fairly common phenomenon for math anxiety, whereas Grothendieck advised uninhibited playfulness to deal with uncertainty. The perceived difficulty of mathematics is a social phenomenon rather than organic comprehension limits on abstraction ceiling.
It is the social aspect of math that is the superintelligent part of it, which transcends the genetic determinist perspective. Civilization advances because education transforms the breakthroughs of genius (which all children have ultimate capability for) into the baseline intuition of the rising children by sharpening their attention. Math is supposed to be a democratization of human understanding, that's why the Greeks were so keen on deduction and why proofs are for systematic communication. If a stupid-ass computer can do math, so can any human being.
> Cognitive disparity is because of compounding investment of attention and metacognition in development [...]
Sure, but exclusively? There are no other factors that don't depend on effort / investment / social context?
I can't take you seriously when you take such an absolutist stance on these things when science has long since accepted nothing complex about humans is 100% nature or 100% nurture (really, shared vs. non-shared environment vs. genetics: but, surely you know this).
There certainly are numerous factors claimed. I believe cognition happens from distinguishing reality and people can always learn better how to do this.
Whereas “abstraction ceiling” is hard science backed by ample literature, not a loose metaphor directly contradicted by several prominent mathematicians who didn’t quit when the going got tough.
I think the fact the Feynman took an IQ test and scored 127 is damning of the entire concept. I think what turns mathematicians off is the thought that psychologists who couldn’t tell you the difference between a scheme and a metric space think they can actually measure who has the capacity to be a mathematician and who doesn’t.
Like, why fucking bother doing anything? Why run the 100 meters at the Olympics, let’s just do some genetic testing and measurements to pick the fastest man in the world. Why teach kids music, let’s just measure hand size and do some sight singing exercises and teach the talented kids piano. This whole nonsense reeks of Gattaca-style quasi-eugenics where people get sorted into profession by people who don’t actually have expertise in any of them. You’re not in the guild, you don’t get to appoint to the guild, and you certainly don’t get to gatekeep who can apply for the guild.
Edit: Dropping slurs when someone compares your views to eugenics is an interesting strategy. Shouldn’t you be at a meetup discussing Curtis Yarvin’s work or something?
"Abstraction ceiling" is just a way to talk about intelligence at at the tails that reveals one of the difficulties / limitations you can encounter when it comes to compressing / abstracting complex mathematical objects. Your objections to the term are facile and clearly stem from an obviously unvocalized intelligence denialism that is simply indefensible today. Also, intelligence != IQ, obviously this is too simplistic.
Everything about your arguments and other posts is similar reductions to retarded extremes (our only options are "eugenics 2.0" or deranged intelligence denialism - there is no room for anything in between, e.g. the idea that base intelligence matters and sets a hard average ceiling on potential, but that effort and other factors might push one slightly above/below this ceiling relative to others with a similar intelligence, and etc). Or alternately you hallucinate things I never said or even remotely implied (e.g. we should gatekeep based on dumb psychology metrics or hand sizes).
Just be honest: you know intelligence is real and matters, but you want to dance around this fact because you find it ideologically inconvenient, or you can't admit you yourself have limits (and lack the courage to realize the obvious social broader consequences of this personal admission).
Dropping slurs after being compared to a eugenicist is an interesting choice. Maybe you would be more comfortable in the comments of Curtis Yarvin’s blog. You’d certainly get less pushback there.
Falling into deranged ideological projection is also an interesting choice - one I chose to mostly ignore. You seem to really obsess a lot about this Yarvin fellow: I tried reading his stuff once and found it intolerable.
I imagine you think calling some of my language choice a "slur" here is some kind of gotcha, when the term I used is specifically one widely disputed as actually being offensive, given it is mostly used now to refer to normal people acting in intellectually deficient ways, and not generally to those with actual learning disabilities that deserve our sympathy. There are studies on this, which you surely are aware of.
If I had referred to your more deranged positions as "smooth-brained halfwit extremes", you likely wouldn't haven't tried to impotently pull this "slur" card, even though the semantics are basically identical. Which basically goes to show that you value irrelevant surfaces over substantial realities, and frankly is perfectly consistent with the midwit intelligence denialism on display in your posts in this exchange.
> This sounds a lot like you may have in fact succumbed to your abstraction ceiling
It sounds more like you're turning a vibes based theory into a tautology.
Hofstadter struggling with math for the first time in graduate school isn't a unique story, nor is his self introspection about this event a good basis for an apparently unfalsifiable theory about human cognition.
We have mountains of evidence that humans differ dramatically in cognitive potential, and more again that often effort / practice can only explain a small amount of the variance in performance in a wide variety of fields. We have basically zero evidence at all that anyone can just learn anything if they try hard enough under the right teacher, and plenty of evidence to the contrary.
Abstraction ceilings are about rates and difficulty of learning, so even if we assumed the (absurd) claim that no one has any fundamental cognitive limits, until we are immortal, being slow enough still creates an effective ceiling.
Intelligence denialism is the incoherent and indefensible position here.
That stuff always gives me such a eugenics 2.0 vibe — no, no, the hierarchy is based on innate cognitive ability now. Gives me the creeps that they're actually in academia pushing that stuff.
> People often hate math because it was not explained to them correctly
Spoiler: this is also why mathematicians hate vibe-math. AIs are outright terrible explainers even when they do have a watertight logical argument—and honestly, this is the load-bearing seam.
It goes beyond "proof vs. exposition": the logical derivations AI comes up with fail to even qualify as human-directed proof because of how terrible they are (far below even the most novice mathematician doing their roughest work) at the exposition part.
> AIs are outright terrible explainers even when they do have a watertight logical argument
I think this only applies to cutting edge mathematics (novel proofs of hard problems). I have seen it reported more than once that such AI proofs are cumbersome to follow.
But in my experience, when it comes to explaining well-established math that is already in the training data, AIs can be very good teachers (at least with recent models). Especially if you use it along with a textbook and ask it about anything that might not be explained well in the textbook.
I have a hatred for people who think they can use this method.
If used incorrectly which it is a great percentage of the time it confuses the student. The person employing the socratic method must actually know the answer and where the student is in their mind. Failure on either account makes it pointless.
Ask anyone unfortunate enough to ask for help on IRC
That only works if the one you're trying to guide can figure it out mostly on their own and is interested in cooperating. Aka does not work for anything below university level.
Intuition happens naturally and people are prone to inducting the wrong conclusions. Proofs provide a framework for rigorously analyzing drawn conclusions such that it can be used to build intuition in others. If math is about sharing the insights gained in a particular class of problems, proofs are the means to getting there.
The math field is confronting something that coders have been dealing with for a few years now, only far more violently. Today's moat for software seems to be that AI can automate tasks but not a full job (yet). But for a large proportion of mathematicians, doing these tasks really was _the_ job. It's the bit they wanted to do, and if they completed a sufficiently difficult set of tasks, they got tenure. Now this model is failing, they frantically need to pivot the role of humans to save their profession from funding cuts.
I remember when "writing code was never the point" became a mantra here. There was truth in it, but removing the coding has certainly taken away a lot of the texture of the work and enjoyment of the craft. Many of us feel this loss as we tech-lead teams of agents as our source of income. I am not optimistic the mathematics pivot is going to work, but I'm certain that most will be depressed with the outcome even if they succeed.
We are all staring at the same existential dread, just seeing it unfold slower. We're being told that utopia is to be obsolete, and that is a jarring idea to contend with.
I feel that you can see quite strongly the truth in “writing code was never the point” when you encounter inevitably at every company the guy who has been around forever but doesn’t seem to be working particularly hard. Their value is (was) no longer in writing code at a furious pace all day. It was having a coherent, intelligible and communicable theory of the software system the company is founded on.
I propose this thought experiment: put all living mathematicians in a very long bus. This bus crashes and they all tragically lose their lives. Can we really say mathematics simply marches onwards with AI alone? Let’s say Anthropic needs a new research result to improve Claude. Are we really already at the point where we burn tokens ad infinitum and arrive at the end of scientific progress in some timely fashion?
I actually have a rather dim view of the "writing code was never the point" line. Not because it's objectively wrong, but because I see it as something we're mostly telling ourselves to feel better about the status quo. Ability to write good code has been highly celebrated (and remunerated) for decades. As it is becoming less relevant, we immediately backtrack and start lionizing the parts where we can still be useful instead. Consider the counterfactual - AI continued to be terrible at writing code, but weirdly better at humans at product decisions, architecture etc. In this universe, saying "coding was never the point" would not be popular.
It also find little solace in it aside from 'well this version of GPT isn't taking your job'. AI labs certainly have no intention for the higher level skills to stay in the human-only domain. The veteran developer with the coherent theory of a large stack is immensely valuable today. But they also don't survive if a company can drop a few coders' salaries on rewriting that stack from scratch - faster, fewer bugs, more coherent, able to react to changing business requirements with more agility etc. I am not saying this is where we are, but I think there is a reasonably good chance this is where our road is leading us.
The counterfactual does not sound appealing in the least. That's basically, AI tells human coders what to do, tells us to re-do our architecture when they don't like it, and generally removes all control and initiative from us. But we get to write a for-loop by ourselves? That's our victory?
Maybe it's just me, but imagining that counterfactual reinforces that "writing code was never the point." At least, for me, I enjoyed writing code because it allowed me to design the architecture rather than having some architect lead or engineering manager above me decide.
As much as I hate it, I don't think we'll ever get a proof of the four color theorem that isn't enumerating cases.
When you have an integral or the sum of an infinite series that comes out to pi, you know there must be some satisfying explanation involving a circle.
Contrast with "Examples of patterns that eventually fail" on math stackexchange[^1]. When a pattern ends at 906150257, you don't really expect the proof for that to be something beautiful. The reason for the exact value of an upper bound is that it isn't smaller and it isn't bigger.
There's a relationship between e, i, pi, and -1 comes from a deeper relationship between complex numbers and rotation.
The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
> The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
Well, the current conjecture is that it hasn't to do with coloring at all, but how many, on a map (partitioning of plane into connected regions), regions can pairwise touch: 4.
(That is: If someone proves Hadwiger's conjecture, then the four colour theorem follows.)
This is, as always from Grant Sanderson, thought provoking and opens new insights.
It occurs to me on reading this that there’s a connection to other computerized mathematical activities.
Occasionally some computer lab in the past would announce that they have computed pi to more digits than ever before; or a new Mersenne prime will be found.
These count as ‘math news’ but they’re of little interest to mathematicians. These computational efforts demonstrate the great power of computers but they do nothing to advance mathematical understanding. Finding a larger Mersenne prime is not surprising to anyone; we’re pretty sure there’s an infinite number of them. Finding the largest Mersenne prime would be the surprise.
So it is with proofs. An LLM might prove some conjecture - Riemann, say or P≠NP. But in general we know that things can be proven and we think those things are probably true, so the existence of a proof doesn’t do much more than producing a new Mersenne prime does.
It’s only if in proving the thing we learned something that there’s actual value in the proof.
Part of the controversy here is that now the skill advantage that some Field Medalist had is much narrower. The fact that fields medals have an age limit implies that it favors brain power over understanding. And that was the guiding light award of the community. So i find it "funny" (and natural) when they are offended by AI.
That is the main "crisis" of mathematics.
In my opinion there has never been a better time to be a mathematitian, and there has never been a better time to be a software builder.
But there has never been a worst time to have the need to prove your economic value as a mathematitian or software developer alone. Because "understanding" is not something you can prove in one afternoon, its something that you prove with a life.
> In my opinion there has never been a better time to be a mathemetician...
As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible. It's like telling a chair-maker during the industrial revolution that there had never been a better time for them, since now they could operate chair-making machines instead of toiling away at the wood themselves. It assumes that they were purely in it for their passion for mass-producing chairs. The majority of mathematicians get into the field because they love problem solving, and the gauntlet thrown down by challenging math tasks.
Many parts of this will never be useful for society on a grander scale - but this is reflected in the finances - pure math is closer in funding-terms to a humanity than to hard science. Now even this is _massively_ under threat, and Tao and co need to pivot quickly to stop this from becoming a bloodbath.
Im an ex mathematitian too. And if i was in academia I would probably have the same reaction. Thats what i say that its the worst time for proving economic value.
But if you are in for theory building and understanding, then you are not constrained anymore by your motivation to grind through countless hours of formal theorem proving. And you do not need to have superhuman formal manipulation skills and memory.
For me mathematics is not the formal system, so LLMs will never be able to do end to end maths.
> As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible.
To throw a counterpoint to this into the writhing cesspit of HN, I'm active in academic mathematics (postdoc) and every one of my collaborators is deeply in love with the field and their jobs. Perhaps the grass is greener on the applied mathematics side of the fence.
IMO this would track, applied mathematics (even e.g. data science, though perhaps calling that applied math is a bit generous / insulting to more serious applied math) is in some ways more exciting now because it is far easier to surface complex / appropriate methods for the task at hand, and you can more confidently explore these methods because the AI sort of "has your back" in catching some of the more obvious beginner errors you make during these explorations. Plus, applied math feels roughly more results- than process-focused, compared to pure math.
IMO the divide here between pure vs. applied math feels a lot like the divide between those who enjoyed coding for the understanding it led to, i.e. the writing itself was the joy, vs. those that primarily coded for the results. I enjoy the creative part of coding, the thought of software jobs just devolving into writing specifications and doing code review very much kills it for me.
Probably true for the dedicated problem solvers (of which Tao is one IMO). But I doubt there's ever been a better time to be a theory builder (more like Peter Scholze, or Grothendieck).
Some up with an idea and leave the system to check it 15 different ways, and see whether you can simplify an existing body of theory. It'd be like having an army of lightning-fast grad students.
This is coming next. There's nothing particularly special about theory building. Successful theory building is always oriented towards solving a problem, because otherwise even humans can easily spam out a bunch of nonsense. This is a real problem that the math community has experienced on multiple occasions pre-AI. I'd go further and say that theory building is irrelevant if it doesn't help solve problems people care about.
I'd even say Scholze is not a great example here. Most of the work he's known for is progression towards the Langlands program - which is very much a problem to be solved, and one I would imagine he'd not be thrilled for an AI to one-shot. I agree that it is somewhat 'up the chain', in the same way that software engineering has not immediately disappeared now that performing coding tasks is largely automatable.
But I also take issue with 'never been a better time' - e.g. is this really the greatest time to be a software engineer? Everyone has AI psychosis and feels like they're a couple of breakthroughs away from being unemployable. The same is even more true in math - we've gone from failing IMO problem 6 last year, to solving NS. The rate of change is formidable, it feels like there may not be many places to hide in a few years.
But these are not very good arguments, because it makes it about the fall of institutions (the funding) and people being miserable for personal reasons rather than prosocial reasons. Tao here clearly suggests that math is not reducible to "problem solving" or "proofs", the valuable part is much more than that framing.
The concerning argument about the status of math would be an outline that it will get destroyed by a process of societal atrophy and there is no turning back, and the AI powers are not a good substitute or replacement for it. If an entire society becomes reliant on these oracle machines then it would be analogous to children never learning arithmetic because they were handed calculators. How could the human race still flourish? We would anthropologically regress. We'd be little better than animals, like the Borg zombies.
That is a much more profound threat than people worrying about their own careers or faculties disappearing like the humanities. This is a serious anthropological reckoning.
If math experts are that freaked out already then basically all of science is soon to follow, decade by decade. "Singularity" comes to mind.
Broadly I agree with this, but I was refuting the statement "there has never been a better time to be a mathematician". You are changing the question to something different here, and using it to say my argument is bad.
I think math is a microcosm for "thought-work" in general. We have been in a symbiotic relationship with capitalism for decades now, where the hope of a well-paid white collar career encourages people to spend time and money to enrich themselves through education. The proliferation of AI cheating at college already signals that employment is the primary goal over intellectual growth, so one imagines this governs what happens next if labor demand disappears.
