Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable understanding it's insane.
Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same unfamiliar sense of total loss like when encountering hard mathematics.
This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane.
Every sub fields (web/kernel/backend/etc.) has a million/bazillion weird words used in a dozen different contexts and if you read a paragraph of even semi technical software text you will feel like an over stuffed turkey.
Even cache could mean the CPU caches, the page cache, a browser cache, a CDN cache, a Redis cache, or imagine the flurry of words we have that have real world meaning. Session, handle, pool, buffer, stream, channel, event, task, worker, or queue. Generally there is some overlapping meaning but often there isn't.
Just be thankful that we don't share the penchant for giving credit to discoverers. Imagine calling a cache a "Murphy/Steinman/Sokolov structure" (made-up names).
I mean, we do for some things, especially algorithms (Boyer-Moore). Probably for the same reason the mathematicians do -- there aren't readily available real-world analogies.
And I won't even mention the branded future, with its "Google HyperZipper String Search" and "OpenAI/Red Bull speedmaxx distributed consensus algorithm"...
This is also true for almost every other field, even within computer science. The only difference is that a lot of people operate at a very surface level without realizing just how much background knowledge they have accumulated. Think about the number of keywords your average SWE is expected to know. It is rather insane.
Nah, math is much harder because there is not just the lingo, but also all the math machinery behind it. Each math definition represents some long process behind it, which builds on another process, etc. The knowledge builds on itself , too much more so than computer science.
At least a lot of those are common words that allow some level of meaning inference with a little bit of adjacent knowledge, like cache and queue make sense with the barest of explanations because their everyday definitions are still applicable. Many of these terms aren’t entirely opaque until you drill down into specific niches.
I had a few moments of this in the past. For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature.
Similarly, at some point somebody pointed out to me "the reason you're confused is that the bold on that variable means it's a matrix"
My favorite moment of this kind was when the teacher said 'Ok, and for the rest of the course we will look at a completely different problem', and the equation he wrote down was exactly the same as before. Except that the letters referred to vectors/matrices now.
That is why Iverson invented APL.
As a notation to get rid of all those inconsistencies in math notation.
And for years he taught math classes with APL on the blackboard without computers.
whether he succeeded, is debatable.
But APL is definitely powerful, succinct and "regular".
In APL you don't infer the operation from the types at all. × is elementwise, +.× is inner product /always/, on scalars, vectors, matrices, whatever. The glyph tells you what happens. Nothing is bold, nothing is inferred, nothing depends on what your professor assumed you'd absorbed.
I've been trying to get into Iversonian languages myself with the book: Calculous on J
A decade or so ago I wondered if the reason maths was hard was the names being optimised for writing by hand. Everything's single letters if they can get away with it, so when mathematicians run out of Latin alphabet, they use Greek, bold, etc.
Even integration's ∫ is a fancy elongated s.
CS version would be e.g. integral(function=some_named_function, from=a, to=b, with_respect_to=argument_of_function), which may be longer, but is less opaque, especially when you get in so deep there's 3 other people in the world who've looked into this specific problem and you had to invent your own operations.
But that's all an outsider's perspective. I stopped with two A-levels in maths and further maths.
I did study math at university for a while. Dropped out eventually. In the beginning I was super annoyed by the brevity and hated it. But after like 3 months it suddenly became natural. I also appreciate the clarity of how mathematicians introduce new ways to write things. That is sometimes even more verbose than some random API docs for a new function…
yes! I really find when computer science ppl start using math notation to describe algorithm very pretentious. we have programming languages in comp sci, we don't need it!
I was reading about Tao's efforts to get more people to use Lean and apparently a big roadblock for people is that Lean uses very specific static typing.
e.g. to use a very simple example on a white board "3" is "overloaded" as:
- the integer 3
- the rational number 3
- the whole number 3
- etc
When you write a proof in Lean, you have to specify the the type of "3" you mean.
Having using Python/Perl and Java over the years, I get that some math folks found handling this daunting or at a minimum friction to getting into using Lean.
LLMs seem to have been a big help here just for the "translate my math notation into a proof" feature.
Sometimes I wonder if mathematics would have been significantly more improved if they hadn't insisted on notating their variables as single letters and also indicated variable types out-of-line (or at all)...
but then I take a look at literally anything the Haskell people do and realize that it probably wouldn't have helped.
>For example, in my quantum class the teacher wrote "H Psi = E Psi" on the board, we all laughed, "just cancel the psi" but it turns out one was a multiplcation and the other was a matrix multiplication (operator) and so we had to learn all new nomenclature.
This is one of the great things about Lean becoming used for more and more mathematics: understanding exactly how an operator/function is defined is just an IDE click or few away. It completely removes the ambiguity present in hand-written proofs, although it still can require a lot of reading to actually meaningfully understand the definitions.
