[RajuVarghese]

In a book on mathematics by Alex Bellos he says that any contemporary cutting-edge research mathematics paper can be understood by no more than 100 mathematicians. The field is getting bigger and bigger and there are less and less mathematicians that have background knowledge sufficient to _understand_it much less critique it. The bar for the word 'understand' may be much higher in mathematics compared to other fields. Comments?

[Hermitian909]

I think the main thing missing from this analysis is how much of the mathematical corpus ends up being "uninteresting" over time.

Take Algebraic Geometry, a field primarily concerned with the question "what are the zeroes of multivariate polynomial equations" e.g. x^2-y+z^3. Papers that could be considered part of the field were published as early as the 16th century but very little is worth reading from before the work of Alexander Groethendieck in the mid 20th century which really formalized the field in the form it is today. The vast improvement in abstractions meant old proofs could be rewritten in a much terser, easier to understand manner with the new machinery.

Similarly, now that we've solved the classification of finite simple groups[0] a lot of the deep expertise mathematicians had surrounding techniques to classify finite groups is no longer needed. A new researcher can learn a small, curated subset of these techniques and fully grasp the field. Since there's no more work to be done, there's no need to build deep intuitions.

More generally a lot of papers in mathematics are attempts to build the machinery to solve some bigger problem and most of those end up not being useful when the field cracks said problem. At which point, all those papers effectively get "trimmed away" since the result they were meant to support does not depend on them, and so no one need read them.

[0] https://en.wikipedia.org/wiki/Classification_of_finite_simpl...

[deadbeef57]

I just want to say that in the case of the classification of finite simple groups, there's actually a bit of a problem. Exactly because "a lot of the deep expertise mathematicians" have left the field; but on the other hand "a small, curated subset of these techniques" is still missing. A small group of experts (all in their 70's or 80's) are currently writing a dozen volumes on this classification, and the rest of the community hopes that they get it done before they pass away.

This proof has a seriously low bus factor at the moment :scared:

[Hermitian909]

Oh you're right! I hadn't been aware that the book series was incomplete, my face is a bit red and I share your concerns. Here's to hoping!

[daxfohl]

In a way that's what makes the question of AI math interesting. Probably AIs will be better than us at math in a couple decades. But that might not change anything because it's still up to humans to determine whether a result is interesting. And it may take us just as long to understand AI-created concepts as to build them up ourselves. Maybe math starts looking more like archaeology at that point. (Though it's arguably archaeology anyway -- the proofs are all "out there", we just have to find them).

[Y_Y]

I like this idea. People often ask if maths is "created" or "discovered". Maybe we can say that it's "excavated" and that we know it's down there somewhere, we just need to put in a few years of shoveling.

[yellowcake0]

There is no way that AI will be better than us at math in a couple of decades.

[daxfohl]

Yeah I think I agree now. They'll probably be better at proving straightforward things that don't require any big new insights, but I don't think machines will ever be any good at determining what kinds of new concepts will be interesting to define and explore. Like a machine isn't going to develop calculus just for fun from base principles.

[daxfohl]

Actually maybe I take that back. I mean, it took how many centuries for humans to invent calculus? Maybe AIs left to their own devices in a reinforcement-learning context could do better.

The interesting thing though, is given free reign to prove whatever they want, they may go off and develop some entirely new field of mathematics and proving really deep stuff, but we just wouldn't recognize it as interesting. Like imagine if such an AI existed 200 years ago and invented Turing machines and proved P != NP, but it wasn't all that great at solving polynomials for whatever reason. We'd have probably thought it was all rubbish and threw it away.

[Agingcoder]

Not really.

There's 'understand', and 'believe you understand', and in other fields (say, software engineering...) you meet lots of people who think they understand, when they don't. I also think that in the SE case, because it's more 'relatable' ('look at my website') , people generally assume more easily that they understand when they don't.

Also if you look at theoretical comp sci results (calculability theory, etc) , or even complex software engineering systems (actual working distributed systems, etc) , I very much doubt that many people (apart from the people who designed and wrote them, and who may even be wrong about them! ) can _actually_ claim to understand them fully. You will understand, to some extent, the 'theorem' (ie the system), but the implementation will remain a black box.

[onion2k]

That sounds like something that'd be true for every field, not just math. The nature of anything that's cutting edge means very few people will have looked at it yet let alone actually understood it.

[RajuVarghese]

I agree that the number of people that can understand a cutting-edge paper will be low. But, in CS for example, the number will be much higher than 100. I would assume that it would be in the 1000's worldwide. Since, it isn't my field I could speculate that cutting-edge biology would be understood by a lot more than 100 persons.