This page contains a list of the four best books on the philosophy of mathematics. Finding good introductory philosophy books can be difficult for two reasons. First, searching google for recommendations usually doesn’t bring up anything useful. Second, phrases like “best books on the philosophy of mathematics” are ambiguous. One person may be looking for a short, beginner friendly introduction, someone else may want a comprehensive academic overview, a third person may be looking for classic works on the philosophy of mathematics. This list tries to account for this ambiguity by recommending different types of books on the philosophy of mathematics. This list was created by compiling recommendations from across the internet and is not a list of personal recommendations. More information about this process is at the end of the post. Here are the best books on the philosophy of mathematics in no particular order:

#### Mathematics: A Very Short Introduction – Timothy Gowers

Category: Short Introduction | Length: 160 pages | Published: 2002

*Publisher description:* The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as “Is it true that mathematicians burn out at the age of 25?”) It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.

#### Philosophy of Mathematics – Øystein Linnebo

Category: General Introduction | Length: 216 pages | Published: 2017

*Publisher description: *A sophisticated, original introduction to the philosophy of mathematics from one of its leading contemporary scholars.

Mathematics is one of humanity’s most successful yet puzzling endeavors. It is a model of precision and objectivity, but appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic yet accessible introduction to the field that is trying to answer that question: the philosophy of mathematics.

Written by Øystein Linnebo, one of the world’s leading scholars on the subject, the book introduces all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. It also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The groundbreaking work of German mathematician and philosopher Gottlob Frege, one of the founders of analytic philosophy, figures prominently throughout the book. Other important thinkers whose work is introduced and discussed include Immanuel Kant, John Stuart Mill, David Hilbert, Kurt Gödel, W. V. Quine, Paul Benacerraf, and Hartry H. Field.

Sophisticated but clear and approachable, this is an essential introduction for all students and teachers of philosophy, as well as mathematicians and others who want to understand the foundations of mathematics.

#### The Oxford Handbook of Philosophy of Mathematics – Stewart Shapiro

Category: Overview | Length: 832 pages | Published: 2007

*Publisher description:* Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas.

This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters–one supportive and one critical.

*The Oxford Handbook of Philosophy of Math and Logic* is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

#### An Historical Introduction to the Philosophy of Mathematics: A Reader – Russell Marcus & Mark McEvoy

Category: Anthology | Length: 848 pages | Published: 2016

*Publisher description: *A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject.

*An Historical Introduction to the Philosophy of Mathematics: A Reader* brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject.

Presenting historical background essential to understanding contemporary trends and a survey of recent work, *An Historical Introduction to the Philosophy of Mathematics: A Reader* is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers.

This list was created by following a method that I’ve found to be useful when searching for introductory philosophy books. It involves:

- browsing required reading lists on university course syllabi
- searching for books using the Open Syllabus Project
- browsing the bibliographies of articles on the Stanford Encyclopedia of Philosophy and Internet Encyclopedia of Philosophy
- searching for recommendations on philosophy forums

The following sources were used to build this list:

**University Course Syllabi:**

- https://philosophy.stanford.edu/sites/default/files/courses/syllabus/phil162-262.pdf
- http://jwood.faculty.unlv.edu/unlv/Phil436XF12syl.htm
- https://www.ucalgary.ca/rzach/567#text

**Bibliographies:**

**Other Recommendations:**

- https://www.reddit.com/r/askphilosophy/comments/1u8ka3/recommended_reading_on_philosophy_of_maths/
- https://www.reddit.com/r/askphilosophy/comments/7vaa0i/good_intro_books_on_phil_of_math/

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