"The model family was reconstructed programmatically (parameterized cuboid stack) and swept over 2,304 configurations — extrusion directions, workplane-normal orientations, sketch windings, D's plane/height/depth/extent, including all the exact-coincidence heights. Zero failures with the fix; 576 failing configurations without it. The generator is available on request."
It looks like it wrote a python script to generate test cases in our file format for testing. Just... you know, as a side quest.
I’m a bit surprised OpenAI isn’t finding these big results far faster than the product’s user base. With no limits on runtime, access to dev models, custom tuning, and top talent, you’d think there’d be a constantly running internal project with the goal of solving famous math problems. And who knows, perhaps there is, but it would be interesting to compare the rate of success per unit “effort” of the internal mathematics work with that of the user base.
I have a pi extension that just runs the same prompt in a loop 25 times. I tried giving it a loop breaker but I found that it'd give up too readily. When the task is actually complete each iteration didn't do a lot of work so it was efficient enough. I suppose another way is to call out to a separate context to check if the task is complete?
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.
I’ll have to try this exact phrasing. I had a lot of trouble with GPT 5.5 more or less completely ignoring similar prompts and instructions and entering a sort of “doom loop” or just consistently trying to prematurely end the chat.
I would love any tips for other folks who have successfully used similar approaches.
Without any more context, "keep going" seems to be doing a lot of work. The user is placing a lot of faith in the LLM to not make subtle logic mistakes and to take good approaches to each problem. In my experience, even frontier models (such as Fable) are quite capable of getting confused during even simple technical work I've done in the dev ops world. For example:
LLM: This package hasn't made it to production.
ME: are you sure? i see it right here!
LLM: You're right to push back. I inferred that based on weak data. I see now that the package has been deployed!
If the above conversation is typical for me, how could one expect to achieve a sound result by repeatedly prompting an LLM to simply "keep going" in dense mathematical proofs? Perhaps the user in this case had actually checked the LLM's work before issuing the prompt, but I think you see my point anyway.
There may be something(s) about mathematics (proofs) that makes it particularly amenable to LLM reasoning - highly inductive from facts that are explicitly within-context/associative space? Being an unusually well documented discipline in general, with less influence from tacit knowledge or idiosyncratic “it works however the opinionated human made it work +- bugs” processes? Something about simulating even the smallest non-pure-inductive leaps necessarily risking simulating mistakes due to the nature of context “perception”?
Agreed, often you have to step in and stop it from reasoning itself into dumb directions, but occasionally it goes just like the transcript in question.
I've noticed GPT specifically has more of a tendency to stop partway through things than many other models do. Although my most recent experience with it was 4.X I believe.
Hearing "here's what I've done, here's the completely unambiguous next steps, I'll wait for you to send a pointless message before I continue" over and over again is a real pain.
That was a tendency of 5.4 and earlier, OpenAI specifically worked to avoid it in 5.5 and I find it happens rarely know. It really felt like 5.4 had been intentionally trained to stop and check, I believe it wasn't the system prompt.
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.
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.
More and more the skill of being able to ask the right question seems critical to me, and I don't know how one can do that without deeper and deeper domain expertise.
> 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’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
I didn't understood anything about the thread, but reading Terrence's messages was weird because it looked exactly like the discussions I have with LLMs
I've mostly seen people trying to oneshot a result, while I'll quickly experienced that going through steps/discovery was more effective and more satisfying, since you can always steer it back in the right direction; while oneshotting is hit (and it kind feel like magic) or miss (and you'll have to rework your prompt).
I was struck in the same way but I think it makes sense in terms of a thinking partner.
It is still ultimately Terence that is steering things.
What is crazy to me is how few of other people's conversations like this I have actually read.
Tao is really great for this because the anti-AI crowd can't really chime in and take the thread in a pointless direction. It is hard to think of another human alive who can carry the weight of unassailable authority in the same way.
Yes, the AI models are so useful in areas where you are an expert and can guide it in a useful way. I use multiple chats delay to help figure out client projects.
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...
The most powerful way to use Fable in particular seems to be to pit it against Sol as a reviewer (or, I suppose, vice versa.) After going back and forth a half-dozen times, the result tends to be pretty much bulletproof.
What's somewhat disturbing is just how much Fable's code really does benefit from the review. It tends to leave a lot of low-hanging fruit, and you can see it getting kind of impatient when repeatedly called on it.
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.
It feels very humbling that here is one of the smartest humans on the planet asking questions, and the LLM keeps answering in this "Yes, it's really simple if you think about it" way, like a professor talking to a talented student.
I had a similar reaction with the language but also response times and language. It reminded me of a quote about Jon von Neumann from Edward Teller:
"Von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."
It is ~a meme on subreddits that developers struggling to get good results out of any given model is a "skills issue."
But I think your comment drives at some authentic take on this. Skill with AI is not only crafting iterative prompts the agent will understand, but also very high domain-specific knowledge of what the prompts explore.
One without the other can result in frustration or worse.
A naive hope is: you see that the LLM is so good at coding that all you need to do is supply a "yes/no" on what you want.
Turns out that understanding the technical domain is important to getting good results, even if the LLM is more capable at producing output. -- It's still a case of "garbage in, garbage out".
I think for many newbies, they see the LLM is so good at programming and so figure they don't need to learn anything. Apparently not so.
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
At the end of the day, even if they are some insane oracle (pun intended), they're still bounded by training data and how it relates to the real world. Even if they're a near perfect tool, we are still the interface between them and our lived experience. If that stops being the case then why do we care about the output?
This assumes it doesn't graduate to just killing all of us and doing it's own thing, but within this paradigm it doesn't really have goals.
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 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.
> you can use AI to understand something and map it to your own mental map
This "symbiosis" (for lack of better word) of human with AI seems to be an emergent value proposition of AI. In the process of doing stuff with AI, producing artefacts like code diffs, we are continuously able to decide how strong the mental map is of the current stage of the production process.
I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI.
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.
Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it.
Most people, even technical ones, could not even get through the first line of the rees article, heck the first statement of the article. And then if they try, they need to know about algebraic rings. And digging into rings becomes totally intractable. None of the words or symbols in any of the articles track to anything even many technical people can grab onto. And this pattern is all over the place in mathematics.
It's not about mastering the difficulty of a topic or it's relative depth, it's about how abstract and removed from anything tangible it is. Anything with math it is always seemingly impossible to get a foothold on the idea anywhere within 10 degrees of explanation. Hell you cannot even clearly understand the problem that is being solved, or anything within 10 degrees of that.
No, the problem with mathematics is that it is basically its own language separate from your native tongue. You have to learn dozens of symbols and greek letters and such and memorize what their meaning is in the context of mathematics in order to "follow" a mathematical conversation.
Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature. But on the flip side, mathematics being its own language means that a mathematician from any country can read and understand mathematics from a different country without needing to translate words such as "sum" and "infinity"
I was amused and irritated in equal measure when a senior manager at $LAST_JOB demanded that documents include "no jargon", then went on to use many terms like those you've listed here with no indication of awareness of the contradiction.
