Butterflies From Beyond Newton’s World

This geometric image, a static reflection of an irreversible and fleeting time, brings to mind another one: Kepler’s ellipse. Thom’s elementary catastrophes, as well as Kepler’s ellipses, attempt to reduce time to space and to understand it through geometry. Whereas Kepler uses the mathematical tools inherited from the Greeks, Thom has the benefit of modern differential topology. Kepler uses Apollonius’s Treatise on Conics. Thom uses singularity theory. However, whereas Kepler’s model leads us into Newton’s world, which is closed upon itself, catastrophe theory is a glance into an open universe. In Newton’s world there is no past and no future, since everything is determined by today’s data. Time holds no surprise in store for whoever can handle the computations. In Thom’s world the future is mostly hidden, and the mathematician inspects the flow of events for forms to recognize and classify, like a butterfly catcher.

Ivar Ekeland
“Mathematics And The Unexpected”