Realism in Mathematics

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics.

When engaged in mathematics, most people tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Penelope Maddy delineates and defends a novel version of mathematical realism that answers the traditional questions and refocuses philosophical attention on the pressing foundational issues of contemporary mathematics.

Realism: Pre-theoretic realism; Realism in philosophy; Realism and truth; Realism in mathematics; Perception and intuition: What is the question?; Perception; Intuition; Godelian Platonism; Numbers: What numbers could not be; Numbers as properties; Frege numbers; Axioms: Reals and sets of reals; Axiomization; Open problems; Competing theories; The challenge; Monism and beyond: Monism; Field's nominalism; Structuralism; Summary; References; Index.