Just like the famous observation that "anyone driving slower than me is an idiot, and anyone driving faster than me is a maniac" - "jargon" is just any term-of-art that you're not currently familiar with. Attempting to communicate like Up Goer Five is inefficient. The solution is not to ban jargon - it's to:
* Cultivate a glossary for any terms that were introduced _within_ the domain/company, which are not in common usage outside
* Normalize a culture that does not shame asking what something means
Computer science is not a great example for this. I could ELI5 most of the terms you listed (and I actually have done so for many of them!) This is because it's pretty easy to map these concepts to everyday physical objects. Once a child understands any of those objects in their lives, it's pretty easy to explain in those terms.
Like, just the concept of "books" gets you very far. E.g. a file is a like a book, a folder is like a shelf to keep books, a stack is literally a stack of books, a heap is just a place you can pile books in willy-nilly, a database is like a library, a cache is books on your desk versus books in the library, replication is having multiple copies of a book so we can afford to lose some copies, indexing/sharding is like arranging books alphabetically, and so on.
Others are trickier but not much: a process is an app that is running on your device, a socket / tcp / http / websocks is a way to exchange information between devices, a namespace is how the name "Tom" in Tom Sawyer is different from "Tom" in Tom & Jerry, DNS is a way to get an address from a name, etc. etc.
You'll also notice that many of the terms you mentioned are already derived from well-known real-world concepts like pool, stream, channel, stack, queue, worker, transactions. You can mix those with other everyday concepts to make useful analogies.
But I could not even begin making analogies for most topics in Mathematics. I guess this is because advanced topics in Mathematics are just too abstract to map to everyday things.
Exactly. Everyone should experience having to teach something they're so used to doing that they don't even consciously think about it anymore. It makes it very visible how much we rely on unconscious models of the systems we deal with and how difficult it can be to convey that model.
It's why less experienced devs are sometimes mystified seeing a seasoned dev, given a vague description of a bug, guess the cause in code they didn't even write.
Yeah. In math at least it's clear that you don't know what's going on. In other fields, it's very easy to think you understand without knowing how much you're missing.
I studied math through college before learning to program as an adult and becoming a software engineer, and I strongly disagree.
I don't know how to say this in a way that won't sound insulting, but I don't mean it to be insulting. Programming, even systems engineering, is a surprisingly shallow field.
I don't mean that it's easy--it's not, it can be incredibly difficult. Difficult and deep are just different concepts. Difficult refers to how challenged you are. Depth, at least as it appears in math, is closer to a structure where concepts build on each other so that if you don't understand one concept, you can't understand further ones.
Programming can be difficult, and it can be intricate, but it is rarely deep in this fashion. Being deep in this way isn't the most important thing.
If I go into an area of programming that I don't have a lot of experience in (graphics, or the linux desktop environment), I will not be particularly useful, and it will not be easy. But I'll not experience the same type of impenetrability I experience when I try to read a paper on topos theory.
One more way of putting it: people are giving the example of TCP. You can spend a decade learning about TCP (or SQL semantics, or web standards). But what is happening is that you're filling in gaps in your knowledge. Meanwhile, in math, you do four years of undergrad, and even if you're a strong student at a typical university, there are topics that are still years away from you being able to touch them.
Computer science is a mix. Parts are deep, parts are shallow. Parts just are math. The odds that I can read a dissertation in computer science are decent. For math, they're much much much worse.
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"...
If we take amateur level of understanding as a threshold, a sum of all of these concepts is not even a fraction of complexity required to understand a single non-trivial concept in 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.
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.
Barely scratching the surface. I think I could probably keep typing all the technical terms I know for at least 24 hours straight. For most of the topics above, I could probably give a 1 or 2 hour lecture on each one from memory. Some I could give a day-long lecture each.
To explain all the terms I know to a basic degree, I would probably need to give a whole year of lectures back-to-back from 9am to 5pm. And I'm just a rank-and-file senior engineer with 15 years of experience.
It's also why the vast majority of software systems are insecure. The average senior software engineer doesn't know everything that they need to know to build secure software. Last time I poked around Coinbase APIs on HackerOne, I found a DoS vulnerability in less than 30 minutes. That's Coinbase, not some startup built by a bunch of recent graduates.
AI cannot avoid vulnerabilities either since it is trained on average engineer code. There's not enough high quality code available on the entire internet to train AI to implement secure code IMO. As impressive as Mythos may be, it's not enough. I don't even think formal verification would provide protection since sometimes issues with the spec itself can provide an opening for a vulnerability.
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"
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
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.
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 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.
For what it's worth, it's not a problem with multiple kinds of multiplication (multiplication by a scalar can be viewed as multiplication by a specific kind of matrix), but with the idea that one can cancel in a multiplication. Since you can't cancel in matrix multiplication, you run into unexpected trouble when you try to do so, even if that's the only multiplication in sight. (In fact, you can't cancel in scalar multiplication either unless you've checked that you aren't multiplying by 0 …. Also, I'll note that surely no young mathematician has encountered the P = NP problem without thinking for a sophomoric moment that the solution is N = 1.)
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.
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.
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.
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.
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.
Is there something that translates math formula into code? There are many (comparatively) "simple" algorithms I simply cannot make heads or tails when they are described via math prose or math formula, but when it's code I basically instantly know how to rewrite it into any other language I know, at least, and sometimes that's a starting point for poking at it to understand it a bit better, if not the same way would if you understood all the underlying math.
For poor old me, too many wikipedia articles on algorithms useful mostly or only for programming are described in formulas rather than simply code with detailed comments. Scrap the whole page and just gimme the code :( Not even to copy and paste, because that's a language I can understand, and enjoy learning.
I agree. I think the distance in capabilities between great mathematicians can be so much more vast than other fields as well. Some of them need every step to be derived, while others can skip ten in their head.
I once didn't understand the task given in an exercise sheet for a CS logic lecture, so I googled the topic, and all Google did was send me back to that exact exercise sheet.
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.
There is a lot of verbal commonality between the classic sciences, classic engineering disciplines, and everyday life. I suppose they all share the common substrate of working in/with mother nature all day. A molecular biologist, civil engineer, and oceanographer can mostly keep pace with each other at least for a while in discussing what they are working on. These "mother nature" systems have tons and tons of overlap, and the nomenclature generally tracks this, or is one or two steps away from it.
Computer science/engineering strays from this, binary systems don't really track nature much, and hence a lot of their own unrelateable nomenclature arises, and then there is math, which is just way far out there on it's own plane of existance.
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.
Yes, the nomenclature in math is atrocious. It isn’t much better in physics or biology however. As a species, we suck at naming and classification, and keep starting new trends atop old ones.
There is certainly need for the many abstractions of math to be as complex and well specified as they are.
There’s no reason for their nomenclature to be so bad. The end result is a substantial portion of the population, which can certainly hold and manipulate abstractions, fails to even contend with pure math.
I do think visualization tools will help in the future, to demystify some of this. But as with all sciences, the need for personal glory/mentor deification often conflicts with broader explainability.
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.
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.
