Mathematical Structuralism is a Kind of Platonism

Abstract

The most important feature of mathematical structuralism is the primary role that patterns have in the interpretations of what mathematics is about. Mathematical structuralists say that the essence of natural number is its relations to other natural numbers. The subject matter of arithmetic is a single abstract structure - w, pattern common to any infinite collection of objects that has a successor relation, a unique initial object, and satisfies the induction principle. This pattern is exemplified by a collection of concrete objects - this is a concrete w sequence. The pattern itself is what has all these particular w sequences in common. What is common is just the abstract w-sequence itself and this is just a first step to the problem, which is from history of philosophy well known as »The Third Man«. Abstract structures-w sequences - are the same items as are in Plato's system ideas (forms). Therefore, we used some hints made by Vlastos about nonidentity (NI) and selfpredication (SP) in the case of analysis of mathematical structuralism. The result of this procedure was devastating for ante rem realism (mathematical structuralism) and we suggested that perhaps in res realism would be a solution for them.