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The cultural divide between mathematics and AI

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Re: The cultural divide between mathematics and AI

#61

Post by mcguire »

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 areas of mathematics."

Re: The cultural divide between mathematics and AI

#63

Post by dang »

Henchman21 wrote: post #15
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…

I’ve worked in tech my entire adult life and boy do I feel this deep in my soul. I have slowly withdrawn from the higher-level tech designs and decision making. I usually disagree with all of it. Useless pursuits made only for resume fodder. Tech decisions made based on the bonus the CTO gets from the vendors (Superbowl tickets anyone?) not based on the suitability of the tech. But absolutely worst of all is the arro…

I'm sure that many of us sympathize, but can you please express your views without fulminating? It makes a big difference to discussion quality, which is why this is in the site guidelines: https://news.ycombinator.com/newsguidelines.html.

It's not just that comments that vent denunciatory feelings are lower-quality themselves, though usually they are. It's that they exert a degrading influence on the rest of the thread, for a couple reasons: (1) people tend to respond in kind, and (2) these comments always veer towards the generic (e.g. "lack of true care for anything and anyone", "just another type of stupidity"), which is bad for curious conversation. Generic stuff is repetitive, and indignant-generic stuff doubly so.

By the time we get further downthread, the original topic is completely gone and we're into "glorification of management over ICs" (https://news.ycombinator.com/item?id=43346257). Veering offtopic can be ok when the tangent is even more interesting (or whimsical) than the starting point, but most tangents aren't like that—mostly what they do is replace a more-interesting-and-in-the-key-of-curiosity thing with a more-repetitive-and-in-the-key-of-indignation thing, which is a losing trade for HN.

Re: The cultural divide between mathematics and AI

#64

Post by [deleted] »

dang wrote: post #63
Henchman21 wrote: post #15

Earlier quoted context omitted.

I’ve worked in tech my entire adult life and boy do I feel this deep in my soul. I have slowly withdrawn from the higher-level tech designs and decision making. I usually disagree with all of it. Useless pursuits made only for resume fodder. Tech decisions made based on the bonus the CTO gets from the vendors (Superbowl tickets anyone?) not based on the suitability of the tech. But absolutely worst of all is the arro…

I'm sure that many of us sympathize, but can you please express your views without fulminating? It makes a big difference to discussion quality, which is why this is in the site guidelines: https://news.ycombinator.com/newsguidelines.html . It's not just that comments that vent denunciatory feelings are lower-quality themselves, though usually they are. It's that they exert a degrading influence on the rest of the th…

[deleted]

Re: The cultural divide between mathematics and AI

#65

Post by nybsjytm »

oersted wrote: post #42
xg15 wrote: post #22

> One question generated particular concern: what would happen if an AI system produced a proof of a major conjecture like the Riemann Hypothesis, but the proof was too complex for humans to understand? Would such a result be satisfying? Would it advance mathematical understanding? The consensus seemed to be that while such a proof might technically resolve the conjecture, it would fail to deliver the deeper understa…

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 that it's very easy to check if a lean-coded proof is correct. But it's hard and time-consuming to formulate most math as lean code. It's something many AI research groups are working on.

Re: The cultural divide between mathematics and AI

#66

Post by jvans »

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…

in poker AI solvers tell you what the optimal play is and it's your job to reverse engineer the principles behind it. It cuts a lot of the guess work out but there's still plenty of hard work left in understanding the why and ultimately that's where the skill comes in. I wonder if we'll see the same in math

Re: The cultural divide between mathematics and AI

#67

Post by ohgr »

throw8404948k wrote: post #16

> This quest for deep understanding also explains a common experience for mathematics graduate students: asking an advisor a question, only to be told, "Read these books and come back in a few months." With AI advisor I do not have this problem. It explains parts I need, in a way I understand. If I study some complicated topic, AI shortens it from months to days. I was somehow mathematically gifted when younger, sadl…

I suspect you probably don't understand it after that. You think you do.

I thought I understood calculus until I realised I didn't. And that took a bit thwack in the face really. I could use it but I didn't understand it.

Re: The cultural divide between mathematics and AI

#68

Post by nonethewiser »

nicf wrote: post #21
lordleft wrote: post #19

Earlier quoted context omitted.

I'm not a mathematician so please feel free to correct me...but wouldn't there still be an opportunity for humans to try to understand why a proof solved by a machine is true? Or are you afraid that the culture of mathematics will shift towards being impatient about this sorts of questions?

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…

>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 understanding was that you were primarily just concerned with understanding why, not being the one to discover why.

Re: The cultural divide between mathematics and AI

#69

Post by optimalsolver »

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?

Re: The cultural divide between mathematics and AI

#70

Post by nicf »

nonethewiser wrote: post #68
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…

>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 being handed the answer, though, even if it's not an original problem. It's the same reason some people like doing sudokus, and why those people wouldn't respond well to being told that they could save a lot of time if they just used a sudoku solver or looked up the answer in the back of the book.

But that's not really what I'm getting at in the sentence you're quoting --- people are still free to solve sudokus even though sudoku solvers exist, and the same would presumably be true of proving theorems in the world we're considering. The thing I'd be most worried about is the destruction of the community of mathematicians. If math were just a fun but useless hobby, like, I don't know, whittling or something, I think there would be way fewer people doing it. And there would be even fewer people doing it as deeply and intensely as they are now when it's their full-time job. And as someone who likes math a lot, I don't love the idea of that happening.

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