This work develops and defends a structural view of the nature of mathematics, which is used to explain a number of striking features of mathematics that have puzzled philosophers for centuries. It rejects the most widely held philosophical view of mathematics (Platonism), according to which mathematics is a science dealing with mathematical objects such as sets and numbers—objects which are believed not to exist in the physical world. Instead, it makes use of the constructibility theory of my earlier work, Constructibility and Mathematical Existence (Oxford University Press, 1990), to develop a view of mathematics that is distinct from Structuralism and yet makes use of some key ideas of Structuralism.
The structural view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. My previous work had a different aim: its goal was to present and develop a new system of mathematics that did not make reference to, or presuppose, mathematical objects. Both works support a nominalistic point of view. However, whereas the earlier book was aimed at creating a new nominalistic system of mathematics, the present work analyzes mathematical systems currently used by scientists to show how such systems are compatible with a nominalistic outlook. The present work also advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam.
I also endeavor, in this book, to present a rationale for the nominalistic outlook that is quite different from those generally put forward by nominalists. I do this, to a great extent, because I believe that serious misunderstandings of the nominalistic outlook have been fostered by the type of rationale for nominalism that is typically discussed in the recent philosophical literature.
A number of criticisms that have been leveled at my constructibility theory by the Structuralists. Since these criticisms have for many years been largely unanswered, they may appear to students and non-specialists to be unanswerable. In this work, the criticisms will be rebutted.