Typicality Reasoning in Probability, Physics, and Metaphysics

Acknowledgments
Contents
List of Figures
List of Tables
1 Introduction
	1.1 Typicality
	1.2 Typicality Explanations
	1.3 The Boltzmannian Framework
	1.4 Brute Facts
	References
Part I Probability
	2 Typicality in Probability Theory
		2.1 Expectation Value and Typical Values
		2.2 Law of Large Numbers
			2.2.1 The N Law
			2.2.2 The Central Limit Theorem
		2.3 Subjective Probabilities and Propensities
			2.3.1 Subjective Probabilities
			2.3.2 Stochastic Laws
		References
	3 Cournot's Principle
		3.1 Formulations of Cournot's Principle
		3.2 On the Rationality of Cournot's Principle
			3.2.1 Moral Certainty
			3.2.2 Does Nature Have to Obey Cournot's Principle?
			3.2.3 Black Swans and Pascal's Wager
			3.2.4 The Lottery Paradox and Rational Belief
				CP and the Stability Theory of Belief
		References
	4 A Typicality Theory of Probability
		4.1 The Coin Toss
			4.1.1 Normal Numbers as a Model for Coin Tossing
				Law of Large Numbers for Rademacher Functions
				Biased Coins
		4.2 Typical Frequencies
		4.3 Probabilities for Singular Events
			4.3.1 Rational Credences from Statistics
		References
	5 The Mentaculus: Typicality Versus Humean Chances
		5.1 Typicality Versus Humean Chances
			5.1.1 A True Regularity Theory of Chance
				Principal Principle Versus Cournot's Principle
				Probability Versus Typicality Measures
		5.2 Epistemology and Metaphysics of Typicality Measures
		5.3 Justification of Typicality Measures
			5.3.1 Stationarity, Uniformity, Symmetry
				A Geometric View of Stationarity
				Invariance Under Symmetries
			5.3.2 How to Choose a Typicality Measure
		5.4 Typicality for Humeans
		References
	6 The Structure of Typicality
		6.1 A Theory of ``Small'' and ``Big'' Sets
		6.2 Criteria for Typicality
		6.3 Typicality Measures
			6.3.1 Equivalence of Typicality Measures
				Absolute Continuity
				Total Variation and Typicality Thresholds
				A Bound from Densities
		References
Part II Physics
	7 From the Universe to Subsystems
		7.1 The Hamiltonian Picture
		7.2 Probabilities in Classical Mechanics
			7.2.1 Ideal Gas: The Maxwell Distribution
			7.2.2 The Coin Toss Again
		7.3 Deterministic Subsystems
			7.3.1 The Stone Throw
		References
	8 Boltzmann's Statistical Mechanics
		8.1 The Second Law of Thermodynamics
			8.1.1 The Typicality Account
			8.1.2 Macroscopic Irreversibility
				Past Hypothesis and the Thermodynamic Arrow
			8.1.3 The Role of the Typicality Measure
			8.1.4 On the Boltzmann Entropy
		8.2 The Status of Macroscopic Laws
			8.2.1 Derivation of Typicality Laws
		8.3 Boltzmann vs. Gibbs
			8.3.1 Empirical Equivalence of Equilibrium Values
			8.3.2 Derivation of the Equilibrium Ensembles
		References
	9 It's Complicated: The Relationship between Physics and Mathematics
		9.1 The Pernicious Influence of Ergodic Theory
		9.2 Proof and Explanation
		References
	10 Boltzmann Equation and the H-Theorem
		10.1 Kinetic Equations
			10.1.1 Molecular Chaos
		10.2 The H-Theorem as a Typicality Result
			10.2.1 The Stoßzahlansatz
			10.2.2 Irreversibility of the Boltzmann Equation
		References
	11 Past Hypothesis and the Arrow of Time
		11.1 The Easy and the Hard Problem of Irreversibility
			11.1.1 The Status of the Past Hypothesis
		11.2 Thermodynamic Arrow Without a Past Hypothesis
			11.2.1 Past Hypothesis and Self-Location
				Predictions and Retrodictions
				The Mystery of Our Low-Entropy Universe
		11.3 Entropy of a Classical Gravitating System
			11.3.1 Typical Evolutions of a Gravitating System
		11.4 Gravity and Typicality from a Relational Point of View
			11.4.1 Shape Complexity and a Gravitational Arrow
			11.4.2 Entropy as an Absolutist Concept
				Conclusion: Can We Dispense with the Past Hypothesis?
		References
	12 Causality and the Arrow of Time
		12.1 Causal Explanations as Typicality Explanations
		12.2 Causal and Epistemic Asymmetry
			12.2.1 Asymmetry of Records
			12.2.2 Asymmetry of Influences
		References
	13 Quantum Mechanics
		13.1 The Measurement Problem
			13.1.1 Connecting the Wave Function to the World
		13.2 Born's Rule and the Measurement Process
			13.2.1 Typicality and Observation
		13.3 Observable Operators as Statistical Book-Keepers
		13.4 Quantum Equilibrium: Probabilities in Bohmian Mechanics
			13.4.1 Bohmian Mechanics
				The Typicality Measure
				Effective Wave Functions for Subsystems
				Quantum Equilibrium
			13.4.2 Absolute Uncertainty
				Why Determinism?
			13.4.3 Thermodynamic Arrow in Bohmian Mechanics
		13.5 Born's Rule in the Many-Worlds Theory
			13.5.1 Probabilities of What?
			13.5.2 Everett's Typicality Argument
			13.5.3 Living and Dying in the Multiverse
		References
Part III Beyond Physics
	14 Other Applications of Typicality
		14.1 Typicality and Well-Posedness
		14.2 Typicality and Fine-Tuning
			14.2.1 The Flatness Problem
			14.2.2 Fine-Tuning of the Natural Constants
		14.3 Typicality in Mathematics
		References
	15 Special Science Laws
		15.1 Ontology of Special Sciences
			15.1.1 Probability and Causation in Special Sciences
		15.2 Special Science Laws as Typicality Laws
			15.2.1 The Hierarchy of Sciences
		15.3 Is Life Atypical?
		References
	16 Typicality and the Metaphysics of Laws
		16.1 What Are the Laws of Nature?
		16.2 Typicality in Metaphysics
			16.2.1 Ontological Possibility
			16.2.2 Typicality and the Case Against Humeanism
		16.3 Typical Humean Worlds Have No Laws
			16.3.1 The Chaitin Model
			16.3.2 From the Toy Model to the Real World
				Finite Systematizations
				Indeterministic Laws
		16.4 On the Uniformity of Nature
		References
A Time-Reversal Invariance
	Newtonian Mechanics
		Electrodynamics
		Quantum Mechanics
		Bohmian Mechanics
		On the Role of Metaphysics
B Proof of Theorems
	B.1 Computation of the Gravitational Entropy
	B.2 Atypicality of Lawfulness Among Possible Humean Worlds
References
Index