How do real mathematicians learn enough

[metric tensor]

This post is inquiring about the process of how one obtains high level math knowledge .How does one produce papers such as this at 24-27 yrs old?

http://books.google.com/books?id=c8...&sa=X&ei=8qXkUZWxEeH9iwLx2YFQ&ved=0CGsQ6AEwCA

I remember taking calculus at 17 or so and reading some books on math but nothing, including my homework even approached anything like this. Is it just 5 years of constant study or do these people learn everything really fast while in HS and enter college with graduate level of knowledge?
How does one go about writing mathematics in this type of rigor? how does one make the jump from just fooling around to producing serious, PHD quality, respectable math like the example given? There seems to be a huge chasm between high-level math in papers versus what is contained in most college courses and textbooks.

 

[lurflurf]

One can learn a lot it 5-10 years. I don't know where your get 24-27 though McKean was it his thirties and Kiyoshi Itō was older than that. You are right that there is a significant difference. Both between an average and unusual student and between a given student and the same student after 10 years hard work.

 

[hsetennis]

I'm not qualified to answer all of your questions, but I can share what I know. The kind of "head start" can make a difference. For example, the mathematician Manjul Bhargava finished his high school math courses by age 14 and his mother was also a mathematician. As a result, he conducted phenomenal research in his twenties. More info: http://en.m.wikipedia.org/wiki/Manjul_Bhargava.

 

[Solcielo L]

Most math courses are taught that math is something you do, such as solving problems. But math isn't just about that. It's a way of thinking, and this isn't normally taught. Mathematicians talk about it as a language. Only those who understand that language can use it to communicate. And I think that's where the problem is: most math teachers aren't mathematicians; they know how to teach students to solve problems but don't know how to teach math as a language. It's easy to take a math teacher licensing test and pass if you've taken a number of math courses but these tests don't require you to understand math as a language.