Have you ever wondered about the foundations of mathematics? What are the original axioms upon which all of current mathematics relies upon? Before reading this book, I thought the answer was established and agreed upon. It turns out however there are several philosophical schools of thought, some of which reject what the majority of mathematicians might accept as proven mathematics. In Pluralism in Mathematics: A New Position in Philosophy of Mathematics, Michéle Friend does a wonderful job of presenting several schools of philosophical thought on the foundation of mathematics. Her conclusion is that pluralism, which is a philosophical position that tolerates all mathematical results accepting them as true unless they are later proven false, is the most reasonable foundation of mathematics. This is in contrast to other positions which reject some of mathematics that are proven using certain techniques, or do not have an analogue in set theory. Other philosophers will reject mathematics that has not been shown to have applications to science; this, of course, rejects a large amount of current mathematics research. I believe that the majority of the mathematics community would agree with the pluralism philosophy, although many might have not known other options existed.
This book is a persuasive argument to consider pluralism as a philosophical foundation of mathematics written primarily for philosophers of mathematics. The author spends the majority of the book contrasting various foundations and arguing in defense of the pluralist philosophy. There is perhaps a secondary audience: interested mathematicians like me who think that philosophical questions like these can be used in the classroom to motivate students to think more deeply about mathematics. Often through discussions on ideas about the foundation of mathematics, students find their own answer to the common question “Why do I need to know this?” Sometimes, it is through determining why we use mathematics as a model to describe our world. Other times, students prefer something more abstract: a system of symbols, formulas and definitions. What is mathematics: a set of objects to be discovered or invented by humans to describe some relationship? The book provides the first fully articulated defense for the pluralist philosophy in mathematics which has been around for some time. I found this book to be an interesting read, and found that I definitely agree most with the pluralist philosophy.
Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle IL. Her training is in computational group theory. More recently she has become interested in ways to introduce undergraduate students to research in abstract algebra through applications.
Part I. Motivating the Pluralist Position from Familiar Positions
Chapter 1. Introduction. The Journey from Realism to Pluralism
Chapter 2. Motivating Pluralism. Starting from Maddy’s Naturalism
Chapter 3. From Structuralism to Pluralism
Chapter 4. Formalism and Pluralism, co-written with Andrea Pedeferri
Part II. Initial Presentation of Pluralism
Chapter 5. Philosophical Presentation of Pluralism
Chapter 6. Using a Formal Theory of Logic Metaphorically
Chapter 7. Rigour in Proof, co-written with Andrea Pedeferri
Chapter 8. Mathematical Fixtures
Part III. Transcendental Presentation of Pluralism
Chapter 9. The Paradoxes of Tolerance and the Transcendental Paradoxes
Chapter 10. Pluralism Towards Pluralism
Part IV. Putting Pluralism to Work. Applications
Chapter 11. A Pluralist Approach to Proof in Mathematics
Chapter 12. Pluralism and Together-Inconsistent Philosophies of Mathematics
Chapter 13. Suggestions for Further Pluralist Enquiry