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…
Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
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Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#192Math 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…
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#193The 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.
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#194Math 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…
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.
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#195This 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.
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.
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#196Earlier 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…
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,…
https://www.jsoftware.com/help/learning/23.htm is the closest i've found, but wondering if i'm missing something perhaps, Julia?
tyvm
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#197How 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!
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#198```
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?
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#199Earlier quoted context omitted.
> If AI researchers cared about scientific thinking, they would be intensely focused on the brains of bees. Basically every academic AI researcher in history was doing what you described. The AI industrialists stopped caring 6 years ago once they realized LLMs seem to have been the only thing in 80 years that actually seems to work at any useful level. There are plenty of pioneering scientists who are either returnin…
> Basically every academic AI researcher in history was doing what you described. That is not true. Alan Turing did not view things that way, his test would say that a dog has zero intelligence. Neither did any of the MIT Lispers. And neither do Lecun or Sutskever or Sutton! They are all focused on human intelligence. None of them are even slightly concerned about an AI which is intelligent before it learns any langu…
??? https://www.youtube.com/watch?v=GvibIstOn_E his arguemtn here is clearly built around using some sort of sensory data to build a model of the world like humans (animals) do. also you clearly decline to mention Lecun who has made this point ad-infinitum
> This isn't true either! Mathematica / Maple / etc are "old-fashioned AI" and they obviously work. The Lisp expert systems were also useful, though less so than an LLM.
i personally find it very strange that non-deep learning AI approaches which essentially boiled down to a giant bundle of if statements, or some very simple statistical modeling were called AI in the first place.
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#200Love this share! Thanks for helping ppl better understand what it really means when folks say "ChatGPT Discovered..."