9 ms·
If math is more than proof, we need to better celebrate the rest of it
- kurthr 12d agoThis 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."
- lacedeconstruct 12d agoDoing 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
- Razengan 12d ago> Now it just means nothing Now it means we can move on to other difficult shit.
- lacedeconstruct 12d agoThere is a finite capacity/time for a human mind to do difficult shit, if it can be slop forked in a microsecond before you even get to flesh it out there is no point
- Razengan 12d agoWell then you have time to invent some new difficult shit. 3Blue1Brown: The last IMO problem AI could not solve: https://www.youtube.com/watch?v=Nbwv5wHQoj0 https://www.youtube.com/watch?v=Nbwv5wHQoj0
- busyant 12d agoAnd what would that be?
- Razengan 12d agoHave you seen the night sky, outside a city?
- jplusequalt 12d agoSpace colonization will never happen.
- Razengan 12d agoPeople like you will always happen.
- jplusequalt 12d agoPeople who understand the vast gulf of technological advancements (some bordering on fantasy) between what we currently have, and what needs to exist for it to happen?
- busyant 11d agoI have. Was that the hard part?
- js8 12d agoToday, the difficult shit is how to get the resources from the people who use them to turn difficult shit into an easy shit. Basically, a war. I am not sure I want people to move on to that.
- vatsachak 12d agoTimothy Gower
- kurthr 12d agoNo. It should have been Gowers' rather than Gower's. The link is: https://gowers.wordpress.com/2026/09/17/why-i-didnt-sign-the-fields-medallists-letter/ https://gowers.wordpress.com/2026/09/17/why-i-didnt-sign-the...
- bonoboTP 12d agoSome 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.
- E-Reverance 12d agoJacob Tsimerman claims [1] we might have superhuman expositors by April, so then what? [1] https://youtu.be/H7_d_sgui6o?t=4436 https://youtu.be/H7_d_sgui6o?t=4436 (timestamped url)
- ViscountPenguin 12d agoIt's all good until we have superhuman appreciators :)
- oliculipolicula 12d agoThat 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
- omnicognate 12d agoThe 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.
- steinwinde 12d agoThanks 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.)
- trhway 12d agoIt 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.
- card_zero 12d agoBasing 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.
- trhway 12d ago>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.
- card_zero 12d agoYes, really. "Positive proof" opposes the concept of falsification. You can only have it within a system formal logic, and science can contain those, but isn't one.
- trhway 12d ago>"Positive proof" opposes the concept of falsification. no. Positive proofs have nothing to do with falsification. They just tell you that there is no point in spending effort on searching for negative proofs and counter examples. They don't prevent nor prohibit you from spending that effort. They just advise you that that effort will be wasted. It is like nobody prevents from experiments to turn lead into gold. Of from searching for a right angled triangle violating Pythagoras.
- random3 12d agoWhile 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.
- freehorse 12d agoThe 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?VersionId=imADDfY7jVza5Xub5KBWV8xhvuwkZyh7#:~:text=MPS%20%2D%201,2023%20Base%20Plan1 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 https://ncses.nsf.gov/pubs/nsb20257/academic-r-d
- D-Machine 12d agoThere'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.
- bell-cot 12d agoLast I knew, at least in America, universities that want their math prof's to do research also expect those prof's to bring in plenty of outside funding. You could argue about the costs of that desk, paper, pencil, and such - but modern "research" universities have evolved into extremely high-overhead operations, and The Beast Must Be Fed.
- fspeech 12d agoI enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human 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).
- Smaug123 12d agoI'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.
- someguynamedq 12d agoFortunately LLMs are smart enough to handle that already.
- fspeech 12d agoLLMs 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.
- ForgotMyUUID 12d agoI’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.
- conmod278 12d ago[flagged]
- Geof25 12d agoPeople 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.
- partyficial 12d agoa good teacher remembers the journey, not just the destination. socratic method exists. almost none follows it.
- smy20011 12d agoEven 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.
- thaumasiotes 12d agoInteresting 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.
- Terr_ 12d ago> 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
- the_af 12d agoA 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.
- thaumasiotes 12d agoSo, you disagree with my comment because you think I'm right? Those students he was reassuring, did they think math was nothing but proofs?
- the_af 12d agoHaha, sort of. I guess I disagree with your claim the majority of people think math is "more" than proof. The majority of people think it's less: they think it's doing high school arithmetic. Most people don't know what a proof or a theorem are. They think math is doing calculations with numbers.
- aborsy 12d agoMr. 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.
- vatsachak 12d agoGr8 b8 m8
- deleted 12d ago[deleted]
- traes 12d ago> 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?
