Mathematics in the age of AI • Hacker News Board

Re: Mathematics in the age of AI

#261

Post by js8 » Thu, Aug 20, 2026, 2:41 PM UTC

aleph_minus_one wrote: post #253

js8 wrote: post #200

Earlier quoted context omitted.

Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm. In any case, I think it's better to read PP as somebody would find a practical, albeit incomprehensible, algorithm for solving NP complete problems. Although I probably disagree with PP, because even a candidate algorithm that mysteriously works without proof would have practical value, so this case is no…

> > If a magic oracle tells you p=np, that's useless. How would that change anything? > Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm. OK, I tell you that P=NP, and that I am a magic oracle. So, you have you psychological boost for finding a practical algorithm for free. :-)

Kinda pointless, because

a) I am already convinced that P=NP

b) You have to convince many other people as well (that you're a magic oracle), because for the effect to work, lot of people would have to work on the problem (or at least spend tokens)

Nevertheless, a plausible magic oracle (such as Lean-verified proof, even if non-constructive and incomprehensible for humans) would convince many to take a 2nd look.

Re: Mathematics in the age of AI

#262

Post by chrisjj » Thu, Aug 20, 2026, 2:51 PM UTC

frozenseven wrote: post #45

qsera wrote: post #11

If the title have said in the age of "LLMs", I might have given it a try.

Out-of-hand dismissal of Terence Tao is certainly a take. And the term "artificial intelligence (AI)" has been the name of the field for 70 years and counting. If anything, "LLM" is a misnomer that's been lingering around since 2018-19. When the term was coined, these systems were relatively small, experimental, and could only produce impractical facsimiles of the English language. This is obviously no longer the cas…

> "LLM" is a misnomer

Google's LLM here says:

Calling a Large Language Model (LLM) an "LLM" is a misnomer ...

Case closed /i

Re: Mathematics in the age of AI

#263

Post by chrisjj » Thu, Aug 20, 2026, 2:54 PM UTC

qsera wrote: post #159

frozenseven wrote: post #123

Earlier quoted context omitted.

We can meaningfully talk about (1) the existence of a field and (2) the said field hitting its stride. The title of Tao's article denotes the latter. And oh, what a stride it is: https://vibemathed.com/stats

Sure, but I expect more precision from people like the author.

Not to mention accuracy. His "age of AI" seems to be no more than "the arrival of artificial intelligence tools that are capable of performing research-level mathematical tasks"

Re: Mathematics in the age of AI

#264

Post by chrisjj » Thu, Aug 20, 2026, 3:12 PM UTC

amoss wrote: post #205

qsera wrote: post #144

Earlier quoted context omitted.

> The "large" is now up to five orders of magnitude larger and "language model" has gone far beyond any simple notion of modeling a singular natural language. Does not matter. It is still an LLM. And I am not the one who is playing word games. You and your idols are, for sake of marketing.

If you place an LLM into a harness, alongside evaluation, feedback and problem decomposition / solution integration - it is still an LLM? There is an LLM acting as a component in a larger system. But that larger system is not an LLM. Calling it an "AI" is indeed an act of marketing as there is no learning / adjustment as we would expect from an intelligence, but calling it an LLM seems to be inaccurate. So what is it…

> If you place an LLM into a harness, alongside evaluation, feedback and problem decomposition / solution integration - it is still an LLM?

Clue: If you put a pig in a poke, is it still a pig?

Re: Mathematics in the age of AI

#265

Post by nilkn » Thu, Aug 20, 2026, 3:24 PM UTC

Jensson wrote: post #180

nilkn wrote: post #110

Earlier quoted context omitted.

Math that humans don't understand but nonetheless allows AI systems to develop breakthroughs in various fields of science, technology, physics, engineering, medicine, etc., would have great value to humanity even if it doesn't help humans understand abstract truth at all. Imagine if humans couldn't understand multivariable calculus, but we had access to an AI system that developed it, it initially seemed useless, the…

But at that point you have full AGI and its not just today's models. Today's models still need humans to understand things since it builds upon human knowledge. When you have full AGI of course you no longer need humans to understand math. > Imagine if humans couldn't understand multivariable calculus, but we had access to an AI system that developed it Developing multivariable calculus requires much more than just s…

I don't find AGI to be a useful technical term, as nobody can agree on what it means. For instance, you used it at least five times here, but you never defined it, and I could point to intellectually credible people who would say we've already reached AGI.

Anyway, if we put the AGI framing aside, I think the main point you're making is that AI mathematics hasn't yet demonstrated the ability to theory-build in the way that the great human mathematicians have (Grothendieck, Scholze, etc.). And I'd agree with you on that. Where we disagree, I suppose, is I think that capability is coming -- I don't see anything that would prevent its development.