It's hard to say where this all leads, but I have very low optimism for higher-level math understanding being something that humans value in the same way in the coming decades.
> In my opinion there has never been a better time to be a mathematitian
I think it’s a great time to be a curious mathematician, especially in a niche field where you’re not competing with hundreds of agents of the best unreleased frontier models.
However it’s a very scary time to be a professional mathematician because publish or perish is going to cause a race to the bottom for cranking out results as fast as AI can let you. [0]
I'm a professional mathematician. Today I proved what for me is a very solid theorem. It's something I had thought about for a few years. With a few weeks of serious use of AI I've found a proof that I am currently trying to write up, but which appears correct. The change in the workflow is enormous, but so is what one can do if one has clear what to do and how to do it.
Not every problem will receive $20M in funding to be solved by AI; for the rest, good human guidance will have to suffice. Labs only pulled this stunt because they wanted to show investors how powerful their models are on their own. But look again at the cost of that army of 10,000 SOTA agents. At the very least, I foresee a need for humans to decide when costly AI resources should be committed to a specific search plan. Grant review remains irreducibly human because it involves choosing which directions to fund and weighing opportunity costs: taking one path forecloses others.
That's not the conclusion. I started using AI after the Jacobian conjecture counterexample and have used a particular problem to learn how to use AI and to explore it's capabilities. I'm not a great mathematician but I'm full faculty with 25+ years of research experience and lots of articles and I just proved in a few weeks something that had resisted my efforts for some years.
The exploration process is much easier now. Ideas are quickly testable and multiple tests can help identify a technical obstruction. The tool requires good guidance and input but as it trains on people like me it will need those less.
At the very least our way of doing things must change. More pessimistic views seem to me defensible.
GP never said it was the conclusion. He was praising you, after you on your own volition decided to comment here to let know others about your experience with AI.
It seems to me you're a very privileged individual, I'd suggest you practice gratitude regularly in your life.
You sound a bit troubled. I was responding to the leading the way part. I'm hardly doing that. I just jumped on the bandwagon. What I do see is a tool that forces me to change behaviors learned over decades and potentially renders useless many of them while facilitating others. That's not all a rosy picture.
As for privilege, don't assume everyone lives and works in the US or gets paid lots of money to be a professor.
I find myself less worried about it than at first. I think what we'll see are that some things are low-hanging fruit and can be solved just by tireless search. Maybe half the millennium and other such high-visibility problems will fall this way.
Others, I think, will be beyond both human and AI. And so what then? Mathematicians just throw in the towel and say it's not worth trying? Of course not. We will continue that pursuit, and as we do, new ideas will arise and new problems will need to be solved. It's math. There is no end.
It's easy to look at the current landscape and see AI ticking off solutions to problems and imagine that soon there will be nothing left. Machines replaced the need for much manual labor, but they also established a basis for an economy that provides the opportunity for more labor. This is the situation with math now. It will take some getting used to. There will be little-to-none pencil-to-paper working out of problems anymore, but there will always be work to do, things to solve, curiosities to unravel. And it will still be professional mathematicians who are the ones most capable of directing that effort. Because, if nothing else, they're the ones whose curiosity is piqued by the problems. Which, let's face it, has been 99% of the motivation for graduate-level math in the first place.
There's the the old question: is math invented or discovered? I think it's both: the problems are invented, and the solutions are discovered. In the age of AI, the discovery part will be greatly affected, but the invention part will remain firmly in the human domain.
One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.
Having to ask the village elder about how to do things meant learning various other incidental lessons. It was reduced when books started to become available.
Having to search in the library also often lead to serendipitously seeing a book on the next shelf and falling in love with a topic you didn't even know about otherwise. Or you had a chat with a librarian asking for advice which books to look in for a topic. Google search eliminated that. With Google search and reading websites, you still had to read or skim the page and may see some other info or click to read the author's About page. Now with AI we get straight to the answer.
I do neither maths nor science with AI but in my experience most models are perfectly willing to burn tokens on a ton of sidequests at the earliest opportunity.
Minor bug and develop whole new field of cryptography are very different scales. I do agree alms are really good with a lot of common bug fix related tasks to the point you have to split commits out. The fact is you recognise it's a minor bug. Who is sitting through the proof of Navier Stokes and going through it and finding connections between itself and other fields? Im not saying LLMs are bad, but I do think understanding is important. Heck, the fact you identified the minor bug suggests you understand the output. Im not so sure the same can be said of a gajillion line lean dump.
I was a PhD student for a while, but I always enjoyed "refactoring" proofs, more than coming up with new ones. Making them simpler, shorter, clearer. Unfortunately, it's not much rewarded.
While I understand and emphatically with Tao's concern I'm afraid it's missing the forest from the trees. Unless you can make a claim that AI will never be able to perform intellectually at the same level as any human at a much lower cost, there's an outstanding utility problem that remains unaddressed.
Sure enough, the AI may not have taste or goals, or many human traits, but that's irrelevant to the much thornier (and much broader than mathematics or even academia) question related to who's getting paid how much and for what.
The funding on mathematics is already one of the lowest accross science [0, 1], and theoretical math funding is probably much smaller than the applied math one already, so that's not even close to how much funding theoretical math gets.
So, we are talking about a field that already does not use that much funding anyway, and most high end theoretical mathematicians probably would make much more money in the industry anyway, so this seems like missing the forest for the tree imo.
There's an old joke about funding, goes something like:
"Why you are always demanding more funding? Why can't you be more like the mathematicians, all they need is a desk, some paper, and a pencil, and a garbage can, and they just do fine. Or how about philosophy, for that matter? They don't even need the garbage can"
I mean, obviously with modern computational mathematics, this doesn't hold so simply, but there is this confound about math research also not getting much funding also because much of it isn't that expensive, relatively speaking.
Last I knew, at least in America, universities that want their math prof's to do research also expect those prof's to bring in plenty of outside funding. You could argue about the costs of that desk, paper, pencil, and such - but modern "research" universities have evolved into extremely high-overhead operations, and The Beast Must Be Fed.
> so this seems like missing the forest for the tree imo.
But is it? Again, regardless of the amount, what is the utility?
What's the end goal of academia (assuming broadly as research with humans) when the answers to the deepest questions become commodities at orders of magnitude higher speed and lower cost?
How are federal grants justified and, forgetting the current hierarchy, how is differentiation made? It's currently based on research output, once that becomes irrelevant what is it? We already have IMO, IOI as competitions and I assume just like Olympics this can be a thing, but it's very remote from research.
Take for example Rene Thom after Alexander Grothendieck overshadowed an entire field
> His technical superiority was crushing. His seminar attracted the whole of Parisian mathematics, whereas I had nothing new to offer. That made me leave the strictly mathematical world and tackle more general notions, like the theory of morphogenesis, a subject which interested me more and led me towards a very general form of 'philosophical' biology.
Today, the difficult shit is how to get the resources from the people who use them to turn difficult shit into an easy shit. Basically, a war. I am not sure I want people to move on to that.
People who understand the vast gulf of technological advancements (some bordering on fantasy) between what we currently have, and what needs to exist for it to happen?
There is a finite capacity/time for a human mind to do difficult shit, if it can be slop forked in a microsecond before you even get to flesh it out there is no point
Some were like this, some weren't. Many people were in it because it's objective and factual and not up for the whims of taste of some established gatekeeper. They weren't in it for art performance reasons or to please the aesthetic judgment of some entrenched mathematician-baron.
I was about to link the "discovery fiction" article [0] by Michael Nielsen, but halfway through saw Grant mention it as an inspiration. It's one of my favorite styles of writing that lets you settle into a cozy read of a narrative around a problem while slowly exposing you to the rigor and nuances of tackling it. It's definitely deserving of a public index of some sort, like one of those "Awesome X" list repositories on GitHub, because I think it's a valuable style that deserves to be curated.
> It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.
> we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems
The core idea seems to me that we should shift the standards for professional evaluation from generating proofs to generating explanations. Makes sense that such a proposal would come from the 3B1B guy, and I actually agree with it, irrespective of AI. But what eludes me is how that could be a defensive mechanism against AI automating humans out of mathematics. AI is likely no less good at producing natural language explanations as it is at generating rigorous proofs. It's telling that even Terrence Tao turned to AI to understand AI-generated results [0]. It seems that the essay doesn't address that issue at all.
It’s a task much harder to RL and much more subjective. I don’t want to say we won’t get there, but let’s just say that LLMs could “write” well enough since gpt3.5 era and I don’t think the pleasantness of the prose improved dramatically since then.
And subjectively the explanation LLMs currently provide are usually horrible, horrible enough that I usually just instruct them to provide me human written literature I can read.
I mean, there's centuries' worth of mathematical prose to train on. But that's presumably already in the training data, so if it isn't good enough today, it might not get better fast enough to keep track with how fast they'll get better by training on formally verified math. But then again, the prose in Terry's conversation I linked above seemed pretty useful. But it's also a problem requiring famously little advanced mathematics.
I have been losing interest in this proof-oriented approach into extremely abstract concepts (what seems to be the core of academic mathematics today). Obviously, proofs are very attractive because they are the closest thing we have to a universal truth (at least under the assumed axioms). Having mechanisms to reliably show a proof, and computational methods to handle complicted proofs is great.
But.. the navier stokes proof was the last straw for me. People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society. And certainly not moving us towards "human understanding". The biologists are the ones working on that, the math folks should try working with them on neuro stuff to understand how human brains can do math at all, given their architecture.
> Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society.
Is it, or is it only the parts you hear about/pay attention to?
> People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Yes, maybe famous solving famous conjectures is just trivia and trophy collecting and the real value is the intuition and techniques you develop along the way to solving them which you can then bring to bear on things like CFD.
Mostly the parts I hear about. But I also communicate with mathematicians on a regular basis and I can see what they are working on, which is representative of the field. It's about 90% "cohomology of abstract Lie groups that morphize into string theory" and about 10% "practical thing that engineers can use to make my cell phone work better". (amusingly, although I literally made up that sentence, it turns out it's not far from something somebody worked on: "The cohomology of compact simple Lie groups and spin groups connects to string theory by providing the topological obstruction classes—specifically the first fractional Pontryagin class ...." OK, kind of emphasizing my point there.
No, the intuition and techniques that get developed to solve the NS conjecture have little or no bearing on the practical details of doing CFD. That's a common story/thread (and I hear the same story in quantitiative biology, my area of expertise), but often times, it's just a loose justification given to justify funding.
Current and future mathematicians can now spend most of their time coming up with problems/conjectures and theories that are hard for a future model versions (6 to 1 year out) to solve them and by itself it can be a new major sub branch of mathematics - “Theory of perplexing frontier models” and who knows it can even open up new dimension of mathematics for mathematicians to explore by themselves (because AI by definition cannot help them here). Now is the time to be excited for mathematics!
Hmmh. I like motivated explanations, but, as acknowledged in the text, this is a subjective thing to measure. What is a great motivated explanation for Tao, might be hard to grasp for me. So I guess judging how well an explanation motivates something depends on two things: 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
There is a third thing: how well does the motivation chime with or go against my current belief system? You would think this is not much of an issue in mathematics, but it can be, and I had my fair share of frustrations because of it.
Anyway, all of the above points to one thing: the best motivated explanation will be generated by an AI, knowing the subject and you in a deep way that no other human will, and being able to interact with you during the explanation.
> 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
I don't think we're discussing pedagogy. Good _research_ exposition is instead related to communicate your intuition and way of seeing things. The conceptualization of a given situation or problem is what is valuable, how you connect it with other stuff, etc. It is then up to you to memorize and interiorize it.
Good research exposition is the same as good "pedagogy" just for a different audience. Both have to consider didactics. When writing a paper, you teach something to the fellow researchers who know less about a thing than you.
Seems to me to be two different sides of the same coin. Pedagogy is about finding a way to communicate to me an idea based on my intuition and seeing things. A research exposition is about presenting the idea in terms of your intuition and seeing things.
The blog post says that the statement "AI really did solve a problem in mathematics." is wrong. But a formal proof showing that Navier-Stokes equations can blow up is certainly such a solution, by AI. There is not much in this world that is more objective than a formal proof, so any disagreement on this is based on how we see the world. Michael Harris will agree with the statement being wrong, Jacob Tsimerman will not.
Another example, Hilbert famously battled Brouwer's view of mathematics. From my point of view, Hilbert was right: intuitionistic logic is certainly interesting; but I like to study it using "normal" (= classical) mathematics.
Finally, my personal frustrations are about how hard it is to publish my work on abstraction logic. I would never have thought it is that difficult, mathematics being objective and all. It seems essential to take out as much motivation out of your paper as possible, because it might offend your reviewers and their belief system. By now my papers come with full Isabelle/HOL formalisations, let's see if that helps.
The value in academics is teaching and research. The value in research is discovery. Proof was a useful function for humans to do towards discovery until recently. Understanding is a useful property insofar as it helps you teach and it is a prerequisite for generating hypothesis. Humans will always be driving discovery, the tooling and focus of work may just be a little different. Attachment to one particular modality of discovery is an aesthetic choice, not a moral one.
1. We humans have managed to take huge amount of information and compress it using a loss function containing some bias we have about the information.
2. We now ask ourselves to decompress the same information with some additional cross-entropy. As a side effect of this process we sometimes spurt out information that may or may not have any meaning since the compression was lossy.
3. Now, we ask ourselves to present this some-what newly decompressed information with brevity in order to understand what we've learned from it.
Knowing that this process is happening on a larger scale, this resurfaces the argument if meaning can be reduced to computation only.
Although some might favor this argument but we are at the risk of anthropomorphizing this process.
The idea presented in the post itself is perspicuous (in Grant Sanderson own words) as he always does.
The last days we are served these high goals about understanding, "digestion" and so on.
But if you look at the practice of present mathematics, in the last 20 years it is all about publishing solutions to problems.
There are famous problems to be solved, there is a hierachy of conjectures to be solved. A quick search here on HN gives pearls like "Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem".
And all of a sudden it turns out that problem solving can be automatized.
So then what will problem solvers do? Well, from now on they will "digest" problems solved by AI.
In a way or another they will find a way to stay on top.
That's the goal, at least, but mathematics as a living practice does not have much to do with these games of power.
“You are right. There is nothing in yesterday’s mathematics that you can prove with
exterior algebra that could not also be proved without it. Exterior algebra is not
meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds
is as worthwile a mathematical enterprise as proving old conjectures.”
Gian-Carlo Rota in “Indiscrete Thoughts”
Yes. AI is a useful tool and we are going to adapt and use it.
The phd student will be forced to publish 10 breaktrough articles, the university department which does not offer "free" access to AI (for its members) will see the its ratings going down, when compared with the other universities.
It will be "use AI or perish" for academic management so on the side of academic management the ones with vision will thrive and the ones without will perish.
But what about the publishers? In the last decades the academic research was made into a feeder for publishers. The main goal of a researcher is to write articles, which are later sold back to other researchers.
This economic system is under big stres now, because for a while at least the academic management and publishers will have contradictory goals.
And that is why this scare which is induced by those who profit the most from the present system.
Arxiv is a good blueprint for what we should build on. Terry Tao gave a recent talk about how "canonicalization" is the key bottleneck. The limitation of Arxiv is that is non-canonical, non-peer reviewed. However, there is no reason that journals should have a monopoly on peer review (it should be noted that peer reviewers aren't even directly compensated for it, granted it is supposedly "priced in" to academic salaries.)
Yes. See ICLR for example, which is a peer-reviewed conference, on the OpenReview platform, with no publisher involved, the papers are just hosted on the OpenReview website. It's a similar situation with ICML. NeurIPS also doesn't involve any traditional publisher. Because why would they? There is an online platform to do the reviews, discussions, uploading the camera-ready etc. There is no real role for IEEE or ACM or Elsevier or Springer in this any more. Machine learning as a field has shown that it's perfectly doable, over many years now.