- aborsy 12d agoYes , 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
- qbit42 12d agoThis post isn't by Tao.
- qtrz-qpo 12d agoI'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-model-initiative/ 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-assisted-science/ https://terrytao.wordpress.com/2026/09/17/a-cern-for-ai-assi...
- vatsachak 12d agoMath 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
- mathisfun123 12d agoYou missed this from last year https://github.com/anthropics/claudes-c-compiler 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
- karmakurtisaani 12d agoThere 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.
- vatsachak 12d agoI said gcc, not a C compiler
- karmakurtisaani 12d agoTrue, it does more than just c. Does it do anything the usual LLMs don't have in their training data tho? I doubt it.
- vatsachak 12d agoNo, not that is does anything more... It's that gcc is an old and reliable piece of software built on abstractions that have stood the test of time.
- mathisfun123 12d ago
- accurrent 12d agoOne 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.
- jochem9 12d agoWe'll get innovations in AI as a side effect.
- blfr 12d agoI 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.
- accurrent 12d agoYeah but do you read through and find if one of those side quests is useful?
- someguynamedq 12d agoYes, why not?
- blfr 12d agoThey're often directly applicable: minor bugs, inconsistencies, missing tests, commonly also stuff I already now (like some infra config details).
- accurrent 12d agoMinor bug and develop whole new field of cryptography are very different scales. I do agree alms are really good with a lot of common bug fix related tasks to the point you have to split commits out. The fact is you recognise it's a minor bug. Who is sitting through the proof of Navier Stokes and going through it and finding connections between itself and other fields? Im not saying LLMs are bad, but I do think understanding is important. Heck, the fact you identified the minor bug suggests you understand the output. Im not so sure the same can be said of a gajillion line lean dump.
- foldr 12d agoI 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.
- someguynamedq 12d agoYou don't think LLMs can translate Ancient Greek poetry or analyze 18th century novels?
- foldr 12d agoOf 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.
- bananaflag 11d agoOn the other hand, STEM will disappear last because, as long as machines won't be able to do something, the study of how to make them do it will be a part of STEM.
- jgord 12d agoIts 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.
- svara 12d agoI 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.
- layer8 12d ago> Tao is speaking Tao isn’t the article author, it’s a guest post.
- traes 12d ago> 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.
- sweezyjeezy 12d agoThe 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.
- ipnon 12d agoI 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?
- sweezyjeezy 12d agoI 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.
- encyclopediai 12d agoThe 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.
- dudeinjapan 12d agoThe 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.
- encyclopediai 12d agoYes. 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.
- bonoboTP 12d ago
- kp995 12d agoIf 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.
- bonoboTP 12d ago> 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.
- hnisjafx40 12d agoTaught proofs too, and plenty of students fake intuition with pattern matching.
- youoy 12d agoPart 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.
- sweezyjeezy 12d ago> 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.
- soVeryTired 12d agoProbably 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.
- someguynamedq 12d agoHow 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
- karmakurtisaani 12d agoI, for one, wish to dedicate my life to improving the wealth and power of the already existing billionaire class.
- practal 12d agoHmmh. 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.
- soVeryTired 12d agoWhat's an example of your belief system conflicting with a motivated example? I'd love to understand that a bit more.
- practal 12d agoOne example is what currently plays out, see the previous guest post on Tao's page: https://terrytao.wordpress.com/2026/09/12/after-math/ 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.
- glimshe 12d agoIt 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")?
- itsalwaysgood 12d agoMathematicians 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?
- glimshe 12d ago
- elendilm 12d agoLogic 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.
- karmakurtisaani 12d agoHate to tell you mate, but this is rather close to stitching together random sentences..
- elendilm 12d agoSadly, 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/ 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)
- karmakurtisaani 12d agoDude, you're just too smart for me.
- elendilm 12d agoBuddy, your lack of good faith is now exposed, which demonstrates zero substance. I doubt you would know that it is very cheap. Ironically, the "stitching random sentences together" has now evidently applied to you.
- derliebej 12d agoSoftware 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.
- alkyon 12d ago> 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 https://www.youtube.com/watch?v=IbGNZQvobkc
- 1223197 12d agoThe 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.
- c7b 12d ago> 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 https://news.ycombinator.com/item?id=49010345
- SirHumphrey 12d agoIt’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.
- c7b 12d agoI 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.
- pcfwik 12d agoIf 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.
- breezybottom 12d agoEvening the playing field would mean those R1 professors now teach five classes a semester for 50k a year instead of doing research.
- 2snakes 12d agoYeah, the early insight is important to preserve in students.
