Earlier quoted context omitted.
I'm from the UK. What's in the image?
Time to get a VPN bucko
I'm in France using a French VPN and I'm being blocked by imgur (been going on for a few years now)
Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
211–220 of 681 posts
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#212Math 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…
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Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#213Earlier 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.
"tcp" can take roughly 3-4 weeks of heads-down dedicated study to have some reasonable familiarity with. Same is true with most of the other concepts. Being able to speak with expertise on that list of topics is 3-4 years of really focused study and work. I think what happens is that people often have passing familiarity with a word or topic and presume knowledge, and years (decades) later they realize they knew almo…
You could get a surface level understanding of TCP, or 99% of topic areas in computing, in less than 4 weeks of study. You could not get a surface level understanding of literally any of the maths in the link in less than 4 weeks of study.
The vast majority of what computers do just isn't that complex. I'm not saying it isn't "complex" just that any reasonably smart person can understand how a computer works and still have other hobbies, basically no one can understand phd level mathematics without dedicating their entire lives to it.
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#214Earlier quoted context omitted.
It’s crazy how much these companies invest in their models but when it comes to UX they do fuckall
UX is still very hard for any startup because top talent almost never work on UX.
What are these "top talent" usually talented at, then?
(I'm not talking about OpenAI/Anthropic at this point, but maybe a <10 people startup.)
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#215This 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.
Without any more context, "keep going" seems to be doing a lot of work. The user is placing a lot of faith in the LLM to not make subtle logic mistakes and to take good approaches to each problem. In my experience, even frontier models (such as Fable) are quite capable of getting confused during even simple technical work I've done in the dev ops world. For example: LLM: This package hasn't made it to production. ME:…
There may be something(s) about mathematics (proofs) that makes it particularly amenable to LLM reasoning - highly inductive from facts that are explicitly within-context/associative space? Being an unusually well documented discipline in general, with less influence from tacit knowledge or idiosyncratic “it works however the opinionated human made it work +- bugs” processes? Something about simulating even the smallest non-pure-inductive leaps necessarily risking simulating mistakes due to the nature of context “perception”?
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#216Math 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…
Math strives to minimize ambiguity, which other fields don't do as much. Non-math fields tend to reuse regular words as jargon (i.e. with specificity of meaning that may fly over the laymen's heads). Social sciences and humanities are most notorious for this, often resulting in non-practitioners not realizing they are out of their depth because they are not looking at symbols from non-Roman alphabets.
> Non-math fields tend to reuse regular words as jargon
Isn’t this the field with a “closed” “set”, an “open” “set”, oh and also a “clopen” “set” for some reason?
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#217Earlier 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…
Just be thankful that we don't share the penchant for giving credit to discoverers. Imagine calling a cache a "Murphy/Steinman/Sokolov structure" (made-up names). I mean, we do for some things, especially algorithms (Boyer-Moore). Probably for the same reason the mathematicians do -- there aren't readily available real-world analogies. And I won't even mention the branded future, with its "Google HyperZipper String S…
Huffman Coding, Turing Machines, Knuth Devices, Bayesian networks, B-tree (named after Bayer), AVL Trees, so many data structures and algorithms, even relatively new ones like Timsort, Jaccard similarity, MapReduce (thankfully but I have seen people call it Dean-Ghemawatt MapReduce in literature).
Well people even name stuff after themselves as well, Fil-C, raylib, etc (I like both Filip and Ray just pointing it out).
Aside: If I butchered some spellings I am sorry. :3
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#218Earlier 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.
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 to cancel to a constant?
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#219Earlier quoted context omitted.
What does "predicting the next token" mean? I ask this every time people say "LLMs are just predicting the next token" and it's maddening that nobody can give a straight answer. Predicting it according to what probability distribution? Every process that produces a sequence of actions (including e.g. a human writing) can be modeled by some probability distribution and therefore their actions are indistinguishable fro…
Yeah that's pretty much what gwern argues here[0]. Or to adapt another proverb: to predict the next token you first need to model the universe. [0] https://gwern.net/scaling-hypothesis#gwern-difference--effic...
> to predict the next token you first need to model the universe
Exactly. The "most likely next" series of tokens, for example, when given the first half of a correct mathematical proof, is the correct rest of the proof. I have never seen anyone define "most likely next token" in such a way that this isn't true.
Re: Terence Tao's ChatGPT conversation about the Jacobian Conjecture counterexample
#220Earlier quoted context omitted.
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…
For those of us with strong verbal processing and weak symbolic/pattern processing, this makes math much more difficult to approach.
Interestingly, discrete math feels the most "verbal" of all the subfields of math I've encountered (I haven't gone very deep). I think this is because notation in discrete math is is somehow closer to compressed prose or logic, whereas other forms of math use notation to fill in for long sequences of symbolic manipulation.
Not sure if that makes sense... I'm curious whether anyone else experiences it that way.