I would expand on this. AI is great for me because it can read the equations I don't understand and turn it into code I can understand. I've worked in science for decades and it's still like pulling teeth to replicate a competitor's paper when they are vague and sloppy with their description (often intentionally).
ugh, I had some text book that used R for a scalar value and (edit: \u{MATHEMATICAL BOLD FRAKTUR CAPITAL R} here) for a matrix that was related to the scalar and I had to go back and re-learn a month of material once I figured out that the font was being used with intent
A term that gets tossed around in math is "mathematical maturity." It's similar to what you see in other fields - e.g. learning how to program, learning how to make music, learning how to cook - that involves many "aha" moments and reshapes your perspective. Math is full of such steps, moreso than most other endeavors, probably because the main limit is the abstract reasoning itself.
Math strives to minimize ambiguity, which other fields don't do as much. Non-math fields tend to reuse regular words as jargon (i.e. with specificity of meaning that may fly over the laymen's heads). Social sciences and humanities are most notorious for this, often resulting in non-practitioners not realizing they are out of their depth because they are not looking at symbols from non-Roman alphabets.
That's something only someone who's never studied advanced math could say. Math notation and jargon can be extremely ambiguous and overloaded. "Normal" has about 20 different meanings.
The abstraction is by necessity. Our puny brains have only a very small working memory. The only way we can reason about many problems is by creating multiple levels of hierarchy. That is actually the essence of what mathematics is.
Isn't it just what you studied in depth? I am not in this area but can understand what's going on "at a high level" here. But I studied no other science formally since the age of 15 (this is possible in the UK school system). So physics and biology just go over my head unless they are sufficiently mathematical.
Yeah it is a lot of simple ideas stacked one on top of the other, but the edifice is so large from some vantages that the building blocks aren't visible, or tractable to think about independently. And sometimes the ideas are very subtle, so you can only develop fluency partly by spending lots of time playing with those blocks by building your own little structures. You also develop fluency by talking to other mathematicians
I like to emphasize that the ideas are usually very simple at their core. Sometimes they map to kinds of objects or reasoning that non-mathematicians use implicitly all the time in their daily lives, mathematicians just have words for them and so are able to use them explicitly.
And I suspect the density of the language/terminology may give the wrong impression about how mathematicians think about the math they are working on. I mean, different people think / experience / practice math differently of course but IME the underlying thought about a particular problem tends to be much looser and concrete than formal math writing would imply.
That more formal language is needed of course because at the end of the day, it is how we communicate our thoughts in the way that other mathematicians can understand them, not to mention how we can check our own thinking
Yeah this is pretty much where I am at. Take the phrase from one of the responses
"The special fiber is the associated graded ring.....and that the filtration admits sufficiently simple homogeneous lifts of the three generators, then one might prove"
In any other context I would at least have some degree of intuition about what is being discussed, but in in math? Absolutely no idea. And usually if I start digging and turning over stones to uncover meaning, I'm just met with even more totally dense code-word language. Unlike other fields were digging is usually quick to relieve ignorance, somehow in math it tends to get worse.
I'm sure I am capable of grasping this if I took the time, and perhaps even what is being discussed it rather intuitive, but the incredibly density of the nomenclatic swamp you have to trudge through for math is totally unrivaled.
The basic problem is that to get to the objects you mention here is at least 3 or 4 years of full-time study away from the kind of math people learn for a typical college degree in science or engineering. If you really want to understand them, to make your "digging" efficient you should probably just get a pure math degree, but setting aside several years to satisfy occasional curiosity is not feasible for most people for various reasons.
One unfortunate feature of published pure math research is that often the ideas are quite accessible and straightforward and don't really require special abstractions or terminology, but those get used anyway because for someone who already has a math PhD it saves a bit of effort.
I agree, the nomenclature is impenetrable, it's like reading software that is not well commented. Perhaps LLMs are very good at "challenging" mathematics because what we perceive as challenging is primarily the language component and not the conceptualization.
It can't be one language, and that's the big problem. It's inescapably a bunch of tiny DSLs. Once you see both the inconsistency and the necessity for inconsistency, it becomes much easier to just roll with it.
This is a very notorious area for dense definitions and concepts that interrelate closely and have to be memorized. Mathematicians from other areas are going to have difficulty but may have some idea of what the concepts try to capture.
Some areas are hard in different ways. I could never quite wrap my head around the way logicians have to think. A clever combinatorial bijection is a work of art you probably can explain to a undergrad class easily but good luck coming up with it. And number theorists will throw the kitchen sink at their problems: no area of math is safe from getting used by them.
People who do this have spent years of their life thinking in this language and studying it, so it is going to be hard. We're also not good at communicating the intuition which for algebraic geometry often comes from other fields.
