Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

Correctamundo. That's EXACTLY what I just wrote about. And how we'll create the next generation of people who know how to evaluate responses when we're creating a generation of people who are increasingly reliant on LLMs to do the work makes for a strange paradox.

https://larsfaye.com/articles/ai-coding-will-prevent-experti...

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 agree. I (15+ years of coding) feel like I'm 100x more productive now using LLMs to code.

But I don't think a beginner would have the same experience. The AI still makes A LOT of stupid mistakes and decisions, but I catch them early enough (sometimes while it's still showing it's reasoning steps), stop the prompt, guide it on the right path, rinse and repeat.

Sometimes I am lazy and give the AI a broader prompt, let it do its thing, and then I come back to see that it spent 90% of the time working on some part/feature/implementation that was not really needed and that it over-engineered the solution.

I rarely write any line of code know or manually change any code, I tell the AI how to do it and what to watch out for. Many times it catches some edge-cases before I even haven to think about them. I do still feel like both me and the AI could miss some edge-cases now, because I'm thinking less about the implementation and what problems can arise, but I feel like 90% of "gotchas" are already engrained in my planning after so many years of coding and problem solving.

> 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.

That is probably not going to be the important difference.

You can't use it for theoretical physics because you have no meaningful question and there is no result that you can do anything with.

In contrast you don't need to be a coder to understand if your to-do list for cats works: You have an idea of what you want. You know the rough shape of what an app is and what it can do. You can put it in front of your can and look at it go. Or not.

> I can't use it for theoretical physics because I can't evaluate the responses.

I think pretty much the opposite. I can ask it to explain to me in ways I understand it. Even drill down the simplest of equations. Since llms have infinite patience. All I need to learn anything is patience.

Indeed that’s the case but most of the programming problems are solved problems and unless you care how things are done someone with less/no knowledge can still make great use of programming to do what it needs to do.

Let’s not pretend you need to be programmer to use AI for programming like mathematicians need to solve math problems. Programming (most of the time)solves real problems rather than abstract constructs.

AI is already good enough to create the next “Facebook”(v1 and maybe v2 as well) without any real programmer. This makes the AI a big enabler and reduces the need for programmers vastly in the early stages of any business.

That being said as more online businesses will flourish these will require actual programmers after they get enough traction so the debate is still on if it will lead to massive layoffs in our industry.

Yeah. I am a professor and I see a lot of other professors saying they are pivoting to "Teaching AI". I tell them I am pivoting harder to teaching fundamentals because using AI is easy. Its the knowledge which is hard.

> 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.

But the AI can teach you to do theoretical physics, by guiding you through materials it selected for you, helping you with questions, etc.

Yes but, AI can still be used for relatively common applications (coding basic pages), and furthermore you can build agentic loops to check more difficult code, or explain the conceptual order of the code, and I imagine over time the domains which will be considered “basic” for the AI will continue to grow, but perhaps so will the abilities of its users…

But someone who is not well versed in coding can evaluate the output simply by running the built app and testing it, which is an oversimplification and omits the tech debt and eventual issues that an experienced developer would catch or restrict the output.

I recently prompted ChatGPT about theoretical physics because I wanted some examples of AI slop. It affirmed my hypothesis that dark energy results from the nonzero VEV of the Higgs field. Wow! I'm so smart! I solved dark energy!

Any actual physicist would probably be able to tell me why that's a category error. I don't know why because I'm not one. But there are actual mathematics underlying a statement like that and I'm 99% sure the maths don't work like that.

By the way I think that's why everyone thinks of so many weird physics ideas more than other fields. It's because things are explained in words that hide math, and you can make hypotheses in words that would be obviously nonsense at the level of maths. Like your boss asking why you don't just recompile the cloud.

My rule of thumb is: Never ask an LLM a question where you don't know what the right answer or wrong answer looks like.

Or in the words of Clickbait articles: MATHMATICIANS SHIVER OF THE IMPLICATIONS OF AI WILL MATHMATICIANS NO LONGER BE NEEDED.

to which betteridge's law of headlines says: NO.

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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.

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.

I am not impressed with that steering of the AI. In the case of Terry Tao, he clearly held a solid understanding of the problem domain and could also understand the outputs of the LLM at an expert level. He asked astute, topical questions that specifically probed areas of the problem that were either unclear or insufficiently specified, and ultimately ended up at a place where (presumably) he both understood the Jacobian Conjecture Counterexample exhaustively, and how it was derived. My impression is that through his learnings during the chat, he became sufficiently well-educated in this subdomain to apply some of these lessons to other mathematical pursuits.

In your counterexample, the human just repeatedly asked the model to "try harder". Not remotely the same.

For Professor Tao, I think that step is reading thru it once. Which is why the tool is more powerful for him than for us.

And I am still waiting for the Apple Reminders AI to sort Milk in the right category…

On my mac if you scroll slightly a blob appears on a slider on the right and you slide it to the top to go to the top. I'm not sure about these hidden user interface features that you only find through trial and error.

if people use it to make money, does this not mean that they providing some sort of service which is valued and hence it improves lifes?!

