Abstract

This book intends to show that, in philosophy of mathematics, radical naturalism (or physicalism), nominalism and strict finitism (which does not assume the reality of infinity in any format, not even potential infinity) can account for the applications of classical mathematics in current scientific theories about the finite physical world above the Planck scale. For that purpose, the book develops some significant applied mathematics in strict finitism, which is essentially quantifier-free elementary recursive arithmetic (with real numbers encoded as elementary recursive Cauchy sequences of rational numbers). Applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the basic applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitisitc system is perhaps surprising in itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. The first chapter of the book contains an informal introduction to its philosophical motivation and technical strategy. The book is intended for students and researchers in philosophy of mathematics. Contents Preface 1 Introduction 2 Strict Finitism 3 Calculus 4 Metric Space 5 Complex Analysis 6 Integration 7 Hilbert Space 8 Semi-Riemann Geometry References Index