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Mathematics in the age of AI

arxiv.org

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Re: Mathematics in the age of AI

#31

Post by a2ff6eeb0 »

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…

Well, that's why we have automated proof checking. And again, I don't think humans will be able to solve problems at a commercial scale in the future.

Maybe we'll have some hobbyist dabblers, but any real progress will be done by machines that skip the human.

Re: Mathematics in the age of AI

#32

Post by akk0 »

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give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published.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…

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.

Re: Mathematics in the age of AI

#33

Post by pfdietz »

bayesnet wrote: post #30
hintymad wrote: post #29

Earlier quoted context omitted.

> 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…

I'm reminded of the joke about the two friends who come across a bear in the woods. When one puts on running shoes, his friend chides him that he can't outrun the bear. He responds, "I don't need to outrun the bear, I just need to outrun you."

AI doesn't have to implement Hilbert's vision and be able to prove everything. I just has to out-prove human mathematicians.

Re: Mathematics in the age of AI

#35

Post by kriro »

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 sure. Especially for counterexamples LLM solutions seem fine. They stop humans from wasting time on pointless things. For proofs it gets more hairy but I think if it is formally verified a proof is a proof. Attribution is a problem (should the person who wrangled the answer out of an LLM get the credit, I guess so).

I think these are non-trivial epistemology and science theory problems.

Re: Mathematics in the age of AI

#36

Post by aleph_minus_one »

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…

> 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.

Just burn lots of tokens on the frontier model of your choice to let the AI find a high-level argument why the four color theorem holds. :-)

--

Seriously: since there exist quite a lot of readers on HN who are both hardcore into AI and mathematical problems: This is a challenge for you.

I am looking forward to seeing an announcement of a novel high-level argument why the four color theorem holds on the first page of HN in at most a month. :-D

Re: Mathematics in the age of AI

#37

Post by TSiege »

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’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?

Re: Mathematics in the age of AI

#38

Post by mohamedkoubaa »

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.

Ive wondered whether a possible outcome of LLM slop is a retvrn to oral wisdom traditions. Ironically that's the most anthropological form of understanding and pedagogy.

Re: Mathematics in the age of AI

#39

Post by mohamedkoubaa »

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…

What if the better routing leads to an outage that the AI can't explain or fix and all the humans who might have understood it were laid off or otherwise unavailable?

Re: Mathematics in the age of AI

#40

Post by FuckButtons »

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 you are free from physical and mental labor, you are in fact, not supplying labor, and are therefore surplus to requirements.
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