I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.
I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.
Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.
And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.
As someone who mostly only applies math, that strikes me as a peculiarly academic take. Intuition is more important for me because it’s what enables me to know what methods are most applicable to whatever practical problem I’m trying to solve. The proof’s purpose is to verify my intuition. It’s just a means to an end. I only take the time to do my own when I can’t confirm what I need from a textbook or paper.
>You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself.
This isn't always the case. Our algebra (or analysis) course focused a lot on proofs for the exam. The result was that a lot of people learned the proofs by heart.
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.
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.
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.
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.
Programming was never about communication. It was always about making the machine do the thing we want. Back in the day, a good game programmer knew which time intervals had writable video memory and which CPU cycles drew which scanlines, and spent more time rearranging the code to hit these timings than to write the actual algorithm. Later programmers (I hesitate to call them good) learned everything there is to learn about OS internals and wrote theoretically nonsensical and invalid code that still worked thanks to those internals, to save CPU cycles and especially memory use. And the next generation of programmers took the principles of late binding and abstraction to the logical extreme and created architectures that cannot be described in words anymore, only in diagrams - but are crazy good for code reuse, traceability and A/B testing.
> Open source programs could be more like motivated explanations of computation.
It is already that. Every time a method/function is created, a structure is defined, a variable is added, a file is created or renamed,… It’s all for the purpose of human communication. The computer only need binary in a single file.
But people feels like they should be able to jumpninto curl code without any understanding of networking, or linux code with no knowlede of computer architecture. Few code are meant for total beginners.
> There’s a wonderful book, How to Prove It by Daniel Velleman
the name sounds familiar but i don't think i have read that one, i did enjoy "introduction to mathematical reasoning" by eccles.
personally my relationship with mathematical proofs has been complicated. it took some work to understand basic proofs (dedekind cuts, ideas vs. instructions with mathematical notation), but all of the theory of computation proofs, which supposedly are difficult for many, were completely intuitively easy for me.
i think mathematicians are facing a similar confusion as computer programmers. the medium used to require precise thinking and the simple act of reading, writing and composing it was a mechanism for thinking and learning. in the llm era, the question is: should there be a new mechanism and if so, what should it look like?
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
> usually by people who are good mathematicians but know close to nothing about teaching.
I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens.
And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
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.
> 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.
Intuition happens naturally and people are prone to inducting the wrong conclusions. Proofs provide a framework for rigorously analyzing drawn conclusions such that it can be used to build intuition in others. If math is about sharing the insights gained in a particular class of problems, proofs are the means to getting there.
The math field is confronting something that coders have been dealing with for a few years now, only far more violently. Today's moat for software seems to be that AI can automate tasks but not a full job (yet). But for a large proportion of mathematicians, doing these tasks really was _the_ job. It's the bit they wanted to do, and if they completed a sufficiently difficult set of tasks, they got tenure. Now this model is failing, they frantically need to pivot the role of humans to save their profession from funding cuts.
I remember when "writing code was never the point" became a mantra here. There was truth in it, but removing the coding has certainly taken away a lot of the texture of the work and enjoyment of the craft. Many of us feel this loss as we tech-lead teams of agents as our source of income. I am not optimistic the mathematics pivot is going to work, but I'm certain that most will be depressed with the outcome even if they succeed.
We are all staring at the same existential dread, just seeing it unfold slower. We're being told that utopia is to be obsolete, and that is a jarring idea to contend with.
I feel that you can see quite strongly the truth in “writing code was never the point” when you encounter inevitably at every company the guy who has been around forever but doesn’t seem to be working particularly hard. Their value is (was) no longer in writing code at a furious pace all day. It was having a coherent, intelligible and communicable theory of the software system the company is founded on.
I propose this thought experiment: put all living mathematicians in a very long bus. This bus crashes and they all tragically lose their lives. Can we really say mathematics simply marches onwards with AI alone? Let’s say Anthropic needs a new research result to improve Claude. Are we really already at the point where we burn tokens ad infinitum and arrive at the end of scientific progress in some timely fashion?
I actually have a rather dim view of the "writing code was never the point" line. Not because it's objectively wrong, but because I see it as something we're mostly telling ourselves to feel better about the status quo. Ability to write good code has been highly celebrated (and remunerated) for decades. As it is becoming less relevant, we immediately backtrack and start lionizing the parts where we can still be useful instead. Consider the counterfactual - AI continued to be terrible at writing code, but weirdly better at humans at product decisions, architecture etc. In this universe, saying "coding was never the point" would not be popular.
It also find little solace in it aside from 'well this version of GPT isn't taking your job'. AI labs certainly have no intention for the higher level skills to stay in the human-only domain. The veteran developer with the coherent theory of a large stack is immensely valuable today. But they also don't survive if a company can drop a few coders' salaries on rewriting that stack from scratch - faster, fewer bugs, more coherent, able to react to changing business requirements with more agility etc. I am not saying this is where we are, but I think there is a reasonably good chance this is where our road is leading us.
As much as I hate it, I don't think we'll ever get a proof of the four color theorem that isn't enumerating cases.
When you have an integral or the sum of an infinite series that comes out to pi, you know there must be some satisfying explanation involving a circle.
Contrast with "Examples of patterns that eventually fail" on math stackexchange[^1]. When a pattern ends at 906150257, you don't really expect the proof for that to be something beautiful. The reason for the exact value of an upper bound is that it isn't smaller and it isn't bigger.
There's a relationship between e, i, pi, and -1 comes from a deeper relationship between complex numbers and rotation.
The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
> The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
Well, the current conjecture is that it hasn't to do with coloring at all, but how many, on a map (partitioning of plane into connected regions), regions can pairwise touch: 4.
(That is: If someone proves Hadwiger's conjecture, then the four colour theorem follows.)
This is, as always from Grant Sanderson, thought provoking and opens new insights.
It occurs to me on reading this that there’s a connection to other computerized mathematical activities.
Occasionally some computer lab in the past would announce that they have computed pi to more digits than ever before; or a new Mersenne prime will be found.
These count as ‘math news’ but they’re of little interest to mathematicians. These computational efforts demonstrate the great power of computers but they do nothing to advance mathematical understanding. Finding a larger Mersenne prime is not surprising to anyone; we’re pretty sure there’s an infinite number of them. Finding the largest Mersenne prime would be the surprise.
So it is with proofs. An LLM might prove some conjecture - Riemann, say or P≠NP. But in general we know that things can be proven and we think those things are probably true, so the existence of a proof doesn’t do much more than producing a new Mersenne prime does.
It’s only if in proving the thing we learned something that there’s actual value in the proof.
Part of the controversy here is that now the skill advantage that some Field Medalist had is much narrower. The fact that fields medals have an age limit implies that it favors brain power over understanding. And that was the guiding light award of the community. So i find it "funny" (and natural) when they are offended by AI.
That is the main "crisis" of mathematics.
In my opinion there has never been a better time to be a mathematitian, and there has never been a better time to be a software builder.
But there has never been a worst time to have the need to prove your economic value as a mathematitian or software developer alone. Because "understanding" is not something you can prove in one afternoon, its something that you prove with a life.
> In my opinion there has never been a better time to be a mathemetician...
As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible. It's like telling a chair-maker during the industrial revolution that there had never been a better time for them, since now they could operate chair-making machines instead of toiling away at the wood themselves. It assumes that they were purely in it for their passion for mass-producing chairs. The majority of mathematicians get into the field because they love problem solving, and the gauntlet thrown down by challenging math tasks.
Many parts of this will never be useful for society on a grander scale - but this is reflected in the finances - pure math is closer in funding-terms to a humanity than to hard science. Now even this is _massively_ under threat, and Tao and co need to pivot quickly to stop this from becoming a bloodbath.
Im an ex mathematitian too. And if i was in academia I would probably have the same reaction. Thats what i say that its the worst time for proving economic value.
But if you are in for theory building and understanding, then you are not constrained anymore by your motivation to grind through countless hours of formal theorem proving. And you do not need to have superhuman formal manipulation skills and memory.
For me mathematics is not the formal system, so LLMs will never be able to do end to end maths.
> As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible.
To throw a counterpoint to this into the writhing cesspit of HN, I'm active in academic mathematics (postdoc) and every one of my collaborators is deeply in love with the field and their jobs. Perhaps the grass is greener on the applied mathematics side of the fence.
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.
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.
> In my opinion there has never been a better time to be a mathematitian
I think it’s a great time to be a curious mathematician, especially in a niche field where you’re not competing with hundreds of agents of the best unreleased frontier models.
However it’s a very scary time to be a professional mathematician because publish or perish is going to cause a race to the bottom for cranking out results as fast as AI can let you. [0]
I'm a professional mathematician. Today I proved what for me is a very solid theorem. It's something I had thought about for a few years. With a few weeks of serious use of AI I've found a proof that I am currently trying to write up, but which appears correct. The change in the workflow is enormous, but so is what one can do if one has clear what to do and how to do it.
Not every problem will receive $20M in funding to be solved by AI; for the rest, good human guidance will have to suffice. Labs only pulled this stunt because they wanted to show investors how powerful their models are on their own. But look again at the cost of that army of 10,000 SOTA agents. At the very least, I foresee a need for humans to decide when costly AI resources should be committed to a specific search plan. Grant review remains irreducibly human because it involves choosing which directions to fund and weighing opportunity costs: taking one path forecloses others.
That's not the conclusion. I started using AI after the Jacobian conjecture counterexample and have used a particular problem to learn how to use AI and to explore it's capabilities. I'm not a great mathematician but I'm full faculty with 25+ years of research experience and lots of articles and I just proved in a few weeks something that had resisted my efforts for some years.
