Contents -- Acknowledgments -- Introduction -- 1. Introduction -- 1.1 Why Would I Have Valued This Book in High School? -- 1.2 Who Else Would Value This Book? -- 1.3 Physics & Biology -- 1.4 Motivation -- 1.5 The Principle of Least Thought . -- 1.6 Measurement -- 1.7 Conceptual Blending -- 1.8 Mental Model of Muscle Contraction -- 1.9 Organization -- 1.10 What is Missing? -- 1.11 What is Original? -- Mathematics -- 2. Ground & Foundation of Mathematics -- 2.1 Introduction -- 2.2 Ground: Discourse & Surface -- 2.2.1 Symbol & Expression -- 2.2.2 Substitution & Rearrangement -- 2.2.3 Diagrams Rule by Diagram Rules -- 2.2.4 Dot & Arrow -- 2.3 Foundation: Category & Functor -- 2.3.1 Category -- 2.3.2 Functor -- 2.3.3 Isomorphism -- 2.4 Examples of Categories & Functors -- 2.4.1 Finite Set -- 2.4.2 Set -- 2.4.3 Exponentiation of Sets -- 2.4.4 Pointed Set -- 2.4.5 Directed Graph -- 2.4.6 Dynamic System -- 2.4.7 Initialized Dynamic System -- 2.4.8 Magma -- 2.4.9 Semigroup -- 2.4.10 Monoid -- 2.4.11 Group -- 2.4.12 Commutative Group -- 2.4.13 Ring -- 2.4.14 Field -- 2.4.15 Vector Space over a Field -- 2.4.16 Ordered Field -- 2.4.17 Topology -- 2.5 Constructions -- 2.5.1 Magma Constructed from a Set -- 2.5.2 Category Constructed from a Directed Graph -- 2.5.3 Category Constructed from a Topological Space -- 3. Calculus as an Algebra of Infinitesimals -- 3.1 Real & Hyperreal -- 3.2 Variable -- 3.2.1 Computer Program Variable -- 3.2.2 Mathematical Variable -- 3.2.3 Physical Variable -- 3.3 Right, Left & Two-Sided Limit -- 3.4 Continuity -- 3.5 Differentiable, Derivative & Differential -- 3.5.1 Partial Derivative -- 3.6 Curve Sketching Reminder -- 3.7 Integrability -- 3.8 Algebraic Rules for Calculus -- 3.8.1 Fundamental Rule -- 3.8.2 Constant Rule -- 3.8.3 Addition Rule -- 3.8.4 Product Rule -- 3.8.5 Scalar Product Rule -- 3.8.6 Chain Rule.

3.8.7 Exponential Rule -- 3.8.8 Change-of-Variable Rule -- 3.8.9 Increment Rule -- 3.8.10 Quotient Rule -- 3.8.11 Intermediate Value Rule -- 3.8.12 Mean Value Rule -- 3.8.13 Monotonicity Rule -- 3.8.14 Inversion Rule -- 3.8.15 Cyclic Rule -- 3.8.16 Homogeneity Rule -- 3.9 Three Gaussian Integrals -- 3.10 Three Differential Equations -- 3.11 Legendre Transform -- 3.12 Lagrange Multiplier -- 4. Algebra of Vectors -- 4.1 Introduction -- 4.2 When is an Array a Matrix? -- 4.3 List Algebra -- 4.3.1 Abstract Row List -- 4.3.2 Set of Row Lists -- 4.3.3 Inclusion of Row Lists -- 4.3.4 Projection of Row Lists -- 4.3.5 Row List Algebra -- 4.3.6 Monoid Constructed from a Set -- 4.3.7 Column List Algebra & Natural Transformation -- 4.3.8 Lists of Lists -- 4.3.8.1 Rows of Columns and Columns of Rows -- 4.3.8.2 Flattening of rows of rows -- 4.4 Table Algebra -- 4.4.1 The Empty and Unit Tables -- 4.4.2 The Set of All Tables -- 4.4.3 Juxtaposition of Tables is a Table -- 4.4.4 Outer Product of Two Lists is a Table -- 4.5 Vector Algebra -- 4.5.1 Category of Vector Spaces & Vector Operators -- 4.5.2 Vector Space Isomorphism -- 4.5.3 Inner Product -- 4.5.4 Vector Operator Algebra -- 4.5.5 Dual Vector Space -- 4.5.6 Double Dual Vector Space -- 4.5.7 The Unique Extension of a Vector Operator -- 4.5.8 The Vector Space of Matrices -- 4.5.9 The Matrix of a Vector Operator -- 4.5.10 Operator Composition & Matrix Multiplication -- 4.5.11 More on Vector Operators -- Particle Mechanics -- 5. Particle Universe -- 5.1 Conservation of Energy & Newton's Second Law -- 5.2 Lagrange's Equations & Newton's Second Law -- 5.3 The Invariance of Lagrange's Equations -- 5.4 Hamilton's Principle -- 5.5 Hamilton's Equations -- 5.6 A Theorem of George Stokes -- 5.7 A Theorem on a Series of Impulsive Forces -- 5.8 Langevin's Trick -- 5.9 An Argument due to Albert Einstein.

