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, includ...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Laos, Nicolas K., 1974- (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: New York : Nova Science Publishers, [2021]
Colección:Mathematics research developments series.
World philosophy series.
Temas:
Mathematics > Philosophy.
Mathématiques > Philosophie.
Mathematics > Philosophy
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.
  • 2.2.5. Matrices of Real Numbers and Vectors
  • Vectors
  • Some Applications of Matrices
  • Input-Output Analysis
  • Linear Programming
  • Game Theory
  • 2.2.6. Analytic Geometry and the Abstract Concept of a Distance
  • Circle
  • Trigonometric Functions
  • Ellipse
  • Hyperbola
  • Parabola
  • Analytic Geometry of Space
  • The Abstract Concept of a Distance
  • 2.3. Topology of Real Numbers
  • 2.3.1. Neighborhoods
  • 2.3.2. Open Sets
  • 2.3.3. Nested Intervals and Cantor's Intersection Theorem
  • 2.3.4. Closure Points and Accumulation Points
  • 2.3.5. Closed Sets
  • 2.3.6. Compactness
  • 2.3.7. Relative Topology and Connectedness
  • 2.4. Sequences of Real Numbers
  • Limit and Convergence of a Sequence
  • Cauchy Sequences and the Completeness of the Real Field
  • Subsequences
  • Monotonic Sequences
  • Hilbert Space
  • Alphabets and Languages
  • 2.5. Infinite Series and Infinite Products
  • 2.6. The Limit of a Function
  • Preliminary Concepts
  • The Limit of a Function
  • 2.7. Continuous Functions
  • Types of Discontinuity
  • 2.8. Complex Numbers
  • 2.9. The Birth and the Development of Infinitesimal Calculus
  • 2.10. Differential Calculus
  • 2.10.1. Derivative
  • Drawing a Tangent Line to the Graph of a Function
  • The Formal Definition of the Derivative of a Function
  • Higher Order Derivatives
  • Table of the Derivatives of Elementary Functions
  • The Differential of a Function
  • A Note about Complex Derivatives
  • 2.10.2. The Basic Theorems of Differential Calculus
  • 2.10.3. Monotonicity, Critical Points, and Extreme Points of a Function
  • 2.10.4. Concave-Up and Concave-Down Functions
  • 2.10.5. Asymptotes of a Function
  • 2.10.6. Steps for Function Investigation and Curve Sketching
  • 2.10.7. Curvature and Radius of Curvature
  • 2.10.8. Differentiation of Multivariable Functions.
  • Differentiation of Composite Functions, Harmonic Functions, and Homogeneous Functions
  • Differentiation of Implicit Functions
  • Jacobian (or Functional) Determinant
  • Mean Value Theorems
  • 2.11. Integral Calculus
  • The Definition of the Integral as the Limit of a Sum
  • The Physical Significance of the Integral
  • Integration of Complex Functions of One Variable
  • 2.12. Standard Integration Techniques
  • Integration by Substitution
  • Integration by Parts
  • 2.13. Reduction Formulas
  • 2.14. Integration of Rational Functions
  • 2.15. Integration of Irrational Functions
  • 2.16. Integration of Trigonometric Functions
  • 2.17. Integration of Hyperbolic Functions
  • 2.18. The Theory of Riemann Integration
  • The Riemann Integral
  • Criteria of Integrability and Methods of Integration
  • Properties of Riemann Integrable Functions
  • The Equivalence of the Definitions of the Integral of a Function
  • Generalized Integrals
  • Riemann Integrability and Sets of Measure Zero
  • The Mean Value Theorems of Integral Calculus and the Fundamental Theorem of Infinitesimal Calculus
  • 2.19. Numerical Integration
  • 2.20. Applications of Integration and Basic Principles of Differential Equations
  • 2.20.1. The Calculation of Areas Using Integrals
  • 2.20.2. The Calculation of the Area between two Arbitrary Curves
  • 2.20.3. The Calculation of the Volume of a Solid of Revolution
  • 2.20.4. The Arc Length of a Curve
  • 2.20.5. Work
  • 2.20.6. Some Basic Applications of Integral Calculus to Economics
  • 2.20.7. A Social Utility Model and Optimal Control
  • 2.20.8. Integration and Ordinary Differential Equations
  • 2.21. Integration of Multivariable Functions
  • 2.22. Vector-Valued Functions
  • Chapter 3
  • Logic, Epistemology, and the Problem of Truth
  • 3.1. Basic Principles of Logic
  • 3.2. Predicate Calculus
  • 3.3. Axiomatic Model Theory.
  • 3.4. Common Sense, Non-Monotonic Logic, and Many-Valued Logic
  • 3.5. Crises in the Foundations of Mathematics and Mathematical Philosophy
  • 3.5.1. The First Crisis in the Foundations of Mathematics
  • 3.5.2. The Second Crisis in the Foundations of Mathematics
  • 3.5.3. Logicism
  • 3.5.4. Axiomatic Set Theory and Category Theory
  • 3.5.5. Intuitionism
  • 3.5.6. Formalism
  • 3.5.7. Conclusions
  • 3.6. The Problem of Empirical Relevance in the Context of Science
  • 3.7. Truth as a Discovery and Truth as an Invention
  • 3.8. Degrees of Truth
  • 3.9. From Logical Values to Moral Values: Ethics and Social Theory from the Perspective of Rational Dynamicity
  • References
  • About the Author
  • Index
  • Blank Page
  • Blank Page.

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