A Systemic Perspective on Cognition and Mathematics

A Systemic Perspective on Cognition and Mathematics

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Features

  • Presents important developments of systems research enabling the uncovering of laws of nature underlying myriad things and events
  • Represents the first time the yoyo model has been employed systematically to the study of the mind and the analysis of the system of mathematics
  • Outlines an approach to rebuild the system of mathematics and attempts to re-establish calculus without using the concept of limits

Summary

This book is devoted to the study of human thought, its systemic structure, and the historical development of mathematics both as a product of thought and as a fascinating case analysis. After demonstrating that systems research constitutes the second dimension of modern science, the monograph discusses the yoyo model, a recent ground-breaking development of systems research, which has brought forward revolutionary applications of systems research in various areas of the traditional disciplines, the first dimension of science. After the systemic structure of thought is factually revealed, mathematics, as a product of thought, is analyzed by using the age-old concepts of actual and potential infinities. In an attempt to rebuild the system of mathematics, this volume first provides a new look at some of the most important paradoxes, which have played a crucial role in the development of mathematics, in proving what these paradoxes really entail. Attention is then turned to constructing the logical foundation of two different systems of mathematics, one assuming that actual infinity is different than potential infinity, and the other that these infinities are the same. This volume will be of interest to academic researchers, students and professionals in the areas of systems science, mathematics, philosophy of mathematics, and philosophy of science.

Table of Contents

PART 1
Basics 1

1 Where everything starts
1.1 Systems research: The second dimension of knowledge
1.1.1 A brief history
1.1.2 Numbers and systems, what is the difference?
1.1.3 Challenges systems science faces
1.1.4 Intuition and playground of systems research
1.2 The background to the systemic yoyo model
1.2.1 The theoretical foundation
1.2.2 The empirical foundations
1.2.3 The social foundations
1.2.4 Unevenness implies spinning
1.3 Problems addressable by using systems thinking and methodology
1.3.1 Kinds of problems systems researchers address
1.3.2 Is 1+1 really 2?
1.3.3 An example of how modern science resolves problems
1.3.4 New frontiers of knowledge
1.4 Some successful applications of systemic yoyos model
1.4.1 Some recent achievements
1.4.2 How a workplace is seen as a spinning field
1.4.3 Fluids in yoyo fields?
1.5 Organization of this book

2 Elementary properties of systemic yoyos
2.1 Quark structure of systemic yoyo fields
2.1.1 The spin of systemic yoyos
2.1.2 The quark structure of systemic yoyos
2.1.3 The field structure of electrons
2.2 Systemic yoyo fields and their structures
2.2.1 The formation of yoyo fields
2.2.2 Classification of yoyo fields
2.2.3 The Coulomb’s law for particle yoyos
2.2.4 The eddy fields of yoyos
2.2.5 Movement of yoyo dipoles in yoyo fields
2.2.5.1 Movement of yoyo dipoles in uniform yoyo fields
2.2.5.2 Movement of yoyo dipoles in uneven yoyo fields
2.2.5.3 The concept of yoyo flux
2.3 States of motion
2.3.1 The first law on state of motion
2.3.2 The second law on state of motion
2.3.3 The third law on state of motion
2.3.4 The fourth law on state of motion
2.3.5 Validity of figurative analysis

PART 2
The mind

3 Human body as a system
3.1 Systems and fundamental properties
3.1.1 Systems: What are they?
3.1.2 Structures and subsystems
3.1.3 Levels
3.2 Looking at human body more traditionally
3.2.1 Basic elements of Chinese traditional medicine
3.2.2 Brief history of Chinese traditional medicine
3.2.3 Modern interest in acupuncture
3.2.4 The meridian system
3.3 System and its environment
3.3.1 Environments
3.3.2 Open systems and close systems
3.3.3 Systems on dynamic sets
3.3.4 Dynamic subsystems
3.3.5 Interactions between systems
3.4 A grand theory about man and nature
3.4.1 Tao Te Ching: The classic
3.4.2 The purpose of Tao Te Ching
3.4.3 Modern systems research in Tao Te Ching
3.4.4 Some final words

