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#mathematics#proof#math#tao#where#more#mathematicians#beauty#lean#theorem
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Discussion (50 Comments)Read Original on HackerNews
From the interviews I've seen with Tao, he's not some AGI maniac, he says things like here's where we can use this tool, here's where it's less likely to be useful. There's a lot of hallucinations, so we need to double check stuff. Most of the stuff the AI produces is nonsense, but there's occasionally a diamond in the rough.
A very tempered attitude, and likely what most sane people are experiencing when using AI.
AI not only provides potential to cause society to become overly dependent on it, but it's being developed by/pushed for by the same fucking people who caused our societies smartphone addiction.
Once you recognize what we've lost already, it's hard to turn off your brain and just compartmentalize this away as a "just a tool". Nothing that is adopted so widely is "just a tool," and thinking of it in those terms eliminates the ability to analyze the potential downstream effects it will cause.
Not sure where you live, but I would guess the West (where we have the luxury to be worried about "smartphone addiction"). I assure you that the net positive of smartphones, especially cheap Androids, have had a significantly more positive effect on society than negative, particularly in the developing world.
> He predicted that in the future, instead of working alone or in small teams of two or three, mathematicians might work on projects with hundreds of other people at a time. And when these collaborations were over, he said — in his modest, understated way — the results might be checked not by human referees but by computers.
Fascinating stuff. My thought has always been that the AI will accelerate individuals and we'll get something like the economy for music or sports (the top few take almost all the revenue) but this may seem like an alternative pathway that might well develop (if only in Mathematics there) where AI systems drop the coordination cost to near zero by making checking cheap.
So far, and I am not foolish enough to say forever, agents are great at operating in the space of checkables and it's hard to get uniqueness out of them (I haven't succeeded in getting a real laugh from Fable) but perhaps there's a whole class of problems that we can now solve by turning humans into the search units. I love it!
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[1] https://youtu.be/cdflu9ZXZGE?si=f1xi65r7kZM8s1JI
We mere mortals (I am a prof. of Maths at Uni) do not.
Instead, you proceed in layers of abstraction. For example
1. the main claim may rest on some set of sub-claims, as well as some local (to teh main claim) work to "patch things together"
2. each of those sub-claims themselves may require other sub-claims + local work, etc
These can be collected into a dependency graph. In lean, this is often called a "blueprint". Here is the blueprint for the formalization of the Polynomial Frieman-Rusza conjecture (now a theorem, by Gowers, Green, Manners, and Tao).
https://teorth.github.io/pfr/blueprint/
This layer of abstractions is (roughly) equivalent a different way to format mathematics. You could remove the Lean component (let alone any AI), and create such a dependency graph for a paper. I would argue this is a clearer way to format mathematics (again, ignoring both the formal verification applications of it, as well as AI).
Any mathematics paper intrinsically has a graph such as this underlying it, and tries to make the various linkages in the graph clear via prose. Prose is only so powerful a way to organize things. I'm sure you're familiar with the way early mathematicians would describe various formula (e.g. the quadratic formula) via prose. It is very hard to understand.
Separately from this dependency-graph perspective, you can do things like
1. add formal verification. Now, each component in the dependency graph is verifiable with high confidence (though harder to write and read). This has some benefits and downsides. Harder to write and read is bad. Being able to have high confidence in the veracity of the result is *very* good. It allows larger collaborations in mathematics. Previously, a large collaboration would require all mathematicians to trust eachother to a large extent. This is (practically) difficult.
2. when each component can now be verified to high accuracy, you can now throw AI at it. I won't extoll the virtue of this. There are parts of it that seem interesting, but many "AI for Math" things currently are stil producing unformalized papers (in prose).
Maybe the main thing I'd say is that this type of "graph structure, with each component trusted" is already implicitly what mathematicians do. You write papers that cite other papers etc. Except now, instead of needing to look for status signals to trust papers (or invest personal effort), you can look for another (honestly fairer) signal to trust papers. So there's a sense in which formalization allows for the democratization of mathematics. I do think there's something beautiful about that.
Why does it need to be beautiful? Once you proved it it's true and you can use its consequences in math, sciences and engineerings.
I am not talking about the supposed "beauty" of a proof (I do not believe in that concept, rather in "elegance", which is not the same), I am talking about the proof itself, and the insights it provides.