The exploration process is much easier now. Ideas are quickly testable and multiple tests can help identify a technical obstruction. The tool requires good guidance and input but as it trains on people like me it will need those less.
At the very least our way of doing things must change. More pessimistic views seem to me defensible.
I find myself less worried about it than at first. I think what we'll see are that some things are low-hanging fruit and can be solved just by tireless search. Maybe half the millennium and other such high-visibility problems will fall this way.
Others, I think, will be beyond both human and AI. And so what then? Mathematicians just throw in the towel and say it's not worth trying? Of course not. We will continue that pursuit, and as we do, new ideas will arise and new problems will need to be solved. It's math. There is no end.
It's easy to look at the current landscape and see AI ticking off solutions to problems and imagine that soon there will be nothing left. Machines replaced the need for much manual labor, but they also established a basis for an economy that provides the opportunity for more labor. This is the situation with math now. It will take some getting used to. There will be little-to-none pencil-to-paper working out of problems anymore, but there will always be work to do, things to solve, curiosities to unravel. And it will still be professional mathematicians who are the ones most capable of directing that effort. Because, if nothing else, they're the ones whose curiosity is piqued by the problems. Which, let's face it, has been 99% of the motivation for graduate-level math in the first place.
There's the the old question: is math invented or discovered? I think it's both: the problems are invented, and the solutions are discovered. In the age of AI, the discovery part will be greatly affected, but the invention part will remain firmly in the human domain.
I was a PhD student for a while, but I always enjoyed "refactoring" proofs, more than coming up with new ones. Making them simpler, shorter, clearer. Unfortunately, it's not much rewarded.
One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.
Having to ask the village elder about how to do things meant learning various other incidental lessons. It was reduced when books started to become available.
Having to search in the library also often lead to serendipitously seeing a book on the next shelf and falling in love with a topic you didn't even know about otherwise. Or you had a chat with a librarian asking for advice which books to look in for a topic. Google search eliminated that. With Google search and reading websites, you still had to read or skim the page and may see some other info or click to read the author's About page. Now with AI we get straight to the answer.
I do neither maths nor science with AI but in my experience most models are perfectly willing to burn tokens on a ton of sidequests at the earliest opportunity.
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.
> so this seems like missing the forest for the tree imo.
But is it? Again, regardless of the amount, what is the utility?
What's the end goal of academia (assuming broadly as research with humans) when the answers to the deepest questions become commodities at orders of magnitude higher speed and lower cost?
How are federal grants justified and, forgetting the current hierarchy, how is differentiation made? It's currently based on research output, once that becomes irrelevant what is it? We already have IMO, IOI as competitions and I assume just like Olympics this can be a thing, but it's very remote from research.
Take for example Rene Thom after Alexander Grothendieck overshadowed an entire field
> His technical superiority was crushing. His seminar attracted the whole of Parisian mathematics, whereas I had nothing new to offer. That made me leave the strictly mathematical world and tackle more general notions, like the theory of morphogenesis, a subject which interested me more and led me towards a very general form of 'philosophical' biology.
Some were like this, some weren't. Many people were in it because it's objective and factual and not up for the whims of taste of some established gatekeeper. They weren't in it for art performance reasons or to please the aesthetic judgment of some entrenched mathematician-baron.
I was about to link the "discovery fiction" article [0] by Michael Nielsen, but halfway through saw Grant mention it as an inspiration. It's one of my favorite styles of writing that lets you settle into a cozy read of a narrative around a problem while slowly exposing you to the rigor and nuances of tackling it. It's definitely deserving of a public index of some sort, like one of those "Awesome X" list repositories on GitHub, because I think it's a valuable style that deserves to be curated.
> It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.
> we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems
The core idea seems to me that we should shift the standards for professional evaluation from generating proofs to generating explanations. Makes sense that such a proposal would come from the 3B1B guy, and I actually agree with it, irrespective of AI. But what eludes me is how that could be a defensive mechanism against AI automating humans out of mathematics. AI is likely no less good at producing natural language explanations as it is at generating rigorous proofs. It's telling that even Terrence Tao turned to AI to understand AI-generated results [0]. It seems that the essay doesn't address that issue at all.
It’s a task much harder to RL and much more subjective. I don’t want to say we won’t get there, but let’s just say that LLMs could “write” well enough since gpt3.5 era and I don’t think the pleasantness of the prose improved dramatically since then.
And subjectively the explanation LLMs currently provide are usually horrible, horrible enough that I usually just instruct them to provide me human written literature I can read.
I mean, there's centuries' worth of mathematical prose to train on. But that's presumably already in the training data, so if it isn't good enough today, it might not get better fast enough to keep track with how fast they'll get better by training on formally verified math. But then again, the prose in Terry's conversation I linked above seemed pretty useful. But it's also a problem requiring famously little advanced mathematics.
I have been losing interest in this proof-oriented approach into extremely abstract concepts (what seems to be the core of academic mathematics today). Obviously, proofs are very attractive because they are the closest thing we have to a universal truth (at least under the assumed axioms). Having mechanisms to reliably show a proof, and computational methods to handle complicted proofs is great.
But.. the navier stokes proof was the last straw for me. People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society. And certainly not moving us towards "human understanding". The biologists are the ones working on that, the math folks should try working with them on neuro stuff to understand how human brains can do math at all, given their architecture.
> Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society.
Is it, or is it only the parts you hear about/pay attention to?
> People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Yes, maybe famous solving famous conjectures is just trivia and trophy collecting and the real value is the intuition and techniques you develop along the way to solving them which you can then bring to bear on things like CFD.
Mostly the parts I hear about. But I also communicate with mathematicians on a regular basis and I can see what they are working on, which is representative of the field. It's about 90% "cohomology of abstract Lie groups that morphize into string theory" and about 10% "practical thing that engineers can use to make my cell phone work better". (amusingly, although I literally made up that sentence, it turns out it's not far from something somebody worked on: "The cohomology of compact simple Lie groups and spin groups connects to string theory by providing the topological obstruction classes—specifically the first fractional Pontryagin class ...." OK, kind of emphasizing my point there.
No, the intuition and techniques that get developed to solve the NS conjecture have little or no bearing on the practical details of doing CFD. That's a common story/thread (and I hear the same story in quantitiative biology, my area of expertise), but often times, it's just a loose justification given to justify funding.
Current and future mathematicians can now spend most of their time coming up with problems/conjectures and theories that are hard for a future model versions (6 to 1 year out) to solve them and by itself it can be a new major sub branch of mathematics - “Theory of perplexing frontier models” and who knows it can even open up new dimension of mathematics for mathematicians to explore by themselves (because AI by definition cannot help them here). Now is the time to be excited for mathematics!
The value in academics is teaching and research. The value in research is discovery. Proof was a useful function for humans to do towards discovery until recently. Understanding is a useful property insofar as it helps you teach and it is a prerequisite for generating hypothesis. Humans will always be driving discovery, the tooling and focus of work may just be a little different. Attachment to one particular modality of discovery is an aesthetic choice, not a moral one.
Hmmh. I like motivated explanations, but, as acknowledged in the text, this is a subjective thing to measure. What is a great motivated explanation for Tao, might be hard to grasp for me. So I guess judging how well an explanation motivates something depends on two things: 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
There is a third thing: how well does the motivation chime with or go against my current belief system? You would think this is not much of an issue in mathematics, but it can be, and I had my fair share of frustrations because of it.
Anyway, all of the above points to one thing: the best motivated explanation will be generated by an AI, knowing the subject and you in a deep way that no other human will, and being able to interact with you during the explanation.
> 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
I don't think we're discussing pedagogy. Good _research_ exposition is instead related to communicate your intuition and way of seeing things. The conceptualization of a given situation or problem is what is valuable, how you connect it with other stuff, etc. It is then up to you to memorize and interiorize it.
Good research exposition is the same as good "pedagogy" just for a different audience. Both have to consider didactics. When writing a paper, you teach something to the fellow researchers who know less about a thing than you.
Seems to me to be two different sides of the same coin. Pedagogy is about finding a way to communicate to me an idea based on my intuition and seeing things. A research exposition is about presenting the idea in terms of your intuition and seeing things.
The blog post says that the statement "AI really did solve a problem in mathematics." is wrong. But a formal proof showing that Navier-Stokes equations can blow up is certainly such a solution, by AI. There is not much in this world that is more objective than a formal proof, so any disagreement on this is based on how we see the world. Michael Harris will agree with the statement being wrong, Jacob Tsimerman will not.
Another example, Hilbert famously battled Brouwer's view of mathematics. From my point of view, Hilbert was right: intuitionistic logic is certainly interesting; but I like to study it using "normal" (= classical) mathematics.
Finally, my personal frustrations are about how hard it is to publish my work on abstraction logic. I would never have thought it is that difficult, mathematics being objective and all. It seems essential to take out as much motivation out of your paper as possible, because it might offend your reviewers and their belief system. By now my papers come with full Isabelle/HOL formalisations, let's see if that helps.
1. We humans have managed to take huge amount of information and compress it using a loss function containing some bias we have about the information.
2. We now ask ourselves to decompress the same information with some additional cross-entropy. As a side effect of this process we sometimes spurt out information that may or may not have any meaning since the compression was lossy.
3. Now, we ask ourselves to present this some-what newly decompressed information with brevity in order to understand what we've learned from it.
Knowing that this process is happening on a larger scale, this resurfaces the argument if meaning can be reduced to computation only.
Although some might favor this argument but we are at the risk of anthropomorphizing this process.