4 The four human endowments
4.1 Self-awareness: The first endowment
4.1.1 The origin of self-awareness (self-consciousness)
4.1.2 Existence of core identity
4.1.3 Self-consciousness and selection of thoughts and actions
4.1.4 Self-awareness and cultural emphasis
4.1.5 Maintenance of self-motivation and self-determination
4.2 Imagination: The second endowment
4.2.1 Mechanism over which imagination works and functions
4.2.2 Formation of philosophical values and beliefs
4.2.3 Imagination more important than knowledge?
4.2.4 How imagination reassembles known ideas/facts for innovative uses
4.2.5 Imagination converts adversities, failures, and mistakes into assets
4.3 Conscience: The third endowment
4.3.1 The functionality of conscience
4.3.2 Conscience, behavior, thought and action
4.3.3 How conscience is affected by culture
4.3.4 Is world conscience possible?
4.4 Free will: The fourth endowment
4.4.1 Systemic mechanism of free will
4.4.2 Rational agents and uncertainty
4.4.3 Laws of nature and causal determinacy
4.4.4 Moral responsibility and free will
4.5 A few final words

5 Character and thought
5.1 Human effectiveness
5.1.1 Systemic yoyo model of character
5.1.2 Attraction of character
5.1.3 Laws that govern effectiveness
5.1.4 Difficulty of breaking loose from undesirable habits
5.1.5 Working with desire
5.2 Thoughts and consequences
5.2.1 Formation of thought
5.2.2 Thoughts and desirable outcomes
5.2.3 Controlling, guiding, and directing the mind
5.2.4 Mental creation and physical materialization
5.2.5 Mental inertia
5.3 Desire
5.3.1 Desire in terms of systemic yoyo model
5.3.2 Origin of desire
5.3.3 Power of desire
5.3.4 Desire and extraordinary capability
5.3.5 Artificial installment of desire
5.4 Enthusiasm and state of mind
5.4.1 Systemic mechanism of enthusiasm
5.4.2 Leadership and enthusiasm
5.4.3 Kindling and maintaining the fire of enthusiasm burning
5.4.4 The working of self suggestion
5.4.5 Importance of self control
5.5 Some final comments

PART 3
Mathematics seen as a systemic flow:A case study

6 A brief history of mathematics
6.1 The start
6.1.1 How basics of arithmetic were naturally applied
6.1.2 Mathematics evolves with human society
6.1.3 Birth of modern mathematics
6.2 First crisis in the foundations of mathematics
6.2.1 A geometrical interpretation of rational numbers
6.2.2 The Hippasus paradox
6.2.3 How the Hippasus paradox was resolved
6.2.4 Three lines of Greek development
6.3 Second crisis in the foundations of mathematics
6.3.1 First problems calculus addressed
6.3.2 Fast emergence of calculus
6.3.3 Building the foundation of calculus
6.3.4 Rigorous basis of analysis
6.4 Third crisis in the foundations of mathematics
6.4.1 Inconsistencies of Naïve set theory
6.4.2 Confronting the difficulty of Naïve set theory
6.4.3 Axiomatic set theory – an alternative
6.4.4 Warnings of the masters

7 Actual and potential infinities
7.1 Paradoxes of the infinite
7.1.1 The infinitely small
7.1.2 The infinitely big
7.1.3 Many and one
7.2 A historical account of the infinite
7.2.1 The infinite
7.2.2 The vase puzzle
7.2.3 A classification of paradoxes
7.3 Descriptive definitions of potential and actual infinite
7.3.1 A historical briefing
7.3.2 An elementary classification
7.3.3 A second angle
7.3.4 Summary of the highlights
7.3.5 Symbolic preparations
7.3.6 A literature review of recent works
7.4 Potential and actual infinities in modern mathematics
7.4.1 Compatibility of both kinds of infinities
7.4.2 Some elementary conclusions

