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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#91

Post by JadeNB »

jhrmnn wrote: post #53
kriro wrote: post #35

Earlier quoted context omitted.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

I can almost see two branches of mathematics developing. One which is human-understandable, the other formally verified. I assume the latter is a strict superset of the former?

Presumably there's not much logical obstruction to all human-understandable math eventually being formalized, although the willingness and ability to commit the requisite enormous amount of time will probably be insurmountable. But definitely that hasn't happened already!

Re: Mathematics in the age of AI

#92

Post by [deleted] »

nilkn wrote: post #48
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give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published.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 believe this rule of thumb will come to fail. The combination of superhuman mathematical reasoning and synthesis in upcoming AI models plus the rapid build-out of scalable formal verification infrastructure means this exponential in math is going to take off quite explosively, and we've barely seen anything yet. Mathematics is going to decisively move beyond human ability fairly soon (within our lifetimes, if not m…

[deleted]

Re: Mathematics in the age of AI

#93

Post by Jblx2 »

a2ff6eeb0 wrote: post #41
TSiege wrote: post #37

Earlier quoted context omitted.

I’m not anti AI but thinking the human brain is obsolete and using it will become a hobby is a dystopian view of the future where no one has any agency anymore. By your logic since our brains provide no value why not just shoot ourselves in the head while we’re at?

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…

Are dolphins obsolete?

Re: Mathematics in the age of AI

#94

Post by gowld »

kriro wrote: post #35
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.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

Math humans can use but not understand is called engineering. -- paraphrase of Bill Hammack, https://engineerguy.com

https://www.youtube.com/@engineerguyvideo

Re: Mathematics in the age of AI

#95

Post by thechao »

zahlman wrote: post #83
kriro wrote: post #35

Earlier quoted context omitted.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

I don't think this is a valid counterpoint at all. Math is cooperative, and comprehension is the point : the proof has value exactly because (and only to that extent) it empowers humans to understand an abstract truth. Chess is competitive: the memorized line has value because it makes you incrementally more likely to defeat your opponent.

I hope I'm remembering this right: a mathematician claims to have a proof for the ABC conjecture, but can't conceive any other mathematician it's right — it's "too weird", so the proof is rejected?

Re: Mathematics in the age of AI

#96

Post by gowld »

timjver wrote: post #78
RandomLensman wrote: post #74

Earlier quoted context omitted.

If the proof is formally verified but impossible to understand how would anyone be able to be sure the formal verification is correct? Complex software is bound to have bugs, no?

The whole point of Lean is that you don't need to understand the entire proof to be sure that it's correct. You only need to understand the definition of the theorem being proven, and you need to trust that the relatively small core of Lean is correct.

Why should you trust that the relatively small core of Lean is correct?

The core of Lean got a lot less correct when a well-meaning AI system probed Lean for corner cases (bugs) that would "prove" a false conjecture. Corner cases so arcane that no human exploit in a proof. Basically, humans are too stupid to break human-created Lean, but the AI is not.

Re: Mathematics in the age of AI

#97

Post by gowld »

akk0 wrote: post #32
czgov wrote: post #9

Earlier quoted context omitted.

I wonder what his views on the 4 color problem are. One can explain it as the computer checked a bunch of cases and all maps reduce to one of these cases. It doesn’t take an expert to state this. Properly explain is an enormous grey area. Soon, I think, there will be proofs of results that are verified in Lean that are so long that no one will be able to “properly explain”. I don’t think they should be discarded. Res…

For an exhaustive search, if you can explain to me: - how to exhaustively list the cases that need to be checked, and why that method is exhaustive - how to check each case, and why that works and then conclude with "we've had a computer do this exhaustive search, and the result came up as X", for me that satisfies completely understanding the proof.

But the "computer" is magic, to you.

I could prove anything by claiming I completed a trivial-to-explain exhaustive search. The only support or refutation would be someone doing their own search. It's a very weak foundation.

We already had the ABC conjecture crisis: A theorem with a human-written proof so complex that no one besides the author can understand it. Some people claim to have refuted it. Most mathematicians are unqualified to decide.

Re: Mathematics in the age of AI

#98

Post by evenhash »

a2ff6eeb0 wrote: post #26
glimshe wrote: post #14

Terence Tao sees a role for AI in science. I'm no genius but he basically described what I've thought all along... We don't need to be "all in" or "all out". It's the old cliche of "if you only have a hammer every problem looks like a nail". Let's not fall into the trap of thinking that our life needs to be 100% about AI or completely devoid of AI. We can really use this thing to make our lives better. Instead of was…

I think it's impossible to be half in. AI will eventually be better at things than people, and people will simply be rocks in the gears of progress. The only thing to do is to be all in, or get run over.

If what you’re saying is true, what’s the use (or even meaning) of being “all in”? You’re a rock either way.

Re: Mathematics in the age of AI

#99

Post by cfiggers »

cfiggers wrote: post #76
metahuman_crumb wrote: post #61

Earlier quoted context omitted.

I suggest "Catching crumbs from the table" by Ted Chiang. Very short piece published in Nature (2000) and well worth a read. Depicts a scenario where modified humans produce science beyond ordinary scientists' comprehension.

This is a theme in Blindsight by Peter Watts as well. In that setting, field experts working at the bleeding edge are so advanced that non-experts literally can't understand what they're saying at all. So there's a whole class of specialists, "synthesists", that specialize in gaining approximate understanding of the experts' work for the purpose of communicating it to outsiders—perhaps wrongly , according to the expe…

What's amusing to me in this context is, summarizing emails and such has for a while been a supposed use case for AI—the LLM serving as the "synthesist" to explain long texts accessibly. But with this math question, a human "synthesist" would be needed to approximately understand the math discovered and programmatically verified by the LLM. So the roles reverse.

Re: Mathematics in the age of AI

#100

Post by qsera »

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…

>This is obviously no longer the case today.

Advancement in capability does not mean the mechanism is the different. The LLM name denotes a very specific mechanism..

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