The idea presented in the post itself is perspicuous (in Grant Sanderson own words) as he always does.
The last days we are served these high goals about understanding, "digestion" and so on.
But if you look at the practice of present mathematics, in the last 20 years it is all about publishing solutions to problems.
There are famous problems to be solved, there is a hierachy of conjectures to be solved. A quick search here on HN gives pearls like "Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem".
And all of a sudden it turns out that problem solving can be automatized.
So then what will problem solvers do? Well, from now on they will "digest" problems solved by AI.
In a way or another they will find a way to stay on top.
That's the goal, at least, but mathematics as a living practice does not have much to do with these games of power.
“You are right. There is nothing in yesterday’s mathematics that you can prove with
exterior algebra that could not also be proved without it. Exterior algebra is not
meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds
is as worthwile a mathematical enterprise as proving old conjectures.”
Gian-Carlo Rota in “Indiscrete Thoughts”
Yes. AI is a useful tool and we are going to adapt and use it.
The phd student will be forced to publish 10 breaktrough articles, the university department which does not offer "free" access to AI (for its members) will see the its ratings going down, when compared with the other universities.
It will be "use AI or perish" for academic management so on the side of academic management the ones with vision will thrive and the ones without will perish.
But what about the publishers? In the last decades the academic research was made into a feeder for publishers. The main goal of a researcher is to write articles, which are later sold back to other researchers.
This economic system is under big stres now, because for a while at least the academic management and publishers will have contradictory goals.
And that is why this scare which is induced by those who profit the most from the present system.
If this suggestion were to come to pass, I wonder how new math PhDs would think about choosing between a 'normal' R1 faculty job vs. the "teaching route" (teaching professorships, lectureships, community college professorships, or SLAC professorships).
It's been my understanding that traditionally the ones who care about "motivated explanations" in this sense go for the latter, but if the research community has now decided they care about teaching and understanding, it might "even the playing field" and make the jobs more similar.
Let's see how long (if it ever happens) it takes for models to generate motivated explanations (possibly done via the Manim library or something like it) along with their Lean proofs. Grant Sanderson is right that this is kind of subjective but so is Art and I'm very enthusiastic about AI generated Art.
> possibly done via the Manim library or something like it
It can be done already: the point is that the motivation and explanation parts are terrible, especially for novel topics where the AI can't just rip off existing content. A Lean proof is at least a verifiable task; you end up with an actual proof that you can work through. A Manim slop video doesn't have that.
I think the thing is.. sorry Grant, but "motivated understanding" won't come from videos, but writing. Solving things too, but a video can only go so far.
I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).
I'd be interested in hearing a field report on this! For example, I can easily imagine that they're great at walking through the proof step by step, explaining background as necessary; but as TFA notes, one of the most important questions is "why is this definition the way it is?", and my bet would be that the Lean is not enough to help the LLMs meaningfully in answering that.
LLMs like even the sota flash models have great range of background math knowledge and have no problem reading and understanding flt level of math. On the other hand you don't want to go through 13 million lines of often repetitive code line by line. Models are great at synthesizing math content out of code. My contribution is to steer it through subjects of most interests to me, drill down into jargons that can be confusing, be creative in using computation for illustration (which coding agents can execute very proficiently) etc.
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.
It would be interesting to see what would happen if we had two competing mathematical institutes, a sort of First/Second Foundations:
1) Rejection of AI for anything but trivial applications while still using computers at their full capacity. Researchers would ensure full human understanding of proofs and methods. This Institute believes on Math as a process of discovery, Mathematicians as explorers/poets/storytellers and not proof machines.
2) Unrestricted, all-embracing use of the latest AI, including potentially research in creating even better AIs as part of the program. These researchers would be okay with not understanding proofs if verified to be correct. This group is focused on rapid problem resolution and believes Mathematicians are theorem creators and provers.
After X years (100?), which one would advance Mathematics and humanity the most (we'd need to define "advance")?
Mathematicians worry about proofs and the intrinsic value of something as elusive as 'understanding'. They are deeply ingrained in the study, deeply concerned with anything effecting the field. Yet they're still emotional beings looking for beauty and meaning in life that might come from an understanding how the universe works purely from a math perspective. I'm glad Mathematicians exist, I certainly can't do that type of work.
And I trust their results: technology wouldn't be possible without advancing our understanding of the world in various fields, including math.
Your idea sounds great for the Mathematicians.
There's a more pragmatic view though, and unrelated to proofs themselves: does understanding a proof help us to advance Humanity in some way?
Do we have better lives afterwards? What if we give up understanding proofs and focus only on results.
In other words, if an AI solves a problem for you, but you don't understand how it works, should you continue building anything on top?
I suppose the results are truly what matter. If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
Your second approach seems good to help figuring out results from both theory and application of math to solve problems.
But also, what if there is no true beauty in Math, the way Dirac and Einstein wanted?
What if these AI brute force proofs are all that's left?
> If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
A lot of medicine is already like this. Shown to work in clinical trials, no complete end to end mechanism understood. They still get approved if the empirical results are strong.
I'll answer one of your points partially: if AI builds a better sorting algorithm and proves its performance characteristics, it's useful. I'd be able to use it to make my programs faster even if I don't/couldn't understand it.
It would be a bit disappointing but still useful and make humanity slightly better.
Software is logic applied to intersubjective truth. It's not physical truth which is the subject of the hard scientific fields such as physics and chemistry, as well as biology for the most part.
I'm not sure that this new approach will be AI-resistant. Why would people not use AI to help in creating the "motivated explanations". Maybe they can't be one shotted today, but AI also makes this easier.
Assume in 2 years we have a heap of these motivated explanations, all as high quality as Grant's videos and the best books. But who will read them? There is limited interest in this genre. Grant reaches a large fraction of this audience but most people really don't want to think about math either way, no matter how good the explanation is.
Indeed, there is now "edutainment slop" online and AI can use 3blue1brown's manim library to copy his style and AI can use blender and video generation to mimic 3d animations of other explainer channels. Today it's still slop, but it may not be for too long. And then people will have to reframe their job until it's "doing X while also farting and burping every now and then", and then a machine will be better at that too eventually.
Also, this new style of doing math will appeal to a different set of people. Many mathematicians aren't super social, they just like to explore a problem on their own. Think Grigori Perelman. They will still face the problem and their temperament may not make it easy to switch to being a communicator.
Last summer Grog was still celebrated and admired for bravely piercing animals with a spear and bringing home the meat. But now Goong made this newfangled arrow and bow thing and any cowardly fool can now shoot animals from a distance. Grog devalued. Grog sad.
I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.
Of course they can. My point is that mathematicians may increasingly find themselves in the same position as academics in the humanities. Mathematicians themselves will be able to see the inherent value of the work they're doing (just as experts on 18th century novels can in their own field), but it will be far less obvious to society at large why their work should be funded.
It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.
A majority? I doubt it, simply because the majority doesn't know what math is at all.
At university level introductory calculus, the person teaching class had to reassure students that math wasn't entirely arithmetic or adding up numbers. He did this because it's a common misunderstanding.
Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".
The theorem thing is invented by human to help other people better understand Math structure in a easier way.
There must be like a 1000 examples of c compilers in git hub alone. No idea why building LLM building one is impressive. It's right there in the training data.
The only concrete step any mathematician on the internet, including on the other AI concern site https://proofsandprompts.com/ , is demanding funding for an academic frontier AI.
Strange that the Poincare conjecture was solved by a hermit without all this AI bullshit. Maybe reject AI, ignore all AI proofs and retreat from the internet.
I fully agree that motivated explanation is more important than proof. This doesn't resolve mathematicians' feelings of existential dread, however. Machines will get better than human mathematicians at motivated explanation in another year.
There will be no more glory in mathematics, but at least the joy of understanding will remain, and it will come without deciphering the tortured proofs that machines output today. Each bit of understanding will come with much less struggle, but this just means we can get more understanding for a given amount of struggle.
The people building AI claim it will surpass human intelligence in all
respects and prerhaps kill
us all. Should we just cease all human activity on the basis of what AI might do in future?
Personally, I doubt AI can surpass a good human explainer because explanation requires empathy, which benefits from being an instance of the kind of entity you are explaining the thing to. That gives you a way of exploring and evaluating the space of possible explanations that isn't available to an LLM.
Thanks for pointing me to this video - it's been interesting to follow the discussion! (I personally don't see that math has lost its purpose at all in the past months. I mean, where would we be, if we were thrown at these AI based mathematical proofs and had no mathematicians and specialists?! Much of this discussion is about a disciplin readjusting its way of work and tasks.)
You get $100 billion for an AI X that generates the proofs, $100 billion for an AI Y that explains the proofs and another $100 billion for an AI Z that reads Y's output and appreciates it.
Humans meanwhile scrub the floor and write blog posts about "what a time to be alive".
That which has received the Mandate of Sapience must be cool (eg any skateboarder skilled enough to impress his mom)
LLMs got It, sometime this year. Terry Tao and Friends have appatently lost It, same time this year. (This is not to claim that the Mandate gets extended to their creators the frontier labs or even Jeff Dean et al. Definitely not their sponsors. Howbout distillers?)
The thing about Mandates--- they are not forever. LLMs can "lose" It. Probably not back to mathematicians -- that'd be atypical (unless they quickly learn to "make their own lightsabers"?) . Likelier: to a scene of humans no-one yet thinks about.
Skateboarders didn't take the Mandate from anyone. So no one takes It from them. There's some karmic law at play
It's too late for this. Much of my work in math has been classified as trivial or best described as not math at all. When I was working on chatbots and described deep learning algorithms to enhance them, it was deemed as a pseudoscience. Mathematicians sound very disingenuous with their backtracking.
