One of the remarkable things about learning math is that steps forward in understanding require a kind of forgetting. Everything always looks simple in retrospect; it’s letting go of your biases that prevent you from learning that is difficult. There are great examples of this in the book Ender’s Game: the main character goes into zero gravity for the first time and realizes that he can let go of his conception of “down.” Simple, but difficult.
I’ve often wondered about why this is so often the case. After all, learning seems constructive in nature, but forgetting and letting go of bias is deconstructive, and seems like it’s the opposite process. Today I had another thought on the topic though. I’ve long suspected that much learning involves a refinement of vision and categorization. Toddlers sometimes go through phases of calling all animals “doggies,” for example, because their categories aren’t sufficiently refined. (I heard one story of a child passing a field and point to the “moo doggies” there.) But this means that we learn through differentiating more deeply between things. In other words, we forget that they seemed alike. They’re actually different.
Of course, so much of math is about seeing similarities between things that look totally dissimilar. In effect, both skills are crucial: we need to disconnect ideas from each other (forget our biases) in order to connect them up to each other in new ways. Learning seems to involve the constructive and deconstructive in equal measure.
Meanwhile, I’m getting pretty hopeful about my thesis. I think I may actually be on the verge of calculating the number of Fourier-Mukai partners of certain K3 surfaces. Just have to go over (and over) the details.