IMO, it's just the notation. Something I've actually found ChatGPT useful for is to create mathematics lessons for me in the form of computer programs. When broken down into a series of readable almost-plain-English steps, it's so much easier to understand. And it's easy to tinker with programs and get a hands-on feel for things quickly.
I'm sure having a compact notation is absolutely invaluable for people who dedicate their lives to maths, but for someone with just a passing interest, I find it more obscuring than helpful. I feel the same way about music notation.
I mentioned this in a sibling comment but even for mathematicians, the intimidating notation and the more formal language might give the wrong impression about how we think about math. Actual thinking and even discussions with other mathematicians tend to be looser and more concrete and tactile, but the notation and language are there in part to act as a sort of lingua franca to help everyone stay on the same page, since everyone thinks at least a little bit differently. It also helps to keep you honest and catch situations where your thinking was muddied, since this language is so specific and writing things down has a funny way of catching things. And good notation goes a long way towards making the simplicity of an idea clear, or completely muddy in the case of bad notation.
It's just language. Mathematicians don't invent notation for fun, they do it because they naturally start thinking at a higher level of abstraction. If you're not thinking at that level then, well, it will be all Greek to you.
That is one of the things that fascinates me most about mathematics compared with other fields, and it led me to discuss the subject with professional mathematicians. The funny thing is that they admitted it is the same for them...stray even slightly outside their own specialized area, and within two or three lemmas, they also feel completely lost.
Math isn’t necessarily hard, but it’s incredibly dense
A simple statement like let f(x) be a continuous function can carry a lot of definitions
In that statement, if you missed the day in class where they covered continuous functions it might not even register that it’s a well defined term
And that’s the most over simplistic example I could think of
As a math major, I remember that being one of the first lessons I learned, that every single word could be carrying a lot of weight so to look things up in detail if I was ever struggling on a problem. One of the oldest entries in my memory.md file
What I am thinking is the way you make it 'keep going' and when you have people of the calibre of Tao doing it I kept thinking how many breakthroughs is he going to cause the LLM to find with his targetted questions :D Amazing that we have the privilege of witnessing a true expert in such a way question the LLM.
Is this the same as Dinitz Theorem[1] which seems to have been proved in 1994? This is the only result I keep stumbling upon when trying to understand the problem formulation
At Mozilla, we had a set of whiteboard tags we could set on bugs, like "[crash]" or "[compat]" or "[leave-open]". That last was used when there were multiple patches attached to the bug, and we wanted to land only some of them without automation closing the bug once they landed. (It's common to have alternate approaches or test cases also attached to the bug, so you normally don't want to wait for all of them to land before closing the bug.)
I started using "[leave-open" for those.
It lasted for a couple of years, until someone went through and "fixed" them all.
Terrance Tao's chatgpt conversation is really interesting for a variety of reasons:
1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result.
2. Terry Tao's questions are very specific and prompts the AI in a useful way, that without high math training you are not going to get the same information out of it. Terry seems to see some aspects of the problem and counter example and uses AI to brute force some parts of it.
I think we have a couple of years of "being good at talking to the LLM about your field of expertise" being a useful human skill, until that too gets washed away
A community of those who distract themselves from the perfectly fixable problems in their lives corruptly self-evaluates. They validate each other's stagnation and unwillingness to move by finding flaws in each day that will enable shutting off the flow of any new data while condemning the world and any actions in it. The lay-z-boy they collectively protect appears as corroboration with a broad population but is in reality a repetition whose independence is meaningless since they are all copies of one system, one kind of person in the same kind of trap.
Immobile. Clogging the Suez with their sandbagging ways. Nothing to add except reasons to stay put. No aspiration. Only cynicism. They deserve nothing but all of our contempt.
It’s endlessly fascinating to read the AI transcript of an expert who _really_ knows how to cut to the chase. It just shows how much you can potentially squeeze out of these models. I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise (emphasis on progression and usage patterns, not absolute skill, obv I don’t match that): short pointed questions that goes all in on the jargon and machinery of the field and steers the llm hard (eg no softballs). I’ve noticed that llms switch their tone and meet you basically more or less on your level.
> I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses.
That is what will happen though to future generations: they won't be able to use it for anything because none of them will have the "decades of coding" experience that you have had the privelege to have without AI.
I encountered something fairly similar working with Claude a few days ago. For a current project I've been fairly hand-wavy with requirements since I was getting good results, but it seemed to be failing hard on some key points, so I started to be more strict with it. Even after the fails were resolved, I've noticed that Claude now behaves differently within that project, carefully checking and rechecking things up front and also looking to me for guidance more often. Mildly irritating, but if it works...
It's crazy how he suggests simplifications over and over and gets led through the finding. Absolutely bonkers how you can use AI to understand something and map it to your own mental map so efficiently, and of course he's most interested in generalizing or finding a simpler sub-result that would explain it.
