1st Edition

# A Systemic Perspective on Cognition and Mathematics

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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.

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

### Biography

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.