Abstract: A continuation of the study of mathematical existence begun in Ontology and the Vicious-Circle Principle (published in 1973); in the present work, Quine's indispensability argument is rebutted by the development of a new nominalistic version of mathematics (the Constructibility Theory) that is specified as an axiomatized theory formalized in a many-sorted first-order language. What is new in the present work is its abandonment of the predicative restrictions of the earlier work and its much greater attention to the applications of mathematics in science and everyday life. The book also contains detailed discussions of rival views (Mathematical Structuralism, Field's Instrumentalism, Burgess's Moderate Realism, Maddy's Set Theoretical Realism, and Kitcher's Ideal Agent account of mathematics), in which many comparisons with the Constructibility Theory are made.

Keywords: John Burgess, Chihara, Constructibility Theory, Hartry Field, Ideal Agent, indispensability argument, Instrumentalism, Philip Kitcher, Penelope Maddy, mathematical existence, Mathematical Structuralism, Moderate Realism, nominalism, philosophy of mathematics, Quine, Set Theoretical Realism, Stewart Shapiro