Just awesome to see new knowledge hit an incredible mind like this. Having these "what if" discussions is what I miss most from JPL and academia.
"I’ve activated Pro. Can you continue to look for a potential geometric explanation of the X_3 ~ A3 miracle that avoids coordinates or other unmotivated constructions ?"
I just do very laconic questions about advanced topics, this seems to prompt it a bit more towards reducing fluff in the answers. But that + the activated pro could be an improvement
Jeez. While I obviously can't talk at all about the math, I've noticed a few things:
a) The model thinks on some questions while straight answers on others. (I wish I'd knew from the questions if this is somehow correlated to hard tasks or "inventive" tasks, but that's way out of my league).
b) The model sometimes pushes back. Again, I'd wish I knew if it was warranted, but I counted 2 instances where it said "yes, but with caveats", one where it said "mostly yes but with this correction" and one where it said "careful here, because x y z".
c) The model did q&a + pdf ingestion + code writing + more q&a + thinking + more q&a, for a looong while, while seemingly staying on topic (at least Terrence Tao seems to think they're still productive, so I'll trust that).
This is what model progress is, not number goes up on xBency or yBencher. Damn.
The "yes, with caveats" thing is boilerplate for both Codex and Claude since this current generation.
It's actually a bit annoying because it primes you to think that the caveats are real, but most of the time it's just something terribly obvious and not a real caveat, but the model probably has some system prompt that tells it to always consider caveats or something like that.
Same as the model starting every reply with a commitment to be "honest". LLMism are fun but I tend to just suppress them via AGENTS.md because they distract me
Similar to how Cypher puts it: I know this is “just” next token inference, matrix mult and just software, ie there’s no “intelligence” there BUT, looking at this convo … damn!
The fascinating this is that the LLM is not acting as a tool here AFAIk, but very much like a colleague.
I have no knowledge of the domain and have only PhD EE level math knowledge, so maybe my bar is too low.
I think the "But this is not intelligence because it is known math" is not a correct argument. It is unknown how the overall higher intelligence of humans works.
What I do notice however is that LLMs are becoming capable of doing an increasing part of the intellectual work I can do, and usually a lot faster.
Just today I presented an agent framework that can take an informal incident statement and propose infrastructure changes to fix it, all evidence backed. This did nothing I could not to, but it did all 5 test cases in 6 - 12 minutes each. I would have found all of the monitoring indications it did, but it would have taken me a day per test case. The LLM also included sass to silly tickets. ("This is not even worth spending monitoring resources on. It's obviously a configuration problem.")
That's how this is reading to me as well. It's just fast at slogging through a certain level of "simple" transformations.
That argument says very little, emergent behavior is a thing in complex systems with billions of parts.
Humans can also be reduced to voltage potentials propagating along of tubes of fat and synapses getting rewired.
What does "predicting the next token" mean? I ask this every time people say "LLMs are just predicting the next token" and it's maddening that nobody can give a straight answer. Predicting it according to what probability distribution? Every process that produces a sequence of actions (including e.g. a human writing) can be modeled by some probability distribution and therefore their actions are indistinguishable from "predicting the next token" emitted by that distribution.
There is clearly intelligence there. We have no way to recognise intelligence other than the appearance of intelligence and this very clearly displays that.
It's also quite clearly different to human intelligence in some notable ways, but not in any that preclude describing it as intelligent. At least for normal non-pedantic definitions of the word.
Everyone uses "intelligence" to mean something slightly different, so for this to be a useful claim to make or refute we need to come up with new, intentionally-pedantic, terms (or new domain-specific definitions for vague existing ones).
Yes, trying to communicate (or watching others try to communicate) about these topics is incredibly frustrating because it's pretty much impossible to make any progress without interrogating people's different definitions, but nobody wants to do that because it would mean being pedantic, splitting hairs, etc.
It's not like this is a new problem. Turing had a definition most of a century ago, he wasn't the first and certainly wasn't the last. I don't think we need new terms necessarily, and I doubt we're all going to agree on a definition tomorrow.
I'd say an entity capable of instructing one of the leading mathematicians of his era is pretty clearly intelligent by any reasonable measure - however it might be arriving at its output.
I'm no intelligence researcher or philosopher; but, I think LLMs make us confront the (IMO, now clear) distinction between cleverness (intuition), reasoning (rational argument), and consciousness. I suspect that we think of "intelligence" as either of the first two welded to the latter. In that vein, I'd say that consciousness may be just another emotion: happiness, sadness, egoness.
We have no way to recognise intelligence other than the appearance of intelligence and this very clearly displays that.
There is about 150 years of cognitive science experimentation in animals that have clarified a little bit how you can actually measure intelligence. Ooorrrrr we can use medieval contempt for scientific thinking and pretend "intelligence" is just some higher intuition that can never be falsified. You know it when you see it, bro! Don't listen to Emily Bender, she's a socialist witch.
