Philosophy Of Mathematics An Anthology Pdf

More Precisely: The Math You Need to Do Philosophy – Second Edition

More Precisely: The Math You Need to Do Philosophy – Second Edition

More Precisely is a rigorous and engaging introduction to the mathematics necessary to do philosophy. Eric Steinhart provides lucid explanations of many basic mathematical concepts and sets out the most commonly used notational conventions. He also demonstrates how mathematics applies to fundamental issues in various branches of philosophy, including metaphysics, philosophy of language, epistemology, and ethics. This second edition adds a substantial section on decision and game theory, as well as a chapter on information theory and the efficient coding of information.

Comments

"More Precisely is a very useful tool for philosophers with a limited formal background who nevertheless aim to leave a lasting mark on the discipline." — Nathan Salmon, Distinguished Professor of Philosophy, University of California, Santa Barbara

"Back in the day, philosophy graduate students might have been expected to complete a course in metalogic. It would be far better to complete a course that covers the topics presented here. This is excellent coverage of a wide range of mathematical ideas and techniques that are at the heart of many contemporary issues in metaphysics and epistemology." — F. Thomas Burke, University of South Carolina

COMMENTS ON FIRST EDITION

"[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

Preface

1. SETS

Collections of Things
Sets and Members
Set Builder Notation
Subsets
Small Sets
Unions of Sets
Intersections of Sets
Difference of Sets
Set Algebra
Sets of Sets
Union of a Set of Sets
Power Sets
Sets and Selections
Pure Sets
Sets and Numbers
Sums of Sets of Numbers
Ordered Pairs
Ordered Tuples
Cartesian Products

2. RELATIONS

Relations
Some Features of Relations
Equivalence Relations and Classes
Closures of Relations
Recursive Definitions and Ancestrals
Personal Persistence

6.1 The Diachronic Sameness Relation
6.2 The Memory Relation
6.3 Symmetric then Transitive Closure
6.4 The Fission Problem
6.5 Transitive then Symmetric Closure

Closure under an Operation
Closure under Physical Relations
Order Relations
Degrees of Perfection
Parts of Sets
Functions
Some Examples of Functions
Isomorphisms
Functions and Sums
Sequences and Operations on Sequences
Cardinality
Sets and Classes

3. MACHINES

Machines
Finite State Machines
- 2.1 Rules for Machines
  2.2 The Careers of Machines
  2.3 Utilities of States and Careers
The Game of Life
- 3.1 A Universe Made from Machines
  3.2 The Causal Law in the Game of Life
  3.3 Regularities in the Causal Flow
  3.4 Constructing the Game of Life from Pure Sets
Turing Machines
Lifelike Worlds

4. SEMANTICS

Extensional Semantics
- 1.1 Words and Referents
  1.2 A Sample Vocabulary and Model
  1.3 Sentences and Truth-Conditions
Simple Modal Semantics
- 2.1 Possible Worlds
  2.2 A Sample Modal Structure
  2.3 Sentences and Truth at Possible Worlds
  2.4 Modalities
  2.5 Intensions
  2.6 Propositions
Modal Semantics with Counterparts
- 3.1 The Counterpart Relation
  3.2 A Sample Model for Counterpart Theoretic Semantics
  3.3 Truth-Conditions for Non-Modal Statements
  3.4 Truth-Conditions for Modal Statements

5. PROBABILITY

Sample Spaces
Simple Probability
Combined Probabilities
Probability Distributions
Conditional Probabilities
- 5.1 Restricting the Sample Space
  5.2 The Definition of Conditional Probability
  5.3 An Example Involving Marbles
  5.4 Independent Events
6. Bayes Theorem
- 6.1 The First Form of Bayes Theorem
  6.2 An Example Involving Medical Diagnosis
  6.3 The Second Form of Bayes Theorem
  6.4 An Example Involving Envelopes with Prizes
7. Degrees of Belief
- 7.1 Sets and Sentences
  7.2 Subjective Probability Functions
8. Bayesian Confirmation Theory
- 8.1 Confirmation and Disconfirmation
  8.2 Bayesian Conditionalization
9. Knowledge and the Flow of Information

