Inner Simplicity vs. Outer Simplicity

Abstract

For me, mathematics is just about understanding. And understanding is a personal and private feeling. However, to appreciate and express this feeling, you need to communicate with others—you need to use language. So there are necessarily two aspects in mathematics: one is very personal, emotional, and internal, and the other is more public and external. Today I want to express this very naïve idea for mathematicians that we should distinguish between two kinds of simplicities. Something could be very simple for me, in my mind, and in my way of knowing mathematics, and yet be very difficult to articulate or write down in a mathematical paper. And conversely, something can be very easy to write down or say in just one sentence of English or French or whatever and nevertheless be all but completely inaccessible to my mind. This basic distinction is something that I believe to be classical, but, nevertheless, we mathematicians conflate the two. We keep forgetting that writing mathematics is not the same as understanding mathematics.