The point of those cognitive science experiments is that they apply to any animal with a brain and plausibly show a real shared concept of "intelligence" that isn't limited to humans. According to this concept, orcas might be smarter than humans, despite their physiological inability to make tools. It's not a "normal, nonpedantic definition" of intelligence because such a definition would be scientifically meaningless.
Indeed, AI's fundamental sin, going back to Alan Turing, is embracing a definition of intelligence that applies to civilized humans, but not to hunter-gatherers, let alone apes, corvids, and cetaceans. Frustratingly, our modern society has two concepts of intelligence:
- an intuitive, social sense of "how smart is this guy?", which is well-understood and, being highly correlated with IQ, a totally pseudoscientific artifact of human psychology
- the poorly-understood scientific concept I mentioned earlier
If AI researchers cared about scientific thinking, they would be intensely focused on the brains of bees. Insteac they love money and sci-fi but have pure contempt for science, even Demis Hassabis. This is why AI researchers have yet to build a robot that navigates real-world 3D space as intelligently as a cockroach. I don't think any of our grandchildren will live to see a computer smarter than a mouse. (It seems like Fable still struggles with small-number arithmetic. Rodents don't.)
> If AI researchers cared about scientific thinking, they would be intensely focused on the brains of bees.
Basically every academic AI researcher in history was doing what you described. The AI industrialists stopped caring 6 years ago once they realized LLMs seem to have been the only thing in 80 years that actually seems to work at any useful level.
There are plenty of pioneering scientists who are either returning to actual AI research (Yann Lecun, Ilya, etc), and plenty who never left (Richard Sutton) who are doing exactly what you are talking about.
> Basically every academic AI researcher in history was doing what you described.
That is not true. Alan Turing did not view things that way, his test would say that a dog has zero intelligence. Neither did any of the MIT Lispers. And neither do Lecun or Sutskever or Sutton! They are all focused on human intelligence. None of them are even slightly concerned about an AI which is intelligent before it learns any language.
> the only thing in 80 years that actually seems to work at any useful level
This isn't true either! Mathematica / Maple / etc are "old-fashioned AI" and they obviously work. The Lisp expert systems were also useful, though less so than an LLM.
There is no intelligence. If anything, this just shows that natural language and mathematics are both fields which are structured in a logically computable way. And if you have a machine that can compute symbolic logic, you can process both natural language and mathematics.
A second corollary is that rational consciousness and thought is less likely to be contained in language than previously thought, because if language is so simple that a machine can process it, it can't contain consciousness.
If natural language was structured in a logically computable way, we'd have had interesting chatbots by the late 80s, basically as soon as a dictionary fit in local RAM, and for the same reason we got compilers.
Da hole raisin y nat-lang be v. hard is dat i kan rite lik dis an it be cool 4 native engrish speekrs 2 unerstand. LLMs are of course fine with this sentence in exactly the way that Zork's engine couldn't be.
The underlying structure of language, which is grammar, is obviously logical. That the symbols used to represent this grammar can be sometimes fuzzy or ambiguous, is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.
It's not "obviously logical", it's a pattern which we mimic to avoid mockery.
example For, semi-randomise I word order can this like, Yoda worse than, and be understood.
> is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.
We had to invent Transformers to be able to do that with reliability anything close to being worth caring about. Transformers have to learn from examples, not be pre-programmed.
I'll have a blog post up tomorrow about it but the Jacobian Conjecture counterexample is a very funny cognitohazard for LLM assistants. It's a paradox for modern LLMs: they have enough math skills such that they can easily compute the Jacobian to formally verify the counterargument, but its own knowledge base is locked prior July 19th 2026 where all it knows is that the Jacobian Conjecture is unsolved and a random chat user providing such a proof is highly unlikely.
I wonder whether when the fact that AIs have started solving conjectures will enter the training data, they will become more confident in their abilities.
Similar to the story of George Dantzig, who was late to class and solved two open problems in statistics because he mistook them for homework, I think the current batch of frontier LLMs are chained up by knowing which problems are supposed to be unsolved. If they're let free (probably via some targeted RLHF) we might get a flurry of solutions to open problems.
But a property of intelligence is to know when to stop, if we treat intelligence as some sort of search and not some a priori intuition of the entire space. Seems kind of hard, if not impossible, to train for specifically that.
I don't understand any of the math here, but I had two thoughts. Soon we'll have explainer agents that translate these according to my level so I can, with effort and interest, follow along and stretch my understanding boundary bit by bit.
Two, at some point AIs will be able to use other context like the fact that this is Terrence Tao and not your average Joe and change how it answers, either in tone or structure.
Fork Tao’s convo and prompt this (with your own math level described).
GPT did a great job of translating Tao’s questions and concepts (e.g. “pre image”) into a progression I could understand.