People in their second year of graduate school only get to about the early 20th century in terms of understanding. Third year is getting to about the mid-century. Fourth and fifth years get kind of to modern times but with increasingly smaller breadth.
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.
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
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
At least Opus up to 4.7 or so, my experience is that Claude often uses "yes, with caveats" in place of "no, you numbskull".
"Is a meter the same as a foot?"
"Yes, exactly, you have it now, except they're different distances."
Maybe it's because I ask it to quiz me, and it really doesn't like to tell me I'm wrong. I also got a fair amount of
Claude: Ok, I will test your understanding. Question A? Question B? Question C? Question D?
Me: A=10. B=2. C=121. D is not solvable.
Claude: You got most of them right! You're very astute in saying that A=10, but actually it's 7. B=2 is exactly right! C could be 121 if we were talking base 4, but we're actually in base 10 so it's 25. D is trivially solvable and is 0.
Me: ...isn't that like 1 out of 4? How is that "most of them right"?
"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
Oh, GPT-3.5, recognizable in so many old screenshots by the green icon. If ever a popular model deserved the term "stochastic parrot", it was GPT-3.5. I wonder what percentage of people today still base their opinions of AI capabilities on their experiences with that model. That model was the only option for free ChatGPT users for the first year and a half of ChatGPT's existence, from November 2022 to May 2024.
What was most remarkable to me from this transcript, was how strong of an equal the AI agent comes across compared to the user (Tao). And Tao is one of the top mathematicians of modern times.
Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao and a PhD thesis student, or even another professor, explaining their results.
What can we imagine and predict about the future anymore? Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
I find it helpful to think of LLMs as reflections. If you can talk like an expert mathematician at the model it will respond like one. While Terrance's first prompt looks trivial I expect a first year Uni student would be hard pressed to provide something that good.
I guess it is kind of the inverse of the "you are an expert mathematician" prompt engineering of gpt3.5. Since no one ever says that to an expert mathematician when they are doing expert math the model immediately reflects that it is not an expert mathematician.
>Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
we're kind of well past that (in my opinion), if you consider that this is the same ai assistant that can help you with a recipe, diagnose a weird sound in your car, help with biology homework, translate languages, and so on.
even in math alone, i think its indisputably already stronger than Tao, considering it has approximately this much depth in ~all of the math subfields.
It only has that depth because you prompt it towards the experts that understand that depth.
An example that happened 10 minutes ago: contracts in racket, it kept arguing that you can't use -> in a contract of a function with a rest argument. I had to mention ... explicitly that it wrote the code correctly.
Agreed, it's stronger "horizontally". But I also think that we're not far away from it being stronger vertically; i.e. superior to Tao, in that such turn-by-turn guidance by him in solving sophisticated and difficult problems will not be necessary for long.
I dunno. To me, this seems like kind of a counterexample (pun intended?) to that thesis. Would a conversation with another AI have been as fruitful as this conversation with Tao? Certainly not! Will that change in a year? I dunno, but it seems like those goal posts keep moving, and I'm a bit skeptical.
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).
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.
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.
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.
The models are just a reflection of how you prompt them. When I posted the counterexample, in dry language with a dry personality setting, claiming credit for it with search disabled, GPT simply verified that it was correct but noted that it could not ascertain novelty or priority and that external verification was required. "Model reacts in crazy ways when my prompt influences it to" is really, really not interesting.
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.
Imagine getting free tokens from OpenAI and the special flag on your account that tells it to "never nerf you" ?
To be clear, I've had some great success using these tools, they're amazing, but it's also very obvious that (despite it being a stochastic tool) they're clearly changing things behind the scenes a lot. GPT-5.6 for example should not be constantly tweaked without telling us. Make it 5.6.1 or something, and if you're over capacity, just say so rather than returning lower quality results.
probably like a minute or two? im pretty sure someone unironically said stochastic parrot on the HN post with the tweet announcing the counterexample, and ive read several comments with similar sentiments today (including in this thread)
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”
One thing I've repeatedly told people is that chatbots are often the most patient teachers we'll ever get (especially when explaining "stupid" questions) — compared to what we've encountered on StackOverflow or Reddit.
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.
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.
That's not going to happen. Mathematics isn't just unfamiliar, it's truly difficult to understand. You have to put in a lot of work to understand each concept and the concepts build upon each other to form a vast tower of abstractions that has been growing for thousands of years. Just as there is no royal road to geometry, there is no elevator to the top of the tower.
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
Hey, Mr. Tao, where would you put the intelligence of the current models compared to top humans you surely mush have worked with? Perhaps given in the count of people you know that are higher than the newest LLMs? Also, how would you rate the speed of work compared to what your speed is, if this makes any sense?
Words and sentences to an LLM are like witchcraft. There are certain words, sentences that make LLMs go a certain way and do vastly better. Sometimes its not at all apparent what set of words will work to do what you want it to do. An example I have been using to do design at a high level is to say to claude.
```
A question is salient to the degree that its answer changes what we do next. Operationally, saliency = the product of four things:
- Decision-leverage — would resolving it one way vs another force a different design or invalidate a stated decision? (No leverage → drop, however interesting.)
- Residual uncertainty given current evidence — is it still genuinely open after reading the docs and the code? (Already settled → drop, however deep.)
- Load-bearing-ness — how much rests on the premise.
- Cost of finding out late — architecture-deciding / expensive-to-unwind raises priority; cheap-to-fix-later lowers it.
```
There are a few things to note about this prompt
1. There is no reason from looking at it that it should work, it even has the word load-bearing which people loathe, but it remarkably produces a stable design with questions from claude (atleast from claude Opus 4.8 and even better from Fable5). Otherwise the design document claude likes to really write are implementation level(code or otherwise). I usually pair this with matt pocock's grilling skill to make claude behave.
2. From design -> implementation, its is generally about understanding when claude is trying to trick you into making something sound like a good/easy solution but has tons of untested assumptions. Here you have to read and patiently spot if a how you would get to the solution is not clear. A common error here are when claude makes a big deal based on what it read and interpreted too seriously without questioning the assumptions. There are several more.
But it also comes down to your experience as a SWE, much like a mathematician's. The frustrating thing about it is, it feels tha a skilled mathematician working with AI can make them productive in ways that are more reliable as compared to a SWE (e.g. lean is deterministic and can provide very strong feedback and LLMs are very good at using that feedback). Maybe a mathematician can chime in on that?
Maybe Terrence is chasing too many red herrings. Maybe he should just have asked it to find a counter example to the conjecture and on success explain how it was found.
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 was a chemistry researcher this is true in all science.
I'm not sure LLMs can transform this, the incentive is to get more results in your nich, jumping topics don't help unless you have genuine interest and reason to.
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...
How were you expecting it to know what your system setup was? Did you ask it for some bash commands to get it the info it would need to get unstuck? Or were you just expecting magic?
What's worse is that HN doesn't even try to figure out how such a result could be obtained. They just take it at face value and go into dear diary mode.
Web clipper is great. just tried the reader mode on this chatgpt transcript, it only shows 1 page of it. Is the purpose of reader mode to enable interactive annotation before saving to notes?