If this suggestion were to come to pass, I wonder how new math PhDs would think about choosing between a 'normal' R1 faculty job vs. the "teaching route" (teaching professorships, lectureships, community college professorships, or SLAC professorships).
It's been my understanding that traditionally the ones who care about "motivated explanations" in this sense go for the latter, but if the research community has now decided they care about teaching and understanding, it might "even the playing field" and make the jobs more similar.
Let's see how long (if it ever happens) it takes for models to generate motivated explanations (possibly done via the Manim library or something like it) along with their Lean proofs. Grant Sanderson is right that this is kind of subjective but so is Art and I'm very enthusiastic about AI generated Art.
> possibly done via the Manim library or something like it
It can be done already: the point is that the motivation and explanation parts are terrible, especially for novel topics where the AI can't just rip off existing content. A Lean proof is at least a verifiable task; you end up with an actual proof that you can work through. A Manim slop video doesn't have that.
I think the thing is.. sorry Grant, but "motivated understanding" won't come from videos, but writing. Solving things too, but a video can only go so far.
I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).
I'd be interested in hearing a field report on this! For example, I can easily imagine that they're great at walking through the proof step by step, explaining background as necessary; but as TFA notes, one of the most important questions is "why is this definition the way it is?", and my bet would be that the Lean is not enough to help the LLMs meaningfully in answering that.
LLMs like even the sota flash models have great range of background math knowledge and have no problem reading and understanding flt level of math. On the other hand you don't want to go through 13 million lines of often repetitive code line by line. Models are great at synthesizing math content out of code. My contribution is to steer it through subjects of most interests to me, drill down into jargons that can be confusing, be creative in using computation for illustration (which coding agents can execute very proficiently) etc.
It would be interesting to see what would happen if we had two competing mathematical institutes, a sort of First/Second Foundations:
1) Rejection of AI for anything but trivial applications while still using computers at their full capacity. Researchers would ensure full human understanding of proofs and methods. This Institute believes on Math as a process of discovery, Mathematicians as explorers/poets/storytellers and not proof machines.
2) Unrestricted, all-embracing use of the latest AI, including potentially research in creating even better AIs as part of the program. These researchers would be okay with not understanding proofs if verified to be correct. This group is focused on rapid problem resolution and believes Mathematicians are theorem creators and provers.
After X years (100?), which one would advance Mathematics and humanity the most (we'd need to define "advance")?
Mathematicians worry about proofs and the intrinsic value of something as elusive as 'understanding'. They are deeply ingrained in the study, deeply concerned with anything effecting the field. Yet they're still emotional beings looking for beauty and meaning in life that might come from an understanding how the universe works purely from a math perspective. I'm glad Mathematicians exist, I certainly can't do that type of work.
And I trust their results: technology wouldn't be possible without advancing our understanding of the world in various fields, including math.
Your idea sounds great for the Mathematicians.
There's a more pragmatic view though, and unrelated to proofs themselves: does understanding a proof help us to advance Humanity in some way?
Do we have better lives afterwards? What if we give up understanding proofs and focus only on results.
In other words, if an AI solves a problem for you, but you don't understand how it works, should you continue building anything on top?
I suppose the results are truly what matter. If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
Your second approach seems good to help figuring out results from both theory and application of math to solve problems.
But also, what if there is no true beauty in Math, the way Dirac and Einstein wanted?
What if these AI brute force proofs are all that's left?
> If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
A lot of medicine is already like this. Shown to work in clinical trials, no complete end to end mechanism understood. They still get approved if the empirical results are strong.
I'll answer one of your points partially: if AI builds a better sorting algorithm and proves its performance characteristics, it's useful. I'd be able to use it to make my programs faster even if I don't/couldn't understand it.
It would be a bit disappointing but still useful and make humanity slightly better.
It's interesting to measure how much we believe in something, and how much trust we've lended in order to have a working model of our reality: enough understanding for us to get around, move about, and be content.
I'm sure you would only trust the improved 'blackbox' AI sorting algorithm after it has been proved out through benchmarks. Once you've seen better, repeatable numbers: your trust would rise and eventually you'd feel confident enough to use the blackbox in other areas of your application. You'd build on top of the trust you lended to the blackbox. And you would continue measuring yourself as you build out, making sure you trust the foundation as you go.
A proper engineering mindset if you ask me, but it's only useful in the physical world when solving physical problems.
The Mathematicians build 'castles in the sky' with vast equations that link up together in shapes that make sense. There's trust being lent to the linking as you go. How do you validate these 'castles in the sky'?
Through understanding. But then, how much understanding is needed? This is where Theory meets Application: and the Article is purely in the Theory territory. Your measure is purely in the Application territory.
The guest posts are from a self selecting group of course, but so far all we have is "inevitability", "adaptation", "exiting times" and, most importantly:
"We want SAIR or the EU shell out $10 billion for a gated AI for privileged academics!"
The last point is particularly troublesome, since the same people were gushing about "democratization by AI" before the N-S proof.
So the subset of mathematicians that is vocal on the internet wants their AI toys, only paid for by the state like in the best academic tradition.
None of these people cares about other professions or wants to slow down the industrialization of academia.
Software is logic applied to intersubjective truth. It's not physical truth which is the subject of the hard scientific fields such as physics and chemistry, as well as biology for the most part.
I'm not sure that this new approach will be AI-resistant. Why would people not use AI to help in creating the "motivated explanations". Maybe they can't be one shotted today, but AI also makes this easier.
Assume in 2 years we have a heap of these motivated explanations, all as high quality as Grant's videos and the best books. But who will read them? There is limited interest in this genre. Grant reaches a large fraction of this audience but most people really don't want to think about math either way, no matter how good the explanation is.
Indeed, there is now "edutainment slop" online and AI can use 3blue1brown's manim library to copy his style and AI can use blender and video generation to mimic 3d animations of other explainer channels. Today it's still slop, but it may not be for too long. And then people will have to reframe their job until it's "doing X while also farting and burping every now and then", and then a machine will be better at that too eventually.
Also, this new style of doing math will appeal to a different set of people. Many mathematicians aren't super social, they just like to explore a problem on their own. Think Grigori Perelman. They will still face the problem and their temperament may not make it easy to switch to being a communicator.
Last summer Grog was still celebrated and admired for bravely piercing animals with a spear and bringing home the meat. But now Goong made this newfangled arrow and bow thing and any cowardly fool can now shoot animals from a distance. Grog devalued. Grog sad.
I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.
Of course they can. My point is that mathematicians may increasingly find themselves in the same position as academics in the humanities. Mathematicians themselves will be able to see the inherent value of the work they're doing (just as experts on 18th century novels can in their own field), but it will be far less obvious to society at large why their work should be funded.
It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.
A majority? I doubt it, simply because the majority doesn't know what math is at all.
At university level introductory calculus, the person teaching class had to reassure students that math wasn't entirely arithmetic or adding up numbers. He did this because it's a common misunderstanding.
Haha, sort of. I guess I disagree with your claim the majority of people think math is "more" than proof. The majority of people think it's less: they think it's doing high school arithmetic. Most people don't know what a proof or a theorem are. They think math is doing calculations with numbers.
Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".
The theorem thing is invented by human to help other people better understand Math structure in a easier way.
There must be like a 1000 examples of c compilers in git hub alone. No idea why building LLM building one is impressive. It's right there in the training data.
No. Starting in a sandbox given only the C specification and assembly code specification build a gcc equivalent (maintainability and performance) from the ground up.
The only concrete step any mathematician on the internet, including on the other AI concern site https://proofsandprompts.com/ , is demanding funding for an academic frontier AI.
Strange that the Poincare conjecture was solved by a hermit without all this AI bullshit. Maybe reject AI, ignore all AI proofs and retreat from the internet.
I fully agree that motivated explanation is more important than proof. This doesn't resolve mathematicians' feelings of existential dread, however. Machines will get better than human mathematicians at motivated explanation in another year.
There will be no more glory in mathematics, but at least the joy of understanding will remain, and it will come without deciphering the tortured proofs that machines output today. Each bit of understanding will come with much less struggle, but this just means we can get more understanding for a given amount of struggle.
The people building AI claim it will surpass human intelligence in all
respects and prerhaps kill
us all. Should we just cease all human activity on the basis of what AI might do in future?
Personally, I doubt AI can surpass a good human explainer because explanation requires empathy, which benefits from being an instance of the kind of entity you are explaining the thing to. That gives you a way of exploring and evaluating the space of possible explanations that isn't available to an LLM.
Thanks for pointing me to this video - it's been interesting to follow the discussion! (I personally don't see that math has lost its purpose at all in the past months. I mean, where would we be, if we were thrown at these AI based mathematical proofs and had no mathematicians and specialists?! Much of this discussion is about a disciplin readjusting its way of work and tasks.)
You get $100 billion for an AI X that generates the proofs, $100 billion for an AI Y that explains the proofs and another $100 billion for an AI Z that reads Y's output and appreciates it.
Humans meanwhile scrub the floor and write blog posts about "what a time to be alive".
That which has received the Mandate of Sapience must be cool (eg any skateboarder skilled enough to impress his mom)
LLMs got It, sometime this year. Terry Tao and Friends have appatently lost It, same time this year. (This is not to claim that the Mandate gets extended to their creators the frontier labs or even Jeff Dean et al. Definitely not their sponsors. Howbout distillers?)
The thing about Mandates--- they are not forever. LLMs can "lose" It. Probably not back to mathematicians -- that'd be atypical (unless they quickly learn to "make their own lightsabers"?) . Likelier: to a scene of humans no-one yet thinks about.
Skateboarders didn't take the Mandate from anyone. So no one takes It from them. There's some karmic law at play
It's too late for this. Much of my work in math has been classified as trivial or best described as not math at all. When I was working on chatbots and described deep learning algorithms to enhance them, it was deemed as a pseudoscience. Mathematicians sound very disingenuous with their backtracking.
I'm afraid that much of mathematicians work is too trivial to be taken seriously and they should just find something completely different to do.
Sadly, your inability to comprehend is noted which leaks your lack of expertise with the subject matter.
For a general overview, assuming good faith and a genuine willingness to learn, refer to https://iep.utm.edu/s-truth/
Its a remarkable intro into propositions, statements and sentences with vivid examples from the works of Tarski, Godel and others as to what constitutes truth.
Pay attention to Tarski’s T-Scheme (sentences and truths)
if language models (current GPT-style or any future model with different architecture) can be viewed as a compression of their output corpus (the corpus it emits by providing random starting native contexts weighted by the model's likelihood of that native context) then one is saying the corpus contains the same information bits as the model, the model just takes less data bits, while the corpus represents inflated data bits.
If learning requires communication of data, one could ponder if it is more effective to transmit learning data as suitable model coefficients instead of corpus monologue or dialogue (say with a teacher).
This may sound absurd, but is it really so far fetched to believe one could formulate "download"-compatible LLM architectures, so that the user or student can play a kind of memory game to train "download"-architected model weights by simple reflex games?
Suppose instead of token vectors we used token matrices or token multivectors as in geometric algebra (not to be confused with algebraic geometry!).
Word2Vec couldn't do general language modelling because it used vectors, but when using token matrices or token multivectors one could postulate the following partition function:
given a dictionary of tokens, and their corresponding (matrix/multivector) M one could define the partition function
exp(-|M1 * M2|^2)
For example it could be alphabet level (to demonstrate the concept):
For every string one can compute the corresponding M in terms of the elementary character matrices:
M_"car" = M_"c" * M_"a" * M_"r"
both matrix products as well as geometric multivector products are associative but not commutative:
But M_"car" != M_"rac" since matrix/geometric products is not commutative.
The result of non-linear but layer-less and order aware architectures could promise the following:
Feasibility of uploading model weights: a random grand tour of 2D projections of the square matrix space, projecting the token positions down to 2 dimensions along an changing axis, a "game" could correctly project 95% of the tokens, requiring the user to identify the incorrectly placed tokens, which jump back to their correct position when clicked.
This means a user can learn (since most of the time tokens are rendered in the correct position of the 2D projected cloud), and as a user learns their performance will go up.
If a user could eventually (after playing for a long time) correct 90% of the token positions on random 2D projected planes, then the user has effectively stored 90% of their coordinates.
Suppose such a user reads text in their mother tongue, then the joint likelihoods in natural text will correspond to matrix products of those weights. In other words the brain will learn that whatever it has learnt playing the token-game aids the brain in predicting a next token.
Once it has learnt to utilize the format (matrix / multivector coordinates), the apparently pointless skill of positioning tokens in the cloud learnt during the game, it would also help predict the next token in languages the user didn't know, say when looking for ingredients on a product package. I believe such a user would swiftly discover they actually know those languages, and I believe transmission of a compressed format would take less time than transmission of the same knowledge in expanded output-corpus form.
I wonder if Terrence Tao has an opinion on such a possibility:
1) does he consider it (im)possible to have token-first formatted LLM's without MLP layers etc reach similar levels of performance as the current crop of LLM's?
2) supposing it were possible, would he predict a user "downloading" weights to his brain by a reflex-game as feasible?
3) does he believe a human brain would be able to make use of the downloaded weights and would the brain notice the utility in predicting tokens?
4) does he believe that once the brain has noticed / learnt how to make use of the model weights, first for predicting the next token in the mother tongue, it would subsequently have learnt how to predict in other tongues?
It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.
Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability. But of course mathematics is all about proof, and for that reason I was wary of it for a very long time.
>Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability
not really. You can consider positive proof as an experiment confirming your theory and the negative proof and counter examples as an experiment falsifying your theory.
Yes, really. "Positive proof" opposes the concept of falsification. You can only have it within a system formal logic, and science can contain those, but isn't one.
>"Positive proof" opposes the concept of falsification.
no. Positive proofs have nothing to do with falsification. They just tell you that there is no point in spending effort on searching for negative proofs and counter examples. They don't prevent nor prohibit you from spending that effort. They just advise you that that effort will be wasted.
It is like nobody prevents from experiments to turn lead into gold. Of from searching for a right angled triangle violating Pythagoras.
Understanding has been the point of mathematics for millenia. The idea that purpose of math is to produce machine-checkable proofs is an entirely modern idea.
> Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.
Ignoring the other ridiculous parts of your comment, isn't this exactly what you're supposed to do? Update your beliefs according to the newest information available?
Yes , you are supposed to do. That doesn’t change the fact that he has no insight into technology, and is a bandwagon person.
If I’m a mathematician, I can write public posts on math. Extensive posts about politics, AI, crypto, … are not useful without expertise in those domains
I'm increasingly seeing people read something in a post that isn't there. GP said "Tao is mobilizing the community".
You can mobilize the community through guest posts that slowly nudge people towards accepting AI, so that they demand funding for AI for academics. Which he himself suggests:
Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.
Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.
Likewise all other technology [ and culture ].
LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.
Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
A moratorium on AI development might be the only way to achieve this preservation of human culture.
> one of the obvious applications of novel mathematical results is in building stronger AI models.
This gets repeated a lot and seems to be one of the primary stated goals of making AI solve math problems, but I still have no idea by what mechanism this is even supposed to happen. I guess they could make some minor improvements to matrix multiplication algorithms or whatever but I don't see what groundbreaking theorem could possibly significantly improve LLMs.
It's the kind of thing where it's sort of expected that you wouldn't know, right?
I think we don't really understand why deep learning works as well as it does, the thinking around that is, as far as I can tell, mostly a collection of empirical observations.
A fundamental theory of learning that can be used to predict optimal network architectures might enable smaller models that consume less energy.
> Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
So many thoughts come to mind at once, they're a jumble in my head rather than a single coherent narrative.
John Henry comes to mind. As does Agent Smith's "I say your civilization because as soon as we started thinking for you, it really became our civilization, which is, of course, what this is all about" monologue in The Matrix. I've not read (or listened to) "With Folded Hands ..." or "The Machine Stops", but I have read the Wikipedia plot summary of both.
Do we want to have comfortable lives, or do we want to serve each other?
