A program of proofs is outlined to show how the linear structure of quantum mechanics can be derived from basic principles. For the dynamics part, three alternatives are reviewed briefly. They include the Kadison theorem for evolution of density matrices and the Segal–Simon theorem for evolution of observables in the Heisenberg picture. The focus is on Wigner’s theorem that gives linear operators for evolution of state vectors. New insights are described that motivate the key assumption about probabilities in that theorem. One is that it is implied by the second law of thermodynamics. Another is that it follows from assumptions that evolution in time preserves mixtures of states and preserves probabilities for what is in the mixture.