Mathematics without Numbers

This book develops a comprehensive alternative to standard Platonism as an approach to fundamental issues in the philosophy of mathematics. It works out a detailed interpretation of mathematics as the investigation of structural possibilities rather than referring to absolute, Platonic objects. After dealing with natural numbers and analysis, the approach is extended to set theory, where it is shown how to dispense with a fixed universe of sets. The final portion of the book is devoted to the topic of applied mathematics, and the problems involved in applying the work to the physical world. - ;Introduction; Chapter 1: The natural numbers and analysis; Introduction; The modal-structural framework: The hypothetical component; The categorical component: An axiom of infinity and a derivation (inspired by Dedekind with help from Frege); Justifying the translation scheme; Justification from within; Extensions; The question of nominalism; Chapter 2: Set theory; Introduction; Informal principles: Many vs. one; The relevant structures; Unbounded sentences: Putnam semantics; Axioms of infinity: Looking back; Axioms of infinity: Climbing up; Appendix; Chapter 3: Mathematics and physical reality; Introduction; The leading ideas; Carrying the mathematics of modern physics: RA2 as a framework; Global solutions; Metaphysical realist commitments? `Synthetic Determination' relations; A role for representation theorems -