Just skimming the answer by Peter Scholze reminds me why I switched from Mathematics to Computer Science: Too much to learn for too little benefit.
While I'm a little skeptical[0] of the tone of this article, I hold out hope this just is the Feynman (is able to communicate complex ideas well) of Math.
Sometimes, I'd like time with someone with advanced knowledge of such things, and be able to shoot my naive questions at them. I'm sometimes shocked by how subjects like this are not just non-intuitive, but outright obs by curious omissions[1] by experts in their explanations.
For example, look on HN/mathOverflow, and ask for an intuitive explanation of something, and you'll often get "there is no such thing, you just have to work through it - math is as simple as it can be". Then, someone eventually provides an explanation that is more intuitive than the norm, proving that it was failure to communicate in simpler terms in the first place.
this article:
"Unlike many mathematicians, he often starts not with a particular problem he wants to solve, but with some elusive concept that he wants to understand for its own sake"
"[Scholze] would never lose himself in the jungle, because he’s never trying to fight the jungle. He’s always looking for the overview, for some kind of clear concept."
I can't find them now, but I'm sure I've read sentiments from mathematicians such as "don't try to get an 'intuition', there is no such thing in advanced math where there are few mappings to real life things, and such metaphors will only create misunderstandings" and "we'd all like a 'map' of mathematics, but math is not neatly ordered like a landscape, it's not possible to even visualise all the links between math fields".
So this guys method is exactly what other mathematicians warn against? Trying to understand?
[0] - He is depicted so far ahead that mathematicians look forward to him entering their field? Even if true, I'm skeptical most would admit something like that - also anything that appears in wired will have a certain minimum of hot-air..
[1] - e.g. the norm distribution id the convergence of the binomial distribution (which can in turn, be constructed), which is why it turns up so often whenever a large number or random variables are involved. I've yet to be taught stats where this is explained, before I realised it was as if norm was just something that appeared by magic.
No, I don't think so.
The "there's no intuition" advice is really good advice for undergraduates or lay people first encountering abstract concepts in pure mathematics. In fact there is an intuition, but the intuition is usually sort of on its own terms -- not so much grounded in physical life experiences. In other words, you will get an intuitive grasp on the ideas but that intuition will probably not be an analogy to your previous experiences in everyday life. When mathematicians say "there's no intuition" they usually mean "there's no physical experience that corresponds to the underlying intuition, and it's not really like anything you've seen before. It's a new experience and you need to experience it on its own terms. Stop trying to ground yourself in classical mechanics or whatever."
But that's quite the mouthful, so better to just say "there's no intuition; just focus on working with the objects and you'll eventually get comfortable".
This quote from the article captures it well, I think:
> "Now I find real numbers much, much more confusing than p-adic numbers. I’ve gotten so used to them that now real numbers feel very strange."
Re: your skepticism... The guy is a once-in-a-generation talent; his constructions were able to vastly simplify multiple very long, very complicated proofs that groups of the top people in this field were working on. This is in a field (algebraic number theory) which is considered one of the more saturated and technically difficult within all of mathematics (admittedly, I am likely biased on this point). That being said, all of his work so far has been in the ballpark of Langlands/p-adic/arithmetic geometry, so I would be surprised if he achieved significant results that strayed too far from this stuff.
I'm not sure what you mean about Feynman; Peter's genius is not so much his ability to communicate complex ideas in a simple way, but rather he was able to come up with constructions (or if you like, abstractions) which compartmentalize the complex ideas in the right way so that they are easier to deal with. To make an analogy with computing, think of the concept of a "thread". Without the concept of a thread, you'd have to do so much manual maintenance that you could never dream of building say Google. Scholze's perfectoid spaces are analogous; their definition would have been understood by mathematicians 50 years ago, but no one really got that this was the right thing to consider.
Feynman was known for being able to explain complex ideas (quantum, physics etc) in easier terms.
According to the article:
""Scholze is known for the clarity of his talks and papers. "I don’t really understand anything until Peter explains it to me" Weinstein said.""
"Scholze makes a point of trying to explain his ideas at a level that even beginning graduate students can follow"
It all depends on what you want out of life.
Youthful idealism -> number theory Mature pragmatism -> pivot to CS
It was entirely luck that my passion was so closely related to the current gold rush.