I'm afraid that much of mathematicians work is too trivial to be taken seriously and they should just find something completely different to do.
Sadly, your inability to comprehend is noted which leaks your lack of expertise with the subject matter.
For a general overview, assuming good faith and a genuine willingness to learn, refer to https://iep.utm.edu/s-truth/
Its a remarkable intro into propositions, statements and sentences with vivid examples from the works of Tarski, Godel and others as to what constitutes truth.
Pay attention to Tarski’s T-Scheme (sentences and truths)
if language models (current GPT-style or any future model with different architecture) can be viewed as a compression of their output corpus (the corpus it emits by providing random starting native contexts weighted by the model's likelihood of that native context) then one is saying the corpus contains the same information bits as the model, the model just takes less data bits, while the corpus represents inflated data bits.
If learning requires communication of data, one could ponder if it is more effective to transmit learning data as suitable model coefficients instead of corpus monologue or dialogue (say with a teacher).
This may sound absurd, but is it really so far fetched to believe one could formulate "download"-compatible LLM architectures, so that the user or student can play a kind of memory game to train "download"-architected model weights by simple reflex games?
Suppose instead of token vectors we used token matrices or token multivectors as in geometric algebra (not to be confused with algebraic geometry!).
Word2Vec couldn't do general language modelling because it used vectors, but when using token matrices or token multivectors one could postulate the following partition function:
given a dictionary of tokens, and their corresponding (matrix/multivector) M one could define the partition function
exp(-|M1 * M2|^2)
For example it could be alphabet level (to demonstrate the concept):
For every string one can compute the corresponding M in terms of the elementary character matrices:
M_"car" = M_"c" * M_"a" * M_"r"
both matrix products as well as geometric multivector products are associative but not commutative:
But M_"car" != M_"rac" since matrix/geometric products is not commutative.
The result of non-linear but layer-less and order aware architectures could promise the following:
Feasibility of uploading model weights: a random grand tour of 2D projections of the square matrix space, projecting the token positions down to 2 dimensions along an changing axis, a "game" could correctly project 95% of the tokens, requiring the user to identify the incorrectly placed tokens, which jump back to their correct position when clicked.
This means a user can learn (since most of the time tokens are rendered in the correct position of the 2D projected cloud), and as a user learns their performance will go up.
If a user could eventually (after playing for a long time) correct 90% of the token positions on random 2D projected planes, then the user has effectively stored 90% of their coordinates.
Suppose such a user reads text in their mother tongue, then the joint likelihoods in natural text will correspond to matrix products of those weights. In other words the brain will learn that whatever it has learnt playing the token-game aids the brain in predicting a next token.
Once it has learnt to utilize the format (matrix / multivector coordinates), the apparently pointless skill of positioning tokens in the cloud learnt during the game, it would also help predict the next token in languages the user didn't know, say when looking for ingredients on a product package. I believe such a user would swiftly discover they actually know those languages, and I believe transmission of a compressed format would take less time than transmission of the same knowledge in expanded output-corpus form.
I wonder if Terrence Tao has an opinion on such a possibility:
1) does he consider it (im)possible to have token-first formatted LLM's without MLP layers etc reach similar levels of performance as the current crop of LLM's?
2) supposing it were possible, would he predict a user "downloading" weights to his brain by a reflex-game as feasible?
3) does he believe a human brain would be able to make use of the downloaded weights and would the brain notice the utility in predicting tokens?
4) does he believe that once the brain has noticed / learnt how to make use of the model weights, first for predicting the next token in the mother tongue, it would subsequently have learnt how to predict in other tongues?
It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.
Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability. But of course mathematics is all about proof, and for that reason I was wary of it for a very long time.
>Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability
not really. You can consider positive proof as an experiment confirming your theory and the negative proof and counter examples as an experiment falsifying your theory.
Understanding has been the point of mathematics for millenia. The idea that purpose of math is to produce machine-checkable proofs is an entirely modern idea.
> Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.
Ignoring the other ridiculous parts of your comment, isn't this exactly what you're supposed to do? Update your beliefs according to the newest information available?
Yes , you are supposed to do. That doesn’t change the fact that he has no insight into technology, and is a bandwagon person.
If I’m a mathematician, I can write public posts on math. Extensive posts about politics, AI, crypto, … are not useful without expertise in those domains
I'm increasingly seeing people read something in a post that isn't there. GP said "Tao is mobilizing the community".
You can mobilize the community through guest posts that slowly nudge people towards accepting AI, so that they demand funding for AI for academics. Which he himself suggests:
Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.
Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.
Likewise all other technology [ and culture ].
LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.
Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
A moratorium on AI development might be the only way to achieve this preservation of human culture.
> one of the obvious applications of novel mathematical results is in building stronger AI models.
This gets repeated a lot and seems to be one of the primary stated goals of making AI solve math problems, but I still have no idea by what mechanism this is even supposed to happen. I guess they could make some minor improvements to matrix multiplication algorithms or whatever but I don't see what groundbreaking theorem could possibly significantly improve LLMs.
> Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
So many thoughts come to mind at once, they're a jumble in my head rather than a single coherent narrative.
John Henry comes to mind. As does Agent Smith's "I say your civilization because as soon as we started thinking for you, it really became our civilization, which is, of course, what this is all about" monologue in The Matrix. I've not read (or listened to) "With Folded Hands ..." or "The Machine Stops", but I have read the Wikipedia plot summary of both.
Do we want to have comfortable lives, or do we want to serve each other?
"Computer" used to be a profession; I grew up around adults bemoaning that "kids these days can't do mental arithmetic", the Pi Zero I've not switched on for probably a year now could beat all humans simultaneously at that (even if everyone was as good as the current world record holder) and yet we still teach arithmetic in schools.
Nobody needs to knit, and yet we do so for fun. Youtube's "Primitive Technology" channel, which has spent around a decade speechlessly making iron from bacterial slime found in a creek, using only clay and sticks and leaves and vines naturally found next to that creek.
Like I said, no coherent narrative. It's been a while since my stream of consciousness became a river delta; usually at worst it only meanders a bit.
I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.
I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.
Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.
And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.
As someone who mostly only applies math, that strikes me as a peculiarly academic take. Intuition is more important for me because it’s what enables me to know what methods are most applicable to whatever practical problem I’m trying to solve. The proof’s purpose is to verify my intuition. It’s just a means to an end. I only take the time to do my own when I can’t confirm what I need from a textbook or paper.
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>You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself.
This isn't always the case. Our algebra (or analysis) course focused a lot on proofs for the exam. The result was that a lot of people learned the proofs by heart.
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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.
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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.
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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.
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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.
Programming was never about communication. It was always about making the machine do the thing we want. Back in the day, a good game programmer knew which time intervals had writable video memory and which CPU cycles drew which scanlines, and spent more time rearranging the code to hit these timings than to write the actual algorithm. Later programmers (I hesitate to call them good) learned everything there is to learn about OS internals and wrote theoretically nonsensical and invalid code that still worked thanks to those internals, to save CPU cycles and especially memory use. And the next generation of programmers took the principles of late binding and abstraction to the logical extreme and created architectures that cannot be described in words anymore, only in diagrams - but are crazy good for code reuse, traceability and A/B testing.
> Open source programs could be more like motivated explanations of computation.
It is already that. Every time a method/function is created, a structure is defined, a variable is added, a file is created or renamed,… It’s all for the purpose of human communication. The computer only need binary in a single file.
But people feels like they should be able to jumpninto curl code without any understanding of networking, or linux code with no knowlede of computer architecture. Few code are meant for total beginners.
> There’s a wonderful book, How to Prove It by Daniel Velleman
the name sounds familiar but i don't think i have read that one, i did enjoy "introduction to mathematical reasoning" by eccles.
personally my relationship with mathematical proofs has been complicated. it took some work to understand basic proofs (dedekind cuts, ideas vs. instructions with mathematical notation), but all of the theory of computation proofs, which supposedly are difficult for many, were completely intuitively easy for me.
i think mathematicians are facing a similar confusion as computer programmers. the medium used to require precise thinking and the simple act of reading, writing and composing it was a mechanism for thinking and learning. in the llm era, the question is: should there be a new mechanism and if so, what should it look like?
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
> usually by people who are good mathematicians but know close to nothing about teaching.
I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens.
And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
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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.
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> 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.
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a good teacher remembers the journey, not just the destination.
socratic method exists. almost none follows it.
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"How To Prove It" is used to initiate people. It was required reading for an introductory class on formal mathematics at university.
Intuition happens naturally and people are prone to inducting the wrong conclusions. Proofs provide a framework for rigorously analyzing drawn conclusions such that it can be used to build intuition in others. If math is about sharing the insights gained in a particular class of problems, proofs are the means to getting there.
> a very precise language of communication
I find that difficult to match to my own experience, in that there is seemingly endless domain specific notation that heavily obscures communication
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The math field is confronting something that coders have been dealing with for a few years now, only far more violently. Today's moat for software seems to be that AI can automate tasks but not a full job (yet). But for a large proportion of mathematicians, doing these tasks really was _the_ job. It's the bit they wanted to do, and if they completed a sufficiently difficult set of tasks, they got tenure. Now this model is failing, they frantically need to pivot the role of humans to save their profession from funding cuts.
I remember when "writing code was never the point" became a mantra here. There was truth in it, but removing the coding has certainly taken away a lot of the texture of the work and enjoyment of the craft. Many of us feel this loss as we tech-lead teams of agents as our source of income. I am not optimistic the mathematics pivot is going to work, but I'm certain that most will be depressed with the outcome even if they succeed.