5.10 An Argument due to Paul Langevin -- Timing Machinery -- 6. Introduction to Timing Machinery -- 6.1 Blending Time & State Machine -- 6.2 The Basic Oscillator -- 6.3 Timing Machine Variable -- 6.4 The Robust Low-Pass Filter -- 6.5 Frequency Multiplier & Di erential Equation -- 6.6 Probabilistic Timing Machine -- 6.7 Chemical Reaction System Simulation -- 6.8 Computer Simulation -- 7. Stochastic Timing Machinery -- 7.1 Introduction -- 7.1.1 Syntax for Drawing Models -- 7.1.2 Semantics for Interpreting Models -- 7.2 Examples -- 7.2.1 The Frequency Doubler of Brian Stromquist -- 7.3 Zero-Order Chemical Reaction -- 7.3.1 Newton's Second Law -- 7.3.2 Gillespie Exact Stochastic Simulation -- 7.3.3 Brownian Particle in a Force Field -- 7.3.3.1 From Random Walk to Smoluchowski's Equation -- 7.3.3.2 A Simpler, Equivalent Stochastic Timing Machine -- Theory of Substances -- 8. Algebraic Thermodynamics -- 8.1 Introduction -- 8.2 Chemical Element, Compound & Mixture -- 8.3 Universe -- 8.4 Reservoir & Capacity -- 8.5 Equilibrium & Equipotentiality -- 8.6 Entropy & Energy -- 8.7 Fundamental Equation -- 8.8 Conduction & Resistance -- 9. Clausius, Gibbs & Duhem -- 9.1 Clausius Inequality -- 9.2 Gibbs-Duhem Equation -- 10. Experiments & Measurements -- 10.1 Experiments -- 10.1.1 Boyle, Charles & Gay-Lussac Experiment -- 10.1.2 Rutherford-Joule Friction Experiment -- 10.1.3 Joule-Thomson Free Expansion of an Ideal Gas -- 10.1.4 Iron-Lead Experiment -- 10.1.5 Isothermal Expansion of an Ideal Gas -- 10.1.6 Reaction at Constant Temperature & Volume -- 10.1.7 Reaction at Constant Pressure & Temperature -- 10.1.8 Théophile de Donder & Chemical Affinity -- 10.1.9 Gibbs Free Energy -- 10.2 Measurements -- 10.2.1 Balance Measurements -- 11. Chemical Reaction -- 11.1 Chemical Reaction Extent, Completion & Realization -- 11.2 Chemical Equilibrium.

11.3 Chemical Formations & Transformations -- 11.4 Monoidal Category & Monoidal Functor -- 11.5 Hess' Monoidal Functor -- Muscle Contraction Research -- 12. Muscle Contraction -- 12.1 Muscle Contraction: Chronology -- 12.1.1 19th Century -- 12.1.2 1930-1939 -- 12.1.3 1940-1949 -- 12.1.4 1950-1959 -- 12.1.5 1960-1969 -- 12.1.6 1970-1979 -- 12.1.7 1980-1989 -- 12.1.8 1990-1999 -- 12.1.9 2000-2010 -- 12.2 Conclusion -- Appendices -- Appendix A Exponential & Logarithm Functions -- Appendix B Recursive Definition of Stochastic Timing Machinery -- B.1 Ordinary Differential Equation: Initial Value Problem -- B.2 Stochastic Differential Equation: A Langevin Equation without Inertia -- B.3 Gillespie Exact Stochastic Simulation: Chemical Master Equation -- B.4 Stochastic Timing Machine: Abstract Theory -- Appendix C MATLAB Code -- C.1 Stochastic Timing Machine Interpreter -- C.2 MATLAB for Stochastic Timing Machinery Simulations -- C.3 Brownian Particle in Force Field -- C.4 Figures. Simulating Brownian Particle in Force Field -- Appendix D Fundamental Theorem of Elastic Bodies -- Bibliography -- Index.