The algebraic uniqueness of the addition of natural numbers

As is known, multiplication is aniteration of addition within the natural numbers. We show that addition is essentially theonly associative binary operation on natural numbers whose iteration is commutative or associative. This result, in which a set determines a structure, is in a sense a converse of the classical one due to Pontrjagin [3] and Kolmogorov [1], in which certain algebraic and topological structures determine the fields of real and complex numbers and of quaternions.