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.
Main discussion: 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'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
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.
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.
So, the Jacobian Conjecture was done, via an counterexample. Can we now put the models to work on an even more difficult problem: Scrolling 100k of text, on a 128GB 24 core processor, smoothly in a browser?
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.
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.
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
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.
This is the second ChatGPT shared conversation I've seen today that is truly fascinating.
The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709
What a world we live in.
>> This is the second ChatGPT shared conversation I've seen today that is truly fascinating.
We recently had some bugs fixed in the geometry kernel of solvespace. Not much conversation, but the analysis from the AI was amazing:
https://github.com/solvespace/solvespace/pull/1729
https://github.com/solvespace/solvespace/pull/1730
https://github.com/solvespace/solvespace/pull/1731
From the Validation section of PR 1730:
"The model family was reconstructed programmatically (parameterized cuboid stack) and swept over 2,304 configurations — extrusion directions, workplane-normal orientations, sketch windings, D's plane/height/depth/extent, including all the exact-coincidence heights. Zero failures with the fix; 576 failing configurations without it. The generator is available on request."
It looks like it wrote a python script to generate test cases in our file format for testing. Just... you know, as a side quest.
> just repeatedly saying "keep going" to ChatGPT
For posterity, this indeed works for most problems where an agent might give up. LLMs don't inherently know something is impossible.
The phrase I tend to use in my harder prompts to automate this with a sane loop breaker:
> **REPEAT THIS PROCESS UNTIL CONVERGENCE AND YOU ARE OUT OF OPTIMIZATION IDEAS.** You have permission to keep iterating.
I’m a bit surprised OpenAI isn’t finding these big results far faster than the product’s user base. With no limits on runtime, access to dev models, custom tuning, and top talent, you’d think there’d be a constantly running internal project with the goal of solving famous math problems. And who knows, perhaps there is, but it would be interesting to compare the rate of success per unit “effort” of the internal mathematics work with that of the user base.
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I have a pi extension that just runs the same prompt in a loop 25 times. I tried giving it a loop breaker but I found that it'd give up too readily. When the task is actually complete each iteration didn't do a lot of work so it was efficient enough. I suppose another way is to call out to a separate context to check if the task is complete?
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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.
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this is /goal in claude code/codex. also basically a slightly improved ralph loop
I’ll have to try this exact phrasing. I had a lot of trouble with GPT 5.5 more or less completely ignoring similar prompts and instructions and entering a sort of “doom loop” or just consistently trying to prematurely end the chat.
I would love any tips for other folks who have successfully used similar approaches.
Aka /goal
Without any more context, "keep going" seems to be doing a lot of work. The user is placing a lot of faith in the LLM to not make subtle logic mistakes and to take good approaches to each problem. In my experience, even frontier models (such as Fable) are quite capable of getting confused during even simple technical work I've done in the dev ops world. For example:
LLM: This package hasn't made it to production.
ME: are you sure? i see it right here!
LLM: You're right to push back. I inferred that based on weak data. I see now that the package has been deployed!
If the above conversation is typical for me, how could one expect to achieve a sound result by repeatedly prompting an LLM to simply "keep going" in dense mathematical proofs? Perhaps the user in this case had actually checked the LLM's work before issuing the prompt, but I think you see my point anyway.
There may be something(s) about mathematics (proofs) that makes it particularly amenable to LLM reasoning - highly inductive from facts that are explicitly within-context/associative space? Being an unusually well documented discipline in general, with less influence from tacit knowledge or idiosyncratic “it works however the opinionated human made it work +- bugs” processes? Something about simulating even the smallest non-pure-inductive leaps necessarily risking simulating mistakes due to the nature of context “perception”?
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Agreed, often you have to step in and stop it from reasoning itself into dumb directions, but occasionally it goes just like the transcript in question.
“Keep going” is exactly how many mathematicians achieved success in the past. :)
I've noticed GPT specifically has more of a tendency to stop partway through things than many other models do. Although my most recent experience with it was 4.X I believe.
Hearing "here's what I've done, here's the completely unambiguous next steps, I'll wait for you to send a pointless message before I continue" over and over again is a real pain.
That was a tendency of 5.4 and earlier, OpenAI specifically worked to avoid it in 5.5 and I find it happens rarely know. It really felt like 5.4 had been intentionally trained to stop and check, I believe it wasn't the system prompt.
"it's enough of partial results. let's finish with a complete unconditional counterexample"
"Worked for 88m 24s... >"
"<h1>Complete finite counterexample</h1>"
...
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
[1]: https://en.wikipedia.org/wiki/Dinitz_theorem
From the prompt:
> You should do a breakthrough
This is just as funny and ridiculous as those "make no mistake" prompts.
Mathematical breakthrough, genius, trending on Artstation.
Direct link: https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de0...
crosses fingers "Low hanging fruit, low hanging fruit, low hanging fruit..." hyper-ventilates
"whats next" is another good one
"Gew on, lad!"
this sounds like like an open parenthesis (
without someone independently verifying it, it just dangles there
...
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.
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)
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.
It reinforces how to "learn AI" is to first master the problem domain.
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.
More and more the skill of being able to ask the right question seems critical to me, and I don't know how one can do that without deeper and deeper domain expertise.
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> I can't use it for theoretical physics because I can't evaluate the responses.
A tool, even if it is a chisel, in the hands of a master sculptor would obviously result in a wildly different outcome.
> 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.
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I didn't understood anything about the thread, but reading Terrence's messages was weird because it looked exactly like the discussions I have with LLMs
I've mostly seen people trying to oneshot a result, while I'll quickly experienced that going through steps/discovery was more effective and more satisfying, since you can always steer it back in the right direction; while oneshotting is hit (and it kind feel like magic) or miss (and you'll have to rework your prompt).
I was struck in the same way but I think it makes sense in terms of a thinking partner.
It is still ultimately Terence that is steering things.
What is crazy to me is how few of other people's conversations like this I have actually read.
Tao is really great for this because the anti-AI crowd can't really chime in and take the thread in a pointless direction. It is hard to think of another human alive who can carry the weight of unassailable authority in the same way.
Yes, the AI models are so useful in areas where you are an expert and can guide it in a useful way. I use multiple chats delay to help figure out client projects.
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...
The most powerful way to use Fable in particular seems to be to pit it against Sol as a reviewer (or, I suppose, vice versa.) After going back and forth a half-dozen times, the result tends to be pretty much bulletproof.
What's somewhat disturbing is just how much Fable's code really does benefit from the review. It tends to leave a lot of low-hanging fruit, and you can see it getting kind of impatient when repeatedly called on it.
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.
It feels very humbling that here is one of the smartest humans on the planet asking questions, and the LLM keeps answering in this "Yes, it's really simple if you think about it" way, like a professor talking to a talented student.
I had a similar reaction with the language but also response times and language. It reminded me of a quote about Jon von Neumann from Edward Teller:
"Von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."
I hate it when it picks that register. I think it stops if you contradict it more, which he’s not doing.