"Computer" used to be a profession; I grew up around adults bemoaning that "kids these days can't do mental arithmetic", the Pi Zero I've not switched on for probably a year now could beat all humans simultaneously at that (even if everyone was as good as the current world record holder) and yet we still teach arithmetic in schools.
Nobody needs to knit, and yet we do so for fun. Youtube's "Primitive Technology" channel, which has spent around a decade speechlessly making iron from bacterial slime found in a creek, using only clay and sticks and leaves and vines naturally found next to that creek.
Like I said, no coherent narrative. It's been a while since my stream of consciousness became a river delta; usually at worst it only meanders a bit.
I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.
I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.
Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.
And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.
As someone who mostly only applies math, that strikes me as a peculiarly academic take. Intuition is more important for me because it’s what enables me to know what methods are most applicable to whatever practical problem I’m trying to solve. The proof’s purpose is to verify my intuition. It’s just a means to an end. I only take the time to do my own when I can’t confirm what I need from a textbook or paper.
> As someone who mostly only applies math, that strikes me as a peculiarly academic take.
Yeah I was talking strictly about preparing students to become pure mathematicians. No opinion here on other goals.
If what you teach is proofs, then wheat you will filter for are students who live proofs.
And if your job is to train people to become mathematicians, that is absolutely what you should be doing.
The explanation of proofs I got in school was terrible. I had no idea that you could actually reduce everything to one of the applications of the axioms until I found metamath's proof explorer.
Of course, it's too much work for most normal purposes, and in school they accepted whatever random breakdown people used inconsistently and never explained why.
Actually understanding that it wasn't about convincing anyone so much as having a chain of reasoning going all the way back to the axioms was something of a revelation for me.
The idea the proofs are the heart and soul of mathematics is an unfortunate unforced error, and will lead to the death of the professions now that machines are better at making proofs.
or memorize a few
Love is more important than breathing. It is and it isn't.
What good is an end you can't reach, or worse, you can reach but it's wrong?
>You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself.
This isn't always the case. Our algebra (or analysis) course focused a lot on proofs for the exam. The result was that a lot of people learned the proofs by heart.
I think intuition is hard to test in a way that feels 'fair'.
You can do it - I doubt you could have got a first when I was at Oxford just by learning and understanding the material, but you should probably have been able to get an upper second. The final part of every question virtually always involved insight, but you'd obviously then have to prove what that insight helped you understand.
If you give people questions like those, there is the risk of complaints about the university not having been taught the material for the exams I guess, or you might find that nobody can answer those harder intuition parts. Certainly most students at Oxford couldn't answer that many of them - you needed to answer about three 'final' parts out of about ten questions say in each three hour exam to get a first and perhaps about 20 percent of students got firsts?
An aside, but tests should be that difficult. Otherwise you aren’t getting any signal at the high end. Even in the face of grade inflation that signal can be translated into grad school recommendations (in the uk case, getting all 10 might even garner a “really not bad.”)
Same here, but I didn't memorize the proofs, I tried to internalize their logic, so I could reconstruct them on demand by just thinking systematically. It did work for me pretty well on my real analysis final exam IIRC (27 years later).
Analogous to the Archimedean Property - there is no approach to teaching mathematics so intrinsically good that it cannot be done poorly enough to yield arbitrarily bad results.
Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and rightly so, certainly not in the way Poincaré was on about.
Maybe it's been done, but I'd like to see an LLM recreate Euclid from questioning without having seen it during training.
> Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that
Yes, I believe that, it's part of what I was implying (I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs)
Appreciate the clarification, even if I disagree!
I think we differ on what "mathematical intuition" is then. I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned.
The Euclid example also shows my bias towards spatial intuition of mathematical concepts (which is deeply unfashionable) but also exposes exactly where at least current LLMs break down; they do the symbol based pattern matching version, but they cannot leap outside of that, at least today.
This feels very related to the issues re: the presence or absence of world models in LLMs. Insofar as they have world models (or "intuitions"), these would seem to have to be primarily verbal-linguistic (or symbolic, when using math). LLM world models are not likely (currently) very spatial, in contrast to e.g. V-JEPA-2 models, which likely do have some basic spatial models (and perhaps "intuitions").
Yes, I think the augmentation of LLMs with (hopefully eventually higher dimensional) world models will prove very interesting for all this.
> I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned.
If you want to catch them, surely you can find proofs they aren't able to produce.
That's easy: basically all the spatial ones.
I used to be a game dev, and one of the interview questions someone came up with consisted of working out the surface area of a variant of Menger sponge to some given level of depth. The bifurcation for people that could do this vs those that couldn't was incredible, and did not follow obvious trends for academic achievement. (The same interview also included the gem "How wide is a pointer?" which also catches a frightening number of people).
I think it's actually the opposite. Intuition is the only thing LLMs can do, which is why they are prone to hallucinating when they can't validate their intuition against reality.
There is an idea that human intuition, expertise, and critical thinking are largely pattern recognition. When you encounter a situation, your brain gives you a plausible starting point, based on what it has experienced before. You then continue with explicit reasoning, which is slow and inefficient, and try to validate your ideas. The more relevant the patterns you have learned are to the situation, the more likely you reach a useful conclusion.
LLMs are largely the same, except that they cannot learn from experience in normal usage. And except that they experience the world only through symbolic data, while the human brain has access to plenty of sensory data.
My hunch (or intuition, hah!) is that intuition is an instinctive mental shortcut required to navigate large problem spaces that can’t entirely fit into our heads.
Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces.
> Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces.
My view is that is certainly true of smaller LLMs but becomes less true as they scale up.
To quote the parent bananaflag in a sub-comment:
> I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs
I think as the sort of spare space adjacent to pure language processing in LLMs grows the probability of the sort of reasoning bananaflag is getting at (or spatial reasoning, or anything else) emerging in that space grows enormously.
One of the questions for AI development over the coming months or years is going to be if deliberately cultivating the architecture of those sub models for specific reasoning types beats any emergent reasoning mechanisms or not.
Hmm we may be talking of two different interpretations of intuition here. I agree that LLM weights contain representations of abstract concepts, as a lot of prior research has shown. This surely includes Mathematical concepts.
But to me that is analogous to what human brains do, and a bit different from intuition. I think of intuition as “heuristics”, typically developed through experience, that may link seemingly unrelated concepts via vague, hard-to-define associations, but which let us make mental leaps (or shortcuts) while reasoning. (Maybe analogous to System 1 / 2 thinking.)
On the other hand, LLMs can do both: build “intuition” from patterns in data AND brute force a huge amount of potentially unrelated concepts. This gets fuzzier when we realize that even these “concepts” themselves are gleaned from patterns in data! But my point is we necessarily have to take shortcuts to scale, whereas machines can scale with hardware.
This is of course a layman theory! But it could explain why these models are progressing so fast.
Yes, in my case "intuition" comes a lot from visualizing things spatially, manipulating them, and being able to capture their properties in equations/proofs, and it's that which is (currently) conspicuously missing when dealing with LLMs. (And may yet appear with world models).
With the alternate view of intuition that many of you are describing it is clear LLMs are somewhat either there or heading there now.
This is an intriguing observation! LLMs were famously bad at spatial reasoning, until Astra which apparently has a huge improvement. I wonder if that has any bearing on the recent jump in Mathematical performance?
One thing that struck me from Dario's last podcast with Dwarkesh was that he said training LLMs on a diverse set of tasks does not make them better just at those tasks, but they get better at unrelated and other tasks overall. What you described could be a concrete example of how that dynamic works!
>Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition
Well yeah they do, obviously.
That has been done, like back in the 1960’s.
LLMs have LLM intuition, not human intuition. (See the movie Her.)
LLM cannot reinvent Euclid from scratch, but a larger system including LLM might.
It does not imply that. He is talking about how people do math. Intuition is what you use when deciding what to try and how to think about things.
Proof is the rigorous outcome.
LLM running probabilistic loop is different kind of process.
The parent comment literally said "You cannot do proof without intuition".
Therefore, according to that logic, an entity producing proofs must have intuition.
Edit to add: the parent commenter has now confirmed my interpretation of their statement.
Your unstated major premise here is that their intent was to make a universal statement about how proofs work and not just talking to humans about how they teach humans.
That premise seems unlikely to be correct.
Why? The entire subject of conversation is triggered by things which are not humans producing proofs.
If it's possible for a machine to produce a proof without intuition then clearly a human could also do it too. (And in fact I'd argue I've seen many people like that, simply very good at pattern matching over memorised items).
Because regardless of the point TFA is making, that interpretation makes less sense for the specific comment. It doesn’t fit with the immediate context, which was a response to a thoughtful comment about how humans do math. And it requires assuming a math professor doesn’t understand a very basic and obvious thing about their area of expertise.
That doesn’t really read as good faith engagement in the discussion. At best, it reads as being so AI pilled that you can’t even fathom that others might want to have a little side discussion about something other than AI.
You realize the commenter has now confirmed my interpretation was right?
What is up with this whole sub thread of obvious hole digging?
That math prof was talking about his stance in discussion between mathematicians long before AI.
Plus, I studied math, I am from that environment. His description matches how math is done by people.
People who are good at pattern matching and memorize are, frankly, shit mathematicians. They are find in fun culture around math, but rarely in actual math. They cant really do it as science.
As a math professor, I care much more about the key idea, heuristics, and motivation than the proof. With the others in place the proof is clear, something an AI or a student can do.
Well, it's knowing when to push and when to not. You probably have an intuition for, I don't know, abstract algebra objects (I don't know your field of specialty :P), without needing to symbolically manipulate all of it, but you developed a deep intuition for them through many proofs and attempts at proofs with them.
Im glad you brought up abstract algebra—that was the one class in my math undergrad that I never developed an intuition for. I learned to do the proofs by pushing symbols around and putting bars on top of them but I never felt like I understood what was happening.
That's leaning into engineering, away from math. Heuristics aren't always accurate. Math history before proof is the history of delusion. Idea, heuristics, and motivation aren't nearly enough for correctness outside of a sandbox.
To agree:
In my experience, proof is the gym reps that allows you to harness strong intuition elsewhere.
In practice as an engineer, intuition is far more useful, eg, being able to “feel” when something is off in our reasoning — but proofs are where I train those same sensibilities on “harder” problems, (eg) details about how to model identity, equality, and equivalence in a formal model.
Engineering is a religion based on faith. Math is the god you follow :-)
Not really: engineering is explicitly empirical compared to other fields — and mathematics serves as ontology for that experience.
There’s not faith involved.
There could be meta-faith, if you insert engineering into
Mathematics, physics, chemistry, astronomy, march in one front [lockstep]. Whichever lags behind is drawn after. Whichever hastens ahead helps on the others...
--Karl Schwarzschild
The problem or nonproblem before (elite) software engineers were pointed at rather bespoke conjectures, depending on one's specific denomination, was that the frontier mathematicians got too far ahead of the others to effectively drag them along (hence Tao's recent fundraising attempt using his one-off compressed-sensing work)
There is also the Experience<->Understanding "wave equation" if you will, codified by the popular engineers' joke about how mistakes/bugs mediate the two
Imho what academia+industry really need are GLM-wielding plumbers cheap but capable enough to find these abstraction leaks between silos. One taxes these plumbers so brutally that their clients can get by on basic tokens (morally speaking, so as not to drive demand in the farflung silos of billions bottles and babes)
There were interdisciplinarian buzzwords but these did not live outside the grant proposal, and probably won't survive better under the reign of Pangram
Similarly, programming is also a precise language of communication. Initially, we focused on direct machine behavior but every abstraction above the hardware (including assembly) has been to make that behavior legible to humans.
Developed notations and shared procedural abstractions have made thinking about computation more intentionally human and source control has established a protocol for conversing with other humans in the language of a program and changes to that program.
The moment just now feels like a neglecting of the idea of communication being central. If the program is a compile target but not sufficiently legible or if the conversation moves too quickly for us to keep up then we retain the effects of computation but loose its meaning as communication. We loose the understanding and the ability to develop and evolve further shared abstractions.
Open source programs could be more like motivated explanations of computation. For open source to survive, maybe we should start to make the distinction between free product distribution and programming as communication and community building.
This feels a bit too optimistic regarding programming. Or perhaps betrays an imprecise definition of programming. To me, programming is about expressing algorithms. But algorithms may be totally illegible and still work fine.
Math on the other hand is exclusively about being understood. It is ideas from math that made algorithms legible and thus made the act of programming an act of communication. If by programming you mean using notation and ideas that were borrowed from mathematics to specify algorithms, then for sure it’s communication, but only inasmuch as it was math first. If you mean only specifying algorithms, then no communication need take place; the executor of the algorithm will deterministically execute it irrespective of its ability to communicate.
Programming was never about communication. It was always about making the machine do the thing we want. Back in the day, a good game programmer knew which time intervals had writable video memory and which CPU cycles drew which scanlines, and spent more time rearranging the code to hit these timings than to write the actual algorithm. Later programmers (I hesitate to call them good) learned everything there is to learn about OS internals and wrote theoretically nonsensical and invalid code that still worked thanks to those internals, to save CPU cycles and especially memory use. And the next generation of programmers took the principles of late binding and abstraction to the logical extreme and created architectures that cannot be described in words anymore, only in diagrams - but are crazy good for code reuse, traceability and A/B testing.
> Open source programs could be more like motivated explanations of computation.
It is already that. Every time a method/function is created, a structure is defined, a variable is added, a file is created or renamed,… It’s all for the purpose of human communication. The computer only need binary in a single file.
But people feels like they should be able to jumpninto curl code without any understanding of networking, or linux code with no knowlede of computer architecture. Few code are meant for total beginners.
> There’s a wonderful book, How to Prove It by Daniel Velleman
the name sounds familiar but i don't think i have read that one, i did enjoy "introduction to mathematical reasoning" by eccles.
personally my relationship with mathematical proofs has been complicated. it took some work to understand basic proofs (dedekind cuts, ideas vs. instructions with mathematical notation), but all of the theory of computation proofs, which supposedly are difficult for many, were completely intuitively easy for me.
i think mathematicians are facing a similar confusion as computer programmers. the medium used to require precise thinking and the simple act of reading, writing and composing it was a mechanism for thinking and learning. in the llm era, the question is: should there be a new mechanism and if so, what should it look like?
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
> usually by people who are good mathematicians but know close to nothing about teaching.
I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens.
And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
I think the problem is partly circular. Most people do not like maths. This includes most primary school teachers - in places I know primary school teachers are not subject specialists so just reflect the population of those with the required level of education in terms of their attitude to maths.
If you do not enjoy a subject, any subject, you cannot make it fun for those you teach. In the case of maths specifically its pretty bad: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%...
My daughter hated maths when I took her out of school at the age of nine. A few years later she was very good at it and enjoying maths and STEM subjects. When she went to a sixth form college[1] she liked it well enough to pick it as one of her A levels[2].
[1] https://en.wikipedia.org/wiki/Sixth_form_college
[2] https://en.wikipedia.org/wiki/A-level
Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
This is tricky, because, in fact, hard math having an intelligence floor is one of the nastier realities of the human condition. Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling (and this term is a common one thrown around in people studying mathematics, because intelligence denial is so obviously false when you do hit your abstraction ceiling).
Most people are correct that they lack the intelligence / mind for a lot of hard math (even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are, and these are just basic undergrad calc).
And yeah, sure, people have different kinds of intelligence and such, but there is still a g-factor, and people of low intelligence almost universally can't do hard math, whereas most people who can do e.g. advanced undergrad math can generally do almost all other advanced undergrad fields reasonably well. The world isn't fair here.
> Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling
Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it. I simply ran out of motivation to pursue higher math, especially when following through on learning and research became more and more labor. (It was always labor; but it was a labor I used to love.) I am far from a prodigy.
> even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are
Disagree. It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind. What your average mind (and student) lacks is exposure to formalism, abstraction, and how these things tie in with what they are familiar with, which is symbolic manipulation. With the exception of geometric proofs (another educational bugbear), they have little context for what formalism is or why it matters.
