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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

#281

Post by agrounds »

steve_adams_86 wrote: post #259
WarmWash wrote: post #224

Earlier quoted context omitted.

> 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…

"The Rees algebra is an algebra over Z[t^−1]" Such a small sentence and yet it means very little to me. I understand some constituent pieces, but I don't understand what Z is here other than a 'ring' and I don't really grasp how t^-1 converts this into a generalized family of algebra. It would take me a lot of effort to understand this and use it practically. I find that fascinating because it really is such a small…

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 structure that you can multiply elements of the algebra together (vector spaces only let you add their elements together). So for example the collection of even integers 2Z is an algebra over the ring of all integers Z. The collection of all polynomials with integer coefficients, Z[t], is another algebra over Z.

This is a great example of how dense language gets in math. There are tons of concepts hiding in the unstated background. Many are quite simple to explain individually, but there are so many of them that an outsider won’t know where to start to tease them apart. There’s a good reason to do it this way though; it would take a very long time to say anything in math without ever increasing levels of information density.

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

#283

Post by 9dev »

WarmWash wrote: post #261
minraws wrote: post #240

Earlier quoted context omitted.

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 compa…

I think you are snagging on thinking this is an observation about difficulty, time-to-mastery, or mental firepower requirements. It's not. It's a plain observation that math exists on mostly it's own path with little to zero overlap with our lived experiences. If mathematics was a vector, it would have similar magnitude to other vectors, but it's direction would be much more removed from the typical knowledge pack, f…

That is one clever metaphor. Thank you, I might steal it.

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

#284

Post by chongli »

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…

Mathematics would be much more approachable if it just used plain English like `sum(0, Infinity, my_func)` instead of a big Greek sigma with nested function nomenclature

First of all, no, mathematics would be far less approachable if it did that. Most of the Greek letters used in mathematics don't have a universal meaning, they're context-specific and defined by convention or just prior to use.

Second of all, mathematics is optimized for hand calculation on paper, not long-term programming and code maintenance. Writing out long names over and over on a whiteboard gets tiring extremely quickly, so mathematicians prefer to stick to single-letter symbols.

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

#285

Post by 2snakes »

nozzlegear wrote: post #242
squidbeak wrote: post #50

Earlier quoted context omitted.

I'd say an entity capable of instructing one of the leading mathematicians of his era is pretty clearly intelligent by any reasonable measure - however it might be arriving at its output.

I think we have wildly different conclusions about what happened here. You see the machine as instructing Terrence Tao, as if it were Plato teaching Socrates about the theory of forms; I see Terrence Tao using the machine to teach himself, like an intelligent student uses a book. In this case, it's just a book that fools us into believing it can think and reason like we do, because it generates language in much the s…

Yeah. Artificial knowledge not artificial intelligence.

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

#286

Post by [deleted] »

agrounds wrote: post #281
steve_adams_86 wrote: post #259

Earlier quoted context omitted.

"The Rees algebra is an algebra over Z[t^−1]" Such a small sentence and yet it means very little to me. I understand some constituent pieces, but I don't understand what Z is here other than a 'ring' and I don't really grasp how t^-1 converts this into a generalized family of algebra. It would take me a lot of effort to understand this and use it practically. I find that fascinating because it really is such a small…

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…

[deleted]

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

#287

Post by jebarker »

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.

I second this and would add that it's really easy to catastrophically forget things in math. I'm pretty sure that most CS knowledge I have I will retain at a level where I won't forget the general ideas and re-reading materials can quickly refresh the details. This is not true for advanced math. I did a pure math PhD and my own thesis is impenetrable to me 20 years later. It would take months, if not years, of focused effort for me to regain the understanding.

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

#288

Post by bmitc »

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…

Capital sigma doesn't always mean sum and certainly lowercase sigma never means sum.

If mathematics used plain language, the ability to meaningfully manipulate and understand would go way down. Proofs would become massively tedius.

Of course, notation is hard. Any good mathematician should put a lot of work into it.

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

#289

Post by elisbce »

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…

I don't think you even understand the problem. The determinant needs to be a non-zero constant AND you need to prove that particular map is not globally injective, meaning you have to find at least two points mapping to the same value. Of course it looks easy when someone shows you the counterexample.

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

#290

Post by AlienRobot »

hyperpape wrote: post #245
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…

I studied math through college before learning to program as an adult and becoming a software engineer, and I strongly disagree. I don't know how to say this in a way that won't sound insulting, but I don't mean it to be insulting. Programming, even systems engineering, is a surprisingly shallow field. I don't mean that it's easy--it's not, it can be incredibly difficult. Difficult and deep are just different concept…

I agree, and I don't even know math.

Most of things that are impenetrable in programming aren't about... programming. They are about some actually complex field like math being applied to programming.

For example, a library that does stuff with geometry. You need to know geometry to understand the program, but the program itself will never be complicated. It's the geometry that is complicated.

In cryptography, it's not the program that is complicated, it's the field of cryptography. In AI, it's statistics.

In graphics programming, math is the most impenetrable part, not programming anything. You can be a very good programmer in the sense that you know how to architect information systems and still fail to write a shader because shader programming requires you to know what a "dot" product is and you haven't heard about that since high school.

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