Musings on Embodied Mathematics

Toronto is under an Orange Warning from Environment Canada. Torontonians don’t always see beauty in the storm; the Montréalais in me still celebrates it. Through the eyes of my youth, a storm is not an inconvenience but a choreography—snow improvising in wind, the city briefly returning to texture and silence. So, I took this picture from my balcony as a break from writing.  It feels like the right moment to think again about Kepler’s  Strena seu de nive sexangula . Mordern textbooks teach that a snowflake is a hexagonal crystal lattice: H2O molecules bond in ways that privilege six directions, spaced 60 degrees apart, and the visible sixfold symmetry follows (six times 60 gives the full 360 degree circle). But on a day like today, what I want to notice is not only the number six. Kepler’s gift to teachers is the way he conducts  his inquiry, how he moves from lived experience to explanation without treating the senses as a childish prelude to “real” knowledge-m...

In this realm, mathematics has always happened in bodies. We count on fingers, trace curves, “hold” an angle open with our hands, and talk our way through an idea using rhythm, emphasis, and pauses. What Susan Gerofsky's paper " Mathematical learning and gesture:  Character viewpoint and observer viewpoint  in students’ gestured graphs of functions",  adds to the conversation is that gestures can be diagnostic of how a learner is relating to a piece of mathematics. In her work on graphing, students’ bodies sometimes stay “ou tside” the representation (as if observing the graph), and sometimes move “from within” it (as if riding the graph). Those are not just performance styles; they can mark different forms of engagements and imaginations. Wittgenstein helps us name what’s at stake: mathematics education is a set of language games, rule-governed ways of using words, symbols, diagrams, and bodies in particular settings. Meaning is not a ghost behind the sign; it is ...

There is a debate about whether mathematics is created or discovered. The issues uncovered by this debate are the prolegomena (προλεγόμενα)—the “pre-sayings” we need before we can speak seriously about embodied mathematics. If mathematics is mainly discovered, then it looks like a mind-independent structure we gradually uncover. If it is created, then it looks like a human practice: a way of stabilizing patterns through perception, action, representation, and agreement. Either way, the debate forces a key question for embodied mathematics: even if mathematical structure is “out there,” how does it become thinkable for creatures like us? Popular culture makes the contrast vivid. In the movie Arrival (2016), is about first contact with aliens. For the depths of inter-gallactic space, twelve massive oblong shaped ships effortlessly hover in locations distributed around the globe. Occasionally they emit undistinct sounds. The central challenge is not a universal constant but a problem of s...