“Ok I have a PhD in financial math and undergrad in engineering math. I have almost zero knowledge of polynomial algebra / geometry, I know what a polynomial is and what roots are but not much beyond that. Could you try and explain to my level what questions the user I the conversation has asked and what the agent has responded with, we can probably go user query by user query to build up”
I want to be able to take a conversation and ask for subconversations as red pen annotations "on the side". The linear nature of the context tends to frustrate this.
I'm not sure an AI will speed things up much. You would probably still need years of layers of foundational understanding to get the advanced material. We don't go through years of school to learn math just because teachers are bad - it's because complex subtle ideas are built on countless other ideas, and aren't necessarily compressible to something every layman can understand.
The years are broad though, the nice thing with AI explanations is that they can go deep quickly, and quite precisely down the path you need for your prior experience.
I’ve had a similar experience using LLMs to have mini personal breakthroughs.
One thing I notice is many models say statements along the lines of “okay we have exhausted this thread it’s diminishing returns from here and we should stop and move on”
It’s funny because I’ve been building a tiny neural network maze solver (23 bytes solves 92.75% of unseen 2D mazes)
When I asked ChatGPT/Fable if we had anymore threads to pull to increase capability and decrease byte size, they both basically said no way - back when I was at ~166 byte models with a ~85% solve rate.
Throughout the experiment I just kept trying different approaches and eventually had 3 mini “breakthroughs” in this particular niche. But if I had listened to the models…
Anyway, these models are amazing to experiment with quickly, but they are dumb as hell and so absolute
The big take away for is the fact that the ONLY reason why chatgpt was able to get to this counterexample was because of the knowledge of the person driving the conversation.
I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.
> I don't think chatgpt could have come to this on its own without the amount of steering he did, which just validates the idea that AI is not a replacement for human expertise but an amplifier.
The problem is that, what happens to human expertise as people start to use AI earlier and earlier in their careers, so that in 50 years? The problem is that Terry Tao spent decades as a mathematician before ever encoutering AI. Of course he and people his age will be able to drive AI somewhat sanely and use it to their advantage.
But as more people grow up with AI, they will likely not reach levels like Terry Tao because their exposure to AI and the temptation to use it will certainly dull raw human intellect over time.
I don't know if I agree with the premise that having access to AI results in dulling human intellect.
I feel like to get to Terry's level you need a combination of passion and aptitude for the subject. People that don't want to learn about a topic will always look for shortcuts, which I think represents the vast majority of people. Terry Tao is quite exceptional, and I think exceptional people will still exist even when the "easy" button is bigger than it's ever been.
> If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. [...] What was conjectured is that this local invertibility property everywhere would mean global invertibility.
> it doesn't overturn much. [...] the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
> From a comment by j2kun https://news.ycombinator.com/item?id=49000833 , someone asked Fable and there was an almost counterexample in 2d but it uses division too. [Instead of f=x^2+7xy they have something like f=x^2+7x/y so it's not a polynomial.] It looks like the new trick was to use a third variable to avoid the division.
I recall several mathematicians (possibly including Terence Tao) mentioning that fields in mathematics have become so specialized and isolated that a conference like the ICM feels more like a collection of mini-conferences. An expert in one area can barely understand a talk in another.
Modern AI feels like a godsend to mathematicians. It helps them break down boundaries and connect concepts in ways a mere mortal couldn't imagine.
I find it amazing how people can use AI to do things that seem hard but yesterday I could not figure out how to install a package on my system. It kept suggesting dependencies that don't exist, and telling me to use functions that are not in the system. The math does not math...
I expect Centaur Field Medals to be awarded. It's not clear how much steering Levent Alpöge did to get the result. In math it's very common to hide the search pattern and show only a simplified path to reach the final result.
Presumably this entire conversation is at least somewhat typical of the way high level mathematicians talk to each other in ego-less fashion? Although, it would be interesting to what extent the model was trained on math conversation as opposed to just analysis and proofs.
The flow of the whole conversation, with Tao guiding and the model calculating, gave me the feel of Tao perhaps talking to himself - just that each of those model responses would have taken him much longer to calculate by hand.
It would be fascinating to hear Tao talk about what he may have learnt from this, and if it suggests approaches to other problems he might not have considered, as well as an analysis of the original Fable counter-example construction.
To me, this shows that extremely talented and qualified mathematicians (can) use frontier-level LLMs to automate their personal grind-y workloads that would otherwise (probably) take more time to accomplish with natural intelligence.
By itself, no consequence. But over time, provided we keep pumping out talented and qualified mathematicians and keep subsidizing costs, we could maybe hit a breakthrough... somewhere... that has real impact.
It's an indicator of AI progress. The solutions aren't especially revolutionary, but no person had been able to solve them after decades of collective attempts.
