Is Mathematics Unnatural?
This October I had the great pleasure of meeting Fields medalist Cedric Villani. Professor Villani gave an illuminating and accessible talk about his innovative work in the study of curvature, and afterwards spent some time hanging out and chatting with a few of the attendees.
Villani is a charismatic and engaging speaker, and he provided a lot of to think about in his talk. One remark that particularly struck me was
“Mathematics, in some sense, will always involve a little pain.”
The idea resonated with me but I was curious what he meant, so I asked him about it. I was a bit surprised when he said that mathematics is unnatural, and unnatural things are always painful.
I pressed him a bit, as I didn’t quite understand. “What are the first things you learn in physics?” he asked. He was alluding to Newton’s Laws, and in particular the law of inertia: An object at rest tends to stay at rest, and an object in motion tends to stay in motion. Villani grabbed a fork from across the table, slammed it down in front of him, and gave it a push. The fork slid a short distance and stopped. “This is absurd!” he said. “It does not stay in motion!”
Physics, that is, the laws of physics, are abstractions of our experiences with the real world. Understanding that when you push something, it will stop, is natural for us; understanding the law of inertia is not. This law is an abstraction of our natural experiences, and as such, is unnatural. He went on to argue that mathematics, too, is a collection of abstractions from our experiences of the real world, and therefore is unnatural.
He made an analogy with speaking and reading: speaking is natural for humans, we are hard-wired for it. But writing is not. It does not come naturally to us. As an abstraction of speaking, writing will always be difficult for humans to learn. It will always involve a little pain. Like mathematics.
Some world-class mathematics, a little philosophy, and a mathematical autograph! All in all, a pretty good evening.