> Eh. I'm a math PhD who fled academia because it was too much for me. But I have never encountered this term "abstraction ceiling" nor did I succumb to it.
This sounds a lot like you may have in fact succumbed to your abstraction ceiling, because in practice, the ceiling manifests as not as it being impossible for you to learn something, but that it would take you years and inordinate effort to master what you notice others mastering easily in just a fraction of the time. You may have not heard the exact term (comes from Douglas Hofstadter), and you may be talking about just the academic busywork, but I find it hard to believe you never encountered discussions about this kind of stuff. I would also politely suggest that unless you are Terry Tao posting under some kind of alt, you most certainly do have an abstraction ceiling (or your own mathematical limits) too.
> It's a notoriously hard subject to teach, and with all the demands placed on e-d in so little time in your average curriculum, it doesn't require appeals to IQ to explain its infamy. With enough motivation and practice, the quantifier alternation is comprehensible to any sound mind
The latter statement is obviously false, but regardless, intelligence explains some of the difficulty, and much other difficulties far more parsimoniously than "everyone could just learn any math if they just tried hard enough and had good enough teachers". E-d is merely an obvious and generally familiar example, and nothing I said really relies on this very specific aspect of maths, obviously. We also shouldn't pretend your (almost certainly false) view of math and intelligence isn't also often harmful to struggling students in its own way.
> Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle - while for myself I felt clumsy. even oafish, wandering painfully up a arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn ( so I was assured), things I felt incapable of understanding the essentials or following through to the end.
(Alexander Grothendieck, Recoltes et Semailles)
Amazing that he managed to keep going after hitting his abstract ceiling in graduate school.
You clearly don't understand the meaning of the term. Grothendieck was almost certainly wrong about his gifts here, and even if not, your mathematical ability and output isn't fully explained by your ability ceiling.
Honestly, the pushback on this post is utterly baffling. Clearly the human mind has limits on what it can comprehend and the rate at which it can learn difficult things. Clearly these limits differ among individuals and are related to intelligence broadly.
Huge proportions of the population struggle to ever even grasp simple fractions, and not for a lack of effort from them or society. Fourth-year undergraduate mathematics is another beast entirely. Pretending the world is otherwise is pure fantasy and also plainly harmful, to the world and people that are unfairly pushed beyond their capabilities.
> Grothendieck was almost certainly wrong about his gifts here, and even if not, your mathematical ability and output isn't fully explained by your ability ceiling
lol, see, it's unfalsifiable. No true abstraction ceiling.
Find me the magical teaching method that can make anyone learn any kind of mathematics at nearly the same rate, and you have falsified the idea that anyone has mathematical limits.
Given we as a society can't even figure out how to do this for educating stuff involving simple fractions, my theory is far superior than whatever exactly it is you think.
That's far different than telling a math PhD whom you've never met that they hit their "abstraction ceiling" based on a 2/3 paragraph comment on hacker news. I'm sure you'd have said a similar thing to a young Grothendieck if he were describing his early struggles in graduate school. You're getting pushback because you were being rude and presumptuous.
Work on your reading comprehension, I made it clear that increased effort is what an abstraction ceiling feels like, but also made it clear that GP could have been talking about the effort of academic busywork.
Let's also not pretend that "you could have learned epsilon delta proofs, you just didn't try hard enough or your teachers weren't competent" or "you just didn't have enough time" and etc. is also not rude and presumptuous. Denying the existence of such limits is equally offensive.
You’re in a kind of compulsive ideology here. This same vein of thought and why it is harmful is described by David Bessis in the book Mathematica: A Secret World of Intuition and Curiosity.
“To stop thinking in terms of “gifts” and “talents,” one has to find an alternate explanation. My way of looking at things, which has served me well throughout my career, was to imagine that creative mathematicians were hackers who had found ways to unlock “hidden modes” of our cognition. Most of the time, they’d done so unwittingly, and were entirely incapable of explaining how.”
It is a phenomenon like child-like mental yoga of attention.
Uh huh. A bunch of quasi-mystical bullshit to justify utterly inept garden variety intelligence denialism (it also just shifts terminology: if we accept your metaphysics, it would still be strange to propose that everyone has exactly equal "hacking" ability).
The hubris and willful ignorance required to imagine that everyone is just equally and infinitely unbounded in their cognitive ability is simply mind-boggling in 2026.
But everyone is as a kid when they learn language and everything else. Some people retain that level of watching, listening, and babbling (hacking) when they don't know what to do. So the task is just to get people back into that mindset. Innate intelligence isn't necessarily only symbolic manipulation (analytical), it also is experiential/creative and practical per Sternberg.
> But everyone is as a kid when they learn language and everything else
Also clearly false by almost all current research.
> So the task is just to get people back into that mindset.
Again, you have no evidence, and this is clearly wrong in cases of mental retardation or brain damage. Modern genetic studies also seem to suggest intelligence is related to a lucky absence of errors / genetic problems that are otherwise inconsequential (or even advantageous) in other domains, so really, your "everyone starts perfectly equal in intellectual ability" is just empirically disconnected and ignorant fantasy.
I haven't talked to a child educational psychologist lately (I imagine that would be quite renewing to spend some time in a kindergarten even if only virtually, don't you?), but I don't think biological claims about elite abstraction is the only way when it comes to explaining more and more with less and less for mathematical education. For example, non-symbolic distinction and indication operations are far more fundamental than symbolic manipulation, and using that is a bottoms-up foundation more in keeping with constructivist understanding exhibited by math exemplars.
Cognitive disparity is because of compounding investment of attention and metacognition in development, preferably in a self-referential non-symbolic universal way because intuition is partly based on sensual metaphors and embodied cognition. Everyone has issues distinguishing ungrounded concepts if they don't have a map of them from their attention previously..
Mental rigidity (aka "fragile perfects") is a fairly common phenomenon for math anxiety, whereas Grothendieck advised uninhibited playfulness to deal with uncertainty. The perceived difficulty of mathematics is a social phenomenon rather than organic comprehension limits on abstraction ceiling.
It is the social aspect of math that is the superintelligent part of it, which transcends the genetic determinist perspective. Civilization advances because education transforms the breakthroughs of genius (which all children have ultimate capability for) into the baseline intuition of the rising children by sharpening their attention. Math is supposed to be a democratization of human understanding, that's why the Greeks were so keen on deduction and why proofs are for systematic communication. If a stupid-ass computer can do math, so can any human being.
> Cognitive disparity is because of compounding investment of attention and metacognition in development [...]
Sure, but exclusively? There are no other factors that don't depend on effort / investment / social context?
I can't take you seriously when you take such an absolutist stance on these things when science has long since accepted nothing complex about humans is 100% nature or 100% nurture (really, shared vs. non-shared environment vs. genetics: but, surely you know this).
There certainly are numerous factors claimed. I believe cognition happens from distinguishing reality and people can always learn better how to do this.
Whereas “abstraction ceiling” is hard science backed by ample literature, not a loose metaphor directly contradicted by several prominent mathematicians who didn’t quit when the going got tough.
I think the fact the Feynman took an IQ test and scored 127 is damning of the entire concept. I think what turns mathematicians off is the thought that psychologists who couldn’t tell you the difference between a scheme and a metric space think they can actually measure who has the capacity to be a mathematician and who doesn’t.
Like, why fucking bother doing anything? Why run the 100 meters at the Olympics, let’s just do some genetic testing and measurements to pick the fastest man in the world. Why teach kids music, let’s just measure hand size and do some sight singing exercises and teach the talented kids piano. This whole nonsense reeks of Gattaca-style quasi-eugenics where people get sorted into profession by people who don’t actually have expertise in any of them. You’re not in the guild, you don’t get to appoint to the guild, and you certainly don’t get to gatekeep who can apply for the guild.
Edit: Dropping slurs when someone compares your views to eugenics is an interesting strategy. Shouldn’t you be at a meetup discussing Curtis Yarvin’s work or something?
"Abstraction ceiling" is just a way to talk about intelligence at at the tails that reveals one of the difficulties / limitations you can encounter when it comes to compressing / abstracting complex mathematical objects. Your objections to the term are facile and clearly stem from an obviously unvocalized intelligence denialism that is simply indefensible today. Also, intelligence != IQ, obviously this is too simplistic.
Everything about your arguments and other posts is similar reductions to retarded extremes (our only options are "eugenics 2.0" or deranged intelligence denialism - there is no room for anything in between, e.g. the idea that base intelligence matters and sets a hard average ceiling on potential, but that effort and other factors might push one slightly above/below this ceiling relative to others with a similar intelligence, and etc). Or alternately you hallucinate things I never said or even remotely implied (e.g. we should gatekeep based on dumb psychology metrics or hand sizes).
Just be honest: you know intelligence is real and matters, but you want to dance around this fact because you find it ideologically inconvenient, or you can't admit you yourself have limits (and lack the courage to realize the obvious social broader consequences of this personal admission).
Dropping slurs after being compared to a eugenicist is an interesting choice. Maybe you would be more comfortable in the comments of Curtis Yarvin’s blog. You’d certainly get less pushback there.
Falling into deranged ideological projection is also an interesting choice - one I chose to mostly ignore. You seem to really obsess a lot about this Yarvin fellow: I tried reading his stuff once and found it intolerable.
I imagine you think calling some of my language choice a "slur" here is some kind of gotcha, when the term I used is specifically one widely disputed as actually being offensive, given it is mostly used now to refer to normal people acting in intellectually deficient ways, and not generally to those with actual learning disabilities that deserve our sympathy. There are studies on this, which you surely are aware of.
If I had referred to your more deranged positions as "smooth-brained halfwit extremes", you likely wouldn't haven't tried to impotently pull this "slur" card, even though the semantics are basically identical. Which basically goes to show that you value irrelevant surfaces over substantial realities, and frankly is perfectly consistent with the midwit intelligence denialism on display in your posts in this exchange.
It’s like someone had Claude binge on Galton, Murray and Yarvin then let it loose on Hacker News.
Well you're just a generally rude person, aren't you.
> This sounds a lot like you may have in fact succumbed to your abstraction ceiling
It sounds more like you're turning a vibes based theory into a tautology.
Hofstadter struggling with math for the first time in graduate school isn't a unique story, nor is his self introspection about this event a good basis for an apparently unfalsifiable theory about human cognition.
We have mountains of evidence that humans differ dramatically in cognitive potential, and more again that often effort / practice can only explain a small amount of the variance in performance in a wide variety of fields. We have basically zero evidence at all that anyone can just learn anything if they try hard enough under the right teacher, and plenty of evidence to the contrary.
Abstraction ceilings are about rates and difficulty of learning, so even if we assumed the (absurd) claim that no one has any fundamental cognitive limits, until we are immortal, being slow enough still creates an effective ceiling.
Intelligence denialism is the incoherent and indefensible position here.
That stuff always gives me such a eugenics 2.0 vibe — no, no, the hierarchy is based on innate cognitive ability now. Gives me the creeps that they're actually in academia pushing that stuff.
I would even go as far as saying that mental conditioning and training is also required, on top of mental capabilities.
> People often hate math because it was not explained to them correctly
Spoiler: this is also why mathematicians hate vibe-math. AIs are outright terrible explainers even when they do have a watertight logical argument—and honestly, this is the load-bearing seam.
It goes beyond "proof vs. exposition": the logical derivations AI comes up with fail to even qualify as human-directed proof because of how terrible they are (far below even the most novice mathematician doing their roughest work) at the exposition part.
> AIs are outright terrible explainers even when they do have a watertight logical argument
I think this only applies to cutting edge mathematics (novel proofs of hard problems). I have seen it reported more than once that such AI proofs are cumbersome to follow.
But in my experience, when it comes to explaining well-established math that is already in the training data, AIs can be very good teachers (at least with recent models). Especially if you use it along with a textbook and ask it about anything that might not be explained well in the textbook.
> AIs are outright terrible explainers
Gemini's explanations are very good.
a good teacher remembers the journey, not just the destination.
socratic method exists. almost none follows it.
>socratic method exists. almost none follows it.
I have a hatred for people who think they can use this method.
If used incorrectly which it is a great percentage of the time it confuses the student. The person employing the socratic method must actually know the answer and where the student is in their mind. Failure on either account makes it pointless.
Ask anyone unfortunate enough to ask for help on IRC
> The person employing the socratic method must actually know the answer and where the student is in their mind.
The socratic method also has a much higher chance of revealing where the student is in their mind.
I sometimes ask more questions than utter new things when trying to explain something.
But that's because I'm trying to focus down and determine exactly where they're at before I just randomly make things worse by accident :-P.
I'm not sure if that's the actual socratic method. But people accuse me of using it. Either way, it does seem to work for me.
That only works if the one you're trying to guide can figure it out mostly on their own and is interested in cooperating. Aka does not work for anything below university level.
"How To Prove It" is used to initiate people. It was required reading for an introductory class on formal mathematics at university.
Intuition happens naturally and people are prone to inducting the wrong conclusions. Proofs provide a framework for rigorously analyzing drawn conclusions such that it can be used to build intuition in others. If math is about sharing the insights gained in a particular class of problems, proofs are the means to getting there.
> a very precise language of communication
I find that difficult to match to my own experience, in that there is seemingly endless domain specific notation that heavily obscures communication
The math field is confronting something that coders have been dealing with for a few years now, only far more violently. Today's moat for software seems to be that AI can automate tasks but not a full job (yet). But for a large proportion of mathematicians, doing these tasks really was _the_ job. It's the bit they wanted to do, and if they completed a sufficiently difficult set of tasks, they got tenure. Now this model is failing, they frantically need to pivot the role of humans to save their profession from funding cuts.
I remember when "writing code was never the point" became a mantra here. There was truth in it, but removing the coding has certainly taken away a lot of the texture of the work and enjoyment of the craft. Many of us feel this loss as we tech-lead teams of agents as our source of income. I am not optimistic the mathematics pivot is going to work, but I'm certain that most will be depressed with the outcome even if they succeed.
We are all staring at the same existential dread, just seeing it unfold slower. We're being told that utopia is to be obsolete, and that is a jarring idea to contend with.
I'm just waiting for "Humanity was never the point, let our AI children transcend and replace us".
Maybe it's time to reject utopia.
I've already seen this sentiment expressed in the wild.
I feel that you can see quite strongly the truth in “writing code was never the point” when you encounter inevitably at every company the guy who has been around forever but doesn’t seem to be working particularly hard. Their value is (was) no longer in writing code at a furious pace all day. It was having a coherent, intelligible and communicable theory of the software system the company is founded on.
I propose this thought experiment: put all living mathematicians in a very long bus. This bus crashes and they all tragically lose their lives. Can we really say mathematics simply marches onwards with AI alone? Let’s say Anthropic needs a new research result to improve Claude. Are we really already at the point where we burn tokens ad infinitum and arrive at the end of scientific progress in some timely fashion?
I actually have a rather dim view of the "writing code was never the point" line. Not because it's objectively wrong, but because I see it as something we're mostly telling ourselves to feel better about the status quo. Ability to write good code has been highly celebrated (and remunerated) for decades. As it is becoming less relevant, we immediately backtrack and start lionizing the parts where we can still be useful instead. Consider the counterfactual - AI continued to be terrible at writing code, but weirdly better at humans at product decisions, architecture etc. In this universe, saying "coding was never the point" would not be popular.
It also find little solace in it aside from 'well this version of GPT isn't taking your job'. AI labs certainly have no intention for the higher level skills to stay in the human-only domain. The veteran developer with the coherent theory of a large stack is immensely valuable today. But they also don't survive if a company can drop a few coders' salaries on rewriting that stack from scratch - faster, fewer bugs, more coherent, able to react to changing business requirements with more agility etc. I am not saying this is where we are, but I think there is a reasonably good chance this is where our road is leading us.
