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

181–190 of 681 posts

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

#181

Post by gus_massa »

khurs wrote: post #152

Can someone ELI5 this? Is this a breakthrough of something or 'kinda interesting'?

Main discussion: https://news.ycombinator.com/item?id=48998362 Probably you shoud read the fist few paragraphs of that post, but it escalates quickly.

I'll cherry pick a few comment I like. I think they are worth reading but I'll quote a highlight of each one.

From kingstnap https://news.ycombinator.com/item?id=49000867

> If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. [...] What was conjectured is that this local invertibility property everywhere would mean global invertibility.

From mswphd https://news.ycombinator.com/item?id=48999959

> it doesn't overturn much. [...] the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.

Can I quote myself? https://news.ycombinator.com/item?id=49007165

> From a comment by j2kun https://news.ycombinator.com/item?id=49000833 , someone asked Fable and there was an almost counterexample in 2d but it uses division too. [Instead of f=x^2+7xy they have something like f=x^2+7x/y so it's not a polynomial.] It looks like the new trick was to use a third variable to avoid the division.

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

#182

Post by HNisCIS »

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…

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.

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

#183

Post by jacobolus »

WarmWash wrote: post #132
vector_spaces wrote: post #89

Earlier quoted context omitted.

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 mathema…

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…

The basic problem is that to get to the objects you mention here is at least 3 or 4 years of full-time study away from the kind of math people learn for a typical college degree in science or engineering. If you really want to understand them, to make your "digging" efficient you should probably just get a pure math degree, but setting aside several years to satisfy occasional curiosity is not feasible for most people for various reasons.

One unfortunate feature of published pure math research is that often the ideas are quite accessible and straightforward and don't really require special abstractions or terminology, but those get used anyway because for someone who already has a math PhD it saves a bit of effort.

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

#184

Post by WithinReason »

ckemere wrote: post #87
axus wrote: post #9

"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

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

#185

Post by khurs »

gus_massa wrote: post #181
khurs wrote: post #152

Can someone ELI5 this? Is this a breakthrough of something or 'kinda interesting'?

Main discussion: https://news.ycombinator.com/item?id=48998362 Probably you shoud read the fist few paragraphs of that post, but it escalates quickly. I'll cherry pick a few comment I like. I think they are worth reading but I'll quote a highlight of each one. From kingstnap https://news.ycombinator.com/item?id=49000867 > If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function f…

Thank you

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

#186

Post by ndriscoll »

ben_w wrote: post #70
dekhn wrote: post #40

Earlier quoted context omitted.

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 i…

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 ma…

Nope, math notations are optimized for reading, not writing (consider that people still use symbols on computers despite it being quite a bit more tedious to type). The conciseness makes it easier for you to see structural patterns and do symbolic manipulation in your mind's eye. Even something basic like the wave equation would become completely illegible with an expanded notation like that.

Same reason why we write 5-3, not subtract(minuend=five, subtrahend=three).

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

#187

Post by Freedom2 »

jephs wrote: post #129
Freedom2 wrote: post #117

Earlier quoted context omitted.

How would you envision ChatGPT paying Terrence?

OpenAI hiring him as a consultant, obviously, which may well be the case.

It's not so obvious. Taking my parent comment literally (as he said "the other way around"), one can assume that ChatGPT the platform itself would pay Terrence, rather than OpenAI.

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

#188

Post by WarmWash »

ccppurcell wrote: post #153
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…

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.

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

#189

Post by EnergyAmy »

superloika wrote: post #112
ben_w wrote: post #93

Earlier quoted context omitted.

If natural language was structured in a logically computable way, we'd have had interesting chatbots by the late 80s, basically as soon as a dictionary fit in local RAM, and for the same reason we got compilers. Da hole raisin y nat-lang be v. hard is dat i kan rite lik dis an it be cool 4 native engrish speekrs 2 unerstand. LLMs are of course fine with this sentence in exactly the way that Zork's engine couldn't be.

The underlying structure of language, which is grammar, is obviously logical. That the symbols used to represent this grammar can be sometimes fuzzy or ambiguous, is no problem for a machine that takes context and probability into account when translating words to the underlying grammar structure.

The idea that grammar is all it takes to process natural language is absolute beans.

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

#190

Post by HNisCIS »

petesergeant wrote: post #136
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…

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.

Post reply on HN