Object, Structure, and Form

Abstract

The main task of this paper is to develop the non-Platonist view of mathematics as a science of structures I have called, borrowing the label from Putnam, `realism with the human face'. According to this view, if by `object' we mean what exists independently of whether we are thinking about it or not, mathematics is a science of patterns (structures), where patterns are neither objects nor properties of objects, but aspects (or aspects of aspects, etc.) of concrete objects which dawn on us when we represent objects (or aspects of... within a given system (of representation). Mathematical patterns, therefore, are real, because they ultimately depend on concrete objects, but are neither objects nor properties of objects, because they are dependent, both metaphysically and epistemically, on systems of representation. Although the article has been written as a presentation of my view of mathematics, and of some of its advantages, the reader should keep in mind that this is essentially a `reply paper', as is shown by the fact that much of it is dedicated to the discussion of some issues which have become the focus of critical attention. Such issues are well expressed by the following questions: am I right in asserting that mathematical patterns are neither objects nor properties of objects? What is the difference, if any, between mathematical patterns and other mind-dependent entities such as the Cleveland Symphony Orchestra? Can mathematical patterns be always assimilated to relations? Can what I call `form of representation' be assimilated to structure? Can the standpoint I take on mathematics, which regards it as a science of patterns, be correctly described as Aristotelian?