Introduction
I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics 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 moves me is a mirage. I must give up until I'm sane and study modestly mathematics, in a relaxed way: playing with my easy problems, reading books, and learning from the basics all I can, without eccesive expectations maybe.
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 young 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)…
Questions
$\mathcal Q_1$ From your($*^1$) experience how can mathematicians deal (or how they usually chose to behave) with this huge mathematical universe and be satisfied from a philosophical($*^2$) point of view?
$\mathcal Q_2$ How can a mathematician interested in the foundations of mathematics be satisfied by an always partial knowledge of the mathematics?
Personal observation and conclusion
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, is in the destiny of mathematicians to abandon the hope of understanding of 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.
Notes
($*^1$) I ask this question HERE on SEMath because I'm looking for human experiences of real mathematicians, that is very important for me in this moment, and I do not think is fair that this question 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.
I must quote Andres Caicedo that was able to express way better than me, maybe, the meaning of my question:
"I think that there may be insights that only working mathematicians
could provide (as opposed to philosophers), and even if there are
wildly differing points of view, seeing them described may be useful.
[…] Of course, it may be that answering the question in detail,
considering as many of its subtleties as possible would just be too
long and unfeasible. That's fine; even providing a few references and
ideas that can potentially be fleshed out would be more useful that
simply dismissing it."
Improved version 05/14/'13:
($*^2$) Following the advice of Gustavo Bandeira I explain better the meaning of "satisfied from a philosophical point of view" in my fisrt question; What I mean is only linked to the mathematicians interested in the foundations, in other words who is interested in that process of generalization of all the mathematics to easier concepts ( like "$\in$" for set theorists or the structuralists for example)
Here the same question on Phylosphy.SE
Best Answer
I'm a student of mathematics at the third year undergraduate level.
Lets begin with what's unsatisfying. Example are when:
etc.
On the other hand, the existence of a huge mathematical universe is not unsatisfying. It's wondrous. I love the fact that there's always more to explore, always harder problems to face... in fact, at any level of difficulty, there's always progress to be made - its like a video game set in an enormously massive world that is just full of secrets, and it just stretches forever.
It can become pretty addictive. I'll leave you with the following quote.