We are all staring at the same existential dread, just seeing it unfold slower. We're being told that utopia is to be obsolete, and that is a jarring idea to contend with.
I'm just waiting for "Humanity was never the point, let our AI children transcend and replace us".
Maybe it's time to reject utopia.
I feel that you can see quite strongly the truth in “writing code was never the point” when you encounter inevitably at every company the guy who has been around forever but doesn’t seem to be working particularly hard. Their value is (was) no longer in writing code at a furious pace all day. It was having a coherent, intelligible and communicable theory of the software system the company is founded on.
I propose this thought experiment: put all living mathematicians in a very long bus. This bus crashes and they all tragically lose their lives. Can we really say mathematics simply marches onwards with AI alone? Let’s say Anthropic needs a new research result to improve Claude. Are we really already at the point where we burn tokens ad infinitum and arrive at the end of scientific progress in some timely fashion?
I actually have a rather dim view of the "writing code was never the point" line. Not because it's objectively wrong, but because I see it as something we're mostly telling ourselves to feel better about the status quo. Ability to write good code has been highly celebrated (and remunerated) for decades. As it is becoming less relevant, we immediately backtrack and start lionizing the parts where we can still be useful instead. Consider the counterfactual - AI continued to be terrible at writing code, but weirdly better at humans at product decisions, architecture etc. In this universe, saying "coding was never the point" would not be popular.
It also find little solace in it aside from 'well this version of GPT isn't taking your job'. AI labs certainly have no intention for the higher level skills to stay in the human-only domain. The veteran developer with the coherent theory of a large stack is immensely valuable today. But they also don't survive if a company can drop a few coders' salaries on rewriting that stack from scratch - faster, fewer bugs, more coherent, able to react to changing business requirements with more agility etc. I am not saying this is where we are, but I think there is a reasonably good chance this is where our road is leading us.
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I posted nine days ago in another thread:
As much as I hate it, I don't think we'll ever get a proof of the four color theorem that isn't enumerating cases.
When you have an integral or the sum of an infinite series that comes out to pi, you know there must be some satisfying explanation involving a circle.
Contrast with "Examples of patterns that eventually fail" on math stackexchange[^1]. When a pattern ends at 906150257, you don't really expect the proof for that to be something beautiful. The reason for the exact value of an upper bound is that it isn't smaller and it isn't bigger.
There's a relationship between e, i, pi, and -1 comes from a deeper relationship between complex numbers and rotation.
The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
[^1]: https://math.stackexchange.com/a/111461
> The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
Well, the current conjecture is that it hasn't to do with coloring at all, but how many, on a map (partitioning of plane into connected regions), regions can pairwise touch: 4.
(That is: If someone proves Hadwiger's conjecture, then the four colour theorem follows.)
This is, as always from Grant Sanderson, thought provoking and opens new insights.
It occurs to me on reading this that there’s a connection to other computerized mathematical activities.
Occasionally some computer lab in the past would announce that they have computed pi to more digits than ever before; or a new Mersenne prime will be found.
These count as ‘math news’ but they’re of little interest to mathematicians. These computational efforts demonstrate the great power of computers but they do nothing to advance mathematical understanding. Finding a larger Mersenne prime is not surprising to anyone; we’re pretty sure there’s an infinite number of them. Finding the largest Mersenne prime would be the surprise.
So it is with proofs. An LLM might prove some conjecture - Riemann, say or P≠NP. But in general we know that things can be proven and we think those things are probably true, so the existence of a proof doesn’t do much more than producing a new Mersenne prime does.
It’s only if in proving the thing we learned something that there’s actual value in the proof.
Part of the controversy here is that now the skill advantage that some Field Medalist had is much narrower. The fact that fields medals have an age limit implies that it favors brain power over understanding. And that was the guiding light award of the community. So i find it "funny" (and natural) when they are offended by AI. That is the main "crisis" of mathematics.
In my opinion there has never been a better time to be a mathematitian, and there has never been a better time to be a software builder.
But there has never been a worst time to have the need to prove your economic value as a mathematitian or software developer alone. Because "understanding" is not something you can prove in one afternoon, its something that you prove with a life.
> In my opinion there has never been a better time to be a mathemetician...
As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible. It's like telling a chair-maker during the industrial revolution that there had never been a better time for them, since now they could operate chair-making machines instead of toiling away at the wood themselves. It assumes that they were purely in it for their passion for mass-producing chairs. The majority of mathematicians get into the field because they love problem solving, and the gauntlet thrown down by challenging math tasks.
Many parts of this will never be useful for society on a grander scale - but this is reflected in the finances - pure math is closer in funding-terms to a humanity than to hard science. Now even this is _massively_ under threat, and Tao and co need to pivot quickly to stop this from becoming a bloodbath.
Im an ex mathematitian too. And if i was in academia I would probably have the same reaction. Thats what i say that its the worst time for proving economic value.
But if you are in for theory building and understanding, then you are not constrained anymore by your motivation to grind through countless hours of formal theorem proving. And you do not need to have superhuman formal manipulation skills and memory.
For me mathematics is not the formal system, so LLMs will never be able to do end to end maths.
> As an ex-mathematician I assure you this is very wrong, and every working mathematician I know right now is completely miserable, and/or trying to flee the field as fast as possible.
To throw a counterpoint to this into the writhing cesspit of HN, I'm active in academic mathematics (postdoc) and every one of my collaborators is deeply in love with the field and their jobs. Perhaps the grass is greener on the applied mathematics side of the fence.
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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.
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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.
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> In my opinion there has never been a better time to be a mathematitian
I think it’s a great time to be a curious mathematician, especially in a niche field where you’re not competing with hundreds of agents of the best unreleased frontier models.
However it’s a very scary time to be a professional mathematician because publish or perish is going to cause a race to the bottom for cranking out results as fast as AI can let you. [0]
[0] https://ev12183725.substack.com/p/a-highly-productive-dark-a...
Spot on. I agree. There has never been a better time to be a software builder or a mathematician.
Seekers whose primary motive is validation instead of understanding are the ones who are getting paranoid.
Thoroughly enjoyed your thoughts. The age limit is a joke if what you care about is true understanding.
I'm a professional mathematician. Today I proved what for me is a very solid theorem. It's something I had thought about for a few years. With a few weeks of serious use of AI I've found a proof that I am currently trying to write up, but which appears correct. The change in the workflow is enormous, but so is what one can do if one has clear what to do and how to do it.
Not every problem will receive $20M in funding to be solved by AI; for the rest, good human guidance will have to suffice. Labs only pulled this stunt because they wanted to show investors how powerful their models are on their own. But look again at the cost of that army of 10,000 SOTA agents. At the very least, I foresee a need for humans to decide when costly AI resources should be committed to a specific search plan. Grant review remains irreducibly human because it involves choosing which directions to fund and weighing opportunity costs: taking one path forecloses others.
Congratulations. You are one of those leading the way, showing how we will adapt and how the world will get better from AI.
That's not the conclusion. I started using AI after the Jacobian conjecture counterexample and have used a particular problem to learn how to use AI and to explore it's capabilities. I'm not a great mathematician but I'm full faculty with 25+ years of research experience and lots of articles and I just proved in a few weeks something that had resisted my efforts for some years.
The exploration process is much easier now. Ideas are quickly testable and multiple tests can help identify a technical obstruction. The tool requires good guidance and input but as it trains on people like me it will need those less.
At the very least our way of doing things must change. More pessimistic views seem to me defensible.
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I find myself less worried about it than at first. I think what we'll see are that some things are low-hanging fruit and can be solved just by tireless search. Maybe half the millennium and other such high-visibility problems will fall this way.
Others, I think, will be beyond both human and AI. And so what then? Mathematicians just throw in the towel and say it's not worth trying? Of course not. We will continue that pursuit, and as we do, new ideas will arise and new problems will need to be solved. It's math. There is no end.
It's easy to look at the current landscape and see AI ticking off solutions to problems and imagine that soon there will be nothing left. Machines replaced the need for much manual labor, but they also established a basis for an economy that provides the opportunity for more labor. This is the situation with math now. It will take some getting used to. There will be little-to-none pencil-to-paper working out of problems anymore, but there will always be work to do, things to solve, curiosities to unravel. And it will still be professional mathematicians who are the ones most capable of directing that effort. Because, if nothing else, they're the ones whose curiosity is piqued by the problems. Which, let's face it, has been 99% of the motivation for graduate-level math in the first place.
There's the the old question: is math invented or discovered? I think it's both: the problems are invented, and the solutions are discovered. In the age of AI, the discovery part will be greatly affected, but the invention part will remain firmly in the human domain.
I was a PhD student for a while, but I always enjoyed "refactoring" proofs, more than coming up with new ones. Making them simpler, shorter, clearer. Unfortunately, it's not much rewarded.
One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.
True.
Having to ask the village elder about how to do things meant learning various other incidental lessons. It was reduced when books started to become available.
Having to search in the library also often lead to serendipitously seeing a book on the next shelf and falling in love with a topic you didn't even know about otherwise. Or you had a chat with a librarian asking for advice which books to look in for a topic. Google search eliminated that. With Google search and reading websites, you still had to read or skim the page and may see some other info or click to read the author's About page. Now with AI we get straight to the answer.
Don't worry, in its final reply the agent will always present one or two things "worth flagging" to keep you engaged
I do neither maths nor science with AI but in my experience most models are perfectly willing to burn tokens on a ton of sidequests at the earliest opportunity.
Yeah but do you read through and find if one of those side quests is useful?
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We'll get innovations in AI as a side effect.
