More Precisely
The Math You Need to Do Philosophy
  • Publication Date: January 29, 2009
  • ISBN: 9781551119090 / 1551119099
  • 208 pages; 6" x 9"

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More Precisely

The Math You Need to Do Philosophy

  • Publication Date: January 29, 2009
  • ISBN: 9781551119090 / 1551119099
  • 208 pages; 6" x 9"

More Precisely provides a rigorous and engaging introduction to the mathematics necessary to do philosophy. It is impossible to fully understand much of the most important work in contemporary philosophy without a basic grasp of set theory, functions, probability, modality and infinity. Until now, this knowledge was difficult to acquire. Professors had to provide custom handouts to their classes, while students struggled through math texts searching for insight. More Precisely fills this key gap.

Eric Steinhart provides lucid explanations of the basic mathematical concepts and sets out most commonly used notational conventions. Furthermore, he demonstrates how mathematics applies to many fundamental issues in branches of philosophy such as metaphysics, philosophy of language, epistemology, and ethics.


“[More Precisely] addresses a need by giving an elementary presentation of a number of technical concepts used in philosophy, which previously were not collected together. It should be especially useful for students preparing for graduate work whose undergraduate training is likely to have skipped over at least some of the concepts that the book covers. The material in each chapter is presented in a very clear and engaging way, without presupposing any background beyond basic high school mathematics.” — Susan Vineberg, Wayne State University

“This is the book I wish I had when I was learning philosophy. It does an excellent job introducing students to the formal tools that philosophers use in an accessible yet rigorous way. From set theory to possible words semantics to probability theory to the mathematics of infinity, all the key concepts are taught in a way that will provide students the foundation for being solid philosophers. Moreover, the concepts are taught in an engaging way, and frequent examples are given to explain why these formal notions are philosophically relevant. I’m going to ask all my incoming graduate students to read this book. And there’s enough material in here that most philosophy professors could learn something from the book as well.” — Bradley Monton, University of Colorado, Boulder

“This is a splendid and innovative book. It explains accurately and accessibly a wide range of technical topics frequently drawn upon in philosophy. Wonderfully informative and clear, this book is a lifeline for students at undergraduate or graduate level.” — Chris Daly, University of Manchester

“This is a great resource! Philosophers have always used the tools of mathematics to make their claims clearer and more precise, even more so since the end of the nineteenth century. Until now we haven’t had a systematic way to acquire those tools. Steinhart’s book remedies the situation, presenting the fundamental ideas thoroughly and comprehensively. Highly recommended for anyone getting into the serious study of philosophy.” — Anthony Dardis, Hofstra University


1. Sets

  1. Collections of Things
  2. Sets and Members
  3. Set Builder Notation
  4. Subsets
  5. Small Sets
  6. Unions of Sets
  7. Intersections of Sets
  8. Difference of Sets
  9. Set Algebra
  10. Sets of Sets
  11. Union of a Set of Sets
  12. Power Sets
  13. Sets and Selections
  14. Pure Sets
  15. Sets and Number
  16. Sums of Sets of Numbers
  17. Ordered Pairs
  18. Ordered Tuples
  19. Cartesian Products

2. Relations

  1. Relations
  2. Some Features of Relations
  3. Equivalence Relations and Classes
  4. Closures of Relations
  5. Recursive Definitions and Ancestrals
  6. Personal Persistence
    1. The Diachronic Sameness Relation
    2. The Memory Relation
    3. Symmetric then Transitive Closure
    4. The Fission Problem
    5. Transitive then Symmetric Closure
  7. Closure Under an Operation
  8. Closure Under Physical Relations
  9. Order Relations
  10. Degrees of Perfection
  11. Parts of Sets
  12. Functions
  13. Some Examples of Functions
  14. Isomorphisms
  15. Functions and Sums
  16. Sequences and Operations on Sequences
  17. Cardinality
  18. Sets and Classes