The counterfactual does not sound appealing in the least. That's basically, AI tells human coders what to do, tells us to re-do our architecture when they don't like it, and generally removes all control and initiative from us. But we get to write a for-loop by ourselves? That's our victory?
Maybe it's just me, but imagining that counterfactual reinforces that "writing code was never the point." At least, for me, I enjoyed writing code because it allowed me to design the architecture rather than having some architect lead or engineering manager above me decide.
I posted nine days ago in another thread:
As much as I hate it, I don't think we'll ever get a proof of the four color theorem that isn't enumerating cases.
When you have an integral or the sum of an infinite series that comes out to pi, you know there must be some satisfying explanation involving a circle.
Contrast with "Examples of patterns that eventually fail" on math stackexchange[^1]. When a pattern ends at 906150257, you don't really expect the proof for that to be something beautiful. The reason for the exact value of an upper bound is that it isn't smaller and it isn't bigger.
There's a relationship between e, i, pi, and -1 comes from a deeper relationship between complex numbers and rotation.
The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
[^1]: https://math.stackexchange.com/a/111461
> The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
Well, the current conjecture is that it hasn't to do with coloring at all, but how many, on a map (partitioning of plane into connected regions), regions can pairwise touch: 4.
(That is: If someone proves Hadwiger's conjecture, then the four colour theorem follows.)
This is, as always from Grant Sanderson, thought provoking and opens new insights.
It occurs to me on reading this that there’s a connection to other computerized mathematical activities.
Occasionally some computer lab in the past would announce that they have computed pi to more digits than ever before; or a new Mersenne prime will be found.
These count as ‘math news’ but they’re of little interest to mathematicians. These computational efforts demonstrate the great power of computers but they do nothing to advance mathematical understanding. Finding a larger Mersenne prime is not surprising to anyone; we’re pretty sure there’s an infinite number of them. Finding the largest Mersenne prime would be the surprise.
So it is with proofs. An LLM might prove some conjecture - Riemann, say or P≠NP. But in general we know that things can be proven and we think those things are probably true, so the existence of a proof doesn’t do much more than producing a new Mersenne prime does.
It’s only if in proving the thing we learned something that there’s actual value in the proof.
Part of the controversy here is that now the skill advantage that some Field Medalist had is much narrower. The fact that fields medals have an age limit implies that it favors brain power over understanding. And that was the guiding light award of the community. So i find it "funny" (and natural) when they are offended by AI. That is the main "crisis" of mathematics.
In my opinion there has never been a better time to be a mathematitian, and there has never been a better time to be a software builder.
But there has never been a worst time to have the need to prove your economic value as a mathematitian or software developer alone. Because "understanding" is not something you can prove in one afternoon, its something that you prove with a life.
> In my opinion there has never been a better time to be a mathemetician...
As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible. It's like telling a chair-maker during the industrial revolution that there had never been a better time for them, since now they could operate chair-making machines instead of toiling away at the wood themselves. It assumes that they were purely in it for their passion for mass-producing chairs. The majority of mathematicians get into the field because they love problem solving, and the gauntlet thrown down by challenging math tasks.
Many parts of this will never be useful for society on a grander scale - but this is reflected in the finances - pure math is closer in funding-terms to a humanity than to hard science. Now even this is _massively_ under threat, and Tao and co need to pivot quickly to stop this from becoming a bloodbath.
Im an ex mathematitian too. And if i was in academia I would probably have the same reaction. Thats what i say that its the worst time for proving economic value.
But if you are in for theory building and understanding, then you are not constrained anymore by your motivation to grind through countless hours of formal theorem proving. And you do not need to have superhuman formal manipulation skills and memory.
For me mathematics is not the formal system, so LLMs will never be able to do end to end maths.
> As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible.
To throw a counterpoint to this into the writhing cesspit of HN, I'm active in academic mathematics (postdoc) and every one of my collaborators is deeply in love with the field and their jobs. Perhaps the grass is greener on the applied mathematics side of the fence.
IMO this would track, applied mathematics (even e.g. data science, though perhaps calling that applied math is a bit generous / insulting to more serious applied math) is in some ways more exciting now because it is far easier to surface complex / appropriate methods for the task at hand, and you can more confidently explore these methods because the AI sort of "has your back" in catching some of the more obvious beginner errors you make during these explorations. Plus, applied math feels roughly more results- than process-focused, compared to pure math.
IMO the divide here between pure vs. applied math feels a lot like the divide between those who enjoyed coding for the understanding it led to, i.e. the writing itself was the joy, vs. those that primarily coded for the results. I enjoy the creative part of coding, the thought of software jobs just devolving into writing specifications and doing code review very much kills it for me.
Probably true for the dedicated problem solvers (of which Tao is one IMO). But I doubt there's ever been a better time to be a theory builder (more like Peter Scholze, or Grothendieck).
Some up with an idea and leave the system to check it 15 different ways, and see whether you can simplify an existing body of theory. It'd be like having an army of lightning-fast grad students.
This is coming next. There's nothing particularly special about theory building. Successful theory building is always oriented towards solving a problem, because otherwise even humans can easily spam out a bunch of nonsense. This is a real problem that the math community has experienced on multiple occasions pre-AI. I'd go further and say that theory building is irrelevant if it doesn't help solve problems people care about.
I'd even say Scholze is not a great example here. Most of the work he's known for is progression towards the Langlands program - which is very much a problem to be solved, and one I would imagine he'd not be thrilled for an AI to one-shot. I agree that it is somewhat 'up the chain', in the same way that software engineering has not immediately disappeared now that performing coding tasks is largely automatable.
But I also take issue with 'never been a better time' - e.g. is this really the greatest time to be a software engineer? Everyone has AI psychosis and feels like they're a couple of breakthroughs away from being unemployable. The same is even more true in math - we've gone from failing IMO problem 6 last year, to solving NS. The rate of change is formidable, it feels like there may not be many places to hide in a few years.
But these are not very good arguments, because it makes it about the fall of institutions (the funding) and people being miserable for personal reasons rather than prosocial reasons. Tao here clearly suggests that math is not reducible to "problem solving" or "proofs", the valuable part is much more than that framing.
The concerning argument about the status of math would be an outline that it will get destroyed by a process of societal atrophy and there is no turning back, and the AI powers are not a good substitute or replacement for it. If an entire society becomes reliant on these oracle machines then it would be analogous to children never learning arithmetic because they were handed calculators. How could the human race still flourish? We would anthropologically regress. We'd be little better than animals, like the Borg zombies.
That is a much more profound threat than people worrying about their own careers or faculties disappearing like the humanities. This is a serious anthropological reckoning.
If math experts are that freaked out already then basically all of science is soon to follow, decade by decade. "Singularity" comes to mind.
Broadly I agree with this, but I was refuting the statement "there has never been a better time to be a mathematician". You are changing the question to something different here, and using it to say my argument is bad.
I think math is a microcosm for "thought-work" in general. We have been in a symbiotic relationship with capitalism for decades now, where the hope of a well-paid white collar career encourages people to spend time and money to enrich themselves through education. The proliferation of AI cheating at college already signals that employment is the primary goal over intellectual growth, so one imagines this governs what happens next if labor demand disappears.
It's hard to say where this all leads, but I have very low optimism for higher-level math understanding being something that humans value in the same way in the coming decades.
The post is not by Tao but by Grant Sanderson, maker of the 3blue1brown YouTube channel.
> In my opinion there has never been a better time to be a mathematitian
I think it’s a great time to be a curious mathematician, especially in a niche field where you’re not competing with hundreds of agents of the best unreleased frontier models.
However it’s a very scary time to be a professional mathematician because publish or perish is going to cause a race to the bottom for cranking out results as fast as AI can let you. [0]
[0] https://ev12183725.substack.com/p/a-highly-productive-dark-a...
Spot on. I agree. There has never been a better time to be a software builder or a mathematician.
Seekers whose primary motive is validation instead of understanding are the ones who are getting paranoid.
Thoroughly enjoyed your thoughts. The age limit is a joke if what you care about is true understanding.
I'm a professional mathematician. Today I proved what for me is a very solid theorem. It's something I had thought about for a few years. With a few weeks of serious use of AI I've found a proof that I am currently trying to write up, but which appears correct. The change in the workflow is enormous, but so is what one can do if one has clear what to do and how to do it.
Not every problem will receive $20M in funding to be solved by AI; for the rest, good human guidance will have to suffice. Labs only pulled this stunt because they wanted to show investors how powerful their models are on their own. But look again at the cost of that army of 10,000 SOTA agents. At the very least, I foresee a need for humans to decide when costly AI resources should be committed to a specific search plan. Grant review remains irreducibly human because it involves choosing which directions to fund and weighing opportunity costs: taking one path forecloses others.
Congratulations. You are one of those leading the way, showing how we will adapt and how the world will get better from AI.
That's not the conclusion. I started using AI after the Jacobian conjecture counterexample and have used a particular problem to learn how to use AI and to explore it's capabilities. I'm not a great mathematician but I'm full faculty with 25+ years of research experience and lots of articles and I just proved in a few weeks something that had resisted my efforts for some years.
The exploration process is much easier now. Ideas are quickly testable and multiple tests can help identify a technical obstruction. The tool requires good guidance and input but as it trains on people like me it will need those less.
At the very least our way of doing things must change. More pessimistic views seem to me defensible.
GP never said it was the conclusion. He was praising you, after you on your own volition decided to comment here to let know others about your experience with AI.
It seems to me you're a very privileged individual, I'd suggest you practice gratitude regularly in your life.
You sound a bit troubled. I was responding to the leading the way part. I'm hardly doing that. I just jumped on the bandwagon. What I do see is a tool that forces me to change behaviors learned over decades and potentially renders useless many of them while facilitating others. That's not all a rosy picture.
As for privilege, don't assume everyone lives and works in the US or gets paid lots of money to be a professor.
>full faculty with 25+ years of research experience
That sounds pretty privileged, money aside. It's always this thing with some people ... they make everything about money.
Anyway, good luck with that "wanting more" thing, it's a road that leads nowhere.
Also, https://www.youtube.com/watch?v=b8gmxT8qj8Y, particularly if any of those 25+ years were funded by public resources.
I find myself less worried about it than at first. I think what we'll see are that some things are low-hanging fruit and can be solved just by tireless search. Maybe half the millennium and other such high-visibility problems will fall this way.
Others, I think, will be beyond both human and AI. And so what then? Mathematicians just throw in the towel and say it's not worth trying? Of course not. We will continue that pursuit, and as we do, new ideas will arise and new problems will need to be solved. It's math. There is no end.
It's easy to look at the current landscape and see AI ticking off solutions to problems and imagine that soon there will be nothing left. Machines replaced the need for much manual labor, but they also established a basis for an economy that provides the opportunity for more labor. This is the situation with math now. It will take some getting used to. There will be little-to-none pencil-to-paper working out of problems anymore, but there will always be work to do, things to solve, curiosities to unravel. And it will still be professional mathematicians who are the ones most capable of directing that effort. Because, if nothing else, they're the ones whose curiosity is piqued by the problems. Which, let's face it, has been 99% of the motivation for graduate-level math in the first place.
There's the the old question: is math invented or discovered? I think it's both: the problems are invented, and the solutions are discovered. In the age of AI, the discovery part will be greatly affected, but the invention part will remain firmly in the human domain.
One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.
True.
Having to ask the village elder about how to do things meant learning various other incidental lessons. It was reduced when books started to become available.
Having to search in the library also often lead to serendipitously seeing a book on the next shelf and falling in love with a topic you didn't even know about otherwise. Or you had a chat with a librarian asking for advice which books to look in for a topic. Google search eliminated that. With Google search and reading websites, you still had to read or skim the page and may see some other info or click to read the author's About page. Now with AI we get straight to the answer.
Don't worry, in its final reply the agent will always present one or two things "worth flagging" to keep you engaged
I do neither maths nor science with AI but in my experience most models are perfectly willing to burn tokens on a ton of sidequests at the earliest opportunity.
Yeah but do you read through and find if one of those side quests is useful?
They're often directly applicable: minor bugs, inconsistencies, missing tests, commonly also stuff I already now (like some infra config details).
Minor bug and develop whole new field of cryptography are very different scales. I do agree alms are really good with a lot of common bug fix related tasks to the point you have to split commits out. The fact is you recognise it's a minor bug. Who is sitting through the proof of Navier Stokes and going through it and finding connections between itself and other fields? Im not saying LLMs are bad, but I do think understanding is important. Heck, the fact you identified the minor bug suggests you understand the output. Im not so sure the same can be said of a gajillion line lean dump.
Yes, why not?
We'll get innovations in AI as a side effect.
I was a PhD student for a while, but I always enjoyed "refactoring" proofs, more than coming up with new ones. Making them simpler, shorter, clearer. Unfortunately, it's not much rewarded.
While I understand and emphatically with Tao's concern I'm afraid it's missing the forest from the trees. Unless you can make a claim that AI will never be able to perform intellectually at the same level as any human at a much lower cost, there's an outstanding utility problem that remains unaddressed. Sure enough, the AI may not have taste or goals, or many human traits, but that's irrelevant to the much thornier (and much broader than mathematics or even academia) question related to who's getting paid how much and for what.
The funding on mathematics is already one of the lowest accross science [0, 1], and theoretical math funding is probably much smaller than the applied math one already, so that's not even close to how much funding theoretical math gets.
So, we are talking about a field that already does not use that much funding anyway, and most high end theoretical mathematicians probably would make much more money in the industry anyway, so this seems like missing the forest for the tree imo.
[0] Table in page 1 in https://nsf-gov-resources.nsf.gov/files/71_fy2025.pdf?Versio...
[1] Figure DISC-13 in https://ncses.nsf.gov/pubs/nsb20257/academic-r-d
There's an old joke about funding, goes something like:
"Why you are always demanding more funding? Why can't you be more like the mathematicians, all they need is a desk, some paper, and a pencil, and a garbage can, and they just do fine. Or how about philosophy, for that matter? They don't even need the garbage can"
I mean, obviously with modern computational mathematics, this doesn't hold so simply, but there is this confound about math research also not getting much funding also because much of it isn't that expensive, relatively speaking.
Last I knew, at least in America, universities that want their math prof's to do research also expect those prof's to bring in plenty of outside funding. You could argue about the costs of that desk, paper, pencil, and such - but modern "research" universities have evolved into extremely high-overhead operations, and The Beast Must Be Fed.
> so this seems like missing the forest for the tree imo.
But is it? Again, regardless of the amount, what is the utility?
What's the end goal of academia (assuming broadly as research with humans) when the answers to the deepest questions become commodities at orders of magnitude higher speed and lower cost?
How are federal grants justified and, forgetting the current hierarchy, how is differentiation made? It's currently based on research output, once that becomes irrelevant what is it? We already have IMO, IOI as competitions and I assume just like Olympics this can be a thing, but it's very remote from research.
Take for example Rene Thom after Alexander Grothendieck overshadowed an entire field
> His technical superiority was crushing. His seminar attracted the whole of Parisian mathematics, whereas I had nothing new to offer. That made me leave the strictly mathematical world and tackle more general notions, like the theory of morphogenesis, a subject which interested me more and led me towards a very general form of 'philosophical' biology.
https://mathshistory.st-andrews.ac.uk/Biographies/Thom/
Now amplify this a few orders of magnitude.
> Tao's concern
The article isn’t by Tao, it’s a guest post by Grant Sanderson (aka 3Blue1Brown).
Indeed. I missed the top row. Thanks for pointing that.
This goes in a necessary direction, from my personal take away of Gower's recent post on the subject.
Mathematics is suffering from Goodhart's Law:
"When a measure becomes a target, it ceases to be a good measure."
Doing something difficult was a signal that you:
a- understood it and all the background information it requires
b- internalized techniques and methods that are helpful in problem solving in general
Now it just means nothing
> Now it just means nothing
Now it means we can move on to other difficult shit.