- deleted 12d ago[deleted]
- contubernio 12d agoI'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.
- getnormality 12d agoCongratulations. You are one of those leading the way, showing how we will adapt and how the world will get better from AI.
- 12ha-22t 12d ago[flagged]
- cindyllm 12d ago[dead]
- contubernio 12d agoThat'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.
- moralestapia 12d agoGP never said it was the conclusion. He was praising you, after you on your own volition decided to comment here to let know others about your experience with AI. It seems to me you're a very privileged individual, I'd suggest you practice gratitude regularly in your life.
- bonoboTP 12d agoI'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.
- bobajeff 12d agoLet'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.
- zozbot234 12d ago> 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.
- bonoboTP 12d agoLast 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.
- dfah-qwes 12d agoWell, 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 https://www.3blue1brown.com/talent The only concrete step any mathematician on the internet, including on the other AI concern site https://proofsandprompts.com/ 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.
- whattheheckheck 12d agoGive teachers and professors 1% of all future earnings of every student
- daxfohl 12d agoI 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.
- lern_too_spel 12d agoI 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.
- kittikitti 12d agoIt'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.
- amelius 12d agoIt's time to stop solving logic problems and start solving the more difficult philosophical problems, like the hard problem of consciousness.
- dekhn 12d agoI 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 $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis). Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society. And certainly not moving us towards "human understanding". The biologists are the ones working on that, the math folks should try working with them on neuro stuff to understand how human brains can do math at all, given their architecture.
- magicalist 12d ago> Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society. Is it, or is it only the parts you hear about/pay attention to? > People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ 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.
- dekhn 12d agoMostly 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.
- someguynamedq 12d agoThe 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.
- beyonddream 12d agoCurrent 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!
- jonesn11 12d agoI 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.
- DoctorOetker 12d agoif 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?
- udbhavs 12d agoI 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 https://michaelnotebook.com/df/index.html
- jameshart 12d agoThis 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.
- pylua 12d agoDumb 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 ?
- js8 12d agoI 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.
- pvillano 12d agoI 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 https://math.stackexchange.com/a/111461
- emil-lp 12d ago> 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.)
- soundworlds 11d agoI this is part of a larger issue of our economics rewarding immediate results, but not good process. Basically, as long as this is the mechanism by which people earn money to stay alive, the world will always be optimized towards results.
- atorodius 11d agoThe whole math and AI debate irks me. I am thinking the same can be said about art. „if art is more than the image …“. But when we debate image generation we dismiss this. So it is different when it is closer to home? Hypocritical IMO
- cbondurant 11d agoI think it says quite a bit that I had already been thinking of Outside In before it was even mentioned in this article. Outside In truly is one of the great works of math communication. I'd argue that its format, a dialog between a novice who is yet still sharp (is able to be the one that makes the connections between ideas) and an expert who guides with pointed questions (fills in background context that is needed for the problem at hand), is one of the most effective ways of formatting mathematical communication. Surreal Numbers by Donald Knuth is another great example of the style I'm referring to.
- Fr0styMatt88 11d agoI remember hating math from late primary school to all through high school. I never learned my times tables properly. They were just absolutely boring AF. What I didn't realize at the time though was that I basically set myself up for failure. When the basic building blocks are hard, everything else on top gets unnecessarily hard. I remember reading Lockhart's Mathematician's Lament and loving it. I discovered that I actually liked math when I was able to let go of the "there is only one way to do things" and realize that math, in a sense, is 'made up but logically consistent' (notwithstanding the arguments about whether math is innate to the universe or invented by man, which I err more on the innate side -- I'm thinking of a slightly different concept here). When I learn something, I always need to know the why. We were never taught that in school. "Just because" has never been a good enough answer for me in anything I want to learn. Learning about the history of the problems, why certain solutions came up as solutions and why they were needed in the first place made everything much more concrete for me. I've been using ChatGPT over the last month or so to re-learn my times tables and basic arithmetic from the ground up. I had no idea what 'automaticity' was (I never developed it; I still would manually count simple multiplications in my head or on my fingers). I had never made the connection between algebra and arithmetic -- at least not deeply. In that you can manipulate arithmetic expressions just the same as algebraic ones if you want (72 - 37, you mean I can subtract 2 from both sides first to make the calculation easier?!?!). Now I'm motivated though, because I know why I'm learning it. I can look at it with my adult brain. I guess it's a bit like the difference between a kid being forced to take piano lessons and an adult that's really motivated to learn and practice because they suddenly discover they want to play. Oh and 3Blue1Brown is amazing!
- jan_m_savage 10d agoActually, there is something like that, given in a book titled: "Did you say mathematics?".