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.
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> so this seems like missing the forest for the tree imo.
But is it? Again, regardless of the amount, what is the utility?
What's the end goal of academia (assuming broadly as research with humans) when the answers to the deepest questions become commodities at orders of magnitude higher speed and lower cost?
How are federal grants justified and, forgetting the current hierarchy, how is differentiation made? It's currently based on research output, once that becomes irrelevant what is it? We already have IMO, IOI as competitions and I assume just like Olympics this can be a thing, but it's very remote from research.
Take for example Rene Thom after Alexander Grothendieck overshadowed an entire field
> His technical superiority was crushing. His seminar attracted the whole of Parisian mathematics, whereas I had nothing new to offer. That made me leave the strictly mathematical world and tackle more general notions, like the theory of morphogenesis, a subject which interested me more and led me towards a very general form of 'philosophical' biology.
https://mathshistory.st-andrews.ac.uk/Biographies/Thom/
Now amplify this a few orders of magnitude.
> Tao's concern
The article isn’t by Tao, it’s a guest post by Grant Sanderson (aka 3Blue1Brown).
Indeed. I missed the top row. Thanks for pointing that.
This goes in a necessary direction, from my personal take away of Gower's recent post on the subject.
Mathematics is suffering from Goodhart's Law:
"When a measure becomes a target, it ceases to be a good measure."
Doing something difficult was a signal that you:
a- understood it and all the background information it requires
b- internalized techniques and methods that are helpful in problem solving in general
Now it just means nothing
> Now it just means nothing
Now it means we can move on to other difficult shit.
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Some were like this, some weren't. Many people were in it because it's objective and factual and not up for the whims of taste of some established gatekeeper. They weren't in it for art performance reasons or to please the aesthetic judgment of some entrenched mathematician-baron.
Timothy Gower
No. It should have been Gowers' rather than Gower's.
The link is:
https://gowers.wordpress.com/2026/09/17/why-i-didnt-sign-the...
I was about to link the "discovery fiction" article [0] by Michael Nielsen, but halfway through saw Grant mention it as an inspiration. It's one of my favorite styles of writing that lets you settle into a cozy read of a narrative around a problem while slowly exposing you to the rigor and nuances of tackling it. It's definitely deserving of a public index of some sort, like one of those "Awesome X" list repositories on GitHub, because I think it's a valuable style that deserves to be curated.
[0]: https://michaelnotebook.com/df/index.html
> It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.
This is really interesting and available here: https://www.youtube.com/watch?v=IbGNZQvobkc
> we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems
The core idea seems to me that we should shift the standards for professional evaluation from generating proofs to generating explanations. Makes sense that such a proposal would come from the 3B1B guy, and I actually agree with it, irrespective of AI. But what eludes me is how that could be a defensive mechanism against AI automating humans out of mathematics. AI is likely no less good at producing natural language explanations as it is at generating rigorous proofs. It's telling that even Terrence Tao turned to AI to understand AI-generated results [0]. It seems that the essay doesn't address that issue at all.
[0] https://news.ycombinator.com/item?id=49010345
It’s a task much harder to RL and much more subjective. I don’t want to say we won’t get there, but let’s just say that LLMs could “write” well enough since gpt3.5 era and I don’t think the pleasantness of the prose improved dramatically since then.
And subjectively the explanation LLMs currently provide are usually horrible, horrible enough that I usually just instruct them to provide me human written literature I can read.
I mean, there's centuries' worth of mathematical prose to train on. But that's presumably already in the training data, so if it isn't good enough today, it might not get better fast enough to keep track with how fast they'll get better by training on formally verified math. But then again, the prose in Terry's conversation I linked above seemed pretty useful. But it's also a problem requiring famously little advanced mathematics.
I have been losing interest in this proof-oriented approach into extremely abstract concepts (what seems to be the core of academic mathematics today). Obviously, proofs are very attractive because they are the closest thing we have to a universal truth (at least under the assumed axioms). Having mechanisms to reliably show a proof, and computational methods to handle complicted proofs is great.
But.. the navier stokes proof was the last straw for me. People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society. And certainly not moving us towards "human understanding". The biologists are the ones working on that, the math folks should try working with them on neuro stuff to understand how human brains can do math at all, given their architecture.
> Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society.
Is it, or is it only the parts you hear about/pay attention to?
> People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).
Yes, maybe famous solving famous conjectures is just trivia and trophy collecting and the real value is the intuition and techniques you develop along the way to solving them which you can then bring to bear on things like CFD.
Mostly the parts I hear about. But I also communicate with mathematicians on a regular basis and I can see what they are working on, which is representative of the field. It's about 90% "cohomology of abstract Lie groups that morphize into string theory" and about 10% "practical thing that engineers can use to make my cell phone work better". (amusingly, although I literally made up that sentence, it turns out it's not far from something somebody worked on: "The cohomology of compact simple Lie groups and spin groups connects to string theory by providing the topological obstruction classes—specifically the first fractional Pontryagin class ...." OK, kind of emphasizing my point there.
No, the intuition and techniques that get developed to solve the NS conjecture have little or no bearing on the practical details of doing CFD. That's a common story/thread (and I hear the same story in quantitiative biology, my area of expertise), but often times, it's just a loose justification given to justify funding.
Dumb question— why doesn’t proof proposal construction (not the solution) in lean get celebrated more ? That seems central to understanding.
If these proofs are so important why is there not a central repositories of the proposal in a formalized language ?
Current and future mathematicians can now spend most of their time coming up with problems/conjectures and theories that are hard for a future model versions (6 to 1 year out) to solve them and by itself it can be a new major sub branch of mathematics - “Theory of perplexing frontier models” and who knows it can even open up new dimension of mathematics for mathematicians to explore by themselves (because AI by definition cannot help them here). Now is the time to be excited for mathematics!
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.
The value in academics is teaching and research. The value in research is discovery. Proof was a useful function for humans to do towards discovery until recently. Understanding is a useful property insofar as it helps you teach and it is a prerequisite for generating hypothesis. Humans will always be driving discovery, the tooling and focus of work may just be a little different. Attachment to one particular modality of discovery is an aesthetic choice, not a moral one.
Hmmh. I like motivated explanations, but, as acknowledged in the text, this is a subjective thing to measure. What is a great motivated explanation for Tao, might be hard to grasp for me. So I guess judging how well an explanation motivates something depends on two things: 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
There is a third thing: how well does the motivation chime with or go against my current belief system? You would think this is not much of an issue in mathematics, but it can be, and I had my fair share of frustrations because of it.
Anyway, all of the above points to one thing: the best motivated explanation will be generated by an AI, knowing the subject and you in a deep way that no other human will, and being able to interact with you during the explanation.
> 1) My way of thinking, and 2) what I already know and how well I recall it in this context.
I don't think we're discussing pedagogy. Good _research_ exposition is instead related to communicate your intuition and way of seeing things. The conceptualization of a given situation or problem is what is valuable, how you connect it with other stuff, etc. It is then up to you to memorize and interiorize it.
Good research exposition is the same as good "pedagogy" just for a different audience. Both have to consider didactics. When writing a paper, you teach something to the fellow researchers who know less about a thing than you.
Seems to me to be two different sides of the same coin. Pedagogy is about finding a way to communicate to me an idea based on my intuition and seeing things. A research exposition is about presenting the idea in terms of your intuition and seeing things.
What's an example of your belief system conflicting with a motivated example? I'd love to understand that a bit more.
One example is what currently plays out, see the previous guest post on Tao's page: https://terrytao.wordpress.com/2026/09/12/after-math/
The blog post says that the statement "AI really did solve a problem in mathematics." is wrong. But a formal proof showing that Navier-Stokes equations can blow up is certainly such a solution, by AI. There is not much in this world that is more objective than a formal proof, so any disagreement on this is based on how we see the world. Michael Harris will agree with the statement being wrong, Jacob Tsimerman will not.
Another example, Hilbert famously battled Brouwer's view of mathematics. From my point of view, Hilbert was right: intuitionistic logic is certainly interesting; but I like to study it using "normal" (= classical) mathematics.
Finally, my personal frustrations are about how hard it is to publish my work on abstraction logic. I would never have thought it is that difficult, mathematics being objective and all. It seems essential to take out as much motivation out of your paper as possible, because it might offend your reviewers and their belief system. By now my papers come with full Isabelle/HOL formalisations, let's see if that helps.
If I have to take the risk of simplifying,
1. We humans have managed to take huge amount of information and compress it using a loss function containing some bias we have about the information.
2. We now ask ourselves to decompress the same information with some additional cross-entropy. As a side effect of this process we sometimes spurt out information that may or may not have any meaning since the compression was lossy.
3. Now, we ask ourselves to present this some-what newly decompressed information with brevity in order to understand what we've learned from it.
Knowing that this process is happening on a larger scale, this resurfaces the argument if meaning can be reduced to computation only.
Although some might favor this argument but we are at the risk of anthropomorphizing this process.
The idea presented in the post itself is perspicuous (in Grant Sanderson own words) as he always does.
> decompress the same information with some additional cross-entropy
What do you mean by this phrase? I know what cross entropy and data compression are.
The last days we are served these high goals about understanding, "digestion" and so on.
But if you look at the practice of present mathematics, in the last 20 years it is all about publishing solutions to problems.
There are famous problems to be solved, there is a hierachy of conjectures to be solved. A quick search here on HN gives pearls like "Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem".
And all of a sudden it turns out that problem solving can be automatized.
So then what will problem solvers do? Well, from now on they will "digest" problems solved by AI.
In a way or another they will find a way to stay on top.