To be fair I don’t think there were too many people really trying to. Symbolically, one could make a parameterization of the Jacobian determinant and then brute force a solution, if one had known such a polynomial existed in only three dimensions.
Oh yes there were. The Jacobian conjecture is "notorious for the large number of published and unpublished false proofs which turned out to contain subtle errors."
It's not quite the Reimann hypothesis, but many prominent mathematicians have spent years working on this problem. Yitang Zhang wrote his PhD thesis on it.
Some materials are readily available on eMazon and aBay, so I've taken the liberty of ordering those for you. Your credit card bill will be a bit high this month, but it'll be worth it. There weren't any sellers for the advanced EUV lithography machines, so I've hacked into the only place on earth that makes them, changed their records and had them ship it to you. Expect to receive a "pinball machine" from Amsterdam, soon. I've instructed the roomba connected to the local network to start assembling stuff while we wait for the other materials. Oh, and you're gonna need a new toaster.
Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, then the Jacobian matrix does not allow for an evaluation of the stability.
Yes—for a continuous-time autonomous system
x
˙
=f(x),f(x
∗
)=0,
this is the standard linearization criterion, with J=Df(x
∗
):
If every eigenvalue of J has strictly negative real part, then x
∗
is locally exponentially asymptotically stable.
If at least one eigenvalue has strictly positive real part, then x
∗
is unstable.
If no eigenvalue has positive real part but at least one has real part 0, linearization is generally inconclusive. Nonlinear terms or a center-manifold analysis are needed.
The last case really can go either way. For example, all three scalar equations below have Jacobian J=0 at x=0:
x
˙
=−x
3
,
x
˙
=x
3
,
x
˙
=0.
Yet 0 is respectively asymptotically stable, unstable, and neutrally stable.
A slightly more precise wording is therefore:
If the spectral abscissa
α(J)=
λ∈σ(J)
max
Reλ
is negative, the equilibrium is locally exponentially stable. If α(J)>0, it is unstable. If α(J)=0, the Jacobian test is inconclusive.
This criterion concerns the Jacobian matrix of a dynamical system at an equilibrium; it is unrelated to the “constant Jacobian determinant” condition in the Jacobian conjecture.
I'm watching how Tao uses AI, and it's interesting.
Expand the entire expression, then change the representation to find the core axis. You can't see the axis from just one perspective, so you change the representation. In programming terms, it's like applying multiple domain models. Then break it down into small contract units. Why is it a Jacobian monomial? Why does x satisfy a cubic equation? And so on.
Then swap out the modeling under a hypothesis, assemble it all back together, and verify it through the equation.
This feels similar to modeling in programming.
Observe the whole -> explore better modeling -> decompose local problem -> verify independently -> reason about the highre level structure -> integrate back into the original problem.
This feels similar to when I receive work from a client and write a programming proposal
It's awesome to publish this kind of thing - great PR at least. Even if you don't understand the details, it's interesting to be able to peek into a technical conversation that a world class mathematician is having about their work with a "colleague". It's also the clearest demonstration I've seen of the vision AI people have about a future with truly intelligent copilots in super technical fields.
Is ChatGPT's interface always this atrociously jittery? Or is it just because this page is getting an HN hug-of-death right now? Every time I try to scroll the whole page goes blank for a few seconds and then re-renders.
I think that's partly because it's a fairly long conversation and partly because of all the LaTeX. There's enough formulas there to fill a small text book, and I doubt their renderer is all that efficient.
It's fun if you ask ChatGPT to guess the identity of its interlocutor :) It will guess math researcher or paper author without hints, but if you give it some additional hints, "this was shared over the internet", "it's someone willing to work with AI", it will guess Terence Tao as the first choice.
/-!
# A counterexample to the Jacobian conjecture in dimension three
We formalize the polynomial map
F : ℂ³ → ℂ³
whose Jacobian determinant is the constant `-2`, but which is not injective.
The final theorem `jacobianConjecture3_false` states the failure of the
polynomial-inverse formulation of the Jacobian conjecture in dimension three.
-/
namespace MvPolynomial
variable {R : Type} {σ : Type}
/-- The formal Jacobian matrix of a family of multivariate polynomials. -/
def jacobianMatrix [CommSemiring R] [DecidableEq σ]
(F : σ → MvPolynomial σ R) :
Matrix σ σ (MvPolynomial σ R) :=
Matrix.of fun i j ↦ pderiv j (F i)
/-- Evaluation of a polynomial map at a point. -/
def evalMap [CommSemiring R]
(F : σ → MvPolynomial σ R) (p : σ → R) : σ → R :=
fun i ↦ eval p (F i)
end MvPolynomial
open MvPolynomial
namespace JacobianCounterexample
variable (K : Type) [Field K]
/--
The three components of the polynomial counterexample.
The variables `X 0`, `X 1`, `X 2` correspond respectively to `x`, `y`, `z`.
