Mathematics for Social Sciences and Arts: Algebraic Modeling (Mathematics in Mind)

Foreword
Preface
	References
Introduction
	References
Contents
Part I Algebraic Thinking and Modeling
	Problematic of Mathematics, Social Sciences, and Arts: A Ubiquitous Constructive Interaction in Algebraic Modeling
		1 Introduction
		2 Mathematics and Society
		3 Algebra, Algebraic Modeling, and Sciences
		4 Mathematics and Arts
		5 Concluding Remarks
		References
	Algebra as a Semiotic Modeling System
		1 Introduction
		2 Modeling Systems Theory: An Outline
		3 Principles
		4 Algebra as a Modeling System
		5 Concluding Remarks
		Glossary
		References
	The “Unreasonable” Effectiveness of Mathematical Modeling
		1 Introduction
		2 Conceptual Integration (Blending) Theory: The Basic Architecture
		3 The Concept of Mathematical Mapping and Small Spatial Stories
			3.1 Small Spatial Stories and the Mathematical Classroom
		4 Concluding Remarks
		Appendix: Glossary of Notation and Definitions
		References
	Algebra and Modeling in Mathematics School Curricula
		1 Introduction
		2 Algebra in the Curriculum
		3 Algebra and Modeling
			3.1 Number Line: A Powerful Model
			3.2 Number Line in Mathematics
			3.3 The Number Line in School Mathematics
			3.4 Complexities in Using a Number Line in School Mathematics
			3.5 Example: A Train Problem
		4 Conclusions
		Appendix: Glossary of Notation and Definitions
		References
	Gödel\'s Incompleteness as an Argument for Dualism
		1 Introduction
		2 Naturalism vs. Dualism
		3 Bare Bones of Incompleteness
		4 Gödel\'s First Incompleteness Theorem
		5 Gödel\'s Second Incompleteness Theorem
		6 Gödelian Dualist Arguments and Their Refutations
		Appendix: Glossary of Notation and Definitions
		References
	Vetoing: Social, Logical and Mathematical Aspects
		1 Introduction
		2 Impossibility through History
		3 Social Decision: Practice and Theory
		4 Social Choice: Traditional Theory
		5 A Method of Axioms Simplification
		6 Impossibilities with Vetoing, Dictatorship and Pareto Rule
		7 Measuring the Veto Power in Voting Systems
		8 A Note on Possible Further Research
		9 Concluding Remarks
		Appendix: Glossary of Notation and Definitions
		References
Part II Semigroups and Other Algebraic Structures in Social Sciences
	Constructive Semigroups with Apartness: A State of the Art
		1 Introduction
		2 The CLASS Case
			2.1 Sets and Relations
				2.1.1 Basic Concepts and Important Examples
				2.1.2 Quotient Sets: Homomorphism and Isomorphism Theorems
				2.1.3 Ordered Sets
			2.2 Semigroups Within CLASS
				2.2.1 Basic Concepts: Definitions, Examples, and Embedding Theorems
				2.2.2 General Structure Results: Quotient Semigroups, Homomorphism, and Isomorphism Theorems
				2.2.3 Ordered Semigroups: Definitions, Ordered Homomorphism and Ordered Isomorphism Theorems
			2.3 Applications and Possible Applications
			2.4 Notes
		3 Within BISH
			3.1 Set with Apartness
				3.1.1 Basic Concepts and Important Examples
				3.1.2 Distinguishing Subsets
				3.1.3 Binary Relations
				3.1.4 Apartness Isomorphism Theorems for Sets with Apartness
				3.1.5 Co-ordered Sets with Apartness
			3.2 Semigroups with Apartness
				3.2.1 Basic Concepts: Definitions, Examples, and Se-embeddings
				3.2.2 Co-quasiorders Defined on a Semigroup
				3.2.3 Apartness Isomorphism Theorems for Semigroups with Apartness
				3.2.4 Co-ordered Semigroup with Apartness
			3.3 Applications and Possible Applications
			3.4 Notes
		4 CLASS and BISH: A Comparative Analysis
		5 Concluding Remarks
		6 Appendix
			6.1 The CLASS Case
			6.2 Within BISH
		References
	Algebraic Approaches to the Analysis of Social Networks
		1 Introduction
		2 Partially Ordered Semigroups from Binary Relations
		3 Semigroup Homomorphisms
		4 Relational Homomorphisms and Semigroup Homomorphisms
		5 Comparing and Analyzing Role Structures
		6 Open Challenges and Opportunities
			6.1 Attention to Sociocultural Dynamics
			6.2 Simultaneous Analysis of Positions and Roles
			6.3 Taking Account of Uncertainty in Relational Data
		7 Concluding Remarks
		Appendix: Glossary of Notation and Definitions
		References
	Relational Systems of Transport Network and Provinces in Ancient Rome
		1 Introduction
		2 RE Transport Network and Provinces
			2.1 Transportation Routes
			2.2 Province Affiliations
		3 Positional Analysis
			3.1 Positional System from Compositional Equivalence
			3.2 Positional System from Formal Concepts
			3.3 Multilevel Positional Systems
		4 Semigroup and Role Structure
			4.1 Decomposition and Quotient Semigroup
				4.1.1 Green\'s Relations
				4.1.2 Congruence Relations
				4.1.3 Factorization
		5 Shared Structure
			5.1 Common Structure Semigroup
		6 Concluding Remarks
		7 Note About Software
		Appendix: Glossary of Notation and Definitions
		References
	Time and Sequence in Networks of Social Interactions
		1 Introduction
		2 Algebraic Representations of Social Networks
			2.1 Algebraic Structures on Relational Networks
		3 Considering Time in Social Network Analysis
			3.1 An Algebra for Temporal Interval Sets
			3.2 Constructing Time-Ordered Walks
			3.3 More General Compositions for Time-Ordered Walks
			3.4 Algebra of Endomorphisms
		4 Conceiving of Centrality Measures in Temporal Social Networks
			4.1 Path-Based Measures: Temporal Implications
			4.2 Brokerage: A Process View
		5 Discussion
		Appendix
		References
	Algebraic Structures and Social Processes
		1 Introduction
		2 Data
		3 Social Structures of Influence
		4 Partial Orderings of Items
		5 Microbelief Representations
		6 Coombs Factorizations
		7 Coombs Factorizations and the Biorder Approach
		8 Conclusion
		Appendix: Glossary
		References