This question has been on my mind for a very long time, and I thought I’d finally ask it here.
When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more quantitive background. He wanted me to have a different life and taught me math accordingly. When I was 11, I taught myself trig. When I was 12, I started taking calculus at my local university. I continued on this track, and finally got to real analysis and abstract algebra at 15. I loved every math course I ever took and found myself breezing through all that was presented to me (the university was not Princeton after all). However, around this time, I came to the conclusion that math was not for me. I decided to try a different path.
Why, you might ask, did I do this? The answer was simple: I didn’t believe I could be a great mathematician. While I thrived taking the courses, I never turned math into a lifestyle. I didn’t come home and do complex questions on a white board. I didn’t read about Euler in my spare time. I also never felt I had a great intuition into problems. Once you showed me how to solve a problem, I was golden. But start from scratch on my own? It seemed like a different story entirely. To make things worse, my sister, who was at Caltech at the time, would call home with stories of all these incredible undergrads who solved the mathematical mysteries of the universe as a hobby. Whenever I mentioned math as a career, she would always issue a strong warning: you’re not like these kids who spend all their time doing math. Think about doing something else.
Over time, I came to agree with this statement. Coincidentally, I got rejected by MIT and Princeton to continue my undergraduate studies there. This crushed me at the time; my dream of studying math at one of the great institutions had ended. Instead, I ended up at Georgia Tech (not terrible by any means, just not what I had envisioned). Being at an engineering school, I thought I’d give aerospace a shot. It had lots of math, right? Not really, or at least not enough for my taste. I went into CS. This was much better, but still didn’t feel quite right. At last, as a sophomore, I felt it was time to get back on track: I’m now doubling majoring in applied math and CS.
My question is, how do I know I’m not making a mistake? There seems to be so many people doing math competitions, research, independent studies, etc, while I just started to take some math courses again. What should I do to test myself and see if I can really make math a career? I apologize for the long and possibly quite subjective post. I’d just really like to hear from math people who know their stuff. Thanks a bunch in advance.
There is a lot of hype surrounding “math competitions”, since “winning” and “competition” can easily be understood in broad cultural terms, whether or not we collectively think of these as either highest goals or as legitimate formative principles. Most professional mathematics and its practice does not resemble competition-math at all, specifically, serious projects have substantial background requirements, may take months or years to complete or partially complete, and have meaning beyond “getting the result before anyone else” (although some people’ve been so indoctrinated in math-as-game that they never recover, and find no other meaning in it).
Also, there is much hype about “the other people” being incredible geniuses, while few sane people will look at themselves and see “a genius”. 🙂 But this is mostly gossip or mythology, in part generated to create a superficial excitement where otherwise there’d be mostly hard work without pop-style-glamour. 🙂 And, of course, there is a common style of “bluffing” and/or never admitting weakness or ignorance, but this is mostly a facade, whose maintenance is most enthusiastic among those most worried about the “game” aspect of mathematics or anything else. Such stuff should not be unquestioningly believed.
To make a living as a mathematician, and to make reasonable contributions, it is not necessary to be a larger-than-life romantic-heroic figure. 🙂 Possibly many of us, occasionally, wish that we were such a figure, but that more comic-book or video-game reality than any human-lifetime reality.
More important than hype is the concrete reality of how one spends one’s days: if you like thinking about mathematical things, perhaps teaching mathematical things, then being a mathematician is a happy job, with or without rich-and-famousness. The purported glamor of “being a great [whatever]” is not a reliable thing to aim toward, since the “process” of practicing the [whatever] is how one will spend one’s days. It is a reasonable analogy, I think, to say that “great musicians” occur by accident among the class of people who really, really enjoy practicing and “jamming” (in whatever genres), rather than people who wish for celebrity but hate practicing. Of course there’re the people who pose as never practicing, but reality belies that cute P.R. pose…