[1] https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994...
We basically subsidize the practice of mathematics as an art form, and if you try to take the artistry away, you might find that the artists don't want to play along. And I guess you can imagine future robo-math production lines without any human involvement, and then LLMs finding applications for the resulting theorems, but it's not possible today.
At the universities I’ve been to (as a student and now faculty), «applied mathematics» and «statistics» have been the two largest divisions. But perhaps that’s a bias from engineering-heavy universities?
*Completely made up statistic.
For any practical application, you are only interested in finite set of concrete identities, so anything beyond that is surplus to requirements, surely?
I do a lot of numerical work in settings where computational efficiency is useful.
In my work, most cases you can do numerically using integration or Monte Carlo sampling or whatever.
It’s slow. It often pays to find a closed-form solution. Even if it’s just a starting point that needs refinement.
To put in terms of the Pythagorean theorem: Proving the Pythagorean theorem gives you a relationship that’s reliable, fast to evaluate, and general. Proving individual tuples gives you none of this.
That doesn’t even touch on how theorems give us a glimpse at deeper structure and truths. Proving a bunch of right-triangle tuples will probably never lead you to the rest of the identities in trig.
“Beauty will save the world”
Supposedly even drowned their member that divulged their existence.
Meh. You can successfully argue that there is no objective anything. It's all just our perception and the emotions we associate with it. We built entire civilizations on subjective notions of good, evil, beauty, and so on. So where do you draw the line between "acceptably subjective" and "too subjective"? And are you sure it's not just a subjective code name for "the thing I don't like"?
Ultimately, people practice mathematics mostly for abstract reasons. It's not a field where you routinely ship products and get rich by meeting market demand. If 99% of contemporary mathematicians don't want to become prompt engineers, there's nothing that makes the transition to AI math inevitable. If not mathematicians, the only party with vested interest in that would be the PR departments of frontier labs.
wait what is the math with no utility
Beautiful explanations are lovely when they exist, but we shouldn't wait for them if we can also find the truth through an ugly method.
A programmer translates a natural-language spec into a machine-readable spec, feeds it to an AI-assisted compiler, and out pops an implementation that's more optimized than any human could ever hope to write, along with a lean proof of its correctness.
It won't be a programmer doing this work, because they will have gone the way of the dodo.
It'll be workers specific to a certain domain (e.g. engineer, architect, accountant) doing this on top of their usual work.
The software industry will collapse.
This is a clever piece reminding people of Tao's pre-AI Lean efforts. Now, however, Tao and especially Gowers are receiving AI money and have AI positions so they are far from unbiased.
Or maybe they have caught Feynman's "computer disease"? Either way, this is a hype piece.
Logic Theorist soon proved 38 of the first 52 theorems in chapter 2 of the Principia Mathematica. The proof of theorem 2.85 was actually more elegant than the proof produced laboriously by hand by Russell and Whitehead (2026-03-20: What is called here Theorem 2.85 is, in fact, numbered as 2.53 in the page 107 of the 1963 Cambridge University Press edition (https://www.uhu.es/francisco.moreno/gii_mac/docs/Principia_M...) and which appears, under the same 2.53 number, on page 112 of the 1910 CUP Edition, according to the digitalization on wikibooks (https://en.wikisource.org/wiki/Russell_%26_Whitehead%27s_Pri...)). Simon was able to show the new proof to Russell himself who "responded with delight".[17] They attempted to publish the new proof in The Journal of Symbolic Logic, but it was rejected on the grounds that a new proof of an elementary mathematical theorem was not notable, apparently overlooking the fact that one of the authors was a computer program.[18][17]
https://en.wikipedia.org/wiki/Logic_Theorist#History
Maybe some people only understand "AI" to mean "LLMs" but, particularly in maths, LLMs ain't going nowhere without a symbolic solver (or a human mathematician) verifying their output.
> Automath ("automating mathematics") is a formal language, devised by Nicolaas Govert de Bruijn starting in 1967, for expressing complete mathematical theories in such a way that an included automated proof checker can verify their correctness.
I was daydreaming about how someone would model symbolic algebra in computer code, and naively thought it would be easy, but the more I thought about it, it seems to get exponentially (pun intended) more complicated.
Is there a Lean/OpenAI connection?