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In my experience as a working mathematician, intuition is crucial.
But I don't know if my experience is universal (e.g. there are mathematicians with different types of thinking, e.g. algebraic thinking vs geometric thinking is an often quoted dichotomy).
My priors do point towards intuition always being super-useful (and also towards intuition usually being somewhat easier to develop than purely mechanistic skills). But my mechanistic skills have always been so-so, people might differ from each other in a rather radical fashion...
1I think any high level thought or movement is intuitive and approximate and not completely trustworthy, including high level thoughts about mathematics.
You find things by looking across long distances, but constructive proof steps only cross short distances. Nothing new is actually found by applying simple rules. Mathematical proofs don't represent a way of thinking, they're artifacts produced after the thought has been done and the realization has been had, they only exist to validate and to discipline (train) the higher-level heuristics you really use when you're navigating the overarching space of mathematics.
I'm not a mathematician, but if someone had told me this when I started undergrad, much more likely I would've been better at it and I would've ended up being a mathematician in that timeline.
I think that the answer has to do with exactly what sense of "learn" you're using. With infinite time you could memorize every math paper, derive exact proofs of all known theorems by brute force, and octuple-check your work. It is less clear to me if you would understand it all in the way it's original discoverers/inventors did, or that you would recognize which of the new theorems you derived were meaningful in the way we see prominent past mathematical results as meaningful.
I know there are some fields of math I have no problem following, and others where seemingly no amount of reading gets me any sense of insight into what is even being discussed. These sometimes but not always maps onto fields that others I talk to consider more complex; it's more a matter of my mind naturally following different kinds of paths than theirs.
At this moment, I don't think it's feasible for any merely human mind to learn all the math in a lifetime, let alone keep up with more new discoveries as they happen. But different people will tend to have the tools to solve different arts of any given puzzle. Maybe one will have them all, maybe we'll need a team, maybe we'll have to have multiple projects done in series. Not even Jeffreysai thought it was necessary or optimal for one person to have all the knowledge and insights.
Did a PhD and post-doc in maths and also have a 10 and 12 year old so have been interested in watching them and their peers learning mathematics at school and have even been in to help at school a little bit.
I think understanding and problem-solving are different, although the former is a prerequisite for the latter. I can imagine understanding areas of mathematics but not having a good ability to solve novel problems.
Abstraction is a massive part of mathematics. Some people seem to find abstraction much easier than others. I don't know to what extent it is a learnable skill, but I think without it more advanced mathematics is almost impossible.
Also some people (and I see this in children) seem to have real blocks with aspects of logic. The difference between 'if' and 'only if' seems to not exist for some people and yet is obvious for others. The idea of mathematical induction likewise I remember really stumping some of the other students in my class. Again, I don't know how learnable this is!
I also suspect there are certain mental models related to number and geometry one acquires early on that either make aspects of maths very easy or very hard depending on whether one acquires them. Jo Boaler's work in this is interesting - I like her 'Mindset Mathematics' activities for school-level mathematics.
I suspect most people who don't get some of these things naturally lose motivation for mathematics making it hard to know how learnable they are. There are examples though of successful research mathematicians who weren't like that, Mirzakhani being an obvious example. Mathematicians do gravitate towards different fields as well - I do know some mathematicians who don't like geometry.
I do find it interesting how from a very early age, there are huge differences in how much different children 'get' mathematics. Either there is something genetic going on or is there are critical things one can do in those first few years that make a difference! I'm sure working memory helps, but I'm inclined to suspect that it is not just that. I have a reasonably good working memory but it wasn't one of my strong points. I was fine at mental arithmetic but not the best person in the class for example, but by the time I was 16-17, I was way better at mathematics than anybody else at school.
I don't know about intuition, but one definitely needs certain innate abilities, otherwise you hit the exponential wall long before you learn sufficiently advanced math. (I wrote about it in https://www.lesswrong.com/posts/kcKZoSvyK5tks8nxA/learning-is-asymptotically-computationally-inefficient.) This is probably not about some simple objective metrics like the amount of working memory, but about hard to define ability to think abstractly about certain topics.
[...] bridge the "gap" between (less-precise proofs backed by advanced intuition) and (precise proofs simple enough for basically anyone to technically "follow").
Meta: Please consider using curly or square brackets ({} or []) for conceptual/grammatic grouping; please avoid overloading parentheses.
(As usual for my questions, the focus here is "[advanced math]... that'll be needed for technical AI alignment".)
Could someone with good-but-not-great working memory, and infinite time, learn
allany known math? Or is there some "inherent intuitive complex nuance" thing (involving e.g. mental visualization) they need?Reductionism would suggest the former (with some caveats), but computational intractability in real life might require the latter anyway. Human brains, paper, and code may not be able to bridge the "gap" between (less-precise proofs backed by advanced intuition) and (precise proofs simple enough for basically anyone to technically "follow").
And, for practical reasons, I wonder how "continuous" that gap is. E.g. how the tradeoff/tractability changes as one's working memory increases, or how the gap changes for different subfields of math.