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
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".
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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:
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.
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.
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 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.
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.
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?
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.
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.
“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”
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.
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.
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.
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.
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.
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]
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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 means we can move on to other difficult shit.
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
And what would that be?
Timothy Gower
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)
It's all good until we have superhuman appreciators :)
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".
I'm pretty sure we already have machines to do both of these.
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.
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.
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.
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
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.
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.
The article isn’t by Tao, it’s a guest post by Grant Sanderson (aka 3Blue1Brown).
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.
Fortunately LLMs are smart enough to handle that already.
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.
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.
a good teacher remembers the journey, not just the destination.
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
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.
The socratic method also has a much higher chance of revealing where the student is in their mind.
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.
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.
I would even go as far as saying that mental conditioning and training is also required, on top of mental capabilities.
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.
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.
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.
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.
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.
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.
Gemini's explanations are very good.
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.
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.
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 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.
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.
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.
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.
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).
LLMs have LLM intuition, not human intuition. (See the movie Her.)
LLM cannot reinvent Euclid from scratch, but a larger system including LLM might.
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.
That has been done, like back in the 1960’s.
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 :-)
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.
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?
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.
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.
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.
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.
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.
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.
Gr8 b8 m8
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
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.
Math academia 2025
Math academia 2026
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.
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.
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?
Yes, why not?
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.
We'll get innovations in AI as a side effect.
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.
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.
Tao isn’t the article author, it’s a guest post.
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.
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.
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?
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.
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.
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”
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.
Taught proofs too, and plenty of students fake intuition with pattern matching.
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.
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.
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.
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.
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.
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 3blue3brown YouTube channel.
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 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...
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.
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.
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.
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.
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.
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?
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).
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..
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.
This is really interesting and available here: https://www.youtube.com/watch?v=IbGNZQvobkc
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.
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
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.
Evening the playing field would mean those R1 professors now teach five classes a semester for 50k a year instead of doing research.
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.
Congratulations. You are one of those leading the way, showing how we will adapt and how the world will get better from AI.
Thank you for this load-bearing comment! We all envision a future with peaceful coexistence, prosperity, democratization and 72 virgins for each.
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.
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.
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.
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.
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.