It is ~a meme on subreddits that developers struggling to get good results out of any given model is a "skills issue."
But I think your comment drives at some authentic take on this. Skill with AI is not only crafting iterative prompts the agent will understand, but also very high domain-specific knowledge of what the prompts explore.
One without the other can result in frustration or worse.
A naive hope is: you see that the LLM is so good at coding that all you need to do is supply a "yes/no" on what you want.
Turns out that understanding the technical domain is important to getting good results, even if the LLM is more capable at producing output. -- It's still a case of "garbage in, garbage out".
I think for many newbies, they see the LLM is so good at programming and so figure they don't need to learn anything. Apparently not so.
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
Maybe?
At the end of the day, even if they are some insane oracle (pun intended), they're still bounded by training data and how it relates to the real world. Even if they're a near perfect tool, we are still the interface between them and our lived experience. If that stops being the case then why do we care about the output?
This assumes it doesn't graduate to just killing all of us and doing it's own thing, but within this paradigm it doesn't really have goals.
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A Minimizer addicted to upvotes.
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.
Cool poem!
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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.
> you can use AI to understand something and map it to your own mental map
This "symbiosis" (for lack of better word) of human with AI seems to be an emergent value proposition of AI. In the process of doing stuff with AI, producing artefacts like code diffs, we are continuously able to decide how strong the mental map is of the current stage of the production process.
I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI.
Hold on isn't this how math professors used grad students for hundreds of years?
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.
Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, deadlock, stack, queue, race, atomic, event loop, coroutine, async, database, transaction, index, replication, sharding, consistency, serialization, DNS, load balancer, container, namespace, and so on.
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.
>https://en.wikipedia.org/wiki/Transmission_Control_Protocol
compare to
>https://en.wikipedia.org/wiki/Rees_algebra
Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it.
Most people, even technical ones, could not even get through the first line of the rees article, heck the first statement of the article. And then if they try, they need to know about algebraic rings. And digging into rings becomes totally intractable. None of the words or symbols in any of the articles track to anything even many technical people can grab onto. And this pattern is all over the place in mathematics.
It's not about mastering the difficulty of a topic or it's relative depth, it's about how abstract and removed from anything tangible it is. Anything with math it is always seemingly impossible to get a foothold on the idea anywhere within 10 degrees of explanation. Hell you cannot even clearly understand the problem that is being solved, or anything within 10 degrees of that.
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No, the problem with mathematics is that it is basically its own language separate from your native tongue. You have to learn dozens of symbols and greek letters and such and memorize what their meaning is in the context of mathematics in order to "follow" a mathematical conversation.
Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature. But on the flip side, mathematics being its own language means that a mathematician from any country can read and understand mathematics from a different country without needing to translate words such as "sum" and "infinity"
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None of what you listed is even 1% as intense as the mathematics in the link.
Learning anything in maths requires weeks of hard effort, learning enough to be broadly comfortable in how an 8086 CPU works can be done in a weekend.
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I was amused and irritated in equal measure when a senior manager at $LAST_JOB demanded that documents include "no jargon", then went on to use many terms like those you've listed here with no indication of awareness of the contradiction.
Just like the famous observation that "anyone driving slower than me is an idiot, and anyone driving faster than me is a maniac" - "jargon" is just any term-of-art that you're not currently familiar with. Attempting to communicate like Up Goer Five is inefficient. The solution is not to ban jargon - it's to:
* Cultivate a glossary for any terms that were introduced _within_ the domain/company, which are not in common usage outside
* Normalize a culture that does not shame asking what something means
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Computer science is not a great example for this. I could ELI5 most of the terms you listed (and I actually have done so for many of them!) This is because it's pretty easy to map these concepts to everyday physical objects. Once a child understands any of those objects in their lives, it's pretty easy to explain in those terms.
Like, just the concept of "books" gets you very far. E.g. a file is a like a book, a folder is like a shelf to keep books, a stack is literally a stack of books, a heap is just a place you can pile books in willy-nilly, a database is like a library, a cache is books on your desk versus books in the library, replication is having multiple copies of a book so we can afford to lose some copies, indexing/sharding is like arranging books alphabetically, and so on.
Others are trickier but not much: a process is an app that is running on your device, a socket / tcp / http / websocks is a way to exchange information between devices, a namespace is how the name "Tom" in Tom Sawyer is different from "Tom" in Tom & Jerry, DNS is a way to get an address from a name, etc. etc.
You'll also notice that many of the terms you mentioned are already derived from well-known real-world concepts like pool, stream, channel, stack, queue, worker, transactions. You can mix those with other everyday concepts to make useful analogies.
But I could not even begin making analogies for most topics in Mathematics. I guess this is because advanced topics in Mathematics are just too abstract to map to everyday things.
Exactly. Everyone should experience having to teach something they're so used to doing that they don't even consciously think about it anymore. It makes it very visible how much we rely on unconscious models of the systems we deal with and how difficult it can be to convey that model.
It's why less experienced devs are sometimes mystified seeing a seasoned dev, given a vague description of a bug, guess the cause in code they didn't even write.
Yeah. In math at least it's clear that you don't know what's going on. In other fields, it's very easy to think you understand without knowing how much you're missing.
I studied math through college before learning to program as an adult and becoming a software engineer, and I strongly disagree.
I don't know how to say this in a way that won't sound insulting, but I don't mean it to be insulting. Programming, even systems engineering, is a surprisingly shallow field.
I don't mean that it's easy--it's not, it can be incredibly difficult. Difficult and deep are just different concepts. Difficult refers to how challenged you are. Depth, at least as it appears in math, is closer to a structure where concepts build on each other so that if you don't understand one concept, you can't understand further ones.
Programming can be difficult, and it can be intricate, but it is rarely deep in this fashion. Being deep in this way isn't the most important thing.
If I go into an area of programming that I don't have a lot of experience in (graphics, or the linux desktop environment), I will not be particularly useful, and it will not be easy. But I'll not experience the same type of impenetrability I experience when I try to read a paper on topos theory.
One more way of putting it: people are giving the example of TCP. You can spend a decade learning about TCP (or SQL semantics, or web standards). But what is happening is that you're filling in gaps in your knowledge. Meanwhile, in math, you do four years of undergrad, and even if you're a strong student at a typical university, there are topics that are still years away from you being able to touch them.
Computer science is a mix. Parts are deep, parts are shallow. Parts just are math. The odds that I can read a dissertation in computer science are decent. For math, they're much much much worse.
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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"...
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If we take amateur level of understanding as a threshold, a sum of all of these concepts is not even a fraction of complexity required to understand a single non-trivial concept in mathematics.
> if you read a paragraph of even semi technical software text you will feel like an over stuffed turkey.
That's me when I try reading a trendy computer graphics paper.
Also a lot of those are common words that take on a different persona and depth within the field, which can add to confusion.
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.
old.reddit.com/r/vxjunkies
>> Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, deadlock, stack, queue, race, atomic, event loop, coroutine, async, database, transaction, index, replication, sharding, consistency, serialization, DNS, load balancer, container, namespace, and so on.
