Mathematical Conversations

In the age of AI, is prediction only a matter of finding a better algorithm? Is chaos the last word on unpredictability? Chaos makes prediction sensitive to errors in the initial conditions, but imagine Laplace’s demon, who knows the present exactly. Starting with Turing’s proof of the undecidability of the halting problem, I will explain why some questions about motion admit no general algorithmic answer: physical systems can compute. We will follow this idea through three systems: water, a billiard ball, and the planets.

First, water. Using contact and cosymplectic geometry, I will explain how to construct Turing-complete solutions of the Euler and Navier–Stokes equations on suitable Riemannian manifolds. The velocity field is stationary, yet the motion of fluid particles can simulate any computer.

Then, a billiard ball. A single ball on a carefully shaped, fixed planar table can carry out any computation. For this table, there is no general algorithm to decide whether the trajectory starting from a given computable position and direction is periodic.

Finally, the planets. Can gravity compute? If it can, what is the smallest number of gravitating bodies capable of simulating any Turing machine? Call this number the Turing scale. Could three suffice? The question is open: it brings Turing into the world of Poincaré, and sends Laplace’s demon back to the heavens it was invented to predict.