The best books on contemporary mathematics for the philosophically inclined

Rocco Gangle Author Of Diagrammatic Immanence: Category Theory and Philosophy
By Rocco Gangle

Who am I?

Philosophy’s core questions have always obsessed me: What is real? What makes life worth living? Can knowledge be made secure? In graduate school at the University of Virginia I was drawn to mathematically formalized approaches to such questions, especially those of C. S. Peirce and Alain Badiou. More recently, alongside colleagues at Endicott College’s Center for Diagrammatic and Computational Philosophy and GCAS College Dublin I have explored applications of diagrammatic logic, category theory, game theory, and homotopy type theory to such problems as abductive inference and artificial intelligence. Philosophers committed to the perennial questions have much to gain today from studying the new methods and results of contemporary mathematics.

I wrote...

Diagrammatic Immanence: Category Theory and Philosophy

By Rocco Gangle,

Book cover of Diagrammatic Immanence: Category Theory and Philosophy

What is my book about?

This book integrates insights from Spinozist metaphysics, Peircean semiotics, and Deleuze’s philosophy of difference using diagrammatic methods in conjunction with the formal operations of category theory. It offers a basic introduction to the methods of category theory allowing philosophers with little or no mathematical training to come to grips with this important field and provides a constructive and rigorous philosophy of diagrammatic reasoning under conditions of immanence. This coordination of immanent metaphysics, diagrammatic methods, and category theoretical mathematics opens a new horizon for contemporary thought.

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The books I picked & why

Synthetic Philosophy of Contemporary Mathematics

By Fernando Zalamea, Zachery Luke Fraser (translator),

Book cover of Synthetic Philosophy of Contemporary Mathematics

Why did I love this book?

Zalamea’s book is the perfect introduction and survey if you want to understand how developments in contemporary mathematics are relevant to current philosophy.

Zalamea’s own approach follows closely in the steps of Peirce, Lautman, and Grothendieck, merging pragmatism, dialectics, and sheaf theory, but he also engages the work of dozens of other key mathematicians and philosophers coming from different points of view, sometimes cursorily, always tantalizingly.

No philosopher can read this book without a quickened heartbeat and eager plans to clear shelf space for some of the many volumes of mathematics and philosophy of mathematics canvassed here by Zalamea.

By Fernando Zalamea, Zachery Luke Fraser (translator),

Why should I read it?

1 author picked Synthetic Philosophy of Contemporary Mathematics as one of their favorite books, and they share why you should read it.

What is this book about?

A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest.

A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest, this book gives the inquisitive non-specialist an insight into the conceptual transformations and intellectual orientations of modern and contemporary mathematics.

The predominant analytic approach, with its focus on the formal, the elementary and the foundational, has effectively divorced philosophy from the real practice of mathematics and the profound conceptual shifts in the discipline over the last century. The first part discusses the…

Book cover of The Joy of Abstraction: An Exploration of Math, Category Theory, and Life

Why did I love this book?

From a strictly philosophical perspective, the emergence of category theory as a unifying paradigm rivaling set theory is probably the most important development in mathematics in the last half-century.

But for philosophers without a lot of mathematical background, learning even its rudiments can be daunting. Among many introductory texts (Lawvere and Schanuel, Awodey, Riehl, Spivak), Cheng’s book stands out as perhaps the friendliest and most accessible.

She does not forego rigor, but she isn’t afraid to put aside precise formalism when necessary for intuition and clearer understanding. Her book takes the reader from mathematical beginnings through category theory’s core constructions to glimpses of higher-order categories (one of Cheng’s areas of expertise).

A mathematically novice philosopher who wants to understand the basics of category theory couldn’t do better.

By Eugenia Cheng,

Why should I read it?

1 author picked The Joy of Abstraction as one of their favorite books, and they share why you should read it.

What is this book about?

Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life - from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material,…

Philosophy and Model Theory

By Tim Button, Sean Walsh,

Book cover of Philosophy and Model Theory

Why did I love this book?

Far too many math books are written in a style so terse and ungenerous that all but the most mathematically gifted readers hardly have a fair chance of understanding.

On the other hand, the discursive style of much philosophy of mathematics gains readability at the expense of formal rigor. Button and Walsh strike the perfect balance in this exceptionally rich introduction to model theory from a distinctively philosophical perspective.

There’s no getting around the fact that the mathematics of model theory is hard going. But this book works through all the relevant proofs in clear and detailed terms (no lazy “we leave this as an exercise for the reader”), and the authors are always careful to motivate each section with well-chosen philosophical concerns right up front.

An Everest, but worth it.

By Tim Button, Sean Walsh,

Why should I read it?

1 author picked Philosophy and Model Theory as one of their favorite books, and they share why you should read it.

What is this book about?

Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics.
But these wide-ranging uses of model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of…

Book cover of Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy

Why did I love this book?

The Univalent Foundations program in foundations of mathematics launched by Voevodsky and others in the past decade and a half has contributed to a promising new paradigm unifying computation, mathematics, logic, and proof theory.

Understanding the core elements of this research program, Homotopy Type Theory, is essential for contemporary philosophers who want to engage directly with current developments in mathematics and computer science.

Corfield is a well-established name in philosophy of mathematics, and this book is the best introduction to Homotopy Type Theory for philosophers.

Working within themes and problematics that will be familiar to philosophers with a basic background in logic, Corfield covers the elementary constructions of homotopy types from a logical point of view and provides plenty of provocative suggestions for how these formal tools might reinvigorate philosophical research today.

By David Corfield,

Why should I read it?

1 author picked Modal Homotopy Type Theory as one of their favorite books, and they share why you should read it.

What is this book about?

"The old logic put thought in fetters, while the new logic gives it wings."

For the past century, philosophers working in the tradition of Bertrand Russell - who promised to revolutionise philosophy by introducing the 'new logic' of Frege and Peano - have employed predicate logic as their formal language of choice. In this book, Dr David Corfield presents a comparable revolution with a newly emerging logic - modal homotopy type theory.

Homotopy type theory has recently been developed as a new foundational language for mathematics, with a strong philosophical pedigree. Modal Homotopy Type Theory: The Prospect of a New…

Book cover of Quantum Computing Since Democritus

Why did I love this book?

Sometimes a book is not just brilliant and thought-provoking but just plain fun.

Scott Aaronson’s Quantum Computing Since Democritus provides a broad overview of philosophical issues arising from computational theory, cryptography, quantum mechanics, and quantum computing. Its style is crisp, clear, and conversational.

It’s the kind of book that can be read a first time briskly for the sheer excitement of it and then a second time with attention to all the details for consistent Eureka! moments of philosophical insight. Fascinating material and an excellent presentation.

By Scott Aaronson,

Why should I read it?

2 authors picked Quantum Computing Since Democritus as one of their favorite books, and they share why you should read it.

What is this book about?

Written by noted quantum computing theorist Scott Aaronson, this book takes readers on a tour through some of the deepest ideas of maths, computer science and physics. Full of insights, arguments and philosophical perspectives, the book covers an amazing array of topics. Beginning in antiquity with Democritus, it progresses through logic and set theory, computability and complexity theory, quantum computing, cryptography, the information content of quantum states and the interpretation of quantum mechanics. There are also extended discussions about time travel, Newcomb's Paradox, the anthropic principle and the views of Roger Penrose. Aaronson's informal style makes this fascinating book accessible…

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