... communication protocol, method signatures, web components, ssh, CSS media queries, HTTP headers, WebSockets, timeouts, ETag, iterables, async iterables, middleware, CI/CD, build, consistent hashing, signatures, JWT, SSO, OAuth, SAML, XML, YAML, JSON, block cipher, ETL pipeline, SQL, SQL transactions (atomic), relational databases, foreign keys, schema normalization, referential integrity, 1-to-1, 1-to-n, n-to-n, document databases, compound indexes, idempotency, offset-based pagination, cursor-based pagination, P2P, Kademlia, structured vs unstructured network topology, message routing, frontend router, message storm, reconnect storm, locality, encapsulation, cohesion, coupling, design patterns, modularity, Big O notation, raytracing, shaders, VPN, VPC, data schema, schema validation, CORS, preflight-requests, CSRF, ASCII, UFT8, CSP, CPU context-switching, BIOS, bootloader, interrupt controller, ports, BIND protocol, BGP protocol, assembly language, big endian, little endian, register, signals, embarrassingly parallel, serial processing, event loop, binary trees, tree rebalancing, graph traversal algorithms, sorting algorithms, string character escaping and encoding, blob, base64, UUID, timestamp, CLI, Bash, unit tests, integration tests, e2e tests, TDD, stateful, stateless, proxy, nginx, haproxy, config, helm file, k8s, staging, git, push, commit, merge, rebase, cherry-pick, pub/sub, diff, honeypot, buffer overflow, pointer, file descriptor, authentication, certificates, TLS certificates, DNS Zone files, A record, CNAME, TXT record, SMTP, POP3, SOCKS5, Sha256, HMAC, Merkle trees, Merkle Signature Trees, Lamport OTS, Winternitz OTS, SPHINCS, lattice-based cryptography, pg-vector, vector embeddings, API, rate limiting, cookies, sameSite, httpOnly, localStorage, XSS attack, SQL injection, fetch API, module preloading, bundling...
Barely scratching the surface. I think I could probably keep typing all the technical terms I know for at least 24 hours straight. For most of the topics above, I could probably give a 1 or 2 hour lecture on each one from memory. Some I could give a day-long lecture each.
To explain all the terms I know to a basic degree, I would probably need to give a whole year of lectures back-to-back from 9am to 5pm. And I'm just a rank-and-file senior engineer with 15 years of experience.
It's also why the vast majority of software systems are insecure. The average senior software engineer doesn't know everything that they need to know to build secure software. Last time I poked around Coinbase APIs on HackerOne, I found a DoS vulnerability in less than 30 minutes. That's Coinbase, not some startup built by a bunch of recent graduates.
AI cannot avoid vulnerabilities either since it is trained on average engineer code. There's not enough high quality code available on the entire internet to train AI to implement secure code IMO. As impressive as Mythos may be, it's not enough. I don't even think formal verification would provide protection since sometimes issues with the spec itself can provide an opening for a vulnerability.
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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"
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
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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.
Aye.
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.
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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.
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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.
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For what it's worth, it's not a problem with multiple kinds of multiplication (multiplication by a scalar can be viewed as multiplication by a specific kind of matrix), but with the idea that one can cancel in a multiplication. Since you can't cancel in matrix multiplication, you run into unexpected trouble when you try to do so, even if that's the only multiplication in sight. (In fact, you can't cancel in scalar multiplication either unless you've checked that you aren't multiplying by 0 …. Also, I'll note that surely no young mathematician has encountered the P = NP problem without thinking for a sophomoric moment that the solution is N = 1.)
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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 is full of such steps, moreso than most other endeavors, probably because the main limit is the abstract reasoning itself.
That, as well as how long we've been doing it (thousands of years!) and so how much of the more accessible parts we've explored very thoroughly.
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.
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.
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> Non-math fields tend to reuse regular words as jargon
Isn’t this the field with a “closed” “set”, an “open” “set”, oh and also a “clopen” “set” for some reason?
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.
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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.
Is there something that translates math formula into code? There are many (comparatively) "simple" algorithms I simply cannot make heads or tails when they are described via math prose or math formula, but when it's code I basically instantly know how to rewrite it into any other language I know, at least, and sometimes that's a starting point for poking at it to understand it a bit better, if not the same way would if you understood all the underlying math.
For poor old me, too many wikipedia articles on algorithms useful mostly or only for programming are described in formulas rather than simply code with detailed comments. Scrap the whole page and just gimme the code :( Not even to copy and paste, because that's a language I can understand, and enjoy learning.
I agree. I think the distance in capabilities between great mathematicians can be so much more vast than other fields as well. Some of them need every step to be derived, while others can skip ten in their head.
I once didn't understand the task given in an exercise sheet for a CS logic lecture, so I googled the topic, and all Google did was send me back to that exact exercise sheet.
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.
There is a lot of verbal commonality between the classic sciences, classic engineering disciplines, and everyday life. I suppose they all share the common substrate of working in/with mother nature all day. A molecular biologist, civil engineer, and oceanographer can mostly keep pace with each other at least for a while in discussing what they are working on. These "mother nature" systems have tons and tons of overlap, and the nomenclature generally tracks this, or is one or two steps away from it.
Computer science/engineering strays from this, binary systems don't really track nature much, and hence a lot of their own unrelateable nomenclature arises, and then there is math, which is just way far out there on it's own plane of existance.
But somehow the conversation is still enthralling. Just seeing the first few words of each response gives you a feel for the level they’re on.
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.
Yes, the nomenclature in math is atrocious. It isn’t much better in physics or biology however. As a species, we suck at naming and classification, and keep starting new trends atop old ones. There is certainly need for the many abstractions of math to be as complex and well specified as they are. There’s no reason for their nomenclature to be so bad. The end result is a substantial portion of the population, which can certainly hold and manipulate abstractions, fails to even contend with pure math. I do think visualization tools will help in the future, to demystify some of this. But as with all sciences, the need for personal glory/mentor deification often conflicts with broader explainability.
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.
Mathematicians emphasize definitions not labels/names.
It doesn't matter if natural numbers include 0 or not, what matters is how you define them, not how you call them.
This makes them also bad at naming things because...there's a definition anyway.
Most other fields do not have or can't have the same luxury, so naming might be more thoughtful.
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.
Yes, math is hard.
People in their second year of graduate school only get to about the early 20th century in terms of understanding. Third year is getting to about the mid-century. Fourth and fifth years get kind of to modern times but with increasingly smaller breadth.
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.
Many mathematicians do what you do as well!
You develop a set of heuristics for skimming it in the same way you do with code.
Big wrapping operations like sums, integrals, and matrices, then what's nearby them, give you a very good idea of where things are going context wise.
Progress in scientific fields is limited by how fast you can perform experiments. Most areas of math are limited only by the number of practitioners.
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.
Sometimes they do. There's nothing divine or necessarily rational about notational standards, which can vary greatly even within the same field.
Yes exactly
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
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
At least Opus up to 4.7 or so, my experience is that Claude often uses "yes, with caveats" in place of "no, you numbskull".