Today, the difficult shit is how to get the resources from the people who use them to turn difficult shit into an easy shit. Basically, a war. I am not sure I want people to move on to that.
And what would that be?
Have you seen the night sky, outside a city?
Space colonization will never happen.
People like you will always happen.
People who understand the vast gulf of technological advancements (some bordering on fantasy) between what we currently have, and what needs to exist for it to happen?
There is a finite capacity/time for a human mind to do difficult shit, if it can be slop forked in a microsecond before you even get to flesh it out there is no point
Well then you have time to invent some new difficult shit.
3Blue1Brown: The last IMO problem AI could not solve:
https://www.youtube.com/watch?v=Nbwv5wHQoj0
Some were like this, some weren't. Many people were in it because it's objective and factual and not up for the whims of taste of some established gatekeeper. They weren't in it for art performance reasons or to please the aesthetic judgment of some entrenched mathematician-baron.
Timothy Gower
No. It should have been Gowers' rather than Gower's.
The link is:
https://gowers.wordpress.com/2026/09/17/why-i-didnt-sign-the...
I was about to link the "discovery fiction" article [0] by Michael Nielsen, but halfway through saw Grant mention it as an inspiration. It's one of my favorite styles of writing that lets you settle into a cozy read of a narrative around a problem while slowly exposing you to the rigor and nuances of tackling it. It's definitely deserving of a public index of some sort, like one of those "Awesome X" list repositories on GitHub, because I think it's a valuable style that deserves to be curated.
[0]: https://michaelnotebook.com/df/index.html
> It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.
This is really interesting and available here: https://www.youtube.com/watch?v=IbGNZQvobkc
> we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems
The core idea seems to me that we should shift the standards for professional evaluation from generating proofs to generating explanations. Makes sense that such a proposal would come from the 3B1B guy, and I actually agree with it, irrespective of AI. But what eludes me is how that could be a defensive mechanism against AI automating humans out of mathematics. AI is likely no less good at producing natural language explanations as it is at generating rigorous proofs. It's telling that even Terrence Tao turned to AI to understand AI-generated results [0]. It seems that the essay doesn't address that issue at all.
[0] https://news.ycombinator.com/item?id=49010345
It’s a task much harder to RL and much more subjective. I don’t want to say we won’t get there, but let’s just say that LLMs could “write” well enough since gpt3.5 era and I don’t think the pleasantness of the prose improved dramatically since then.
And subjectively the explanation LLMs currently provide are usually horrible, horrible enough that I usually just instruct them to provide me human written literature I can read.
I mean, there's centuries' worth of mathematical prose to train on. But that's presumably already in the training data, so if it isn't good enough today, it might not get better fast enough to keep track with how fast they'll get better by training on formally verified math. But then again, the prose in Terry's conversation I linked above seemed pretty useful. But it's also a problem requiring famously little advanced mathematics.
I have been losing interest in this proof-oriented approach into extremely abstract concepts (what seems to be the core of academic mathematics today). Obviously, proofs are very attractive because they are the closest thing we have to a universal truth (at least under the assumed axioms). Having mechanisms to reliably show a proof, and computational methods to handle complicted proofs is great.
But.. the navier stokes proof was the last straw for me. People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society. And certainly not moving us towards "human understanding". The biologists are the ones working on that, the math folks should try working with them on neuro stuff to understand how human brains can do math at all, given their architecture.
> Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society.
Is it, or is it only the parts you hear about/pay attention to?
> People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Yes, maybe famous solving famous conjectures is just trivia and trophy collecting and the real value is the intuition and techniques you develop along the way to solving them which you can then bring to bear on things like CFD.
Mostly the parts I hear about. But I also communicate with mathematicians on a regular basis and I can see what they are working on, which is representative of the field. It's about 90% "cohomology of abstract Lie groups that morphize into string theory" and about 10% "practical thing that engineers can use to make my cell phone work better". (amusingly, although I literally made up that sentence, it turns out it's not far from something somebody worked on: "The cohomology of compact simple Lie groups and spin groups connects to string theory by providing the topological obstruction classes—specifically the first fractional Pontryagin class ...." OK, kind of emphasizing my point there.
No, the intuition and techniques that get developed to solve the NS conjecture have little or no bearing on the practical details of doing CFD. That's a common story/thread (and I hear the same story in quantitiative biology, my area of expertise), but often times, it's just a loose justification given to justify funding.
Dumb question— why doesn’t proof proposal construction (not the solution) in lean get celebrated more ? That seems central to understanding.
If these proofs are so important why is there not a central repositories of the proposal in a formalized language ?
How about we stop moralizing technology so much and start focusing on how we want to spend our time in the real world which now contains it
I, for one, wish to dedicate my life to improving the wealth and power of the already existing billionaire class.
Current and future mathematicians can now spend most of their time coming up with problems/conjectures and theories that are hard for a future model versions (6 to 1 year out) to solve them and by itself it can be a new major sub branch of mathematics - “Theory of perplexing frontier models” and who knows it can even open up new dimension of mathematics for mathematicians to explore by themselves (because AI by definition cannot help them here). Now is the time to be excited for mathematics!
Hmmh. I like motivated explanations, but, as acknowledged in the text, this is a subjective thing to measure. What is a great motivated explanation for Tao, might be hard to grasp for me. So I guess judging how well an explanation motivates something depends on two things: 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
There is a third thing: how well does the motivation chime with or go against my current belief system? You would think this is not much of an issue in mathematics, but it can be, and I had my fair share of frustrations because of it.
Anyway, all of the above points to one thing: the best motivated explanation will be generated by an AI, knowing the subject and you in a deep way that no other human will, and being able to interact with you during the explanation.
> 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
I don't think we're discussing pedagogy. Good _research_ exposition is instead related to communicate your intuition and way of seeing things. The conceptualization of a given situation or problem is what is valuable, how you connect it with other stuff, etc. It is then up to you to memorize and interiorize it.
Good research exposition is the same as good "pedagogy" just for a different audience. Both have to consider didactics. When writing a paper, you teach something to the fellow researchers who know less about a thing than you.
Seems to me to be two different sides of the same coin. Pedagogy is about finding a way to communicate to me an idea based on my intuition and seeing things. A research exposition is about presenting the idea in terms of your intuition and seeing things.
What's an example of your belief system conflicting with a motivated example? I'd love to understand that a bit more.
One example is what currently plays out, see the previous guest post on Tao's page: https://terrytao.wordpress.com/2026/09/12/after-math/
The blog post says that the statement "AI really did solve a problem in mathematics." is wrong. But a formal proof showing that Navier-Stokes equations can blow up is certainly such a solution, by AI. There is not much in this world that is more objective than a formal proof, so any disagreement on this is based on how we see the world. Michael Harris will agree with the statement being wrong, Jacob Tsimerman will not.
Another example, Hilbert famously battled Brouwer's view of mathematics. From my point of view, Hilbert was right: intuitionistic logic is certainly interesting; but I like to study it using "normal" (= classical) mathematics.
Finally, my personal frustrations are about how hard it is to publish my work on abstraction logic. I would never have thought it is that difficult, mathematics being objective and all. It seems essential to take out as much motivation out of your paper as possible, because it might offend your reviewers and their belief system. By now my papers come with full Isabelle/HOL formalisations, let's see if that helps.
The value in academics is teaching and research. The value in research is discovery. Proof was a useful function for humans to do towards discovery until recently. Understanding is a useful property insofar as it helps you teach and it is a prerequisite for generating hypothesis. Humans will always be driving discovery, the tooling and focus of work may just be a little different. Attachment to one particular modality of discovery is an aesthetic choice, not a moral one.
If I have to take the risk of simplifying,
1. We humans have managed to take huge amount of information and compress it using a loss function containing some bias we have about the information.
2. We now ask ourselves to decompress the same information with some additional cross-entropy. As a side effect of this process we sometimes spurt out information that may or may not have any meaning since the compression was lossy.
3. Now, we ask ourselves to present this some-what newly decompressed information with brevity in order to understand what we've learned from it.
Knowing that this process is happening on a larger scale, this resurfaces the argument if meaning can be reduced to computation only.
Although some might favor this argument but we are at the risk of anthropomorphizing this process.
The idea presented in the post itself is perspicuous (in Grant Sanderson own words) as he always does.
> decompress the same information with some additional cross-entropy
What do you mean by this phrase? I know what cross entropy and data compression are.
The last days we are served these high goals about understanding, "digestion" and so on.
But if you look at the practice of present mathematics, in the last 20 years it is all about publishing solutions to problems.
There are famous problems to be solved, there is a hierachy of conjectures to be solved. A quick search here on HN gives pearls like "Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem".
And all of a sudden it turns out that problem solving can be automatized.
So then what will problem solvers do? Well, from now on they will "digest" problems solved by AI.
In a way or another they will find a way to stay on top.
That's the goal, at least, but mathematics as a living practice does not have much to do with these games of power.
With hammers do we build our mud huts more easily and sit back to rot? Or do we build more complicated structures and do it more quickly?
“You are right. There is nothing in yesterday’s mathematics that you can prove with exterior algebra that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwile a mathematical enterprise as proving old conjectures.” Gian-Carlo Rota in “Indiscrete Thoughts”
The issue here is not AI--it's academic papermill culture and paywalled journals.
AI gives us greater freedom to "stop and smell the roses", explore hidden structures, etc in mathematics. It is a dream come true for curious minds.
Yes. AI is a useful tool and we are going to adapt and use it.
The phd student will be forced to publish 10 breaktrough articles, the university department which does not offer "free" access to AI (for its members) will see the its ratings going down, when compared with the other universities.
It will be "use AI or perish" for academic management so on the side of academic management the ones with vision will thrive and the ones without will perish.
But what about the publishers? In the last decades the academic research was made into a feeder for publishers. The main goal of a researcher is to write articles, which are later sold back to other researchers.
This economic system is under big stres now, because for a while at least the academic management and publishers will have contradictory goals.
And that is why this scare which is induced by those who profit the most from the present system.
Academia can work fine with open access conferences and Arxiv preprint as machine learning and computer vision and other computer science fields show.
Arxiv is a good blueprint for what we should build on. Terry Tao gave a recent talk about how "canonicalization" is the key bottleneck. The limitation of Arxiv is that is non-canonical, non-peer reviewed. However, there is no reason that journals should have a monopoly on peer review (it should be noted that peer reviewers aren't even directly compensated for it, granted it is supposedly "priced in" to academic salaries.)
Yes. See ICLR for example, which is a peer-reviewed conference, on the OpenReview platform, with no publisher involved, the papers are just hosted on the OpenReview website. It's a similar situation with ICML. NeurIPS also doesn't involve any traditional publisher. Because why would they? There is an online platform to do the reviews, discussions, uploading the camera-ready etc. There is no real role for IEEE or ACM or Elsevier or Springer in this any more. Machine learning as a field has shown that it's perfectly doable, over many years now.
If this suggestion were to come to pass, I wonder how new math PhDs would think about choosing between a 'normal' R1 faculty job vs. the "teaching route" (teaching professorships, lectureships, community college professorships, or SLAC professorships).
It's been my understanding that traditionally the ones who care about "motivated explanations" in this sense go for the latter, but if the research community has now decided they care about teaching and understanding, it might "even the playing field" and make the jobs more similar.
Yeah, the early insight is important to preserve in students.
Evening the playing field would mean those R1 professors now teach five classes a semester for 50k a year instead of doing research.
Taught proofs too, and plenty of students fake intuition with pattern matching.
Let's see how long (if it ever happens) it takes for models to generate motivated explanations (possibly done via the Manim library or something like it) along with their Lean proofs. Grant Sanderson is right that this is kind of subjective but so is Art and I'm very enthusiastic about AI generated Art.
> possibly done via the Manim library or something like it
It can be done already: the point is that the motivation and explanation parts are terrible, especially for novel topics where the AI can't just rip off existing content. A Lean proof is at least a verifiable task; you end up with an actual proof that you can work through. A Manim slop video doesn't have that.
I think the thing is.. sorry Grant, but "motivated understanding" won't come from videos, but writing. Solving things too, but a video can only go so far.
I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).
I'd be interested in hearing a field report on this! For example, I can easily imagine that they're great at walking through the proof step by step, explaining background as necessary; but as TFA notes, one of the most important questions is "why is this definition the way it is?", and my bet would be that the Lean is not enough to help the LLMs meaningfully in answering that.
LLMs like even the sota flash models have great range of background math knowledge and have no problem reading and understanding flt level of math. On the other hand you don't want to go through 13 million lines of often repetitive code line by line. Models are great at synthesizing math content out of code. My contribution is to steer it through subjects of most interests to me, drill down into jargons that can be confusing, be creative in using computation for illustration (which coding agents can execute very proficiently) etc.
Fortunately LLMs are smart enough to handle that already.
It would be interesting to see what would happen if we had two competing mathematical institutes, a sort of First/Second Foundations:
1) Rejection of AI for anything but trivial applications while still using computers at their full capacity. Researchers would ensure full human understanding of proofs and methods. This Institute believes on Math as a process of discovery, Mathematicians as explorers/poets/storytellers and not proof machines.
2) Unrestricted, all-embracing use of the latest AI, including potentially research in creating even better AIs as part of the program. These researchers would be okay with not understanding proofs if verified to be correct. This group is focused on rapid problem resolution and believes Mathematicians are theorem creators and provers.
After X years (100?), which one would advance Mathematics and humanity the most (we'd need to define "advance")?
Mathematicians worry about proofs and the intrinsic value of something as elusive as 'understanding'. They are deeply ingrained in the study, deeply concerned with anything effecting the field. Yet they're still emotional beings looking for beauty and meaning in life that might come from an understanding how the universe works purely from a math perspective. I'm glad Mathematicians exist, I certainly can't do that type of work.
And I trust their results: technology wouldn't be possible without advancing our understanding of the world in various fields, including math.
Your idea sounds great for the Mathematicians.
There's a more pragmatic view though, and unrelated to proofs themselves: does understanding a proof help us to advance Humanity in some way?
Do we have better lives afterwards? What if we give up understanding proofs and focus only on results.
In other words, if an AI solves a problem for you, but you don't understand how it works, should you continue building anything on top?
I suppose the results are truly what matter. If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
Your second approach seems good to help figuring out results from both theory and application of math to solve problems.
But also, what if there is no true beauty in Math, the way Dirac and Einstein wanted?
What if these AI brute force proofs are all that's left?
> If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
A lot of medicine is already like this. Shown to work in clinical trials, no complete end to end mechanism understood. They still get approved if the empirical results are strong.
I'll answer one of your points partially: if AI builds a better sorting algorithm and proves its performance characteristics, it's useful. I'd be able to use it to make my programs faster even if I don't/couldn't understand it.
It would be a bit disappointing but still useful and make humanity slightly better.
It's interesting to measure how much we believe in something, and how much trust we've lended in order to have a working model of our reality: enough understanding for us to get around, move about, and be content.
I'm sure you would only trust the improved 'blackbox' AI sorting algorithm after it has been proved out through benchmarks. Once you've seen better, repeatable numbers: your trust would rise and eventually you'd feel confident enough to use the blackbox in other areas of your application. You'd build on top of the trust you lended to the blackbox. And you would continue measuring yourself as you build out, making sure you trust the foundation as you go.
A proper engineering mindset if you ask me, but it's only useful in the physical world when solving physical problems.
The Mathematicians build 'castles in the sky' with vast equations that link up together in shapes that make sense. There's trust being lent to the linking as you go. How do you validate these 'castles in the sky'?
Through understanding. But then, how much understanding is needed? This is where Theory meets Application: and the Article is purely in the Theory territory. Your measure is purely in the Application territory.
I don't see the point. No one is proposing a complete rejection of AI tools.
Many people here most certainly are (not the majority though).
The guest posts are from a self selecting group of course, but so far all we have is "inevitability", "adaptation", "exiting times" and, most importantly:
"We want SAIR or the EU shell out $10 billion for a gated AI for privileged academics!"
