The cultural divide between mathematics and AI • Hacker News Board

Re: The cultural divide between mathematics and AI

#101

Post by jebarker » Thu, Mar 13, 2025, 12:56 AM UTC

solveit wrote: post #98

godelski wrote: post #93

Earlier quoted context omitted.

> the primary aim isn't really to find out whether a result is true but why it's true. I'm honestly surprised that there are mathematicians that think differently (my background[0]). There are so many famous mathematicians stating this through the years. Some more subtle like Poincare stating that math is not the study of numbers but the relationship between them, while others far more explicit. This sounds more like…

> I'd argue that the biggest reason machines are black boxes are because no one is bothering to look inside of them. People do look, but it's extremely hard. Take a look at how hard the mechanistic interpretability people have to work for even small insights. Neel Nanda[1] has some very nice writeups if you haven't already seen them. [1]: https://www.neelnanda.io/mechanistic-interpretability

The problem is that mechanistic interpretability is a lot like neuroscience or molecular biology, i.e. you're trying to understand huge complexity from relatively crude point measurements (no offense intended to neuroscientists and biologists). But AI wants publishable results yesterday. I often wonder whether the current AI systems will stay around long enough for anyone to remain interested in understanding why they ever worked.

Re: The cultural divide between mathematics and AI

#102

Post by gsf_emergency_2 » Thu, Mar 13, 2025, 12:57 AM UTC

rssoconnor wrote: post #88

nicf wrote: post #17

Earlier quoted context omitted.

The Four Color Theorem is a great example! I think this story is often misrepresented as one where mathematicians didn't believe the computer-aided proof. Thurston gets the story right: I think basically everyone in the field took it as resolving the truth of the Four Color Theorem --- although I don't think this was really in serious doubt --- but in an incredibly unsatisfying way. They wanted to know what underlyin…

Is the proof of the Four Colour Theorem really that unsatisfying? The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED. To verify this computational fact one uses a (very) glorified pocket calculator.

Fwiw fairly recently posted to HN was progress towards a more satisfying (read, likely mind-blowing) proof:

https://blog.tanyakhovanova.com/2024/11/foams-made-out-of-fe...

This is also nice because only pre-1600 tech involved

Re: The cultural divide between mathematics and AI

#103

Post by gsf_emergency_2 » Thu, Mar 13, 2025, 12:58 AM UTC

rssoconnor wrote: post #99

godelski wrote: post #97

Earlier quoted context omitted.

I think you're missing the core component. We care __WHY__ the theorem is true. To be honest, the __IF__ part matters a lot less. The thing is that the underlying reasoning (the logic) is what provides real insights. This is how we recognize other problems that are similar or even identical. The steps in between are just as important, and often more important. I'll give an example from physics. (If you're unsatisfied…

I do think the why that the Four Colour Theorem is true is captured my statement. The reason why it is true is because there exists some finite unavoidable and reducible set of configurations. I'm fairly sure that people are only getting hung up on the size of this finite set, for no good reason. I suspect that if the size of this finite set were 2, instead of 633, and you could draw these unavoidable configuration o…

If it were size 2, we could more easily make sure that the answer is definitely mind-blowing.

Have programmers given up on wanting their mind blown by unbelievable simplicity?

Re: The cultural divide between mathematics and AI

#104

Post by gsf_emergency_2 » Thu, Mar 13, 2025, 1:04 AM UTC

godelski wrote: post #97

rssoconnor wrote: post #88

Earlier quoted context omitted.

Is the proof of the Four Colour Theorem really that unsatisfying? The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED. To verify this computational fact one uses a (very) glorified pocket calculator.

I think you're missing the core component. We care __WHY__ the theorem is true. To be honest, the __IF__ part matters a lot less. The thing is that the underlying reasoning (the logic) is what provides real insights. This is how we recognize other problems that are similar or even identical. The steps in between are just as important, and often more important. I'll give an example from physics. (If you're unsatisfied…

Mind-blowing result from a different attempt to prove four color theorem

https://blog.tanyakhovanova.com/2024/11/foams-made-out-of-fe...

Re: The cultural divide between mathematics and AI

#105

Post by dullcrisp » Thu, Mar 13, 2025, 1:05 AM UTC

diamondage wrote: post #91

nicf wrote: post #71

Earlier quoted context omitted.

Hm, good question. It depends on what you mean. If you're asking about restricting which theorems we try to prove, then we definitely are cutting ourselves off from vast swathes of math space, and we're doing it on purpose! The article we're responding to talks about mathematicians developing "taste" and "intuition", and this is what I think the author meant --- different people have different tastes, of course, but…

So Godel proved that there are true theorems that are unprovable. My hunch is that there is a fine grained version of this result <-- that there is a some distribution on the length of the proof for any given conjecture. If true that would mean that we better get used to dealing with long nasty proofs because they are a necessary part of mathematics...perhaps even, in some kind of Kolmogorov complexity-esque fashion,…

I’m not sure that’s quite true. Say the proof of proposition P requires a minimum of N symbols. You could prove it in one paper that’s N symbols long and nobody can read, or you can publish k readable papers, with an average length on the order of N/k symbols, and develop a theory that people can use.

I think even if N is quite large, that just means it may take decades or millennia to publish and understand all k necessary papers, but maybe it’s still worth the effort even if we can get the length-N paper right away. What are you going to do with a mathematical proof that no one can understand anyway?

