Abstract: This book provides comprehensive and accessible coverage of the disciplines of philosophy of mathematics and philosophy of logic. After an introduction, the book begins with a historical section, consisting of a chapter on the modern period, Kant and his intellectual predecessors, a chapter on later empiricism, including Mill and logical positivism, and a chapter on Wittgenstein. The next section of the volume consists of seven chapters on the views that dominated the philosophy and foundations of mathematics in the early decades of the 20th century: logicism, formalism, and intuitionism. They approach their subjects from a variety of historical and philosophical perspectives. The next section of the volume deals with views that dominated in the later twentieth century and beyond: Quine and indispensability, naturalism, nominalism, and structuralism. The next chapter in the volume is a detailed and sympathetic treatment of a predicative approach to both the philosophy and the foundations of mathematics, which is followed by an extensive treatment of the application of mathematics to the sciences. The last six chapters focus on logical matters: two chapters are devoted to the central notion of logical consequence, one on model theory and the other on proof theory; two chapters deal with the so-called paradoxes of relevance, one pro and one contra; and the final two chapters concern second-order logic (or higher-order logic), again one pro and one contra.

Keywords: mathematics, logic, Immanuel Kant, empiricism, John Stuart Mill, logical positivism, Wittgenstein, logicism, formalism, intuitionism, Quine, indispensability, naturalism, nominalism, structuralism, predicativity, logical consequence, paradoxes of relevance, second-order logic, higher-order logic