Structuralism reconsidered

The properties and relations that perform a role in mathematical reasoning arise from the basic relations that obtain among mathematical objects. It is in terms of these basic relations that mathematicians identify the objects they intend to study. The way in which mathematicians identify these objects has led some philosophers to draw metaphysical conclusions about their nature. These philosophers have been led to claim that mathematical objects are positions in structures or akin to positions in patterns. This article retraces their route from (relatively uncontroversial) facts about the identification of mathematical objects to high metaphysical conclusions. Beginning with the natural numbers, how are they identified? The mathematically significant properties and relations of natural numbers arise from the successor function that orders them; the natural numbers are identified simply as the objects that answer to this basic function. But the relations (or functions) that are used to identify a class of mathematical objects may often be defined over what appear to be different kinds of objects.