Re: The cultural divide between mathematics and AI

#106

Post by skinner_ » Thu, Mar 13, 2025, 1:07 AM UTC

nybsjytm wrote: post #60

EigenLord wrote: post #50

Is it really a culture divide or is it an economic incentives divide? Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. What differs are the incentives. Math rewards openness because there's no real concept of a "competitive edge", you're incentivized to freely…

> Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. There is a major caveat here. Most 'serious math' in AI papers is wrong and/or irrelevant! It's even the case for famous papers. Each lemma in Kingma and Ba's ADAM optimization paper is wrong, the geometry in…

Amazing! I looked into your ADAM claim, and it checks out. Thanks! Now I'm curious. I you have the time, could you please follow up with the 'etc...'?

Re: The cultural divide between mathematics and AI

#107

Post by FilosofumRex » Thu, Mar 13, 2025, 1:12 AM UTC

troymc wrote: post #79

Sniffnoy wrote: post #74

> As Gauss famously said, there is "no royal road" to mathematical mastery. This is not the point, but the saying "there is no royal road to geometry" is far older than Gauss! It goes back at least to Proclus, who attributes it to Euclid.

I never understood that quote until recently. The story goes that the (royal) pharaoh of Egypt wanted to learn geometry, but didn't want to have to read Euclid. He wanted a faster route. But, "there is no royal road to geometry."

The last Egyptian pharaoh was Nectanebo II, who ruled from 358 to approximately 340 BC. Alexander founded Alexandria in 331 BC as the crown jewel of his empire where Euclid wrote his magnum opus, The Elements in 300 BC!

Unless the royal pharaoh of Egypt, refers to Ptolemy I Soter, Macedonian general who was the first Ptolemaic Kingdom ruler of Egypt after Alexander's death.

Re: The cultural divide between mathematics and AI

#108

Post by zmgsabst » Thu, Mar 13, 2025, 1:14 AM UTC

jebarker wrote: post #100

nicf wrote: post #21

Earlier quoted context omitted.

Well, it depends on exactly what future you were imagining. In a world where the model just spits out a totally impenetrable but formally verifiable Lean proof, then yes, absolutely, there's a lot for human mathematicians to do. But I don't see any particular reason things would have to stop there: why couldn't some model also spit out nice, beautiful explanations of why the result is true? We're certainly not there…

> The thing that makes me sad about these conversations is that the people I talk to sometimes don't seem to have any appreciation for the thing they say they want to dismantle Yes! This is what frustrates my about the pursuit of AI for the arts too.

This seems obviously untrue: why would they be replicating it if they didn’t want it?

I see both cases as people who aren’t well served by the artisanal version attempting to acquire a better-than-commoditized version because they want more of that thing to exist. We regularly have both things in furniture and don’t have any great moral crisis that chairs are produced mechanistically by machines. To me, both things sound like “how dare you buy IKEA furniture — you have no appreciation of woodwork!”

Maybe artisanal math proofs are more beautiful or some other aesthetic concern — but what I’d like is proofs that business models are stable and not full of holes constructed each time a new ML pipeline deploys; which is the sort of boring, rote work that most mathematicians are “too good” to work on. But they’re what’s needed to prevent, eg, the Amazon 2018 hiring freeze.

That’s the need that, eg, automated theorem proving truly solves — and mathematicians are being ignored (much like artist) by people they turn up their noses at.

Re: The cultural divide between mathematics and AI

#109

Post by godelski » Thu, Mar 13, 2025, 1:17 AM UTC

rssoconnor wrote: post #99

godelski wrote: post #97

Earlier quoted context omitted.

I think you're missing the core component. We care __WHY__ the theorem is true. To be honest, the __IF__ part matters a lot less. The thing is that the underlying reasoning (the logic) is what provides real insights. This is how we recognize other problems that are similar or even identical. The steps in between are just as important, and often more important. I'll give an example from physics. (If you're unsatisfied…

I do think the why that the Four Colour Theorem is true is captured my statement. The reason why it is true is because there exists some finite unavoidable and reducible set of configurations. I'm fairly sure that people are only getting hung up on the size of this finite set, for no good reason. I suspect that if the size of this finite set were 2, instead of 633, and you could draw these unavoidable configuration o…

I really doubt this. I mean mathematicians spent decades trying to answer if the number 2 exists. People spend a lot of time on what seems fairly mundane and frankly, the results are quite beneficial. What's incredible or mind blowing is really just about your perspective, it is really just about your choice to wonder more. https://www.youtube.com/shorts/lcQAWEqPmeg

Re: The cultural divide between mathematics and AI

#110

Post by zmgsabst » Thu, Mar 13, 2025, 1:18 AM UTC

nicf wrote: post #70

nonethewiser wrote: post #68

Earlier quoted context omitted.

>But I don't see any particular reason things would have to stop there: why couldn't some model also spit out nice, beautiful explanations of why the result is true? Oh… I didnt anticipate this would bother you. Would it be fair to say that its not that you like understanding why its true, because you have that here, but that you like process of discovering why? Perhaps thats what you meant originally. But my underst…

This is an interesting question! You're giving me a chance to reflect a little more than I did when I wrote that last comment. I can only speak for myself, but it's not that I care a lot about me personally being the first one to discover some new piece of mathematics. (If I did, I'd probably still be doing research, which I'm not.) There is something very satisfying about solving a problem for yourself rather than b…

CNCs and other technology haven’t destroyed woodworking. There’s whole communities on YouTube — with a spectrum from casual to hobbyist to artisanal to industrial.

Why would mathematics be different than woodworking?

Do you believe there’s a limited demand for mathematics? — my experience is quite the opposite, that we’re limited by the production capacity.