That's the goal, at least, but mathematics as a living practice does not have much to do with these games of power.
With hammers do we build our mud huts more easily and sit back to rot? Or do we build more complicated structures and do it more quickly?
“You are right. There is nothing in yesterday’s mathematics that you can prove with exterior algebra that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwile a mathematical enterprise as proving old conjectures.” Gian-Carlo Rota in “Indiscrete Thoughts”
The issue here is not AI--it's academic papermill culture and paywalled journals.
AI gives us greater freedom to "stop and smell the roses", explore hidden structures, etc in mathematics. It is a dream come true for curious minds.
Yes. AI is a useful tool and we are going to adapt and use it.
The phd student will be forced to publish 10 breaktrough articles, the university department which does not offer "free" access to AI (for its members) will see the its ratings going down, when compared with the other universities.
It will be "use AI or perish" for academic management so on the side of academic management the ones with vision will thrive and the ones without will perish.
But what about the publishers? In the last decades the academic research was made into a feeder for publishers. The main goal of a researcher is to write articles, which are later sold back to other researchers.
This economic system is under big stres now, because for a while at least the academic management and publishers will have contradictory goals.
And that is why this scare which is induced by those who profit the most from the present system.
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If this suggestion were to come to pass, I wonder how new math PhDs would think about choosing between a 'normal' R1 faculty job vs. the "teaching route" (teaching professorships, lectureships, community college professorships, or SLAC professorships).
It's been my understanding that traditionally the ones who care about "motivated explanations" in this sense go for the latter, but if the research community has now decided they care about teaching and understanding, it might "even the playing field" and make the jobs more similar.
Yeah, the early insight is important to preserve in students.
Evening the playing field would mean those R1 professors now teach five classes a semester for 50k a year instead of doing research.
Taught proofs too, and plenty of students fake intuition with pattern matching.
Let's see how long (if it ever happens) it takes for models to generate motivated explanations (possibly done via the Manim library or something like it) along with their Lean proofs. Grant Sanderson is right that this is kind of subjective but so is Art and I'm very enthusiastic about AI generated Art.
> possibly done via the Manim library or something like it
It can be done already: the point is that the motivation and explanation parts are terrible, especially for novel topics where the AI can't just rip off existing content. A Lean proof is at least a verifiable task; you end up with an actual proof that you can work through. A Manim slop video doesn't have that.
I think the thing is.. sorry Grant, but "motivated understanding" won't come from videos, but writing. Solving things too, but a video can only go so far.
I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).
I'd be interested in hearing a field report on this! For example, I can easily imagine that they're great at walking through the proof step by step, explaining background as necessary; but as TFA notes, one of the most important questions is "why is this definition the way it is?", and my bet would be that the Lean is not enough to help the LLMs meaningfully in answering that.
LLMs like even the sota flash models have great range of background math knowledge and have no problem reading and understanding flt level of math. On the other hand you don't want to go through 13 million lines of often repetitive code line by line. Models are great at synthesizing math content out of code. My contribution is to steer it through subjects of most interests to me, drill down into jargons that can be confusing, be creative in using computation for illustration (which coding agents can execute very proficiently) etc.
Fortunately LLMs are smart enough to handle that already.
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.
It would be interesting to see what would happen if we had two competing mathematical institutes, a sort of First/Second Foundations:
1) Rejection of AI for anything but trivial applications while still using computers at their full capacity. Researchers would ensure full human understanding of proofs and methods. This Institute believes on Math as a process of discovery, Mathematicians as explorers/poets/storytellers and not proof machines.
2) Unrestricted, all-embracing use of the latest AI, including potentially research in creating even better AIs as part of the program. These researchers would be okay with not understanding proofs if verified to be correct. This group is focused on rapid problem resolution and believes Mathematicians are theorem creators and provers.
After X years (100?), which one would advance Mathematics and humanity the most (we'd need to define "advance")?
Mathematicians worry about proofs and the intrinsic value of something as elusive as 'understanding'. They are deeply ingrained in the study, deeply concerned with anything effecting the field. Yet they're still emotional beings looking for beauty and meaning in life that might come from an understanding how the universe works purely from a math perspective. I'm glad Mathematicians exist, I certainly can't do that type of work.
And I trust their results: technology wouldn't be possible without advancing our understanding of the world in various fields, including math.
Your idea sounds great for the Mathematicians.
There's a more pragmatic view though, and unrelated to proofs themselves: does understanding a proof help us to advance Humanity in some way?
Do we have better lives afterwards? What if we give up understanding proofs and focus only on results.
In other words, if an AI solves a problem for you, but you don't understand how it works, should you continue building anything on top?
I suppose the results are truly what matter. If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
Your second approach seems good to help figuring out results from both theory and application of math to solve problems.
But also, what if there is no true beauty in Math, the way Dirac and Einstein wanted?
What if these AI brute force proofs are all that's left?
> If AI solved cancer, disease, anything that lowers quality of life, but you have no idea how it did it: is that good enough?
A lot of medicine is already like this. Shown to work in clinical trials, no complete end to end mechanism understood. They still get approved if the empirical results are strong.
I'll answer one of your points partially: if AI builds a better sorting algorithm and proves its performance characteristics, it's useful. I'd be able to use it to make my programs faster even if I don't/couldn't understand it.
It would be a bit disappointing but still useful and make humanity slightly better.
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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).
Software is logic applied to intersubjective truth. It's not physical truth which is the subject of the hard scientific fields such as physics and chemistry, as well as biology for the most part.
So no, software is much less than science.
It's time to stop solving logic problems and start solving the more difficult philosophical problems, like the hard problem of consciousness.
I'm not sure that this new approach will be AI-resistant. Why would people not use AI to help in creating the "motivated explanations". Maybe they can't be one shotted today, but AI also makes this easier.
Assume in 2 years we have a heap of these motivated explanations, all as high quality as Grant's videos and the best books. But who will read them? There is limited interest in this genre. Grant reaches a large fraction of this audience but most people really don't want to think about math either way, no matter how good the explanation is.
Indeed, there is now "edutainment slop" online and AI can use 3blue1brown's manim library to copy his style and AI can use blender and video generation to mimic 3d animations of other explainer channels. Today it's still slop, but it may not be for too long. And then people will have to reframe their job until it's "doing X while also farting and burping every now and then", and then a machine will be better at that too eventually.
Also, this new style of doing math will appeal to a different set of people. Many mathematicians aren't super social, they just like to explore a problem on their own. Think Grigori Perelman. They will still face the problem and their temperament may not make it easy to switch to being a communicator.
Last summer Grog was still celebrated and admired for bravely piercing animals with a spear and bringing home the meat. But now Goong made this newfangled arrow and bow thing and any cowardly fool can now shoot animals from a distance. Grog devalued. Grog sad.
I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.
You don't think LLMs can translate Ancient Greek poetry or analyze 18th century novels?
Of course they can. My point is that mathematicians may increasingly find themselves in the same position as academics in the humanities. Mathematicians themselves will be able to see the inherent value of the work they're doing (just as experts on 18th century novels can in their own field), but it will be far less obvious to society at large why their work should be funded.
Interesting headline.
It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.
> don't even include proofs in their mental concept of what math involves
Technically, the largest majority are the people who go: "What are proofs?" :P
A majority? I doubt it, simply because the majority doesn't know what math is at all.
At university level introductory calculus, the person teaching class had to reassure students that math wasn't entirely arithmetic or adding up numbers. He did this because it's a common misunderstanding.
So, you disagree with my comment because you think I'm right?
Those students he was reassuring, did they think math was nothing but proofs?
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Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".
The theorem thing is invented by human to help other people better understand Math structure in a easier way.
Give teachers and professors 1% of all future earnings of every student
Math academia 2025
> Sorry, only epic problem solvers allowed here
Math academia 2026
> We were more than just problem solvers
I think people are overblowing this though. Wake me up when GPT-whatever writes gcc from scratch, then by the Curry-Howard I'd be impressed
You missed this from last year
https://github.com/anthropics/claudes-c-compiler
Also I dunno why you should be impressed by this - gcc isn't anything near eg navier stokes
There must be like a 1000 examples of c compilers in git hub alone. No idea why building LLM building one is impressive. It's right there in the training data.
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Okay tell me why that's not even close to gcc. You can use an LLM
Well, so Tao now invites literal industry boosters to lure mathematicians into a pro-AI stance. This is the guest poster:
https://www.3blue1brown.com/talent
The only concrete step any mathematician on the internet, including on the other AI concern site https://proofsandprompts.com/ , is demanding funding for an academic frontier AI.
Strange that the Poincare conjecture was solved by a hermit without all this AI bullshit. Maybe reject AI, ignore all AI proofs and retreat from the internet.
I fully agree that motivated explanation is more important than proof. This doesn't resolve mathematicians' feelings of existential dread, however. Machines will get better than human mathematicians at motivated explanation in another year.
There will be no more glory in mathematics, but at least the joy of understanding will remain, and it will come without deciphering the tortured proofs that machines output today. Each bit of understanding will come with much less struggle, but this just means we can get more understanding for a given amount of struggle.
Jacob Tsimerman claims [1] we might have superhuman expositors by April, so then what?
[1] https://youtu.be/H7_d_sgui6o?t=4436 (timestamped url)
The people building AI claim it will surpass human intelligence in all respects and prerhaps kill us all. Should we just cease all human activity on the basis of what AI might do in future?
Personally, I doubt AI can surpass a good human explainer because explanation requires empathy, which benefits from being an instance of the kind of entity you are explaining the thing to. That gives you a way of exploring and evaluating the space of possible explanations that isn't available to an LLM.
