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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

#231

Post by minraws »

B-Con wrote: post #223
Jtarii wrote: post #194

Earlier quoted context omitted.

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.

I think one of the key differences is that math is abstract whereas CS is relatively concrete. CS examples are often easy to picture and understand the motivation for. You can use tools to visualize or play around with them and test them. Math gets abstract so fast you have to spend a week of research to even understand the problem statement. The the motivations themselves can be completely unclear until you have a l…

I think so is upper level CS, there are fields in CS that are foreign to me too, there is a lot of depth in CS, computing is a very deep field for instance ML research although may seem simple isn't quite so intuition based as people make it out to be. Similarly there are dozens of topics where sophisticated research happens where we don't interact with at all as regular software developers.

Every slice has so much depth to it, in Maths it all seems like all of it is required at once but in computing it feels like so little is needed to get started which I honestly feel like is failure of our modern education systems.

But yes Computers being so easily accessible and compilers, documentation and libraries have made computer science so easy to get started with.

Imagine having to implement your own network layer to communicate with someone, you would have had to understand ip, tcp, network layer to an extent like http and etc. and then you finally would have been able to communicate.

In maths that's our reality for a lot of the field, there aren't good libraries, interfaces to help skip the unnecessary details. Hopefully AI might solve it I don't know though. It's fun to hope for it.

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

#232

Post by vovavili »

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…

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.

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

#233

Post by traes »

hyperhello wrote: post #218
Legend2440 wrote: post #59

Earlier quoted context omitted.

Oh yes there were. The Jacobian conjecture is "notorious for the large number of published and unpublished false proofs which turned out to contain subtle errors." It's not quite the Reimann hypothesis, but many prominent mathematicians have spent years working on this problem. Yitang Zhang wrote his PhD thesis on it.

I shouldn’t, but: F1 = x^3y^3z + 3x^2y^4 + 3x^2y^2z + 7xy^3 + 3xyz + 4y^2 + z F2 = 3x^3y^2z + 9x^2y^3 + 6x^2yz + 12xy^2 + 3xz + y F3 = -x^3z - 3x^2y + 2x That’s the counterexample. Low integer coefficients, power 7 in three variables. If someone said it was there, couldn’t we all have written a pretty simple brute force solution for the search space, especially with the constraints that the symbolic determinant had t…

Honestly just try it. You'll figure out the problem very quickly.

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

#234

Post by eh_why_not »

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 and a PhD thesis student, or even another professor, explaining their results.

What can we imagine and predict about the future anymore? Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.

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

#235

Post by natsucks »

michaelchisari wrote: post #12
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…

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.

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.

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

#236

Post by mrpf1ster »

umvi wrote: post #227
minraws wrote: post #151

Earlier quoted context omitted.

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…

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 sigm…

Your translation only makes intuitive sense to you because you are well versed in programming.

I suspect if I showed a non-technical person with no background in either math or programming they would think both are nonsense until you explained it to them

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

#237

Post by neurahawk »

wiether wrote: post #208
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’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 I didn't understood anything about the thread, but reading Terrence's messages was weird because it looked exactly like the discussions I have with LLMs I've mostly seen people trying to oneshot a result, while I'll quickly experienced that going through steps/discovery wa…

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.

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

#238

Post by siva7 »

Jtarii wrote: post #194
minraws wrote: post #151

Earlier quoted context omitted.

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…

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.

As a counterpoint, i've worked with enough math PhD's over my career who couldn't wrap their head intuitively around many concepts from software engineering, while others had no problems in doing so even from folks without any stem background. We often undererstimate how much field knowledge we aquire over the years and overestimate how easy it is for others to catch up.

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

#240

Post by minraws »

WarmWash wrote: post #224
minraws wrote: post #151

Earlier quoted context omitted.

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…

> 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. A…

You are comparing TCP a relatively basic topic in the grand scheme of computing with Rees_algebra which is fairly specialized, we could take a simpler topic more foundational and clearer to understand and compare them.

I can understand that this feels like one is so much more complicated part of it is also how the articles were written, wikipedia is not known for quality maths explanations.

But beyond that this comparison to me feels unfair.

Let's take Euclidean algorithm or just modular arthimetic for example what a lot of computing even is based on I feel like that's a fairer comparison. No?

Perhaps that's too easy but I just find this specific comparison very unfair to both Math's intuitive-ness and Computing's complexity. Perhaps I am the one being delusional.

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