8 Are actual and potential infinity the same?
8.1 Yes, actual and potential infinite are the same!
8.1.1 Mathematical induction
8.1.2 A brief history of mathematical induction
8.1.3 What is implied temporally by the inductive step
8.2 No, actual and potential infinite are different!
8.2.1 The Littlewood-Ross paradox
8.2.2 The paradox of Thompson’s Lamp
8.2.3 The paradox of Wizard and Mermaid
8.3 A pair of hidden contradictions in the foundations of mathematics
8.3.1 Implicit convention #1 in modern mathematical system
8.3.2 Implicit convention #2 in modern mathematical system
8.3.3 Some plain explanations
8.4 Role of paradoxes in the development of mathematics
8.4.1 Paradoxes in the history of mathematics
8.4.1.1 Paradoxes related numbers
8.4.1.2 Paradoxes related logarithms
8.4.1.3 Paradoxes related continuity
8.4.1.4 Paradoxes related power series
8.4.1.5 Paradoxes related geometric curves
8.4.1.6 Axiom of choice
8.4.2 Modification of mathematical induction and expected impacts

9 Inconsistencies of modern mathematics
9.1 An inconsistency of countable infinite sets
9.1.1 Terminology and notes
9.1.2 An inconsistency with countable infinite sets
9.2 Uncountable infinite sets under ZFC framework
9.2.1 An inconsistency of uncountable infinite sets
9.2.2 Several relevant historical intuitions
9.3 The phenomena of Cauchy Theater
9.3.1 Spring sets and Cauchy Theater
9.3.2 Special Cauchy Theater in naïve and modern axiomatic set theories
9.3.3 Transfinite spring sets and transfinite Cauchy Theaters
9.3.4 Transfinite Cauchy Theater phenomena in the ZFC framework
9.4 Cauchy Theater and diagonal method
9.5 Return of Berkeley paradox
9.5.1 Calculus and theory of limits: A history recall
9.5.2 Abbreviations and notes
9.5.3 Definability and realizability of limit expressions
9.5.4 New Berkeley paradox in the foundation of mathematical analysis
9.6 Inconsistency of the natural number system
9.6.1 Notes and abbreviations
9.6.2 The proof of inconsistency of N
9.6.3 Discussion and explanations

PART 4
Next stage of mathematics as a systemic field of thought
 
10 Calculus without limit
10.1 An overview
10.1.1 Fundamental relationship between parts and whole
10.1.2 The tangent problem
10.1.3 Monotonicity of functions
10.1.4 Area under parabola
10.2 Derivatives
10.2.1 Differences and ratio of differences
10.2.2 A and B functions
10.2.3 Estimation inequalities
10.2.4 Derivatives
10.3 Integrals
10.3.1 The fundamental theorem of calculus
10.3.2 Some applications of definite integrals
10.3.3 Taylor series
10.4 Real number system and existences
10.4.1 Characteristics of real number system
10.4.2 Existence of inverse functions
10.4.3 Existence of definite integrals
10.5 Sequences, series, and continuity
10.5.1 Limits of sequences and infinite series
10.5.2 Limits of functions
10.5.3 Pointwise continuity
10.5.4 Pointwise differentiability
10.6 Riemann integrals and integrability
10.6.1 The concept of Riemann Integrals
10.6.2 Riemann integrability and uniqueness of integral systems

11 New look at some historically important paradoxes
11.1 The necessary symbolism
11.1.1 Proposition and its negation
11.1.2 Two layers of thinking
11.1.3 Judgmental statements
11.2 Self-referential paradoxes and contradictory fallacies
11.2.1 Plato-Socrates’ paradox
11.2.2 The liar’s paradox
11.2.3 The barber’s paradox
11.3 Paradoxes of composite contradictory type
11.3.1 Russell’s paradox
11.3.2 The catalogue paradox
11.3.3 Richard’s paradox
11.4 Structural characteristics of contradictory paradoxes
11.4.1 Characteristics of simple contradictory paradoxes
11.4.2 Characteristics of composite contradictory paradoxes
11.5 Self-referential propositions
11.5.1 Evaluation of self-referential propositions
11.5.2 Self-substituting propositions
11.5.3 Violation of the law of identity
11.5.4 Violation of the law of contradiction

