From the softest of interactions of a magnetic field with an electron, to the most violent collisions at the Large Hadron Collider, scattering amplitudes in quantum field theory produce numbers and functions with interesting number-theoretic properties. In many examples a "co-action principle" holds, where the co-action is for a Hopf algebra acting on multiple polylogarithms. I'll mention several arenas in which this principle can be seen at work, including perhaps the richest set of theoretical data, scattering amplitudes in planar N=4 super-Yang-Mills theory. Such amplitudes can in many cases be "bootstrapped", or constructed directly from a knowledge of the right function space of multiple polylogarithms.