Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The 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. Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He 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. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.
A Structural Account of Mathematics will be required reading for anyone working in this field.
"The book has a clear structure and is very well written. Apart from the themes described here, the book contains a wealth of material. Throughout the book many positions relevant to the debate between realists and anti-realists are discussed. It is impossible in a short space to do justice to all
the points made in this book. Suffice it to say that it certainly makes a substantial contribution to the debate on realism in mathematics and is a 'must read' for anyone interested in this debate."--International Studies in the Philosophy of Science
1: Five Puzzles in Search of an Explanation
2: Geometry and Mathematical Existence
3: The Van Inwagen Puzzle
6: Minimal Anti-Nominalism
7: The Constructibility Theory
8: Constructible Structures
11: Field's Account of Mathematics and Metalogic
Appendix A: Some Doubts about Hellman's Views
Appendix B: Balaguer's Fictionalism