mathematical structuralism the view that the subject of any branch of mathematics is a structure or structures. The slogan is that mathematics is the science of structure. Define a ‘natural number system’ to be a countably infinite collection of objects with one designated initial object and a successor relation that satisfies the principle of mathematical induction. Examples of natural number systems are the Arabic numerals and an infinite sequence of distinct moments of time. According to structuralism, arithmetic is about the form or structure common to natural number systems. Accordingly, a natural number is something like an office in an organization or a place in a pattern. Similarly, real analysis is about the real number structure, the form common to complete ordered fields. The philosophical issues concerning structuralism concern the nature of structures and their places. Since a structure is a one-over-many of sorts, it is something like a universal. Structuralists have defended analogues of some of the traditional positions on universals, such as realism and nominalism. See also MATHEMATICAL INDUCTION , PEANO POSTULATES , PHILOSOPHY OF MATHEMATIC. S.Sha.