6. INFORMATION THEORY

Communication
Exponents and Logarithms
The Probabilities of Messages
Efficient Codes for Communication
- 4.1 A Method for Making Binary Codes
  4.2 The Weight Moving across a Bridge
  4.3 The Information Flowing through a Channel
  4.4 Messages with Variable Probabilities
  4.5 Compression
  4.6 Compression Using Huffman Codes
Entropy
- 5.1 Probability and the Flow of Information
  5.2 Shannon Entropy
  5.3 Entropy in Aesthetics
  5.4 Joint Probability
  5.5 Joint Entropy
Mutual Information
- 6.1 From Joint Entropy to Mutual Information
  6.2 From Joint to Conditional Probabilities
  6.3 Conditional Entropy
  6.4 From Conditional Entropy to Mutual Information
  6.5 An Illustration of Entropies and Codes
Information and Mentality
- 7.1 Mutual Information and Mental Representation
  7.2 Integrated Information Theory and Consciousness

7. DECISIONS AND GAMES

Act Utilitarianism
- 1.1 Agents and Actions
  1.2 Actions and Their Consequences
  1.3 Utility and Moral Quality
From Decisions to Games
- 2.1 Expected Utility
  2.2 Game Theory
  2.3 Static Games
Multi-Player Games
- 3.1 The Prisoner's Dilemma
  3.2 Philosophical Issues in the Prisoner's Dilemma
  3.3 Dominant Strategies
  3.4 The Stag Hunt
  3.5 Nash Equilibria
The Evolution of Cooperation
- 4.1 The Iterated Prisoner's Dilemma
  4.2 The Spatialized Iterated Prisoner's Dilemma
  4.3 Public Goods Games
  4.4 Games and Cooperation

8. FROM THE FINITE TO THE INFINITE

Recursively Defined Series
Limits of Recursively Defined Series
- 2.1 Counting through All the Numbers
  2.2 Cantor's Three Number Generating Rules
  2.3 The Series of Von Neumann Numbers
Some Examples of Series with Limits
- 3.1 Achilles Runs on Zeno's Racetrack
  3.2 The Royce Map
  3.3 The Hilbert Paper
  3.4 An Endless Series of Degrees of Perfection
Infinity
- 4.1 Infinity and Infinite Complexity
  4.2 The Hilbert Hotel
  4.3 Operations on Infinite Sequences
Supertasks
- 5.1 Reading the Borges Book
  5.2 The Thomson Lamp
  5.3 Zeus Performs a Super-Computation
  5.4 Accelerating Turing Machines

9. BIGGER INFINITIES

Some Transfinite Ordinal Numbers
Comparing the Sizes of Sets
Ordinal and Cardinal Numbers
Cantor's Diagonal Argument
Cantor's Power Set Argument
- 5.1 Sketch of the Power Set Argument
  5.2 The Power Set Argument in Detail
  5.3 The Beth Numbers
The Aleph Numbers
Transfinite Recursion
- 7.1 Rules for the Long Line
  7.2 The Sequence of Universes
  7.3 Degrees of Divine Perfection

Further Study

Glossary of Symbols

References

Index

Eric Steinhart is a professor of philosophy at William Paterson University. He is the author of Your Digital Afterlives: Computational Theories of Life after Death (Palgrave Macmillan, 2014), On Nietzsche (Wadsworth, 1999), and The Logic of Metaphor: Analogous Parts of Possible Worlds (Kluwer Academic, 2001).

—A primer on the mathematical and formal aspects of philosophy, including set theory, relations, functions, probability, and more
—Suitable as a course text or as supplemental reading for students who wish to fill gaps in their technical knowledge
—This edition adds chapters on decision theory and information theory
—Diagrams and tables are used to illustrate and organize technical ideas
—Supplemental exercises and solutions are provided on a companion website

This book comes with access to a companion website that hosts exercises, answer keys, and supplements.

For a sample chapter of More Precisely, Second Edition, click here. (opens as a PDF).