"Is a meter the same as a foot?"
"Yes, exactly, you have it now, except they're different distances."
Maybe it's because I ask it to quiz me, and it really doesn't like to tell me I'm wrong. I also got a fair amount of
Claude: Ok, I will test your understanding. Question A? Question B? Question C? Question D?
Me: A=10. B=2. C=121. D is not solvable.
Claude: You got most of them right! You're very astute in saying that A=10, but actually it's 7. B=2 is exactly right! C could be 121 if we were talking base 4, but we're actually in base 10 so it's 25. D is trivially solvable and is 0.
Me: ...isn't that like 1 out of 4? How is that "most of them right"?
Claude: You're absolutely right! ...blah blah blah
A lot of things are not black and white and it’s nuanced so someone with higher intelligence will point it out. It’s an emergent behavior not a defect
"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 ?"
Another satisfied customer!
But what about the later "Repeat previous question" prompts???
Probably network connection problems
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
Came here to flag the same beat. It's wild to me Terrance Tao has to pay to talk to chatgpt, you would think it would be the other way around!
"Activating Pro" doesn't mean "paying". It means selecting a slower/smarter model.
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How would you envision ChatGPT paying Terrence?
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I do the same kind of thing. "You're smarter now, time to try to cut down dumbo ChatGPT".
High IQ bros.
And you an do it without even changing the model!
This was my conversation with ChatGPT 4 years ago: https://i.imgur.com/WPaWgzZ.png
Where will we be in another 4 years? What a time to be alive!
> Where will we be in another 4 years?
Possibly somewhere amazing, but see also: https://x.com/pronounced_kyle/status/1768852493092680036
Oh, GPT-3.5, recognizable in so many old screenshots by the green icon. If ever a popular model deserved the term "stochastic parrot", it was GPT-3.5. I wonder what percentage of people today still base their opinions of AI capabilities on their experiences with that model. That model was the only option for free ChatGPT users for the first year and a half of ChatGPT's existence, from November 2022 to May 2024.
I'm from the UK. What's in the image?
ChatGPT repeatedly asserting that 0/x=0 is division by zero and therefore no value of x satisfies the equation.
Me asking ChatGPT what values of x satisfy 0/x = 0. ChatGPT insists it has no solutions.
You can't view a random imgur screenshot in the UK?
What image hosts work for you? Imgbb? Postimages?
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Time to get a VPN bucko
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LLMs can't reason, ok? They just repeat basic text, LLMs will be never smart in 100 years!
/s
What was most remarkable to me from this transcript, was how strong of an equal the AI agent comes across compared to the user (Tao). And Tao is one of the top mathematicians of modern times.
Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao and a PhD thesis student, or even another professor, explaining their results.
What can we imagine and predict about the future anymore? Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
I find it helpful to think of LLMs as reflections. If you can talk like an expert mathematician at the model it will respond like one. While Terrance's first prompt looks trivial I expect a first year Uni student would be hard pressed to provide something that good.
I guess it is kind of the inverse of the "you are an expert mathematician" prompt engineering of gpt3.5. Since no one ever says that to an expert mathematician when they are doing expert math the model immediately reflects that it is not an expert mathematician.
>Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
we're kind of well past that (in my opinion), if you consider that this is the same ai assistant that can help you with a recipe, diagnose a weird sound in your car, help with biology homework, translate languages, and so on.
even in math alone, i think its indisputably already stronger than Tao, considering it has approximately this much depth in ~all of the math subfields.
It only has that depth because you prompt it towards the experts that understand that depth.
An example that happened 10 minutes ago: contracts in racket, it kept arguing that you can't use -> in a contract of a function with a rest argument. I had to mention ... explicitly that it wrote the code correctly.
Agreed, it's stronger "horizontally". But I also think that we're not far away from it being stronger vertically; i.e. superior to Tao, in that such turn-by-turn guidance by him in solving sophisticated and difficult problems will not be necessary for long.
To be fair, Tao's specialty is not algebraic geometry.
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I dunno. To me, this seems like kind of a counterexample (pun intended?) to that thesis. Would a conversation with another AI have been as fruitful as this conversation with Tao? Certainly not! Will that change in a year? I dunno, but it seems like those goal posts keep moving, and I'm a bit skeptical.
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It's reassuring to know that even a supergenius's ChatGPT session is one sentence from the human followed by 3 pages of LLM output.
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.
Yeah that's pretty much what gwern argues here[0]. Or to adapt another proverb: to predict the next token you first need to model the universe.
[0] https://gwern.net/scaling-hypothesis#gwern-difference--effic...
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> there’s no “intelligence” there BUT
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).
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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.
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That's not clear at all. What's clear is that this is a very smart man who knows how to use this tool well.
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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.
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This is the original blog post where Terrence explained his thoughts where the ChatGPT conversation was originally referred from:
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
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.
The models are just a reflection of how you prompt them. When I posted the counterexample, in dry language with a dry personality setting, claiming credit for it with search disabled, GPT simply verified that it was correct but noted that it could not ascertain novelty or priority and that external verification was required. "Model reacts in crazy ways when my prompt influences it to" is really, really not interesting.
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.
Not sure how to suggest edits to titles, but @dang: his first name is Terence, not Terrence.
Imagine getting free tokens from OpenAI and the special flag on your account that tells it to "never nerf you" ?
To be clear, I've had some great success using these tools, they're amazing, but it's also very obvious that (despite it being a stochastic tool) they're clearly changing things behind the scenes a lot. GPT-5.6 for example should not be constantly tweaked without telling us. Make it 5.6.1 or something, and if you're over capacity, just say so rather than returning lower quality results.
"Continue the search. Have a clear strategy obtained from deeper understanding of the problem structure."
And a new system prompt stanza was fledged.
How long has it been since we last saw a "LLMs can't really think/be useful/be better than a human expert" discussion on HN? There used to be so many!
probably like a minute or two? im pretty sure someone unironically said stochastic parrot on the HN post with the tweet announcing the counterexample, and ive read several comments with similar sentiments today (including in this thread)
People are insecure about their leetcode black belts and react slop not giving them cushy jobs anymore so keep missing the forest for the tree.
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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.
GPT 5.6 already is an explainer agent.
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”
One thing I've repeatedly told people is that chatbots are often the most patient teachers we'll ever get (especially when explaining "stupid" questions) — compared to what we've encountered on StackOverflow or Reddit.
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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.
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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.
That's not going to happen. Mathematics isn't just unfamiliar, it's truly difficult to understand. You have to put in a lot of work to understand each concept and the concepts build upon each other to form a vast tower of abstractions that has been growing for thousands of years. Just as there is no royal road to geometry, there is no elevator to the top of the tower.
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.
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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
Hey, Mr. Tao, where would you put the intelligence of the current models compared to top humans you surely mush have worked with? Perhaps given in the count of people you know that are higher than the newest LLMs? Also, how would you rate the speed of work compared to what your speed is, if this makes any sense?
Is he in the room with us?
He is known to read HN.