Re: Mathematics in the age of AI

#266

Post by a2ff6eeb0 » Thu, Aug 20, 2026, 3:39 PM UTC

copemaxxxing wrote: post #141

a2ff6eeb0 wrote: post #41

Earlier quoted context omitted.

What's wrong with sitting on the beach with a bottle of wine for eternity, with no need to do anything, knowing that all your needs and desires will be automatically taken care of? I don't think you can be coherently pro-AI without thinking that the human brain will be obsolete, unless you believe in some inherent magic that the brain is imbued with. The only other option is that you haven't thought through the long…

That's not gonna happen. What's gonna happen is you'll be in forever slavery. You think those in power will let you enjoy your life?

Why do you think any humans would stay in power? The AI will know what's good for you better than other people would.

Re: Mathematics in the age of AI

#267

Post by a2ff6eeb0 » Thu, Aug 20, 2026, 3:40 PM UTC

fooker wrote: post #90

a2ff6eeb0 wrote: post #25

I don't know why anyone should care about understanding the results if the AI is better at math than us. It'd be like demanding that human mathematicians are banned from publishing until their cats understand the theorems. If Amazon uses AI math to come up with better routing, the cats can benefit from cheaper delivery fees just as much as humans can. No understanding needed. The human brain is being obsoleted, soon…

Your analogy is great. What happens when it's the cats who get to decide what's published? Not an ideal scenario, but that's exactly the situation here. Mathematicians decide what gets reviewed and published in a a top journal. In the long run, this can and should make journals obsolete.

Exactly, the cats aren't going to be part of any real decision making soon enough.

Re: Mathematics in the age of AI

#268

Post by feoren » Thu, Aug 20, 2026, 4:11 PM UTC

jbstack wrote: post #239

sonicrocketman wrote: post #8

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

I don't agree with this rule of thumb at all. Let's say tomorrow someone comes up with a formally verified proof that a major encryption algorithm underpinning the security of the internet can be trivially broken, but they can't explain it. You're saying it should be kept under wraps and not published? Tao is essentially saying that the only value in a proof is its ability to be understood, but that's wrong. A proof…

There has already been a formally verified proof of the Collatz Conjecture. The AI agent formally verified it by exploiting previously undiscovered bugs in Lean. The Collatz Conjecture is still unsolved.

> Let's say tomorrow someone comes up with a formally verified proof that a major encryption algorithm underpinning the security of the internet can be trivially broken, but they can't explain it. You're saying it should be kept under wraps and not published?

Absolutely. It could also be exploiting bugs in the verifier. Even if not -- even if that proof were correct and entirely written by humans, care should still be taken in how such knowledge is published. I'd want to give trusted parties a chance to try to fix the issue before letting it be known by black-hats, for instance.

Re: Mathematics in the age of AI

#269

Post by frozenseven » Thu, Aug 20, 2026, 6:13 PM UTC

chrisjj wrote: post #262

frozenseven wrote: post #45

Earlier quoted context omitted.

Out-of-hand dismissal of Terence Tao is certainly a take. And the term "artificial intelligence (AI)" has been the name of the field for 70 years and counting. If anything, "LLM" is a misnomer that's been lingering around since 2018-19. When the term was coined, these systems were relatively small, experimental, and could only produce impractical facsimiles of the English language. This is obviously no longer the cas…

> "LLM" is a misnomer Google's LLM here says: Calling a Large Language Model (LLM) an "LLM" is a misnomer ... Case closed /i

Well, suit yourself. At the rate we're going, the Millennium Prize Problems will start falling before the year is over.

Re: Mathematics in the age of AI

#270

Post by fultonn » Thu, Aug 20, 2026, 6:56 PM UTC

energy123 wrote: post #131

fultonn wrote: post #62

Earlier quoted context omitted.

People say similar things about automation of software engineering. Different, but similar. I'm deeply suspicious. I do not yet have a concise statement for why, but a lot of literature on the sociology of knowledge work sort of points at my thoughts. Section 5 of the Thurston article cited by Tao touches the elephant. Raduchel's article on the economics of software [2] also touches it. I've tried to put words to thi…

Many eminent mathematicians did their best work while not talking about it with anyone, sometimes in isolation. Newton's calculus, Perelman's poincare, much of Grothendiek's work, Wiles's fermat, Ramanujan's earlier days. That's not all top mathematicians as you can look at Von Neumann as a sociable counter-example. But it shows that discussion of your current ideas is not a requirement. Grothendiek goes so far as to…

> eminent

I would use a different adjective: exceptional.

There are not not times where lone geniuses produce amazing output. But most of humanity's progress over the last few thousand years (or at least certainly the last few hundred) resulting from a different type of work.