Skip to content

Is Mathematics Unnatural?

villani notebookThis 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.


  1. dusanmal says:

    Evidence to the contrary: what are the earliest known human expressions about understanding the World? – highly abstract and simplified petroglyphs. Yes, humans see the real World and real animals and objects but as first step of understanding we use our power of simplification to reduce those real world complex sets of things to some basic representations. Petroglyph stick figures are how human mind works naturally and are as natural as law of inertia or Math’. Math’ is how human mind works.
    As for “hard” part… that’s life. Life is naturally hard. Look at the leopard after he chased antelope – at the edge of life hard.

    • MrHonner says:

      I’m not sure I’d consider petroglyphs to be abstractions, or even the result of abstraction. And in any case, I wouldn’t consider them comparable to the abstractions of experience to theories. Petroglyphs are symbols; the abstractions of science are logical systems.

  2. Grant says:

    I think this is a cornerstone of any attempt to teach for understanding. Almost every modern idea that matters is counter-intuitive. A plane that goes on forever; only 1 force slowing an object down and bringing it back to earth once it leaves the hand and is thrown up; the theory of natural selection; correlation vs. causality; chance in a world of meaning, Divide one little fraction by another and get a large number, etc.

    That’s why we ask teachers in UbD to identify and design mindful of the predictable misconceptions.

    • MrHonner says:

      Almost every modern idea that matters is counter-intuitive.

      As is often the case, you’ve given me something to ponder, Grant. Thanks! I’ll get back to you when I have a defensible sequence of counter-examples. :-)

      And yes, excellent point about how this relates to instructional design and the need to plan for, if not confront at the outset, student misconceptions.

Leave a Reply

− 2 = six