How do Michael Resnik and Stewart Shapiro explain mathematical structures?

In the philosophy of mathematics, Michael Resnik and Stewart Shapiro developed a nontraditional version of mathematical Platonism known as structuralism.12

Their central thesis is that the primary objects of mathematical study are structures or patterns—such as geometric spaces or set-theoretic hierarchies—rather than individual entities. Under this view, what we typically call "mathematical objects," such as the number 4, are not objects in the ordinary sense. Instead, they are merely positions within a structure.12

Resnik and Shapiro illustrate this using the analogy of a baseball defense. A "shortstop" is not an object with inherent physical properties like height or weight; it is a role or position within a larger pattern that can be filled by different players. Similarly, the number 4 is simply the fourth position in the positive integer pattern. Consequently, numbers lack internal properties; their only characteristics are structural, defined entirely by the relations they bear to other positions within the same system.12