A Course of Philosophy and Mathematics : Toward a General Theory of Reality

"The nature of this book is fourfold: First, it provides comprehensive education in ontology, epistemology, logic, and ethics. From this perspective, it can be treated as a philosophical textbook. Second, it provides comprehensive education in mathematical analysis and analytic geometry, including significant aspects of set theory, topology, mathematical logic, number systems, abstract algebra, linear algebra, and the theory of differential equations. From this perspective, it can be treated as a mathematical textbook. Third, it makes a student and a researcher in philosophy and/or mathematics capable of developing a holistic approach to reality, of undertaking interdisciplinary endeavors, of understanding (and possibly contributing to) advances and research projects in different academic disciplines, and of having more sources of inspiration and pleasure. From this perspective, it can be treated as a contribution to pedagogy and as an attempt to refresh and, indeed, revitalize modern philosophy. Fourth, it seeks to defend, refresh, and enrich philosophical and scientific structuralism and dynamical philosophy (known also as dynamism). From this perspective, this book can be treated as a research monograph on structuralism and dynamism, tackling the fundamental problems of reality, truth, and consciousness. In this context, Nicolas Laos expounds and proposes: (i) the concepts of dynamized time and dynamized space; (ii) a theory and method that he calls the "dialectic of rational dynamicity"; and (iii) his attempt to consider the fundamental problems of philosophy and science from the perspective of the dialectic of rational dynamicity. Thus, this book pertains to every field that is controlled by the function of consciousness, namely, being, knowing, and acting. The philosophy of rational dynamicity, as the author explains in this book, is a way of contemplating the laws of motion of nature, history, and spirit"-- Provided by publisher Contents 9 Prolegomena by Giuliano di Bernardo 11 Preface 13 The Scope and the Structure of this Project 13 Acknowledgments 15 Chapter 1 17 Philosophy, Science, and The Dialectic of Rational Dynamicity 17 1.1. The Meaning of Philosophy and Preliminary Concepts 17 1.2. The Abstract Study of a Being 24 1.2.1. Epistemological Presuppositions 25 1.2.2. The Significance and the Presence of a Being 28 1.2.3. The Knowledge of a Being 40 Structuralism in Physics 53 Newton’s Three Laws of Kinematics 54 Newton’s Law of Universal Gravitation 55 Conservation of Mass and Energy 56 Laws of Thermodynamics 58 Electrostatic Laws 60 Quantum Mechanics 64 Structuralism in Biology 69 Structuralism in Linguistics 71 Philosophical Structuralism and Hermeneutics 72 1.2.4. The Modes of Being 77 1.3. The Dialectic of Rational Dynamicity 79 1.3.1. Dynamized Time 79 1.3.2. Dynamized Space and the Problem of the Extension of the Quantum Formalism 89 1.3.3. Consciousness, the World, and the Dialectic of Rational Dynamicity 100 1.3.4. Matter, Life, and Consciousness 124 Chapter 2 137 Foundations of Mathematical Analysis and Analytic Geometry 137 2.1. Sets, Relations, and Groups 137 2.1.2. Basic Operations on Sets 141 Applications of Set Theory to Probability Theory 144 2.1.3. Relations 147 2.1.4. Groups 151 2.2. Number Systems, Algebra, and Geometry 156 2.2.1. Axiomatic Number Theory 157 The System of Natural Numbers 157 Principle of Mathematical Induction 158 Recursion 159 Properties of the System of Natural Numbers 160 Enumeration 163 Order in N and Ordinal Numbers 169 Division 171 2.2.2. The Set of Integral Numbers 171 2.2.3. The Set of Rational Numbers 173 2.2.4. The Set of Real Numbers 174 Dedekind Algebra 175 R as a Field 179 The Absolute Value of a Real Number 182 Exponentiation and Logarithm 183 Properties of the System of the Real Numbers 185 2.2.5. Matrices of Real Numbers and Vectors 187 Vectors 192 Some Applications of Matrices 196 Input–Output Analysis 196 Linear Programming 198 Game Theory 199 2.2.6. Analytic Geometry and the Abstract Concept of a Distance 203 Circle 208 Trigonometric Functions 209 Ellipse 211 Hyperbola 213 Parabola 215 Analytic Geometry of Space 217 The Abstract Concept of a Distance 219 2.3. Topology of Real Numbers 223 2.3.1. Neighborhoods 225 2.3.2. Open Sets 227 2.3.3. Nested Intervals and Cantor’s Intersection Theorem 228 2.3.4. Closure Points and Accumulation Points 229 2.3.5. Closed Sets 231 2.3.6. Compactness 233 2.3.7. Relative Topology and Connectedness 236 2.4. Sequences of Real Numbers 237 Limit and Convergence of a Sequence 238 Cauchy Sequences and the Completeness of the Real Field 240 Subsequences 242 Monotonic Sequences 243 Hilbert Space 245 Alphabets and Languages 245 2.5. Infinite Series and Infinite Products 246 2.6. The Limit of a Function 255 Preliminary Concepts 255 The Limit of a Function 259 2.7. Continuous Functions 264 Types of Discontinuity 267 2.8. Complex Numbers 279 2.9. The Birth and the Development of Infinitesimal Calculus 288 2.10. Differential Calculus 289 2.10.1. Derivative 289 Drawing a