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Table of Contents
3.6.1 Inexact Solutions and Mathematical Freedom
3.6.2 Formal Calculus of Variations
3.7 Philosophical Opportunities
3.7.1 Beautiful Mathematics
3.7.2 Responsible Fictionalism
3.7.3 Calculation
3.7.4 Ontology of Mathematics
3.8 Appendix: Symbolic Vector Calculus
3.9 Appendix: The Laplacian
3.10 Appendix: Some Weak Derivative Calculations
3.10.1 Going Farther Into delta left parenthesis x right parenthesis(x)
References
4 Fishbones, Wheels, Eyes, and Butterflies: Heuristic Structural Reasoning in the Search for Solutions to the Navier-Stokes Equations
Intro
Preface
Contents
Contributors
1 Introduction
References
2 How Mathematics Figures Differently in Exact Solutions, Simulations, and Physical Models
2.1 Introduction
2.2 Exact Solutions of (Differential) Equations
2.3 Simulations
2.3.1 Simulations of (Differential) Equations of Mathematical Physics
2.3.2 Agent-Based Simulations
2.4 Role of Mathematics in Experimental Physical Models
2.4.1 Physical Analogies in the Early Twentieth Century
2.4.2 The Method of Physically Similar Systems
2.5 Conclusion
References
3 Fluid Mechanics for Philosophers, or Which Solutions Do You Want for Navier-Stokes?
3.1 How the Equation Says What it Says
3.1.1 How a Fluid Moves: the Material Derivative
3.1.2 The Millennium Prize version of Navier-Stokes
3.1.3 Invariance
3.2 The Question of Solutions
3.2.1 Contrast the Question of Numbers
3.2.2 What are Solutions to PDEs?
3.3 Weak Derivatives
3.3.1 The Problem of Multiplication
3.4 Weak Solutions to Differential Equations
3.4.1 Distributions
3.5 The Weak Navier-Stokes Equation
3.6 Relations to Physics and Engineering
4.1 The Argument from Successful Theories
4.2 Are Theories Based on Differential Equations Successful?
4.3 Working to Formulate and Solve an Equation
4.3.1 Finding Exact Solutions to the Navier-Stokes Equations
4.4 Heuristic Reasoning: What's In an Equation?
References
5 Black Hole Coalescence: Observation and Model Validation
5.1 Introduction
5.2 Modelling and Observing Binary Black Hole Mergers
5.2.1 Template-Based Searches for Gravitational Waves
5.2.2 Bayesian Parameter Estimation
5.2.3 Modelling Black Hole Coalescence
5.2.4 Model-Dependent Observation
5.3 Model Validation and Circularity
5.3.1 Worry 1: Theory-Laden Observation of Gravitational Waves
5.3.2 Worry 2: Theory-Laden Observation of Binary Black Hole Mergers
5.3.3 Modelling Bias
5.3.4 Theoretical Bias
5.4 Model Validation with the LIGO-Virgo Observations
5.5 Conclusion
References
3.6.2 Formal Calculus of Variations
3.7 Philosophical Opportunities
3.7.1 Beautiful Mathematics
3.7.2 Responsible Fictionalism
3.7.3 Calculation
3.7.4 Ontology of Mathematics
3.8 Appendix: Symbolic Vector Calculus
3.9 Appendix: The Laplacian
3.10 Appendix: Some Weak Derivative Calculations
3.10.1 Going Farther Into delta left parenthesis x right parenthesis(x)
References
4 Fishbones, Wheels, Eyes, and Butterflies: Heuristic Structural Reasoning in the Search for Solutions to the Navier-Stokes Equations
Intro
Preface
Contents
Contributors
1 Introduction
References
2 How Mathematics Figures Differently in Exact Solutions, Simulations, and Physical Models
2.1 Introduction
2.2 Exact Solutions of (Differential) Equations
2.3 Simulations
2.3.1 Simulations of (Differential) Equations of Mathematical Physics
2.3.2 Agent-Based Simulations
2.4 Role of Mathematics in Experimental Physical Models
2.4.1 Physical Analogies in the Early Twentieth Century
2.4.2 The Method of Physically Similar Systems
2.5 Conclusion
References
3 Fluid Mechanics for Philosophers, or Which Solutions Do You Want for Navier-Stokes?
3.1 How the Equation Says What it Says
3.1.1 How a Fluid Moves: the Material Derivative
3.1.2 The Millennium Prize version of Navier-Stokes
3.1.3 Invariance
3.2 The Question of Solutions
3.2.1 Contrast the Question of Numbers
3.2.2 What are Solutions to PDEs?
3.3 Weak Derivatives
3.3.1 The Problem of Multiplication
3.4 Weak Solutions to Differential Equations
3.4.1 Distributions
3.5 The Weak Navier-Stokes Equation
3.6 Relations to Physics and Engineering
4.1 The Argument from Successful Theories
4.2 Are Theories Based on Differential Equations Successful?
4.3 Working to Formulate and Solve an Equation
4.3.1 Finding Exact Solutions to the Navier-Stokes Equations
4.4 Heuristic Reasoning: What's In an Equation?
References
5 Black Hole Coalescence: Observation and Model Validation
5.1 Introduction
5.2 Modelling and Observing Binary Black Hole Mergers
5.2.1 Template-Based Searches for Gravitational Waves
5.2.2 Bayesian Parameter Estimation
5.2.3 Modelling Black Hole Coalescence
5.2.4 Model-Dependent Observation
5.3 Model Validation and Circularity
5.3.1 Worry 1: Theory-Laden Observation of Gravitational Waves
5.3.2 Worry 2: Theory-Laden Observation of Binary Black Hole Mergers
5.3.3 Modelling Bias
5.3.4 Theoretical Bias
5.4 Model Validation with the LIGO-Virgo Observations
5.5 Conclusion
References