3. Machines

  1. Machines
  2. Finite State Machines
    1. Rules for Machines
    2. The Careers of Machines
    3. Utilities of States and Careers
  3. Networks of Machines
    1. Interacting Machines
    2. Machines in the Network
    3. Configurations in the Network
    4. The History of a Network of Machines
    5. Mechanical Metaphysics
  4. The Game of Life
    1. A Universe Made from Machines
    2. The Causal Law in the Game of Life
    3. Regularities in the Causal Flow
  5. Turing Machines

4. Semantics

  1. Extensional Semantics
    1. Words and Referents
    2. A Sample Vocabulary and Model
    3. Sentences and Truth-Conditions
  2. Simple Modal Semantics
    1. Possible Worlds
    2. A Sample Modal Structure
    3. Sentences and Truth at Possible Worlds
    4. Modalities
    5. Intensions
    6. Propositions
  3. Modal Semantics with Counterparts
    1. The Counterpart Relation
    2. A Sample Model for Counterpart Theoretic Semantics
    3. Truth-Conditions for Non-Modal Statements
    4. Truth-Conditions for Modal Statements

5. Probability

  1. Sample Spaces
  2. Simple Probability
  3. Combined Probabilities
  4. Probability Distributions
  5. Conditional Probabilities
    1. Restricting the Sample Space
    2. The Definition of Conditional Probability
    3. An Example Involving Marbles
    4. Independent Events
  6. Bayes Theorem
    1. The First Form of Bayes Theorem
    2. An Example Involving Medical Diagnosis
    3. The Second Form of Bayes Theorem
    4. An Example Involving Envelopes with Prizes
  7. Degrees of Belief
    1. Sets and Sentences
    2. Subjective Probability Functions
  8. Bayesian Confirmation Theory
    1. Confirmation and Disconfirmation
    2. Bayesian Conditionalization
  9. Knowledge and the Flow of Information

6. Utilitarianism

  1. Act Utilitarianism
    1. Agents and Actions
    2. Actions and their Consequences
    3. Utility and Moral Quality
  2. Expected Utility
  3. World Utilitarianism
    1. Agents and their Careers
    2. Careers and Compatible Worlds
    3. Runs and Prefixes
    4. Runs and the Worlds Open to Them
    5. The Utilities of Worlds
    6. Optimal Open Worlds
    7. Actions and their Moral Qualities

7. From the Finite to the Infinite

  1. Recursively Defined Series
  2. Limits of Recursively Defined Series
    1. Counting Through All the Numbers
    2. Cantor’s Three Number Generating Rules
    3. The Series of Von Neumann Numbers
  3. Some Examples of Series with Limits
    1. Achilles Runs on Zeno’s Racetrack
    2. The Royce Map
    3. The Hilbert Paper
    4. An Endless Series of Degrees of Perfection
  4. Infinity
    1. Infinity and Infinite Complexity
    2. The Hilbert Hotel
    3. Operations on Infinite Sequences
  5. Supertasks
    1. Reading the Borges Book
    2. The Thomson Lamp
    3. Zeus Performs a Super-Computation
    4. Accelerating Turing Machines

8. Bigger Infinities

  1. Some Transfinite Ordinal Numbers
  2. Comparing the Sizes of Sets
  3. Ordinal and Cardinal Numbers
  4. Cantor’s Diagonal Argument
  5. Cantor’s Power Set Argument
    1. Sketch of the Power Set Argument
    2. The Power Set Argument in Detail
    3. The Beth Numbers
  6. The Aleph Numbers
  7. Transfinite Recursion
    1. Rules for the Long Line
    2. The Sequence of Universes
    3. Degrees of Divine Perfection

Further Study
Glossary of Symbols

Eric Steinhart is an Associate Professor of Philosophy at William Paterson University. He is the author of On Nietzsche (Wadsworth 1999) and The Logic of Metaphor: Analogous Parts of Possible Worlds (Kluwer Academic 2001).

The instructor site includes exercises for chapters 1, 2, 7, and 8, as well as supplemental sections on relations, recursive definitions, and other topics. An access code to the website is included with all examination copies.