-/
def F : Fin 3 → MvPolynomial (Fin 3) K :=
![
(1 + X 0 X 1) ^ 3 * X 2
+ X 1 ^ 2 * (1 + X 0 * X 1)
* (C 4 + C 3 * (X 0 * X 1)),
X 1
+ C 3 * X 0 * (1 + X 0 * X 1) ^ 2 * X 2
+ C 3 * X 0 * X 1 ^ 2
* (C 4 + C 3 * (X 0 * X 1)),
C 2 * X 0
- C 3 * X 0 ^ 2 * X 1
- X 0 ^ 3 * X 2
]
/--
The formal Jacobian determinant of `F` is the constant polynomial `-2`.
-/
theorem jacobianDet_F :
jacobianDet (F K) = C (-2) := by
simp only [
jacobianDet,
jacobianMatrix,
det_fin_three,
of_apply,
F,
cons_val_zero,
cons_val_one,
cons_val_two,
head_cons,
tail_cons,
map_add,
map_sub,
Derivation.map_one_eq_zero,
pderiv_mul,
pderiv_pow,
pderiv_C,
pderiv_X_self,
pderiv_X_of_ne,
ne_eq,
Fin.reduceEq,
not_false_eq_true
]
simp only [map_neg, map_ofNat]
ring
variable {K}
/--
The point `(0, 0, -1/4)` maps to `(-1/4, 0, 0)`.
-/
theorem evalMap_F_p0 :
evalMap (F K) ![0, 0, -(1 / 4)] =
![-(1 / 4), 0, 0] := by
funext i
fin_cases i <;> simp [evalMap, F]
/--
Provided `2 ≠ 0`, the point `(1, -3/2, 13/2)` also maps to
`(-1/4, 0, 0)`.
-/
theorem evalMap_F_p1 (h2 : (2 : K) ≠ 0) :
evalMap (F K) ![1, -(3 / 2), 13 / 2] =
![-(1 / 4), 0, 0] := by
have h4 : (4 : K) ≠ 0 :=
(by norm_num : (2 : K) * 2 = 4) ▸ mul_ne_zero h2 h2
funext i
fin_cases i <;>
simp [evalMap, F] <;>
field_simp [h4] <;>
ring
end JacobianCounterexample
open JacobianCounterexample
/--
The Jacobian determinant of the displayed map over `ℂ` is a unit.
Indeed, it is the nonzero constant `-2`.
-/
theorem F_jacobian_isUnit :
IsUnit (jacobianDet (F ℂ)) := by
rw [jacobianDet_F]
exact
(isUnit_iff_ne_zero.mpr
(by norm_num : (-2 : ℂ) ≠ 0)).map C
/--
The polynomial map `F : ℂ³ → ℂ³` is not injective.
-/
theorem F_not_injective :
¬ Injective (evalMap (F ℂ)) := by
intro hInjective
have hp :
(![0, 0, -(1 / 4)] : Fin 3 → ℂ) =
![1, -(3 / 2), 13 / 2] :=
hInjective
((evalMap_F_p0 (K := ℂ)).trans
(evalMap_F_p1 (K := ℂ) (by norm_num)).symm)
exact zero_ne_one (congrFun hp 0)
/--
The injectivity consequence of the dimension-three Jacobian conjecture
is false over `ℂ`.
-/
theorem unitJacobian_does_not_imply_injective :
¬ ∀ P : Fin 3 → MvPolynomial (Fin 3) ℂ,
IsUnit (jacobianDet P) →
Injective (evalMap P) := by
intro h
exact F_not_injective (h (F ℂ) F_jacobian_isUnit)
/-!
We now formulate the polynomial-inverse version explicitly.
-/
/-- Polynomial self-maps of affine three-space over `ℂ`. -/
abbrev PolyMap3 :=
Fin 3 → MvPolynomial (Fin 3) ℂ
/--
A polynomial map has a polynomial two-sided inverse, viewed as functions
on `ℂ³`.
-/
def HasPolynomialInverse (P : PolyMap3) : Prop :=
∃ Q : PolyMap3,
LeftInverse (evalMap Q) (evalMap P) ∧
RightInverse (evalMap Q) (evalMap P)
/--
The polynomial-inverse formulation of the Jacobian conjecture in
dimension three.
-/
def JacobianConjecture3 : Prop :=
∀ P : PolyMap3,
IsUnit (jacobianDet P) →
HasPolynomialInverse P
/--
The Jacobian conjecture in dimension three is false.
-/
theorem jacobianConjecture3_false :
¬ JacobianConjecture3 := by
intro hJC
unfold JacobianConjecture3 at hJC
apply unitJacobian_does_not_imply_injective
intro P hP
rcases hJC P hP with ⟨Q, hleft, _⟩
exact hleft.injective