Tangent Line to the Graph of a Function 290 The Formal Definition of the Derivative of a Function 294 Higher Order Derivatives 300 Table of the Derivatives of Elementary Functions 301 The Differential of a Function 302 A Note about Complex Derivatives 303 2.10.2. The Basic Theorems of Differential Calculus 303 2.10.3. Monotonicity, Critical Points, and Extreme Points of a Function 312 2.10.4. Concave-Up and Concave-Down Functions 317 2.10.5. Asymptotes of a Function 318 2.10.6. Steps for Function Investigation and Curve Sketching 320 2.10.7. Curvature and Radius of Curvature 320 2.10.8. Differentiation of Multivariable Functions 322 Differentiation of Composite Functions, Harmonic Functions, and Homogeneous Functions 326 Differentiation of Implicit Functions 328 Jacobian (or Functional) Determinant 329 Mean Value Theorems 330 2.11. Integral Calculus 341 The Definition of the Integral as the Limit of a Sum 343 The Physical Significance of the Integral 344 Integration of Complex Functions of One Variable 346 2.12. Standard Integration Techniques 346 Integration by Substitution 346 Integration by Parts 347 2.13. Reduction Formulas 348 2.14. Integration of Rational Functions 349 2.15. Integration of Irrational Functions 350 2.16. Integration of Trigonometric Functions 351 2.17. Integration of Hyperbolic Functions 352 2.18. The Theory of Riemann Integration 352 The Riemann Integral 353 Criteria of Integrability and Methods of Integration 359 Properties of Riemann Integrable Functions 363 The Equivalence of the Definitions of the Integral of a Function 367 Generalized Integrals 369 Riemann Integrability and Sets of Measure Zero 370 The Mean Value Theorems of Integral Calculus and the Fundamental Theorem of Infinitesimal Calculus 373 2.19. Numerical Integration 378 2.20. Applications of Integration and Basic Principles of Differential Equations 379 2.20.1. The Calculation of Areas Using Integrals 379 2.20.2. The Calculation of the Area between two Arbitrary Curves 379 2.20.3. The Calculation of the Volume of a Solid of Revolution 381 2.20.4. The Arc Length of a Curve 383 2.20.5. Work 385 2.20.6. Some Basic Applications of Integral Calculus to Economics 385 2.20.7. A Social Utility Model and Optimal Control 386 2.20.8. Integration and Ordinary Differential Equations 387 2.21. Integration of Multivariable Functions 395 2.22. Vector-Valued Functions 400 Chapter 3 405 Logic, Epistemology, and the Problem of Truth 405 3.1. Basic Principles of Logic 405 3.2. Predicate Calculus 408 3.3. Axiomatic Model Theory 413 3.4. Common Sense, Non-Monotonic Logic, and Many-Valued Logic 417 3.5. Crises in the Foundations of Mathematics and Mathematical Philosophy 420 3.5.1. The First Crisis in the Foundations of Mathematics 420 3.5.2. The Second Crisis in the Foundations of Mathematics 425 3.5.3. Logicism 428 3.5.4. Axiomatic Set Theory and Category Theory 433 3.5.5. Intuitionism 442 3.5.6. Formalism 447 3.5.7. Conclusions 453 3.6. The Problem of Empirical Relevance in the Context of Science 455 3.7. Truth as a Discovery and Truth as an Invention 461 3.8. Degrees of Truth 464 3.9. From Logical Values to Moral Values: Ethics and Social Theory from the Perspective of Rational Dynamicity 468 References 481 About the Author 499 Index 501 Blank Page 2 Blank Page 509 Intro -- Contents -- Prolegomena by Giuliano di Bernardo -- Preface -- The Scope and the Structure of this Project -- Acknowledgments -- Chapter 1 -- Philosophy, Science, and The Dialectic of Rational Dynamicity -- 1.1. The Meaning of Philosophy and Preliminary Concepts -- 1.2. The Abstract Study of a Being -- 1.2.1. Epistemological Presuppositions -- 1.2.2. The Significance and the Presence of a Being -- 1.2.3. The Knowledge of a Being -- Structuralism in Physics -- Newton's Three Laws of Kinematics -- Newton's Law of Universal Gravitation -- Conservation of Mass and Energy -- Laws of Thermodynamics -- Electrostatic Laws -- Quantum Mechanics -- Structuralism in Biology -- Structuralism in Linguistics -- Philosophical Structuralism and Hermeneutics -- 1.2.4. The Modes of Being -- 1.3. The Dialectic of Rational Dynamicity -- 1.3.1. Dynamized Time -- 1.3.2. Dynamized Space and the Problem of the Extension of the Quantum Formalism -- 1.3.3. Consciousness, the World, and the Dialectic of Rational Dynamicity -- 1.3.4. Matter, Life, and Consciousness -- Chapter 2 -- Foundations of Mathematical Analysis and Analytic Geometry -- 2.1. Sets, Relations, and Groups -- 2.1.2. Basic Operations on Sets -- Applications of Set Theory to Probability Theory -- 2.1.3. Relations -- 2.1.4. Groups -- 2.2. Number Systems, Algebra, and Geometry -- 2.2.1. Axiomatic Number Theory -- The System of Natural Numbers -- Principle of Mathematical Induction -- Recursion -- Properties of the System of Natural Numbers -- Enumeration -- Order in N and Ordinal Numbers -- Division -- 2.2.2. The Set of Integral Numbers -- 2.2.3. The Set of Rational Numbers -- 2.2.4. The Set of Real Numbers -- Dedekind Algebra -- R as a Field -- The Absolute Value of a Real Number -- Exponentiation and Logarithm -- Properties of the System of the Real Numbers