The last point is particularly troublesome, since the same people were gushing about "democratization by AI" before the N-S proof.
So the subset of mathematicians that is vocal on the internet wants their AI toys, only paid for by the state like in the best academic tradition.
None of these people cares about other professions or wants to slow down the industrialization of academia.
Software is logic applied to intersubjective truth. It's not physical truth which is the subject of the hard scientific fields such as physics and chemistry, as well as biology for the most part.
So no, software is much less than science.
It's time to stop solving logic problems and start solving the more difficult philosophical problems, like the hard problem of consciousness.
I'm not sure that this new approach will be AI-resistant. Why would people not use AI to help in creating the "motivated explanations". Maybe they can't be one shotted today, but AI also makes this easier.
Assume in 2 years we have a heap of these motivated explanations, all as high quality as Grant's videos and the best books. But who will read them? There is limited interest in this genre. Grant reaches a large fraction of this audience but most people really don't want to think about math either way, no matter how good the explanation is.
Indeed, there is now "edutainment slop" online and AI can use 3blue1brown's manim library to copy his style and AI can use blender and video generation to mimic 3d animations of other explainer channels. Today it's still slop, but it may not be for too long. And then people will have to reframe their job until it's "doing X while also farting and burping every now and then", and then a machine will be better at that too eventually.
Also, this new style of doing math will appeal to a different set of people. Many mathematicians aren't super social, they just like to explore a problem on their own. Think Grigori Perelman. They will still face the problem and their temperament may not make it easy to switch to being a communicator.
Last summer Grog was still celebrated and admired for bravely piercing animals with a spear and bringing home the meat. But now Goong made this newfangled arrow and bow thing and any cowardly fool can now shoot animals from a distance. Grog devalued. Grog sad.
I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.
You don't think LLMs can translate Ancient Greek poetry or analyze 18th century novels?
Of course they can. My point is that mathematicians may increasingly find themselves in the same position as academics in the humanities. Mathematicians themselves will be able to see the inherent value of the work they're doing (just as experts on 18th century novels can in their own field), but it will be far less obvious to society at large why their work should be funded.
Interesting headline.
It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.
> don't even include proofs in their mental concept of what math involves
Technically, the largest majority are the people who go: "What are proofs?" :P
A majority? I doubt it, simply because the majority doesn't know what math is at all.
At university level introductory calculus, the person teaching class had to reassure students that math wasn't entirely arithmetic or adding up numbers. He did this because it's a common misunderstanding.
So, you disagree with my comment because you think I'm right?
Those students he was reassuring, did they think math was nothing but proofs?
Haha, sort of. I guess I disagree with your claim the majority of people think math is "more" than proof. The majority of people think it's less: they think it's doing high school arithmetic. Most people don't know what a proof or a theorem are. They think math is doing calculations with numbers.
Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".
The theorem thing is invented by human to help other people better understand Math structure in a easier way.
Give teachers and professors 1% of all future earnings of every student
Math academia 2025
> Sorry, only epic problem solvers allowed here
Math academia 2026
> We were more than just problem solvers
I think people are overblowing this though. Wake me up when GPT-whatever writes gcc from scratch, then by the Curry-Howard I'd be impressed
You missed this from last year
https://github.com/anthropics/claudes-c-compiler
Also I dunno why you should be impressed by this - gcc isn't anything near eg navier stokes
There must be like a 1000 examples of c compilers in git hub alone. No idea why building LLM building one is impressive. It's right there in the training data.
I said gcc, not a C compiler
True, it does more than just c. Does it do anything the usual LLMs don't have in their training data tho? I doubt it.
No, not that is does anything more...
It's that gcc is an old and reliable piece of software built on abstractions that have stood the test of time.
by that measure an LLM can "build" you gcc by doing cp -R lol
No. Starting in a sandbox given only the C specification and assembly code specification build a gcc equivalent (maintainability and performance) from the ground up.
Brother I have no idea what your goalposts are supposed to measure in your mind but I do know they're meaningless for everyone else.
Okay tell me why that's not even close to gcc. You can use an LLM
Well, so Tao now invites literal industry boosters to lure mathematicians into a pro-AI stance. This is the guest poster:
https://www.3blue1brown.com/talent
The only concrete step any mathematician on the internet, including on the other AI concern site https://proofsandprompts.com/ , is demanding funding for an academic frontier AI.
Strange that the Poincare conjecture was solved by a hermit without all this AI bullshit. Maybe reject AI, ignore all AI proofs and retreat from the internet.
I fully agree that motivated explanation is more important than proof. This doesn't resolve mathematicians' feelings of existential dread, however. Machines will get better than human mathematicians at motivated explanation in another year.
There will be no more glory in mathematics, but at least the joy of understanding will remain, and it will come without deciphering the tortured proofs that machines output today. Each bit of understanding will come with much less struggle, but this just means we can get more understanding for a given amount of struggle.
Jacob Tsimerman claims [1] we might have superhuman expositors by April, so then what?
[1] https://youtu.be/H7_d_sgui6o?t=4436 (timestamped url)
The people building AI claim it will surpass human intelligence in all respects and prerhaps kill us all. Should we just cease all human activity on the basis of what AI might do in future?
Personally, I doubt AI can surpass a good human explainer because explanation requires empathy, which benefits from being an instance of the kind of entity you are explaining the thing to. That gives you a way of exploring and evaluating the space of possible explanations that isn't available to an LLM.
Thanks for pointing me to this video - it's been interesting to follow the discussion! (I personally don't see that math has lost its purpose at all in the past months. I mean, where would we be, if we were thrown at these AI based mathematical proofs and had no mathematicians and specialists?! Much of this discussion is about a disciplin readjusting its way of work and tasks.)
You get $100 billion for an AI X that generates the proofs, $100 billion for an AI Y that explains the proofs and another $100 billion for an AI Z that reads Y's output and appreciates it.
Humans meanwhile scrub the floor and write blog posts about "what a time to be alive".
> Humans meanwhile scrub the floor and write blog posts about "what a time to be alive".
I'm pretty sure we already have machines to do both of these.
It's all good until we have superhuman appreciators :)
That which has received the Mandate of Sapience must be cool (eg any skateboarder skilled enough to impress his mom)
LLMs got It, sometime this year. Terry Tao and Friends have appatently lost It, same time this year. (This is not to claim that the Mandate gets extended to their creators the frontier labs or even Jeff Dean et al. Definitely not their sponsors. Howbout distillers?)
The thing about Mandates--- they are not forever. LLMs can "lose" It. Probably not back to mathematicians -- that'd be atypical (unless they quickly learn to "make their own lightsabers"?) . Likelier: to a scene of humans no-one yet thinks about.
Skateboarders didn't take the Mandate from anyone. So no one takes It from them. There's some karmic law at play
It's too late for this. Much of my work in math has been classified as trivial or best described as not math at all. When I was working on chatbots and described deep learning algorithms to enhance them, it was deemed as a pseudoscience. Mathematicians sound very disingenuous with their backtracking.
I'm afraid that much of mathematicians work is too trivial to be taken seriously and they should just find something completely different to do.
Logic is the foundational weapon operating on sentences.
The act of stitching together, a series of sentences as true is what logic is.
If you make the stitching as airtight as possible, congratulations, you are in the realm of math.
If you are stitching together reasonably similiar to how the masses do, congratulations you have common sense.
If you stitch together completely random sentences, you are in the realm of nonsense and you may be classified as a retard.
The weapon is the same. The discipline differs and hence the effort to produce the chain.
So I am not at all worried about LLMs producing math proofs.
Godel with his incompleteness theorem helps one sleep easy. Rest assured no LLM can fly above Godel Incompleteness theorem.
There will always be statements that are true. So yes, it is time to celebrate.
Hate to tell you mate, but this is rather close to stitching together random sentences..
Sadly, your inability to comprehend is noted which leaks your lack of expertise with the subject matter.
For a general overview, assuming good faith and a genuine willingness to learn, refer to https://iep.utm.edu/s-truth/
Its a remarkable intro into propositions, statements and sentences with vivid examples from the works of Tarski, Godel and others as to what constitutes truth.
Pay attention to Tarski’s T-Scheme (sentences and truths)
Dude, you're just too smart for me.
Buddy, your lack of good faith is now exposed, which demonstrates zero substance. I doubt you would know that it is very cheap.
Ironically, the "stitching random sentences together" has now evidently applied to you.
Symbolic manipulation relationship with narrative selection for transformational bridges of understanding.
if language models (current GPT-style or any future model with different architecture) can be viewed as a compression of their output corpus (the corpus it emits by providing random starting native contexts weighted by the model's likelihood of that native context) then one is saying the corpus contains the same information bits as the model, the model just takes less data bits, while the corpus represents inflated data bits.
If learning requires communication of data, one could ponder if it is more effective to transmit learning data as suitable model coefficients instead of corpus monologue or dialogue (say with a teacher).
This may sound absurd, but is it really so far fetched to believe one could formulate "download"-compatible LLM architectures, so that the user or student can play a kind of memory game to train "download"-architected model weights by simple reflex games?
Suppose instead of token vectors we used token matrices or token multivectors as in geometric algebra (not to be confused with algebraic geometry!).
Word2Vec couldn't do general language modelling because it used vectors, but when using token matrices or token multivectors one could postulate the following partition function:
given a dictionary of tokens, and their corresponding (matrix/multivector) M one could define the partition function
exp(-|M1 * M2|^2)
For example it could be alphabet level (to demonstrate the concept):
For every string one can compute the corresponding M in terms of the elementary character matrices:
M_"car" = M_"c" * M_"a" * M_"r"
both matrix products as well as geometric multivector products are associative but not commutative:
(M_"c" * M_"a") * M_"r" = M_"c" * M_"a" * M_"r" = M_"c" * (M_"a" * M_"r")
But M_"car" != M_"rac" since matrix/geometric products is not commutative.
The result of non-linear but layer-less and order aware architectures could promise the following:
Feasibility of uploading model weights: a random grand tour of 2D projections of the square matrix space, projecting the token positions down to 2 dimensions along an changing axis, a "game" could correctly project 95% of the tokens, requiring the user to identify the incorrectly placed tokens, which jump back to their correct position when clicked.
This means a user can learn (since most of the time tokens are rendered in the correct position of the 2D projected cloud), and as a user learns their performance will go up.
If a user could eventually (after playing for a long time) correct 90% of the token positions on random 2D projected planes, then the user has effectively stored 90% of their coordinates.
Suppose such a user reads text in their mother tongue, then the joint likelihoods in natural text will correspond to matrix products of those weights. In other words the brain will learn that whatever it has learnt playing the token-game aids the brain in predicting a next token.
Once it has learnt to utilize the format (matrix / multivector coordinates), the apparently pointless skill of positioning tokens in the cloud learnt during the game, it would also help predict the next token in languages the user didn't know, say when looking for ingredients on a product package. I believe such a user would swiftly discover they actually know those languages, and I believe transmission of a compressed format would take less time than transmission of the same knowledge in expanded output-corpus form.
I wonder if Terrence Tao has an opinion on such a possibility:
1) does he consider it (im)possible to have token-first formatted LLM's without MLP layers etc reach similar levels of performance as the current crop of LLM's?
2) supposing it were possible, would he predict a user "downloading" weights to his brain by a reflex-game as feasible?
3) does he believe a human brain would be able to make use of the downloaded weights and would the brain notice the utility in predicting tokens?
4) does he believe that once the brain has noticed / learnt how to make use of the model weights, first for predicting the next token in the mother tongue, it would subsequently have learnt how to predict in other tongues?
It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.
Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability. But of course mathematics is all about proof, and for that reason I was wary of it for a very long time.
Fortunately this is not basing science on proof. It's just making it much easier to do proofs when that is what we choose to do
>Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability
not really. You can consider positive proof as an experiment confirming your theory and the negative proof and counter examples as an experiment falsifying your theory.
Yes, really. "Positive proof" opposes the concept of falsification. You can only have it within a system formal logic, and science can contain those, but isn't one.
>"Positive proof" opposes the concept of falsification.
no. Positive proofs have nothing to do with falsification. They just tell you that there is no point in spending effort on searching for negative proofs and counter examples. They don't prevent nor prohibit you from spending that effort. They just advise you that that effort will be wasted.
It is like nobody prevents from experiments to turn lead into gold. Of from searching for a right angled triangle violating Pythagoras.
... within a system formal logic.
Understanding has been the point of mathematics for millenia. The idea that purpose of math is to produce machine-checkable proofs is an entirely modern idea.
Mr. Tao is an excellent politician. Lots of awards and texts, yet no major problem solved.
It seems now that NS is solved he is mobilizing the community to convince taxpayers continue to pay even though AI may do a better job in his work.
Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.
> Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.
Ignoring the other ridiculous parts of your comment, isn't this exactly what you're supposed to do? Update your beliefs according to the newest information available?
Yes , you are supposed to do. That doesn’t change the fact that he has no insight into technology, and is a bandwagon person.
If I’m a mathematician, I can write public posts on math. Extensive posts about politics, AI, crypto, … are not useful without expertise in those domains
Gr8 b8 m8
This post isn't by Tao.
I'm increasingly seeing people read something in a post that isn't there. GP said "Tao is mobilizing the community".
You can mobilize the community through guest posts that slowly nudge people towards accepting AI, so that they demand funding for AI for academics. Which he himself suggests:
https://terrytao.wordpress.com/2026/09/18/sairs-open-math-mo...
He is founder of SAIR.
The EU version is to demand a CERN for AI:
https://terrytao.wordpress.com/2026/09/17/a-cern-for-ai-assi...
It just looked to me like the GP hadn't read the article and assumed it was a post by Tao. You are right about Tao of course, no objections there.
Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.
Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.
Likewise all other technology [ and culture ].
LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.
Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
A moratorium on AI development might be the only way to achieve this preservation of human culture.
I want to agree with this, but I have a hard time seeing how it can be done.
Tao is speaking of a very particular kind of mathematics, that done out of pure curiosity.
But maths, even at the highest levels, often finds applications sooner or later.
It will be economically impossible to justify boycotting correct mathematics that no humans understand on grounds only of purity.
This may happen very soon: one of the obvious applications of novel mathematical results is in building stronger AI models.
> one of the obvious applications of novel mathematical results is in building stronger AI models.
This gets repeated a lot and seems to be one of the primary stated goals of making AI solve math problems, but I still have no idea by what mechanism this is even supposed to happen. I guess they could make some minor improvements to matrix multiplication algorithms or whatever but I don't see what groundbreaking theorem could possibly significantly improve LLMs.
It's the kind of thing where it's sort of expected that you wouldn't know, right?
I think we don't really understand why deep learning works as well as it does, the thinking around that is, as far as I can tell, mostly a collection of empirical observations.
A fundamental theory of learning that can be used to predict optimal network architectures might enable smaller models that consume less energy.
> Tao is speaking
Tao isn’t the article author, it’s a guest post.
> Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
So many thoughts come to mind at once, they're a jumble in my head rather than a single coherent narrative.
John Henry comes to mind. As does Agent Smith's "I say your civilization because as soon as we started thinking for you, it really became our civilization, which is, of course, what this is all about" monologue in The Matrix. I've not read (or listened to) "With Folded Hands ..." or "The Machine Stops", but I have read the Wikipedia plot summary of both.
Do we want to have comfortable lives, or do we want to serve each other?
"Computer" used to be a profession; I grew up around adults bemoaning that "kids these days can't do mental arithmetic", the Pi Zero I've not switched on for probably a year now could beat all humans simultaneously at that (even if everyone was as good as the current world record holder) and yet we still teach arithmetic in schools.
Nobody needs to knit, and yet we do so for fun. Youtube's "Primitive Technology" channel, which has spent around a decade speechlessly making iron from bacterial slime found in a creek, using only clay and sticks and leaves and vines naturally found next to that creek.
Like I said, no coherent narrative. It's been a while since my stream of consciousness became a river delta; usually at worst it only meanders a bit.