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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#51

Post by BeetleB »

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…

The essay On Proof and Progress in Mathematics by a Field's medalist is worth reading:

https://arxiv.org/abs/math/9404236

He wrote it in 1994.

He writes about how he almost "destroyed" a subdiscipline in mathematics by becoming so good at it that he outclassed everyone. PhD students were advised to stay away from the whole field.

When he discovered this, he realized his error was that he was focusing on producing results, and not focusing on explaining his thought process. It's that thought process that is valuable in advancing the frontier - results alone won't do it. It didn't matter how many theorems he proved, if he was the only one who had the mental framework in mind on how to think about the whole field.

I'm sure we've come across abstruse books where every theorem has a rabbit being pulled out of a hat, whereas other readers find it intuitive. It's because the latter has developed a mental model for the discipline, and you haven't.

So he set about slowing down, and focusing on holding lots of seminars where he worked with other mathematicians to explain the thought process. Eventually others started publishing proofs of key theorems.

When people publish in a journal, they are not merely doing it to show the result. They are having a conversation with other mathematicians. If they cannot explain their own proof, they're not having a conversation.

This is why even decades after the Four Color Theorem was proved, plenty of mathematicians don't consider it "mathematics".

Re: Mathematics in the age of AI

#52

Post by a2ff6eeb0 »

BeetleB wrote: post #51
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…

The essay On Proof and Progress in Mathematics by a Field's medalist is worth reading: https://arxiv.org/abs/math/9404236 He wrote it in 1994. He writes about how he almost "destroyed" a subdiscipline in mathematics by becoming so good at it that he outclassed everyone. PhD students were advised to stay away from the whole field. When he discovered this, he realized his error was that he was focusing on producing res…

I don't understand why people are so fixated on the minds doing the mathematics being made out of meat. It seems obvious that soon, minds made of meat aren't going to be able to keep up.

Useful thought, rather than hobbyist thought, seems destined to be the exclusive domain of silicon.

Re: Mathematics in the age of AI

#53

Post by jhrmnn »

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…

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?

Re: Mathematics in the age of AI

#54

Post by TMWNN »

highfrequency wrote: post #7

Terence Tao's quote about AI's math proofs is relatable outside of pure math: "the writing very often dwells at length on trivialities while passing briefly through — or even actively obscuring — the most interesting and novel portions of the argument."

>Terence Tao's quote about AI's math proofs is relatable outside of pure math: "the writing very often dwells at length on trivialities while passing briefly through — or even actively obscuring — the most interesting and novel portions of the argument."

I noticed a long time ago, that the more people focus on trivialities like typos when arguing against someone online, the more compelling the original argument is. Basically, bikeshedding.

The most compelling evidence of the compelling nature of the original argument is when the most-upvoted reply is a joke or a meme. That's when you really know that those responding have nothing else to say. It's a white flag being run up, or the dog turning over and exposing its belly.

Re: Mathematics in the age of AI

#55

Post by cubefox »

pfdietz wrote: post #28
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 problem with that rule of thumb is that unless there's some status/reward for completing the result, it won't happen. People will just put up the formally verified result and call it a day, and there's no incentive for them or anyone else to clean things up. We'll end up with incomprehensible math because comprehensibility isn't rewarded. No one is going to get a Fields Medal, or tenure, for digesting someone els…

> The problem with that rule of thumb is that unless there's some status/reward for completing the result, it won't happen.

He says it shouldn't be able to published if they can't explain it. Publishing it is the reward.

Re: Mathematics in the age of AI

#56

Post by BeetleB »

a2ff6eeb0 wrote: post #52
BeetleB wrote: post #51

Earlier quoted context omitted.

The essay On Proof and Progress in Mathematics by a Field's medalist is worth reading: https://arxiv.org/abs/math/9404236 He wrote it in 1994. He writes about how he almost "destroyed" a subdiscipline in mathematics by becoming so good at it that he outclassed everyone. PhD students were advised to stay away from the whole field. When he discovered this, he realized his error was that he was focusing on producing res…

I don't understand why people are so fixated on the minds doing the mathematics being made out of meat. It seems obvious that soon, minds made of meat aren't going to be able to keep up. Useful thought, rather than hobbyist thought, seems destined to be the exclusive domain of silicon.

> I don't understand why people are so fixated on the minds doing the mathematics being made out of meat.

I don't follow - are you surprised that mathematicians have social rules on how they interact with others?

You're definitely welcome to set up a journal that takes whatever types of papers you deem acceptable. It's not like they're preventing the dissemination of information by taking this stance.

Personally, I wouldn't hire a SW engineer who only showcases output from LLMs, and can't explain the code it wrote.

Re: Mathematics in the age of AI

#57

Post by BeetleB »

pfdietz wrote: post #28
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 problem with that rule of thumb is that unless there's some status/reward for completing the result, it won't happen. People will just put up the formally verified result and call it a day, and there's no incentive for them or anyone else to clean things up. We'll end up with incomprehensible math because comprehensibility isn't rewarded. No one is going to get a Fields Medal, or tenure, for digesting someone els…

> People will just put up the formally verified result and call it a day, and there's no incentive for them or anyone else to clean things up.

The incentive will be to be able to publish in a top tier journal. I suspect what Tao is advocating for is having journals reject such manuscripts.

> No one is going to get a Fields Medal, or tenure, for digesting someone else's results.

I'm sure no one gets a Field's Medal if others can't digest their results.

Re: Mathematics in the age of AI

#58

Post by cubefox »

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…

> If Amazon uses AI math to come up with better routin

Most research mathematics is pure mathematics which is completely useless. No routing algorithms. It's only relevant because we (or at least mathematicians) are interested in it. So an AI producing incomprehensible proofs would be completely pointless. That's why Tao insists on the importance of human understanding.

Re: Mathematics in the age of AI

#59

Post by lacker »

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 think the mathematicians are going to be able to make that work, because journals are already struggling to keep up with their review load, and AI seems like it will make that harder. So a solution that involves "journals will do a lot more effort to review each paper" doesn't seem practical.

It would work better as a bar for hiring, rather than as a bar for publishing.

Re: Mathematics in the age of AI

#60

Post by cubefox »

paulpauper wrote: post #21
rramach wrote: post #10

Terence argues that explanation of results ("understanding") will be the new bottleneck in math research but I am not sure this is the real bottleneck for progress. Understanding was critical for the field to progress when only humans were involved but if humans are not needed to make progress, I wonder if we split into two worlds: an AI math-world where amazing new results continue at a rapid pace bottlenecked only…

It will be the same as before: some effort will go into checking proofs and the other into creating them. AI speeds up both.

You clearly didn't read the paper.
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