> few people use it to improve people's lives.

Did Tao do that?

It always is about money. As long as there's money it will always be about it. LLMs are big techs attempt of shifting money from the people further up, by getting rid of jobs. We're all not angry enough because some of yall are so delusional about the whole thing, by thinking we will get some kind of Utopia.

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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.

Maybe it's just me, but does anyone else feel goofy speaking to a computer this way?

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

> 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.

On the one hand, agents have done this sort of thing for a year+, if you pushed them to check their work. On the other, I absolutely can feel Fable and Sol have crossed a threshold where they can be trusted far more than before. Huge difference between plans written by Opus or Fable.

Accumulated AI slop can simply be cleaned up by better models. Real cost of technical debt is shrinking due to the the inflationary devaluation of code!

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.

It's going to be pretty crazy when GPT-6 comes out, since the rumors are it's quite a bit smarter and wiser than GPT-5.6.

That’s probably not interesting anymore, but 5.6 wasn’t able to name the conjecture when presented the notation only, but confirmed the proof and when told what it was, agreed it works. Much less psychosis than when given the Jacobian counterexample, at least.

Mathematical breakthrough, genius, trending on Artstation.

The day is coming when some *BIG* problem is solved by AI just because someone jokingly asks about it.

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.

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.

)

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.

A lot of “advanced” concepts are simple once you find the right way to frame them.

Terrance Tao is GOATed.

That’s what I got from reading about a third of that exchange.

I mean that’s clear from reading anything about the dude, but to _experience_ it so to speak is different.

A true mark of brilliance is someone who makes something difficult seem easy by framing it just right.

In this instance ChaGPT is just expanding upon his prompts. Not to diminish, it’s amazing what it’s capable of.

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.

It is a skill issue. Such developers have to take a problem, isolate it and explain it to someone/something else with enough context to be able to work with it.

It is perplexing how many developers lack this basic skill, some of them borderline lack theory of mind and are incapable to understand that other people can't see the unspoken part in their heads.

>It is ~a meme on subreddits that developers struggling to get good results out of any given model is a "skills issue."

*GIT GUD* (at prompting)

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.

He is well known for the large numbers of collaborators he has worked with to develop new math in all sorts of fields. Grad students in general wouldn’t be sharp enough to benefit maximally from his skills, but other math professors.

Cool poem!

> 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.

I do get what you are saying and agree with you in principle, as I have noticed the same. But that said, this same way of thinking can really apply to any abstraction. It really just depends on the level you're working at. A software architect might know the nitty gritty details of how each service works, but really those implementation details don't matter if the abstractions are handled well enough, so in theory they don't really need to know those details as long as "the pieces fit". But of course the catch 22 there is you can't build good abstractions that can fit together well if you don't understand well enough the underlying details

"used" is a bit loaded, but this is just how people talk to eachother when they are thinking really hard about something they want to solve. So, I'm fairly sure it goes back as far as speech. It's just debugging - "What if it was caused by xyz", "What if we tried abc"

>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.

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"

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.

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.

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

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.

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"...

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.

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.

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.

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.

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.

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.

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 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.)

>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.

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.

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.

> 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 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.

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.

Sometimes they do. There's nothing divine or necessarily rational about notational standards, which can vary greatly even within the same field.

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!

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.

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.

To be fair, Tao's specialty is not algebraic geometry.

Isn't LLM doing something like argmax{x} P(x|context) where x is the next word, and context is the words before including your instructions? Thus, the LLM output is heavily depending on your instructions, and if you input an expert context it would produce an expert result. Consequently, domain expertise has a significant bearing on what you can achieve from LLM.

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

This has to be one of the more annoying characteristics of the recent models.

Every answer is met with "But did you think about X", or "One more thing to consider" and it almost starts to feel manic as you dig down that hole.

Multiple times I've had to literally say "Stop with the follow-ups and suggestions. Just answer the questions clearly".

GPT-3.5 was so nice because the safeguards were so easy to bypass and it wasn't lobotomized by RLHF. Newer models really don't hit the same spot and I'm not sure what I liked so much about that initial release. I look back at conversation logs I saved and I just can't replicate them nowadays.

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?

firefox has a built in VPN now. I wonder if it would work for you.

Time to get a VPN bucko

Probably network connection problems

"Activating Pro" doesn't mean "paying". It means selecting a slower/smarter model.

How would you envision ChatGPT paying Terrence?

And you an do it without even changing the model!

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.

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.

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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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...

A property of lack of intelligence though is stopping too soon. The wisdom to differentiate is probably rare at the extremes.

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.

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 don’t think the comparison is fair. Deep Blue was an opponent in a highly bounded game. Mathematics has no bounds, and modern agents are specifically designed and trained NOT to be opponents.

He is known to read HN.

UX is still very hard for any startup because top talent almost never work on UX.

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?

Thank you

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.

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.

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 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.

Has the conversation that originally led to the counterexample been released? Surely it was similar to this one, also driven by a highly knowledgeable mathematician.

No AI would have been able to find it if asked to "Find a counterexample to the Jacobian conjecture. Make no mistakes".