I have zillions of things that I don't know in mathematics. I feel I would never know any of them completely. Especially after this age (26)... and I immediately run away since I am a perfectionist. The field I would like to deeply specialize is the analytic number theory. I wonder whether there are things that you dont know or would never learn at all. How do you feel about yourself for these things? What would you advice for a coward man like me? I like mathematics so much but most of the questions here look like written in an alien language to me!
closed as primarily opinion-based by Antonio Vargas, Claude Leibovici, RecklessReckoner, Hawk, Asaf Karagila May 10 '14 at 6:47
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I've been riddled with the very same and more grave doubts for a long time. Unfortunately, SE has a history of closing such questions for being too subjective. I hope they don't close this one. Now, on to my answer.
"Coward". I love how you describe the situation, because I've been a first-rate coward all these 29 years of my life. Not only in mathematics, but in life as well. Will I make enough money? Will I turn out to be a genius? Will people call my bluff? Am I fit for higher studies? And so on.
Time and again I've tried to have a stable job and earn money, but failed. Because that was not my motivation. Also, time and again I tried to do math, but failed -- because there was just so much to do, and it seemed like I'd never "deserve" to explore more. For instance, I wanted to learn discrete mathematics, but it had some parts of calculus. For that, I needed of course to learn calculus, but also, trigonometry. And for that, plane geometry. Which is all fine, except that I let myself be waylaid by black and white thinking. I reasoned that I until I knew ALL about geometry, that until I was as perfect as Euclid himself, there was no point moving on to trigonometry. You can well imagine where that left me.
People used to tell me to "enjoy mathematics" and not think too much, but I could never manage my mind well. It seemed too overwhelming. I'm not going to say "and then, something changed overnight". Something did change, but it took time. It was a slow realization, but I came to the conclusion that something was wrong with my mind and went to see a psychiatrist. Turned out I was a victim of Dysthymia, which made me lose interest and hope. Since the medication has started (which is rather mild, thankfully), I've had an almost miraculous effect on my mind. I've been able to minimize my paranoia and focus on mathematics a lot more, and life is fun.
But that's not the whole of the solution. I also realized that there was tremendous detail in mathematics, which can be very dry if you're not driven by something. Some people (like Russell) were fired up with the idea of finding paradoxes in set theory, so they kept working tirelessly at it. Others, like Ramanujan, seemed to produce math out of thin air. I'm neither of these types, so what do I do?
I've found that narrowing my focus helps. First, I needed to identify what I enjoyed the most in math, even if it seems to have no job prospects. For me, it has to be fractals, complex variables and stuff, because of the excellent visualizations involved. Then I'd like to do 3D math and simulating curves and surfaces in R or Octave. For that I'll need a powerful laptop, which I'll upgrade in the days to come. My path now is clear: enough of geometry, calculus (and maybe vectors) and reach fractals as quickly as possible. I can sidestep and go deeper as needed, which will be more scary, but will at least keep me moving and motivated. So I'd say: find out which area you like the most, and pursue it from books that are very well written.
Finally, consider what the others are saying: nobody can know all there is to know, and nobody can be wrong all the time. Even the likes of Euler were wrong about several of their assumptions, and their towering intellects were the result of years of toiling. But we're not competing; we have to focus on enjoyment.
Once you have got a hold on the area you admire in math, you will become confident and start tackling other areas that appeared too difficult or pointless at first. But please, whatever you do, don't be in a hurry. Make sure internet, phone, Facebook, TV, etc., is kept to a minimum. You can't internalize mathematics by allocating so many hours in a week. It has to be total immersion.
If I, at 29, can begin again with fresh energy, so can you!
Hope there was some help.
P.S.: Remember, always focus on enjoyment!