I don't study Mathematics at university, and I probably will never understand of what mathematics consists of in all its aspects. But I love to find structures and links betwen ideas, and to ask myself philosophical questions about foundations of mathematics. If I have understood well is what is happening with the "categorical" point of view, there is some kind of great unification of the mathematics from a structural point of view.
Obviously I understand that for humans it is impossible to reach a total understanding of something, and that what moves me is a mirage. I must give up until I'm sane and study modestly mathematics, in a relaxed way: playing with easy problems, reading books, and learning from the basics all I can, without excessive demands on myself.
More I understand and more new interesting mathematical topics I discover... but the facts that overwhelm me are the acceleration of this process and the fact that there are a lot of new theories; and that this implies that this growth will be even faster.
Probably I, as an amateur mathematician, don't have a chance to come at new ideas in the foundational research (or even understand the actual point of view since it is changing so fast)... but
Q1: From your experience how can mathematicians deal (or how they usually chose to behave) with this huge mathematical universe and be satisfied from a philosophical point of view?
Q2: How can a mathematician interested in the foundation of mathematics be satisfied by an always partial knowledge of the mathematics?
Is maybe possible that in 100 years tha amount of mathematical knowledge will be so huge that even Mathematicians will be totally overwhelmed from it? In other words, is possible that will be impossible for anyone to research new things because the knowledge required will take like 80 or 90 years of hard study? I say this because I think that humans has a limit to the speed of learning.
If this is possible, should mathematicians abandon the hope of understanding mathematics?
I remember a quote of John von Neumann:
"Young man, in mathematics you don't understand things. You just get used to them."
I have read many stories about his math skills, and how he was a genius... anyways from my point of view... I feel very sad when I read this quote.
Is really this the ultimate nature of mathematics?
I'm searching for human experience of real mathematicians, that is very important for me in this moment, I asked this question on SEMath but it was closed in 40 seconds...is a soft question, as many others questions, but it has philosophical contenents too and I think it deserves at least a chance, thanks in advance.
(Link to a closed crosspost on SE Mathematics)