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The cultural divide between mathematics and AI

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Re: The cultural divide between mathematics and AI

#71

Post by nicf »

optimalsolver wrote: post #69
nicf wrote: post #8

I'm a former research mathematician who worked for a little while in AI research, and this article matched up very well with my own experience with this particular cultural divide. Since I've spent a lot more time in the math world than the AI world, it's very natural for me to see this divide from the mathematicians' perspective, and I definitely agree that a lot of the people I've talked to on the other side of thi…

If the shortest proof for some theorem is several thousand pages long and beyond the ability of any biological mind to comprehend, would mathematicians not care about it? Which is to say, if you only concern yourself with theorems which have short, understandable proofs, aren't you cutting yourself off from vast swathes of math space?

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 most conceivable true mathematical statements are ones that everyone would agree are completely uninteresting; they're things like "if you construct these 55 totally unmotivated mathematical objects that no one has ever cared about according to these 18 random made-up rules, then none of the following 301 arrangements are possible."

If you're talking about questions that are well-motivated but whose answers are ugly and incomprehensible, then a milder version of this actually happens fairly often --- some major conjecture gets solved by a proof that everyone agrees is right but which also doesn't shed much light on why the thing is true. In this situation, I think it's fair to describe the usual reaction as, like, I'm definitely happy to have the confirmation that the thing is true, but I would much rather have a nicer argument. Whoever proved the thing in the ugly way definitely earns themselves lots of math points, but if someone else comes along later and proves it in a clearer way then they've done something worth celebrating too.

Does that answer your question?

Re: The cultural divide between mathematics and AI

#72

Post by zozbot234 »

nybsjytm wrote: post #65
oersted wrote: post #42

Earlier quoted context omitted.

Serious theorem-proving AIs always write the proof in a formal syntax where it is possible to check that the proof is correct without issue. The most popular such formal language is Lean, but there are many others. It's just like having a coding AI, it may write some function and you check if it compiles. If the AI writes a program/proof in Lean, it will only compile if the proof is correct. Checking the correctness…

> Checking the correctness of proofs is a much easier problem than coming up with the proof in the first place. Just so this isn't misunderstood, not so much cutting-edge math is presently possible to code in lean. The famous exceptions (such as the results by Clausen-Scholze and Gowers-Green-Manners-Tao) have special characteristics which make them much more ground-level and easier to code in lean. What's true is th…

> The famous exceptions (such as the results by Clausen-Scholze and Gowers-Green-Manners-Tao) have special characteristics which make them much more ground-level and easier to code in lean.

"Special characteristics" is really overstating it. It's just a matter of getting all the prereqs formalized in Lean first. That's a bit of a grind to be sure, but the Mathlib effort for Lean has the bulk of the undergrad curriculum and some grad subjects formalized.

I don't think AI will be all that helpful wrt. this kind of effort, but it might help in some limited ways.

Re: The cultural divide between mathematics and AI

#75

Post by psunavy03 »

mcguire wrote: post #61
nicf wrote: post #8

I'm a former research mathematician who worked for a little while in AI research, and this article matched up very well with my own experience with this particular cultural divide. Since I've spent a lot more time in the math world than the AI world, it's very natural for me to see this divide from the mathematicians' perspective, and I definitely agree that a lot of the people I've talked to on the other side of thi…

Many years ago I heard a mathematician speaking about some open problem and he said, "Sure, it's possible that there is a simple solution to the problem using basic techniques that everyone has just missed so far. And if you find that solution, mathematics will pat you on the head and tell you to run off and play. "Mathematics advances by solving problems using new techniques because those techniques open up new area…

That seems like a justification that is right on the knife's edge of being a self-licking ice cream cone.

Re: The cultural divide between mathematics and AI

#76

Post by analog31 »

woah wrote: post #20

> Perhaps most telling was the sadness expressed by several mathematicians regarding the increasing secrecy in AI research. Mathematics has long prided itself on openness and transparency, with results freely shared and discussed. The closing off of research at major AI labs—and the inability of collaborating mathematicians to discuss their work—represents a significant cultural clash with mathematical traditions. Th…

It involves math at a research level, but from what I've observed, people in industry with engineering job titles make relatively little use of math. They will frequently tell you with that sheepish smile: "Oh, I'm not really a math person." Students are told with great confidence by older engineers that they'll never use their college math after they graduate.

Not exactly AI by today's standards, but a lot of the math that they need has been rolled into their software tools. And Excel is quite powerful.

Re: The cultural divide between mathematics and AI

#77

Post by NooneAtAll3 »

I feel like this rumbling can be summarized as "Ai is engineering, not math" - and suddenly a lot of things make sense

Why Ai field is so secretive? Because it's all trade secrets - and maybe soon to become patents. You don't give away precisely how semiconductor fabs work, only base research level of "this direction is promising"

Why everyone is pushed to add Ai in? Because that's where the money is, that's where the product is.

Why Ai needs results fast? Because it's production line, and you create and design stuff

Even the core distinction mentioned - that Ai is about "speculation and possibility" - that's all about tool experimenting and prototyping. It's all about building and constructing. Aka Engineering/Technology letters of STEM

I guess next step is to ask "what to do next?". IMO, math and Ai fields should realise the divide and slowly diverge, leaving each other alone on an arm's length. Just as engineers and programmers (not computer scientists) already do

Re: The cultural divide between mathematics and AI

#78

Post by coffeeaddict1 »

m0llusk wrote: post #18

> The last mathematicians considered to have a comprehensive view of the field were Hilbert and Poincaré, over a century ago. Henri Cartan of the Bourbaki had not only a more comprehensive view, but a greater scope of the potential of mathematical modeling and description

I would also add Grothendieck to that list.

Re: The cultural divide between mathematics and AI

#79

Post by troymc »

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."

Re: The cultural divide between mathematics and AI

#80

Post by umutisik »

If AI can prove major theorems, it will likely by employing similar heuristics as the mathematical community employs when searching for proofs and understanding. Studying AI-generated proofs, with the help of AI to decipher contents will help humans build that 'understanding' if that is desired.

An issue in these discussions is that mathematics is both an art, a sport, and a science. And the development of AI that can build 'useful' libraries of proven theorems means different things for each. The sport of mathematics will be basically over. The art of mathematics will thrive as it becomes easier to explore the mathematical world. For the science of mathematics, it's hard to say, it's been kind of shaky for ~50 years anyway, but it can only help.

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