As many of you know, I am currently on pre-tenure leave. I spent the Fall in Paris doing research at Jussieu (thanks in large part to the AWM mentoring travel grant), and I’m currently stationed at the University of Texas at Austin, where I am continuing my Paris project and working on a couple of others. It has definitely been a different experience for me to work exclusively on research and to have no teaching responsibilities. Even in graduate school, I taught a class or graded every single semester. I am definitely loving the chance to dedicate myself to doing research, but this is inevitably accompanied by a not-so-fun feeling of inadequacy and ignorance. I have been thinking about this feeling a lot lately, and I thought I would share some of my thoughts on this post (even though this will not be new to many of you).
In the Fall, for example, I spent about a month trying to prove something that had essentially already been proven by someone else. The proof was not written down or published anywhere, but the mathematician who had proved it had given me a sketch of how he had done it. I had proved some of the steps, but was stuck on the last one. One day, I decided to try something different (not using his sketch). It took me that one day to finish it. Now, if I had been a bit more confident in my skills, maybe I could have saved that month’s work. On the other hand, maybe it was thinking about this problem for so long that made the last minute change so simple. Maybe I just needed to get used to these ideas before I could figure it out. So sometimes I do believe the struggle, although bad for one’s self-esteem in the short term, really pays off. In fact, I believe that in large part our success as mathematicians has more to do with our ability to keep working on a problem even after failing numerous times than with our “intelligence”.
I once told a friend that working on research always makes me feel stupid, and she laughed at me. She, a humanist, couldn’t understand how I, a mathematician, could possibly think I was bad at math. Obviously, I’m not “bad at math” in the traditional sense (although I don’t believe anyone is, but that is part of a different story). But the thing is, the more math you know, the more difficult interesting problems are. As a matter of fact, I think that the feeling of inadequacy is always there, because you’re trying to solve a difficult problem. The difference is that what is difficult to me is not the same as what is difficult to a second grade math student, but I can still understand how the second-grader feels. I think Einstein perfectly captured this idea when he said: “Do not worry about your difficulties in mathematics. I can assure you, mine are still greater.” So even Einstein, one of the most popular representatives of “genius”, felt inadequate doing mathematics.
Of course, like I said earlier, being good at mathematics has more to do with perseverance than genius (at least, in my opinion). This actually leads me to a not-at-all-famous quote by a friend’s Ph.D. advisor: “To be good at math research, it’s more important to really love it and to work really hard than to be smart. You really love math and work really hard.” I don’t think she realized the backhandedness of her compliment, and truly meant it as a regular compliment (my friend, however, keeps bringing this up as both the nicest and meanest thing anyone has ever said to him). But I believe this to be true, and in fact this is the only thing that keeps me working hard at math, even though many times I feel like an idiot. First of all, I’m not really an idiot (although I don’t think I would call anyone that… well, OK, if they made me really angry maybe). But most importantly, my success does not depend so much on my intelligence (or any other innate feature that is hard to change), but on my ability to keep working on and learning about the problem until I find a solution (and sometimes even after… many times I’ve found that my proof is actually wrong or needs some fixing).
So today, instead of struggling with the computation I’ve been trying to do for the past few weeks, I have decided to write about how I feel when I struggle with a computation. I guess it’s not the most productive use of my frustration, but I do feel a little better after writing this down. I guess now I should get back to it.
So, dear readers, how do you feel about your difficulties with mathematics (assuming you have any)?