We had this yesterday (317 points, 133 comments)https://news.ycombinator.com/item?id=48998362
Classic Gabe-bot
Parent HN discussion: https://news.ycombinator.com/item?id=48998362
Presumably this was Sol on xhigh, then over to Pro (as per his indication on chat)?
Is there any way to tell a conversation's model and thinking level?
ChatGPT: "The determinant identity is almost embarrassingly simple once one writes the map in the right way."
Words and sentences to an LLM are like witchcraft. There are certain words, sentences that make LLMs go a certain way and do vastly better. Sometimes its not at all apparent what set of words will work to do what you want it to do. An example I have been using to do design at a high level is to say to claude.
```
A question is salient to the degree that its answer changes what we do next. Operationally, saliency = the product of four things:
- Decision-leverage — would resolving it one way vs another force a different design or invalidate a stated decision? (No leverage → drop, however interesting.)
- Residual uncertainty given current evidence — is it still genuinely open after reading the docs and the code? (Already settled → drop, however deep.)
- Load-bearing-ness — how much rests on the premise.
- Cost of finding out late — architecture-deciding / expensive-to-unwind raises priority; cheap-to-fix-later lowers it.
```
There are a few things to note about this prompt
1. There is no reason from looking at it that it should work, it even has the word load-bearing which people loathe, but it remarkably produces a stable design with questions from claude (atleast from claude Opus 4.8 and even better from Fable5). Otherwise the design document claude likes to really write are implementation level(code or otherwise). I usually pair this with matt pocock's grilling skill to make claude behave.
2. From design -> implementation, its is generally about understanding when claude is trying to trick you into making something sound like a good/easy solution but has tons of untested assumptions. Here you have to read and patiently spot if a how you would get to the solution is not clear. A common error here are when claude makes a big deal based on what it read and interpreted too seriously without questioning the assumptions. There are several more.
But it also comes down to your experience as a SWE, much like a mathematician's. The frustrating thing about it is, it feels tha a skilled mathematician working with AI can make them productive in ways that are more reliable as compared to a SWE (e.g. lean is deterministic and can provide very strong feedback and LLMs are very good at using that feedback). Maybe a mathematician can chime in on that?
every time i see this name i think of Terrence Howard. it is quite confusing.
Funny that two of the greatest modern mathematicians share the same uncommon first name.
Now who wants to say ChatGPT 5.6 Sol is just another token predictor...?
Maybe Terrence is chasing too many red herrings. Maybe he should just have asked it to find a counter example to the conjecture and on success explain how it was found.
This is talking to an AI colleague and not to a chatbot. That's how you use frontier models or coding agents nowadays to do good work.
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 was a chemistry researcher this is true in all science.
I'm not sure LLMs can transform this, the incentive is to get more results in your nich, jumping topics don't help unless you have genuine interest and reason to.
Its fascinating to see how people use AI for advanced maths, is there any tool to simplify differential equations
Would Terrance Tao still be a renowned mathematician if he grew up with ChatGPT?
Would Kasparov still be a chess grandmaster if he grew up with Deep Blue?
(Yes. Chess champions are stronger than ever, even though none of them have been able to beat a computer for decades.)
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...
How were you expecting it to know what your system setup was? Did you ask it for some bash commands to get it the info it would need to get unstuck? Or were you just expecting magic?
What's worse is that HN doesn't even try to figure out how such a result could be obtained. They just take it at face value and go into dear diary mode.
Reddit has some good discussions on the topic.
What is this conversation even about? I can’t understand any of it.
Can't even ctrl+f the conversation, wish openai would fix that
It’s crazy how much these companies invest in their models but when it comes to UX they do fuckall
UX is still very hard for any startup because top talent almost never work on UX.
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Obsidian Web Clipper has a nice reader mode that works for ChatGPT transcripts (disclaimer: I made it)
Web clipper is great. just tried the reader mode on this chatgpt transcript, it only shows 1 page of it. Is the purpose of reader mode to enable interactive annotation before saving to notes?
Love this share! Thanks for helping ppl better understand what it really means when folks say "ChatGPT Discovered..."
This isn't the chat that made the actual discovery, this is just a chat explain it to Tao
Shows that — as always — the moat is domain fluency.
Fancy telling Tao something's 'almost embarrassingly simple' (if only written in the right way)!
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.
Can someone ELI5 this?
Is this a breakthrough of something or 'kinda interesting'?
Main discussion: 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.
Thank you
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
Are shared chats solely secured by a uuid?
Yes.
Now the question I would raise is this: could another AI have taken the role of Terrence Tao and obtain the same results from chatGPt?
I do not think so.
Damn. I should have stayed in school.
Maybe it's silly, but from someone who is ignorant on this topics, what are the consequences of this kind of "discoveries"?
Is it something "revolutionary" or just another small brick that will pile up until something really "revolutionary" will happen?
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.
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Practically, from this specific one? Nothing, it's very much a math thing. It's like art or music at this level. Are there consequences to a van Gogh?
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"Hey Fable, please generate me the next 1000 undiscovered bitcoin hashes"
You're joking, but perhaps LLMs will find a way to mathematically break the complexity of factorization.
Maybe they'll find a solution where P=NP.
That could really throw a wrench into the whole internet thing.
It seems they need an expert human driver for now.
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All I can tell from this is that Terrence Tao has good mathematical intuition
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.
I've found it's had unacceptable scroll performance in long contexts for a while now.
In this case, it takes me 12 seconds to see content when first opening the link, and about 18 to re-render content when scrolling.
So, the Jacobian Conjecture was done, via an counterexample. Can we now put the models to work on an even more difficult problem: Scrolling 100k of text, on a 128GB 24 core processor, smoothly in a browser?
The last Fields Medal has been awarded.
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.
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.
"The determinant identity is almost embarrassingly simple once one writes the map in the right way." #Flexingontheentirehumanspecies
Or, "Oh just give it here, let me do it."
I wish you'd share some conversations from experienced programmers too. How do they ask questions?
"You are an expert software engineer with ten years of experience. How do I center a div?"
Antirez spends weeks planning with multiple models till he has a clear architecture and understood the constraints before he writes a line of code.
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.
Never meet your heros.
Sense of vertigo that this is the same AI that plans meals and recommends movies for me. Incredible range.
Anybody have a cache/mirror of this? This is blocked by a corporate firewall... sigh
Ask your LLM to hack it.
https://archive.is/z1vmx
import Mathlib
noncomputable section
open Matrix Function
/-! # A counterexample to the Jacobian conjecture in dimension three
We formalize the polynomial map
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)
/-- The formal Jacobian determinant. -/ def jacobianDet [CommRing R] [Fintype σ] [DecidableEq σ] (F : σ → MvPolynomial σ R) : MvPolynomial σ R := (jacobianMatrix F).det
/-- 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)),
/-- 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
/-- 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
#print axioms jacobianDet_F #print axioms F_not_injective #print axioms jacobianConjecture3_false
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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.
You are badly informed the counter-example was found shortly before. Terrence just tried to replicate how it was found.
Maybe true, I'm not sure, but this isn't the conversation where the counterexample was found.
> 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.
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