Typicality Reasoning in Probability, Physics, and Metaphysics

✍ Scribed by Dustin Lazarovici

• Publisher: Palgrave Macmillan
• Year: 2023
• Tongue: English
• Leaves: 380
• Series: New Directions in the Philosophy of Science
• Category: Library

✦ Synopsis

This book provides a comprehensive investigation into the concept of typicality and its significance for physics and the philosophy of science. It identifies typicality as a fundamental way of reasoning, central to how natural laws explain and are tested against phenomena. The book discusses various applications of typicality to foundational questions in physics and beyond.These include:

a unified interpretation of objective probabilities in classical mechanics and quantum mechanics

• a detailed discussion of Boltzmann's statistical mechanics, entropy, and the second law of thermodynamics
• a novel account of the asymmetry of causation and the arrow of time

Finally, the book turns to the question: "What are laws of nature"? It argues that typicality extends to a powerful way of reasoning in metaphysics that can and should inform our commitments about the fundamental ontology of the world. On this basis, it develops an argument against the Humean best system account, according to which laws of nature are merely an efficient summary of contingent regularities.

✦ Table of Contents

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'' andBig'' 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