From Pythagoreans to Hegel, and beyond, this book gives a brief overview of the history of the notion of graphs and introduces the main concepts of graph theory in order to apply them to philosophy. In addition, this book presents how philosophers can use various mathematical notions of order. Throughout the book, philosophical operations and concepts are defined through examining questions relating the two kinds of known infinities – discrete and continuous – and how Woodin's approach can influence elements of philosophy.
We also examine how mathematics can help a philosopher to discover the elements of stability which will help to build an image of the world, even if various approaches (for example, negative theology) generally cannot be valid. Finally, we briefly consider the possibilities of weakening formal thought represented by fuzziness and neutrosophic graphs. In a nutshell, this book expresses the importance of graphs when representing ideas and communicating them clearly with others.
Author(s): Daniel Parrochia
Series: Mathematics and Statistics
Edition: 1
Publisher: Wiley-ISTE
Year: 2023
Language: English
Pages: 262
City: London
Cover
Title Page
Copyright Page
Contents
Introduction
Chapter 1. Graphs
1.1. Graph theory: a brief history
1.2. Basic definitions
1.3. Different types of graphs
1.4. More on the list of graphs
1.5. Graphs and vertices
1.6. Some operations on graphs
1.7. Graph isomorphisms
1.7.1. Self-complementary graphs
1.7.2. Properties of self-complementary graphs
1.8. Symmetric and asymmetric graphs
1.9. Extremal graphs
1.10. Independence, non-separability, reconstruction conjecture
Chapter 2. Philosophical Graphs
2.1. Ancientmappings
2.2. Chinese tetragrams
2.3. Pythagorism and pentagram
2.4. n-grams and some figures of the world
2.5. Graphs and classical systematicity
2.6. Towards a new kind of systematicity
2.6.1. Non-Hamiltonian and non-Eulerian philosophies
2.6.2. Pancyclic graphs and Metahegelianism
2.7. Non-pythagorism and arrangement of lines
2.7.1. Levi graphs of line arrangements
2.7.2. Line arrangements of curve lines
2.7.3. Hyperbolic graphs
Chapter 3. Order and Its Philosophical Use
3.1. Themathematical notion of order: a brief history
3.2. The idea of “well-ordering”
3.3. Quasiorders (or preorders)
3.4. Partial orders
3.4.1. The notion of well partial order
3.4.2. Linear extension of a poset
3.4.3. Well partially orderings
3.5. Trees
3.5.1. Azoo of infinite trees
3.5.2. Ordinal infinite classifications
3.6. Moral problems in a finite world
3.7. Order versus circularity
3.8. Conclusion
Chapter 4. Towards a Formal Philosophy
4.1. Asenjo’s systems and Dubarle’s formalization of Hegelianism
4.1.1. Asenjo’s systems and Dubarle’s case
4.1.2. Dubarle, Parmenides’ thought and Hegel
4.1.3. Projective algebras
4.2. Some criticisms
4.3. Porphyry and the neoplatonist mode of thought
4.4. Avariant of Dubarle’s formalism
4.5. Quasi-Hegelian systems
4.6. Philosophical thinking and finite projective geometry
4.7. Other algebras for philosophical thinking
4.8. Models derived from geometry and algebraic geometry
4.9. Conclusion
Chapter 5. Philosophical Transformations
5.1. The paradox of ametasystem
5.2. In search of an algebra
Chapter 6. Concepts and Topology
6.1. Formal concepts
6.2. Fuzzy concepts
6.3. The case of philosophical concepts
Chapter 7. The Problem of the Infinite
7.1. The arithmetic of infinite cardinals
7.2. The question of large cardinals
7.3. Woodin’s program
7.3.1. Woodin I: CH would be false
7.3.2. Woodin II: CH may be true
7.3.3. Ultimate L andCH
7.4. Infinite and philosophy
Chapter 8. In Search for a New Philosophy
8.1. The finite case
8.2. The infinite case
Chapter 9. Extension of Structuralism and Negative Theology
9.1. Complementarity graphs
9.2. Order relation, ordered set
9.3. Graphs associated with apartially ordered set
9.4. Complementarity and incomparability graphs of a poset
9.5. Boolean representation of a poset
9.6. Case of lattices
9.6.1. Case of Boolean lattices
9.6.2. Generalization: Boolean lattices as n-cubes
9.7. Consequences for negative theology
Chapter 10. From Fuzzy Graphs to Neutrosophic Graphs
10.1. Fuzzy sets
10.2. Fuzzy graphs
10.3. Intuitionistic fuzzy set theory
10.4. Neutrosophy
10.5. Single-valued neutrosophic sets and graphs
Conclusion
References
Index
EULA