12 An attempt to rebuild the system of mathematics
12.1 Mathematical system of potential infinite
12.1.1 Preparation
12.1.1.1 The division principle of the background world
12.1.1.2 On constructing mathematics of potential infinite
12.1.2 Formal basis of logical systems
12.1.2.1 Introduction
12.1.2.2 Deduction system of PIMS’s propositional logic
12.1.2.3 Deduction system of PIMS’s predicate logic
12.1.3 A meta theory of logical basis
12.1.3.1 Reliability of FPIN
12.1.3.2 Potentially maximum harmonicity in PIMS+
12.1.3.3 Properties of potentially maximum harmonic sets
12.1.3.4 Model M∗ and the completeness of FPIN
12.1.4 The set-theoretic foundation
12.1.4.1 Basics of the PIMS-Se
12.1.4.2 Axioms of the PIMS-Se
12.1.4.3 Properties and more axioms
12.1.5 Some comments
12.2 Problem of infinity between predicates and infinite sets
12.2.1 Expressions for a variable to approach its limit
12.2.2 Actually infinite rigidity of natural number set and medium transition
12.3 Intension and structure of actually infinite, rigid sets
12.3.1 Relation between predicates and sets in infinite background world
12.3.2 Constructive mode of actually infinite rigid sets without constraints
12.3.3 Constructive mode of actually infinite rigid sets with constraints
Afterword A systemic yoyo model prediction

PART 5

Appendices: A close look at the systemic yoyo model

Appendix A Theoretical foundation of the yoyo model
A.1 Blown-ups: Moments of evolutions
A.2 Properties of transitional changes
A.3 Quantitative infinity
A.4 Equal quantitative movements and effects

Appendix B Empirical evidence of the yoyo model
B.1 Circulations in fluids
B.2 Informational infrastructure
B.3 Silent human evaluations

Bibliography
Subject index

Author Bio(s)

Dr. Jeffrey Yi-Lin Forrest, also known as Yi Lin, holds all his educational degrees (BS, MS, and PhD) in pure mathematics from Northwestern University (China) and Auburn University (USA) and had one year of postdoctoral experience in statistics at Carnegie Mellon University (USA). Currently, he is a guest and specially appointed professor in economics, finance, systems science, and mathematics at several major universities in China, including Huazhong University of Science and Technology, National University of Defense Technology, Nanjing University of Aeronautics and Astronautics, and a tenured professor of mathematics at the Pennsylvania State System of Higher Education (Slippery Rock campus). Since 1993, he has been serving as the president of the International Institute for General Systems Studies, Inc. Along with various professional endeavors he organized, Dr. Forrest has had the honor to mobilize scholars from over 80 countries representing more than 50 different scientific disciplines.
Over the years, he has served on the editorial boards of 11 professional journals, including Kybernetes: The International Journal of Cybernetics, Systems and Management Sciences, Journal of Systems Science and Complexity, International Journal of General Systems, and Advances in Systems Science and Applications. And, he is the editor of the book series entitled "Systems Evaluation, Prediction and Decision-Making", and the editor of the book series “Communications in Cybernetics, Systems Science and Engineering”, both published by Taylor and Francis with the former since 2008 and the latter since 2011.

Some of Dr. Forrest’s research was funded by the United Nations, the State of Pennsylvania, the National Science Foundation of China, and the German National Research Center for Information Architecture and Software Technology.

Professor Jeffrey Forrest’s professional career started in 1984 when his first paper was published. His research interests are mainly in the area of systems research and applications in a wide-ranging number of disciplines of the traditional science, such as mathematical modeling, foundations of mathematics, data analysis, theory and methods of predictions of disastrous natural events, economics and finance, management science, philosophy of science, etc. As of the end of the summer of 2013, he had published over 300 research papers and over 40 monographs and edited special topic volumes by such prestigious publishers as Academic Press, Elsevier, Kluwer Academic, Springer, Taylor and Francis, Wiley, World Scientific, and others. Throughout his career, Dr. Jeffrey Forrest’s scientific achievements have been recognized by various professional organizations and academic publishers. In 2001, he was inducted into the honorary fellowship of the World Organization of Systems and Cybernetics.

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