Thanks for pointing me to this video - it's been interesting to follow the discussion! (I personally don't see that math has lost its purpose at all in the past months. I mean, where would we be, if we were thrown at these AI based mathematical proofs and had no mathematicians and specialists?! Much of this discussion is about a disciplin readjusting its way of work and tasks.)
You get $100 billion for an AI X that generates the proofs, $100 billion for an AI Y that explains the proofs and another $100 billion for an AI Z that reads Y's output and appreciates it.
Humans meanwhile scrub the floor and write blog posts about "what a time to be alive".
> Humans meanwhile scrub the floor and write blog posts about "what a time to be alive".
I'm pretty sure we already have machines to do both of these.
It's all good until we have superhuman appreciators :)
That which has received the Mandate of Sapience must be cool (eg any skateboarder skilled enough to impress his mom)
LLMs got It, sometime this year. Terry Tao and Friends have appatently lost It, same time this year. (This is not to claim that the Mandate gets extended to their creators the frontier labs or even Jeff Dean et al. Definitely not their sponsors. Howbout distillers?)
The thing about Mandates--- they are not forever. LLMs can "lose" It. Probably not back to mathematicians -- that'd be atypical (unless they quickly learn to "make their own lightsabers"?) . Likelier: to a scene of humans no-one yet thinks about.
Skateboarders didn't take the Mandate from anyone. So no one takes It from them. There's some karmic law at play
It's too late for this. Much of my work in math has been classified as trivial or best described as not math at all. When I was working on chatbots and described deep learning algorithms to enhance them, it was deemed as a pseudoscience. Mathematicians sound very disingenuous with their backtracking.
I'm afraid that much of mathematicians work is too trivial to be taken seriously and they should just find something completely different to do.
Logic is the foundational weapon operating on sentences.
The act of stitching together, a series of sentences as true is what logic is.
If you make the stitching as airtight as possible, congratulations, you are in the realm of math.
If you are stitching together reasonably similiar to how the masses do, congratulations you have common sense.
If you stitch together completely random sentences, you are in the realm of nonsense and you may be classified as a retard.
The weapon is the same. The discipline differs and hence the effort to produce the chain.
So I am not at all worried about LLMs producing math proofs.
Godel with his incompleteness theorem helps one sleep easy. Rest assured no LLM can fly above Godel Incompleteness theorem.
There will always be statements that are true. So yes, it is time to celebrate.
Hate to tell you mate, but this is rather close to stitching together random sentences..
Sadly, your inability to comprehend is noted which leaks your lack of expertise with the subject matter.
For a general overview, assuming good faith and a genuine willingness to learn, refer to https://iep.utm.edu/s-truth/
Its a remarkable intro into propositions, statements and sentences with vivid examples from the works of Tarski, Godel and others as to what constitutes truth.
Pay attention to Tarski’s T-Scheme (sentences and truths)
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Symbolic manipulation relationship with narrative selection for transformational bridges of understanding.
if language models (current GPT-style or any future model with different architecture) can be viewed as a compression of their output corpus (the corpus it emits by providing random starting native contexts weighted by the model's likelihood of that native context) then one is saying the corpus contains the same information bits as the model, the model just takes less data bits, while the corpus represents inflated data bits.
If learning requires communication of data, one could ponder if it is more effective to transmit learning data as suitable model coefficients instead of corpus monologue or dialogue (say with a teacher).
This may sound absurd, but is it really so far fetched to believe one could formulate "download"-compatible LLM architectures, so that the user or student can play a kind of memory game to train "download"-architected model weights by simple reflex games?
Suppose instead of token vectors we used token matrices or token multivectors as in geometric algebra (not to be confused with algebraic geometry!).
Word2Vec couldn't do general language modelling because it used vectors, but when using token matrices or token multivectors one could postulate the following partition function:
given a dictionary of tokens, and their corresponding (matrix/multivector) M one could define the partition function
exp(-|M1 * M2|^2)
For example it could be alphabet level (to demonstrate the concept):
For every string one can compute the corresponding M in terms of the elementary character matrices:
M_"car" = M_"c" * M_"a" * M_"r"
both matrix products as well as geometric multivector products are associative but not commutative:
(M_"c" * M_"a") * M_"r" = M_"c" * M_"a" * M_"r" = M_"c" * (M_"a" * M_"r")
But M_"car" != M_"rac" since matrix/geometric products is not commutative.
The result of non-linear but layer-less and order aware architectures could promise the following:
Feasibility of uploading model weights: a random grand tour of 2D projections of the square matrix space, projecting the token positions down to 2 dimensions along an changing axis, a "game" could correctly project 95% of the tokens, requiring the user to identify the incorrectly placed tokens, which jump back to their correct position when clicked.
This means a user can learn (since most of the time tokens are rendered in the correct position of the 2D projected cloud), and as a user learns their performance will go up.
If a user could eventually (after playing for a long time) correct 90% of the token positions on random 2D projected planes, then the user has effectively stored 90% of their coordinates.
Suppose such a user reads text in their mother tongue, then the joint likelihoods in natural text will correspond to matrix products of those weights. In other words the brain will learn that whatever it has learnt playing the token-game aids the brain in predicting a next token.
Once it has learnt to utilize the format (matrix / multivector coordinates), the apparently pointless skill of positioning tokens in the cloud learnt during the game, it would also help predict the next token in languages the user didn't know, say when looking for ingredients on a product package. I believe such a user would swiftly discover they actually know those languages, and I believe transmission of a compressed format would take less time than transmission of the same knowledge in expanded output-corpus form.
I wonder if Terrence Tao has an opinion on such a possibility:
1) does he consider it (im)possible to have token-first formatted LLM's without MLP layers etc reach similar levels of performance as the current crop of LLM's?
2) supposing it were possible, would he predict a user "downloading" weights to his brain by a reflex-game as feasible?
3) does he believe a human brain would be able to make use of the downloaded weights and would the brain notice the utility in predicting tokens?
4) does he believe that once the brain has noticed / learnt how to make use of the model weights, first for predicting the next token in the mother tongue, it would subsequently have learnt how to predict in other tongues?
It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.
Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability. But of course mathematics is all about proof, and for that reason I was wary of it for a very long time.
Fortunately this is not basing science on proof. It's just making it much easier to do proofs when that is what we choose to do
>Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability
not really. You can consider positive proof as an experiment confirming your theory and the negative proof and counter examples as an experiment falsifying your theory.
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Understanding has been the point of mathematics for millenia. The idea that purpose of math is to produce machine-checkable proofs is an entirely modern idea.
Mr. Tao is an excellent politician. Lots of awards and texts, yet no major problem solved.
It seems now that NS is solved he is mobilizing the community to convince taxpayers continue to pay even though AI may do a better job in his work.
Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.
> Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.
Ignoring the other ridiculous parts of your comment, isn't this exactly what you're supposed to do? Update your beliefs according to the newest information available?
Yes , you are supposed to do. That doesn’t change the fact that he has no insight into technology, and is a bandwagon person.
If I’m a mathematician, I can write public posts on math. Extensive posts about politics, AI, crypto, … are not useful without expertise in those domains
Gr8 b8 m8
This post isn't by Tao.
I'm increasingly seeing people read something in a post that isn't there. GP said "Tao is mobilizing the community".
You can mobilize the community through guest posts that slowly nudge people towards accepting AI, so that they demand funding for AI for academics. Which he himself suggests:
https://terrytao.wordpress.com/2026/09/18/sairs-open-math-mo...
He is founder of SAIR.
The EU version is to demand a CERN for AI:
https://terrytao.wordpress.com/2026/09/17/a-cern-for-ai-assi...
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Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.
Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.
Likewise all other technology [ and culture ].
LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.
Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
A moratorium on AI development might be the only way to achieve this preservation of human culture.
I want to agree with this, but I have a hard time seeing how it can be done.
Tao is speaking of a very particular kind of mathematics, that done out of pure curiosity.
But maths, even at the highest levels, often finds applications sooner or later.
It will be economically impossible to justify boycotting correct mathematics that no humans understand on grounds only of purity.
This may happen very soon: one of the obvious applications of novel mathematical results is in building stronger AI models.
> one of the obvious applications of novel mathematical results is in building stronger AI models.
This gets repeated a lot and seems to be one of the primary stated goals of making AI solve math problems, but I still have no idea by what mechanism this is even supposed to happen. I guess they could make some minor improvements to matrix multiplication algorithms or whatever but I don't see what groundbreaking theorem could possibly significantly improve LLMs.
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> Tao is speaking
Tao isn’t the article author, it’s a guest post.
> Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
So many thoughts come to mind at once, they're a jumble in my head rather than a single coherent narrative.
John Henry comes to mind. As does Agent Smith's "I say your civilization because as soon as we started thinking for you, it really became our civilization, which is, of course, what this is all about" monologue in The Matrix. I've not read (or listened to) "With Folded Hands ..." or "The Machine Stops", but I have read the Wikipedia plot summary of both.
Do we want to have comfortable lives, or do we want to serve each other?
"Computer" used to be a profession; I grew up around adults bemoaning that "kids these days can't do mental arithmetic", the Pi Zero I've not switched on for probably a year now could beat all humans simultaneously at that (even if everyone was as good as the current world record holder) and yet we still teach arithmetic in schools.
Nobody needs to knit, and yet we do so for fun. Youtube's "Primitive Technology" channel, which has spent around a decade speechlessly making iron from bacterial slime found in a creek, using only clay and sticks and leaves and vines naturally found next to that creek.
Like I said, no coherent narrative. It's been a while since my stream of consciousness became a river delta; usually at worst it only meanders a bit.