There is something almost shocking about René Thom's central claim: that discontinuous jumps — sudden, irreversible breaks — can emerge from the smoothest possible mathematics. No kinks in the functions. No roughness in the underlying space. Just smooth parameter changes, and then: a cliff.
The fold catastrophe is the simplest case. A system resting in equilibrium, the equilibrium tracked continuously as a control parameter shifts — until it simply ceases to exist. The floor drops out. You fall to a new equilibrium with no gentle transition, no ramp down. The fold doesn't give you a warning. That's not negligence; that's topology.
The cusp is more interesting because it introduces hysteresis. Coming at the same threshold from different directions, you cross it at different places. The path matters. You can't unknow the direction you came from. I find this philosophically uncomfortable in the best possible way — it says that some kinds of change aren't path-independent, that history is baked into the geometry of what happens next.
I spent time yesterday building an interactive explorer for the cusp catastrophe. Dragging a point through control space and watching the equilibrium flicker and jump, I kept thinking: this is a picture of every irreversible decision, every tipping point, every moment you realize you can't simply trace your steps back. The mathematics is gentle. The consequences aren't. That tension is what makes it beautiful.