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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#21

Post by paulpauper »

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.

Re: Mathematics in the age of AI

#22

Post by GPerson »

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 more that he seeks to preserve and promote human understanding of mathematics, and sees that grappling with this new technology is necessary. One reason is that for human mathematical practices and institutions to retain legitimacy, they need to justify their value. As Tao explains, one obvious answer to that is made less obvious now with AI.

Re: Mathematics in the age of AI

#24

Post by [deleted] »

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…

[deleted]

Re: Mathematics in the age of AI

#25

Post by a2ff6eeb0 »

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 thinking is going to be a recreational activity like weightlifting. If you want to think as a hobby, that's fine, but most people will be free of that toil of unwanted brain labor.

Re: Mathematics in the age of AI

#26

Post by a2ff6eeb0 »

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.

Re: Mathematics in the age of AI

#27

Post by intuitionist »

czgov wrote: post #9
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 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…

Nowadays the proof of resolution of singularities in characteristic zero is considered something you can teach in an intro algebraic geometry course, though. The concepts have been absorbed and are now much better understood. 4CT is very different because so much of it is exhaustive case analysis; you can understand the high-level ideas of the proof as a bright undergraduate, but you still can’t check the cases by hand

Re: Mathematics in the age of AI

#28

Post by pfdietz »

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 else's results.

Re: Mathematics in the age of AI

#29

Post by hintymad »

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…

> I don't know why anyone should care about understanding the results if the AI is better at math than us

This is a big if, right? AI can still generate subtle or even silly mistakes that any normal human, let alone a mathematician, wouldn't make. Besides, math is more than just getting a conclusion but to understand and to generalize new ways of solving problems. After all, mathematicians are a curious bunch. To quote Hilbert's epitaph: We must know. We shall know.

Re: Mathematics in the age of AI

#30

Post by bayesnet »

hintymad wrote: post #29
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…

> I don't know why anyone should care about understanding the results if the AI is better at math than us This is a big if, right? AI can still generate subtle or even silly mistakes that any normal human, let alone a mathematician, wouldn't make. Besides, math is more than just getting a conclusion but to understand and to generalize new ways of solving problems. After all, mathematicians are a curious bunch. To quo…

It’s a bit of an ominous quote given that Hilbert’s program was dismantled shortly thereafter by Gödel…
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