In this talk we propose a definition of commutativity for Boolean subalgebras. We will concentrate on two cases: atomic Boolean algebras whose theory can be expressed in terms of commuting equivalence relations, and Boolean sigma-algebras which is the stochastic analogue of the first case. In each case, we deal with a pair of commuting Boolean subalgebras, and characterize such a pair by a structure theorem. We study the properties of lattices of commuting Boolean algebras, and develop a proof theory for such lattices.