Functorial Semiotics for Creativity in Music and Mathematics (Computational Music Science)

Preface
Contents
Part I Orientation
	1 Motivation and Background
		1.1 Local and Global Contents
		1.2 The Artificial Intelligence Problem
		1.3 Mathematical Concept Constructions
			1.3.1 Size of Categories
		1.4 Some Ontology
			1.4.1 Ontology: Where, Why, and How
			1.4.2 Oniontology: Facts, Processes, and Gestures
Part II General Concepts
	2 Semiotics
		2.1 Generalities about Signs
			2.1.1 The Problematic Fundamental Position of Semiotics in Human Behavior
			2.1.2 Definition of Signs
		2.2 De Saussure and Peirce: the Semiotic Architecture
			2.2.1 Pierce
			2.2.2 de Saussure
			2.2.3 Hjelmslev
			2.2.4 Barthes
		2.3 De Saussure’s Six Dichotomies
			2.3.1 Signifier/Signified
			2.3.2 Arbitrary/Motivated
				2.3.2.1 The Digital Approach, Sampling
			2.3.3 Syntagm/Paradigm
			2.3.4 Speech/Language
				2.3.4.1 Speech and Language Examples: Bach and Schönberg
			2.3.5 Synchrony/Diachrony
			2.3.6 Lexem/Shifter
				2.3.6.1 Semiotics in Music Performance: the Example of Celibidache’s Ideas
		2.4 The Babushka Principle in Semiotics: Connotation, Motivation, and Metatheory
			2.4.1 The Structural Consequences of the Babushka Principle
	3 Functorial Semantics Category
		3.1 The Overall Construction
			3.1.1 The Basic Digraph
			3.1.2 The Basic Category
		3.2 Quotient Categories
		3.3 Semiotics for the Yoneda Lemma
		3.4 Constructing Colimits of Representable Functors
		3.5 Functorial Filters
			3.5.1 Functors for Filters
			3.5.2 Filter Equivalence
			3.5.3 Remarks on Full Filters
			3.5.4 Filters and Grothendieck Topologies
		3.6 Functors and Trees
		3.7 The Creative Evolution of Semiotic Categories: Time in Categories?
	4 Examples
		4.1 A Classical Example from Music
		4.2 Pointers
		4.3 ZF Set Theory
		4.4 A Second Example from Music
		4.5 The Role of Signification
			4.5.1 An Elementary First Example of Signification
			4.5.2 The Case of Logical Signs
		4.6 Forms and Denotators as Signs
			4.6.1 Forms
			4.6.2 Denotators
				4.6.2.1 A Musical Denotator Example: Cadence
			4.6.3 Examples of Non-representable Functors and Their Semiotics
			4.6.4 A Musical Content Filter: Catastrophe Modulation in Beethoven’s Op. 106
			4.6.5 Colimits, Filters and Beethoven’s Op. 109
		4.7 Artificial Conceptual Frameworks
			4.7.1 ANNs
			4.7.2 Artificial Conceptual Networks, ACNs
		4.8 Examples
			4.8.1 A Functorial Example with Tensor Products
			4.8.2 A Functorial Example with Simple Denotators
			4.8.3 Yoneda
			4.8.4 The Recursion Theorem
			4.8.5 RUBATO Networks
			4.8.6 A Melody Creator
	5 Semantic and Expressive Topology
		5.1 Semantic Topology of H-jets
			5.1.1 Limits
		5.2 Expressive Topology of H-jets
Part III Semantic Math
	6 Concept Mathematics
	7 Yoneda
		7.1 Yoneda’s Lemma as a Semiotic Statement
		7.2 The Bidual Lifting of the Yoneda Construction
		7.3 A Concrete Separating Functor of Semantic Significance
			7.3.1 Semantic Classes
	8 Semantic Representations
	9 Cech Cohomology
		9.1 Spaces of Functions
			9.1.1 Representing Filters within Function Space Functors
		9.2 Global Filters and a First Chech Cohomology Theory
		9.3 A Second Cohomological Approach
			9.3.1 Hjelmslev-Yoneda Functors
			9.3.2 Cech Cohomology
			9.3.2.1 Extensions of Functors
	10 Semiotic Classification of Creative Strategies
		10.1 The General Method of Creativity
		10.2 The Three Basic Strategies in Creativity
			10.2.1 Type (1) Walls
				10.2.1.1 Albert Einstein’s Critique of the Newtonian Time Concept
				10.2.1.2 Cecil Taylor’s Critique of the Elementary Components in Jazz Improvisation
				10.2.1.3 Counterpoint
				10.2.1.4 Creativity for Denotators and Similar Signs
				10.2.1.5 Semantic Topology for Type (1) Problems
			10.2.2 Type (2) Walls
				10.2.2.1 Introducing Integers, Rationals, and Real Numbers
				10.2.2.2 Dodecaphonic Composition
				10.2.2.3 Sins and Jesus
			10.2.3 Some General Ideas for Type (2) Creativity
				10.2.3.1 Abel’s General Method
				10.2.3.2 The Idea of a Conceptual Problem Ideal
				10.2.3.3 Conceiving Quotient Solution Semiotics
				10.2.3.4 The Lesson Learned
				10.2.3.5 A Functorial Approach
				10.2.3.6 Grothendieck Topologies for the Semantic Scheme
				10.2.3.7 Expressive Topology for Type (2) Problems
				10.2.3.8 Limited H-jets
				10.2.3.9 Content Search, Topology, and Manin’s Suggestion
				10.2.3.10 Grothendieck’s Coconut Metaphor and Wiles’ Solution
			10.2.4 Type (3) Walls
				10.2.4.1 Ludwig van Beethoven’s Type (3) Creativity in the Sonata Hammerklavier op. 106, Allegro
				10.2.4.2 John Coltrane’s Type (3) Creativity
				10.2.4.3 A Different Example of Type (3): The Continuum Hypothesis
				10.2.4.4 Type (3) Walls: Elimination beyond Combinatorial Efforts?
				10.2.4.5 Expressive Topology for Type (3) Problems
				10.2.4.6 Grothendieck’s Credo
				10.2.4.7 Type (3) and Length of Proofs
		10.3 In Search of A Global Geometric Perspective
			10.3.1 A Classical Type (3) to Type (2) Switch
			10.3.2 Galois’ Miracle
		10.4 The Deep Mathematical Architecture: Objects, Structures, Concepts
			10.4.1 New Objects Needed?
			10.4.2 Objects, Structures, Conceptopoi
			10.4.3 Concepts and Structures
			10.4.4 A First Synthesis?
			10.4.5 Doing Conceptual Mathematics
			10.4.6 Conceptual Aspects of the Goldbach Conjecture
Part IV Applications and Consequences
	11 Applications and Consequences
		11.1 The Semiotic Power of Music
			11.1.1 Program and Absolute Music
			11.1.2 The Reference Architecture and Consistency in Absolute Music
		11.2 Implementation Issues
			11.2.1 Object-Oriented Implementation in Java
Part V Conclusions
	12 Conclusions and Perspectives
		12.1 Intelligence and Creativity Advancement with Mathematics
			12.1.1 Physics
			12.1.2 Linguistics
			12.1.3 Quantum Mechanics
			12.1.4 Music
				12.1.4.1 The Role and Importance of Semiotics in Musical Creativity
				12.1.4.2 Psychological Aspects of Semiotic Activity for Creativity
Part VI References, Index
	References
	Index