Imagine I took a small handful of pennies and showed them to you clumped together, then spread the same collection out into a wider array; you know, without having to think about it, that the pennies are worth the same amount either way. It’s the same group of pennies, after all. To children, this isn’t obvious. The psychologist Piaget noticed that many of the the “obvious” things we know are not obvious to children until they encounter them in certain ways at certain ages. Moreover, there seems to be something irreducible about the leap in brain development. One day, a child can’t see that the pennies are worth the same amount, the next day it’s obvious, and he can’t imagine that he ever didn’t know.
As we grow up, we have less and less of these Piagetian type leaps. Learning becomes something more incremental. If we want to learn a piece of music, we practice it, and watch ourselves know it better, in tiny increments. The practice leads to our better understanding something.
But there are still Piagetian leaps that take place, I would argue, and they’re quite thrilling when they happen. I remember taking real analysis in college: essentially most of the course is centered on a techincal point of drawing a series of increasingly smaller circles around points in order to define a very precise notion of what “continuity” means. I remember it being baffling at the time. But by the time I was taking the following course, it felt so obvious that it was almost like second nature. I couldn’t imagine not understanding the principle.
I’ve discussed before how the nature of these Piagetian leaps forward make math hard to teach: the teacher puts the problem forward and feels like it must be easy to see, because he can’t imagine being unable to see it; the student can’t see how to begin thinking about it, because in some way, you can’t see the breakthough until, magically, you do. What’s astonishing, though, is how this continues as long as you’re studying math. I’m looking at incredibly abstract notions, abstractions piling on abstractions. There are moments when the whole idea escapes me completely. There is also the almost constant feeling like I’ll never be able to see what it is I want to see, or prove what it is I want to prove. It’s impossible to imagine knowing (and knowing naturally) what it is I don’t know now.
This is where faith is required in mathematics. I have to keep throwing myself at what I don’t understand. When I don’t understand what I don’t understand, I have to keep searching: it’s like hunting around an obstacle course for the right brick wall to throw yourself against. But even though I can’t see progress at times, I have to assume that something is happening inside my head. I’ve repeated the process of total confusion yielding, suddenly, to total understanding so many times that I have to believe that it will keep happening.
Keeping this kind of faith is challenging sometimes. Not knowing is such a vast feeling. Even more, the creative insight is so slippery that many mathematicians, I think, have an anxiety in them akin to writers: do I have another good idea in me?
Back to work, back to work… I will continue to believe that something is building inside me, though I cannot see it.