Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample • Hacker News Board

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#601

Post by Antipode » Thu, Jul 23, 2026, 1:28 PM UTC

looofooo0 wrote: post #502

godelski wrote: post #427

Earlier quoted context omitted.

> What a world we live in. It's a really interesting world. You can spam GPT to get novel math results but here I am trying to scroll up to the beginning of the conversation and 5 minutes in I still don't know if I'm near the top yet. Scroll... wait for render... scroll... wait for render... repeat... We live in a world where there's so much crazy technology but few people use it to make products better or to improve…

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?!

There are a multitude of methods of obtaining money that do not benefit society.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#602

Post by lanstin » Thu, Jul 23, 2026, 1:39 PM UTC

theplumber wrote: post #464

michaelchisari wrote: post #12

Earlier quoted context omitted.

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.

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.…

That seems only partially true. I have no doubt it could crank out a UI like Facebook, but having it work for high volume is less well represented in the training data. Things that seems obvious to us, like sharding, using asynchronous messaging systems to provide more durability at the cost of instant consistency, traffic shedding, intelligent clients, conscious decisions on CAP properties in the light of user experience. etc., are not going to be requested by that non-technical founder.

Facebooks early years were dominated by a concern for users per engineer ratio; the rather florid style of LLMs suggest that they will generate so many systems of such a high complexity you will get Hadoop levels of non-application support needed - forget self healing, it’ll need constant LLM spend just to keep running at scale.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#603

Post by lanstin » Thu, Jul 23, 2026, 1:41 PM UTC

tempfile wrote: post #543

eru wrote: post #459

Earlier quoted context omitted.

You can't hallucinate a working lean proof.

You absolutely can. How do you know your "working lean proof" actually proves the theorem you intended it to?

One of the concerns of the new LLM made lean proofs is ensuring they are using standard MathLib formulations in the theorem, so (quoting something in I longer recall the source of) a Grothendieck scheme is indeed what the reader and world know as a Grothendieck scheme.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#604

Post by lanstin » Thu, Jul 23, 2026, 1:42 PM UTC

MeteorMarc wrote: post #578

lukebuehler wrote: post #8

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): sho…

I think the questions are sufficiently suggestive to induce hallucinations. Still a lot of manual work to check all these claims.

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.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#605

Post by lanstin » Thu, Jul 23, 2026, 1:47 PM UTC

stephbook wrote: post #595

godelski wrote: post #427

Earlier quoted context omitted.

> What a world we live in. It's a really interesting world. You can spam GPT to get novel math results but here I am trying to scroll up to the beginning of the conversation and 5 minutes in I still don't know if I'm near the top yet. Scroll... wait for render... scroll... wait for render... repeat... We live in a world where there's so much crazy technology but few people use it to make products better or to improve…

> few people use it to improve people's lives. Did Tao do that?

The architecture of our modern technological society lies on top of pure maths. One cannot say “resolving the physicality of the Navier-Stokes equation” will lead to x, y or Zed human improvements, but for hundreds of years it has. It is like Bach making his fugues for beauty and pleasure but occasionally you get chip technology or mathematical ecology or GPS falling out of it.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#606

Post by lanstin » Thu, Jul 23, 2026, 1:54 PM UTC

inigyou wrote: post #514

looofooo0 wrote: post #502

Earlier quoted context omitted.

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?!

No, money and value have been decorrelated for a long time. At least 5 years and arguably since Reagan.

Like it was so correlated back in 1600? Under the Romans? But perhaps you just mean that the increased financialization of the US economy means that the percentage of economic wealth devoted to actions without value is increasing? Sort of a modern trajectory to 1789 with the foppish nobles not realizing how much everyone else can get along without them?

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#607

Post by DigitResort » Thu, Jul 23, 2026, 1:55 PM UTC

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Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#608

Post by lanstin » Thu, Jul 23, 2026, 2:04 PM UTC

Xcelerate wrote: post #340

minimaxir wrote: post #106

Earlier quoted context omitted.

> 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 mathem…

Had you heard of these “famous unsolved problems” before now? If you aren’t a trained mathematician, you won’t know where to point the LLM towards achievable goals. Having it solve some Erdos problems is pretty different than a Millenium prize, and even some of those might be unknown to the general non-mathematical public. Finding counter examples, unattended, to various algebraic geometry conjectures is amazing but still a ways off from the a definitive disposition of the Riemann hypothesis, P vs NP, the Navier-Stokes problem (although I wouldn’t be shocked if Professor Tao put an LLM towards a counter example for that based on his technique of encoding machines into initial conditions of PDEs).

Although if I were a mathematical quantum physicist I might be trying to prompt one into a % mathematical field formalized quantum field theory. Weirdly this sort of obvious step has not been accomplished in the last 100 years.

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#609

Post by aleph_minus_one » Thu, Jul 23, 2026, 2:04 PM UTC

StilesCrisis wrote: post #587

aleph_minus_one wrote: post #549

Earlier quoted context omitted.

> Having studied CS and maths to post-grad, I think you exaggerate. Although a CS course might use these tools, they didn't in my experience go into explaining or defining them. I studied computer science (and mathematics) in Germany. I am very certain that this was taught to computer science students, even though (compared to the lectures for math students) the lecturer did not get very deep into these topics. > mos…

Do you think weed-out courses are a good thing? If these courses are so important, they should be taught in a way that students can understand. If they made it to college, and can program, they're clearly smart and motivated.

> If they made it to college, and can program, they're clearly smart and motivated.

You clearly write from the perspective of the US-American university system.

In Germany, basically everybody can enroll into a computer science program at a university, assuming the person has a Abitur (Allgemeine Hochschulreife) certificate (these terms are difficult to translate into English) from the grammar school [1]. So, "have it made to college" is like "not having been a complete failure in school". [2]

So, making it to the university is no achievement in Germany, and also no sign of motivation either.

> If these courses are so important, they should be taught in a way that students can understand.

These courses are taught in a way that students can understand, but not in a way where you can afford to slack off.

It is basically a consensus in Germany that a university is clearly a wrong place for you if you are incapable of closing knowledge gaps on your own (for example by reading books from the library), and you don't have the self-motivation to sit over the lecture material for hours to finally understand it.

So yes, I would say that among the possible options, weed-out courses in mathematics are in my opinion likely the least bad one.

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[1] In years where there was an insane demand for places at the university to study computer science such as during the dot-com bubble, there were some restrictions (numerus clausus), but for computer science, this was always the exception to the rule.

[2] There exist good reasons for puns like "Abitur: nichts gerafft und doch geschafft" (Abitur: Didn't get a thing, yet still passed) or "A-bier-tur" (a portmenteau of "Abitur" and "beer", which suggests that even pupils who are more into drinking than learning typically get their Abitur certificate).

Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

#610

Post by divyekapoor » Thu, Jul 23, 2026, 2:05 PM UTC

buzzin_ wrote: post #349

jongjong wrote: post #301

Earlier quoted context omitted.

>> 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,…

Nice list. I like it how you also included some made up terms. For completeness, those are not real: ETag, SAML, CSP, BGP, haproxy, helm file, k8s, SOCKS5, Lamport OTS, Winternitz OTS, SPHINCS, lattice-based, pg-vector, sameSite, httpOnly. (Or, at least, i have no idea what those are ::)

Not at all! All of these are real. ETag: expiry tag on HTTP(s), SAML: auth protocol, CSP: constraint sat, BGP: border gateway protocol, haproxy: load balancer, helm: k8s package manager, SOCKS5: http proxy protocol, sameSite/httpOnly: configs for HTTP security… and so on.

Never mind. The answer before me did a much better job.