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.
Author(s): Nicolas Laos
Series: Mathematics Research Developments
Publisher: Nova Science Publishers
Year: 2021
Language: English
Pages: 508
City: New York
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
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