Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample

chatgpt.com

Post reply on HN
Print view View original

331–340 of 681 posts

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

#331

Post by pfdietz »

traes wrote: post #229
hyperhello wrote: post #41

Earlier quoted context omitted.

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.

This is not true at all. The parameter space is absolutely MASSIVE. The counterexample is a degree 7 polynomial in 3 variables, which means 360 coefficients. There's no particular way to bound these coefficients or even the degree or number of variables apriori, but assume you somehow did. Also assume you were confident that it would work with integer coefficients bounded from -12 to 12. Now you have to iterate over…

Here the search wouldn't have been chosing the coefficients independently. Note that one intermediate variable is a polynomial in the input variables, and it is used in other polynomials. A search over expressions like the ones in the counterexample would have a much smaller search space.

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

#332

Post by jameshart »

avisser wrote: post #297
jameshart wrote: post #292

Earlier quoted context omitted.

> 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. Careful, I think you might be committing an https://xkcd.com/2501/ error. What even is a protocol? What is a host? What is a ‘stream of octets’? Wiki helpfully tells you octets are also known as ‘bytes’.

My argument is that all the vocabulary for computer science are things. Even if they are virtual, they are tangible. You can draw a picture and label a box "bytes". Nothing in the ChatGPT conversation is tangible. It's all in the realm of concepts.

A ‘Byte’ is not a concrete thing and the fact you think it is speaks to the degree to which you have immersed yourself in a mental model which thinks of ‘information’ as if it is a real concrete thing, to the extent that you don’t even realize the levels of conceptual abstraction you needed to build in order to internalize what a ‘byte’ is.

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

#333

Post by scubbo »

minraws wrote: post #151
WarmWash wrote: post #30

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…

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???, dea…

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

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

#334

Post by [deleted] »

ImaCake wrote: post #325
eh_why_not wrote: post #234

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…

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 mathema…

[deleted]

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

#335

Post by [deleted] »

WarmWash wrote: post #30

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…

[deleted]

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

#336

Post by aleph_minus_one »

ChrisClark wrote: post #303
agrounds wrote: post #281

Earlier quoted context omitted.

Z is the ring of integers, t is a formal variable allowing us to discuss polynomials whose coefficients are in some ring. That’s what R[t] means: the ring of polynomials of the formal variable t with coefficients in R. Adding in t^-1 lets us include inverted terms like 2t^-3. An algebra over a ring (call it S so we don’t confuse it with R from the previous paragraph) is a like a vector space over S, with the added st…

But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean? His point is the terms are dense too

> But... what is a ring? What is a formal variable? What is a vector space? What does "algebra over the ring" mean?

All these terms were taught to computer science (and of course math, physics, ...) students as part of getting their degree in computer science, because these concepts are important for many algorithms.

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

#338

Post by Frost1x »

metanonsense wrote: post #315
ecshafer wrote: post #31

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…

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."

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

#339

Post by jongjong »

ftjftjft wrote: post #314
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,…

a janitor could do the same though

They could drag it out by telling stories of situations they faced but information density would be low and repetitive. As someone who cleaned boats professionally for a year in my early days, I'm pretty sure I could convey all I know about cleaning in under an hour. That's very different from one year of back-to-back lectures.

Cleaning boats, the learning tapered off after a few weeks. With software, I'm 15 years in and it barely tapered at all and it's much more intense. I've been pulling nights and weekends too.

With software knowledge, it would be high information density with little to no repetition. Every piece would provide useful concrete knowledge which would serve to increase technical capabilities and/or security.

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

#340

Post by Xcelerate »

minimaxir wrote: post #106
napoleoncomplex wrote: post #21

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.

> 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.
Post reply on HN