Welcome to the first installment of my series on Eric Steinhart’s More Precisely: The Math You Need to Do Philosophy! This will be a series covering in depth the content of Steinhart’s book. Think of this series like an extended crash course. I will systematically proceed through the book’s main sections, explicating and expounding upon the essential concepts and examples contained therein.
If you plumb through the annals of philosophy journals, it won’t be long before you stumble across abstruse and technical concepts like denumerability, proper parthood, infinite supertasks, and Bayes’ Theorem. Such mathematically formalized notions play essential roles in countless philosophical debates — causal and temporal finitism, existential inertia, God’s existence, the nature of the mind, the realism/antirealism debate in the philosophy of science, and so on. It stands to reason, then, that a solid understanding of the mathematical concepts underlying such debates is of paramount significance.
Steinhart’s book does precisely that: as the author himself describes his project, “More Precisely is a mathematics book for philosophers.” It introduces the formal tools used in philosophy and illustrates the applications of those tools with examples from both classical and contemporary philosophy.
The book starts with set theory, as almost all branches of philosophy employ it in some form or other. It then turns in the second chapter to relations and functions. Next up is an introduction to machines — a topic with critical relevance to philosophy of mind and computer science. Such machines aren’t industrial devices but are instead formal structures used to describe a lawful or regular pattern of activity.
Steinhart then turns in chapter four to the mathematics implicated in the philosophy of language. In particular, he explores formal semantic theories and possible worlds semantics (which itself has immense bearing on metaphysics, causation, and modality). Chapters five and six cover probability theory (like Bayesian epistemology and confirmation theory) and information theory, respectively. Steinhart next discusses the application of mathematics to ethics and ethical frameworks like utilitarianism. In particular, he focuses on decision theory and game theory. Finally, Steinhart last considers the wonderful world of infinity, tackling questions like: is the mind finitely or infinitely complex? Can infinitely many tasks be done in finite time? What does it mean to say that God is infinite? In mathematical terms, Steinhart explains recursion, countable infinity, infinite cardinalities, and transfinite recursion.
Steinhart illustrates the applications of the mathematics contained in the book with philosophical examples. Importantly, he doesn’t take a stance on the soundness of the positions involved in such examples; rather, what he is solely aiming to show that an understanding of math is required as a precondition for the analysis and evaluation of such philosophical positions and debates. Steinhart is engaged in an expositional — not evaluative — conceptual narrative.
“Our hope,” writes Steinhart, “is that learning the mathematics we present in More Precisely will help you to do philosophy. You’ll be better equipped to read technical philosophical articles. Articles and ideas that once might have seemed far too formal will become easy to understand. And you’ll be able to apply these concepts in your own philosophical thinking and writing.”
Here are some resources pertaining to the book if you would like to follow along with my blog series:
- This is the Table of Contents for the Second Edition of More Precisely.
- This site contains supplemental exercises to the book, available in three formats: with no answers provided, with some of the answers provided, and with all of the answers provided.
- These are the chapter abstracts for the book.
- You can follow the link below to download supplementary materials and exercises associated with the chapters of More Precisely.
- Finally, Guarded Acumen Archives was kind enough to provide readers of this blog with an online version. It can be found on the website below.