The Arithmoí Library
Arithmoí Book 1, by Jason Glowney

Masteries

Arithmoí Book 1

Essential to the undertaking is that mathematics be the means of conveying the topic. The synthesis presented in the book starts by first bringing to light the historical human biases pervasive in mathematics, physics, and metaphysics subject matter. Anthropomorphism, meaning attributing human emotions, traits, and intents to non-human forms, is a deeply rooted inclination in the human psyche. Evidence for this abounds when we consider religion and mythos, where divine entities took human forms, traits, and qualities. How the human sensorium interprets phenomena lends itself to the tendency to overemphasize what is physically perceived while abandoning the concern of understanding the holos. Anthropocentrism, also known as homocentrism, is another closely related belief system that places humans at the center of the Universe. A prime example of this in the past was the geocentric model, which put the Earth at the center of the solar system during a time that preceded any understanding of what galaxies and the Universe were. Some still believe that the observer plays a fundamental role in the observed quantum mechanical system, and this should be considered a form of anthropocentrism.

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Contents

  1. Introduction
  2. Acknowledgments
  3. 1.1 Matrix Group Theory

  4. 1.1.1 Matrix Group Theory
  5. 1.1.2 Matrix Group Properties
  6. 1.1.3 General Linear Group
  7. 1.1.4 Special Linear Group
  8. 1.1.5 Orthogonal Groups
  9. 1.1.6 The Linear Transformation
  10. 1.1.7 Group Exponentiation
  11. 1.1.8 The Singular Value Decomposition
  12. 1.1.9 The Matrix Transpose
  13. 1.1.10 Calculating the SVD
  14. 1.1.11 The Commutator
  15. 1.1.12 Complex Powers and Logarithms
  16. 1.1.13 Matrix Powers and Logarithms
  17. 1.1.14 Symmetric and Dihedral Groups
  18. 1.1.15 The SymmetricSymmetric Groups
  19. 1.1.16 The Dihedral Groups
  20. 1.1.17 The Symmetric/Dihedral Isomorphism
  21. 1.1.18 The Symmetric/DihedralSymmetric/Dihedral Group ∈3ℝ\in 3\mathbb{R}
  22. 1.1.19 The DihedralGroup and Symmetry Transformations ∈3ℂ\in 3\mathbb{C}
  23. 1.1.20 The U(3)U(3) Symmetry Group
  24. 1.1.21 Polygonal Rotations in U(3)U(3)
  25. 1.1.22 Hilbert Spaces
  26. 1.1.23 Geometric Visualization of the Inner Product
  27. 1.1.24 The Cartan Subalgebra of U(3)
  28. 1.1.25 The Anti-Diagonal Symmetry of the Cartan Subalgebra
  29. 1.1.26 The Symmetric/Dihedral Cone Group
  30. 1.1.27 Projective Forms
  31. 1.2 Tangent Spaces and Differential Forms

  32. 1.2.1 Tangent Spaces and Differential Forms
  33. 1.2.2 The Cyclic Trigonometric Functions and their Derivatives
  34. 1.2.3 The Dihedral Differential Forms ∈3ℂ\in 3\mathbb{C}
  35. 1.2.4 The Dihedral Differential Group ∈3ℝ\in 3\mathbb{R}
  36. 1.2.5 The Dihedral Cone Differential Group ∈3ℂ\in 3\mathbb{C}
  37. 1.2.6 The Lie Algebra and Differential Forms
  38. 1.2.7 Tangent Spaces
  39. 1.2.8 Dihedral Group Tangent Spaces
  40. 1.2.9 Numeri Mirabiles
  41. 1.2.10 Summary
  42. 1.3 Curvature and Light

  43. 1.3.1 Curvature and Light
  44. 1.3.2 The Space of Angles ∈ℂ\in \mathbb{C}
  45. 1.3.3 Vector Representations in Complex-Angled Space
  46. 1.3.4 Curvature
  47. 1.3.5 Circular Curvature and the Wave Equation
  48. 1.3.6 Hyperbolic Curvature
  49. 1.3.7 Complex Curvature
  50. 1.3.8 Geometric Algebra
  51. 1.3.9 The Asymptotic Light Functions
  52. 1.3.10 The Curvature of Light
  53. 1.3.11 Summary
  54. 1.4 Matrix Group Conic Sections

  55. 1.4.1 Matrix Group Conic Sections
  56. 1.4.2 SU(2) as the Inscribed Sphere of the Light Cone
  57. 1.4.3 The Tangent Space of the Cone
  58. 1.4.4 Null Curves and the Cone Group
  59. 1.4.5 Number Forms, Odd and Even Partition Geometry, and Minimal Elements
  60. 1.4.6 The Conic Group
  61. 1.4.7 The Axis Group
  62. 1.4.8 The 𝒴\mathcal{Y} Subgroup of the Symmetric Conic Group
  63. 1.4.9 The SU(2) Subgroup of the Symmetric Conic Group
  64. 1.4.10 The Pauli Matrices Subgroup of the Symmetric Conic Group
  65. 1.4.11 Geometric Visualization of Matrix Group Conics
  66. 1.4.12 Summary
  67. 1.5 Tangent Space Angles and Path-Connectedness in the Symmetric Cone Group

  68. 1.5.1 Tangent Space Angles and Path-Connectedness in the Symmetric Cone Group
  69. 1.5.2 The Symmetric Cone Group Lie Algebra
  70. 1.5.3 The Matrix Logarithm
  71. 1.5.4 The Baker-Campbell-Hausdorff Formula
  72. 1.5.5 The Matrix-Matrix Exponential
  73. 1.5.6 The Matrix-Matrix Exponential and Smoothing the Tangent Space
  74. 1.5.7 Smoothing the Minimal Element Basis
  75. 1.5.8 The Infinite Symmetry Group and the Subtangent Space
  76. 1.5.9 Subtangent Vectors
  77. 1.5.10 Summary
  78. 1.6 The Singularity

  79. 1.6.1 The Singularity
  80. 1.6.2 Infinite Differentiation and the Matrix Logarithm
  81. 1.6.3 Matrix-Based Infinite Integration and Differentiation
  82. 1.6.4 Harmonic Logarithms and the Infinite Subtangent Space
  83. 1.6.5 Outer Products and Infinities
  84. 1.6.6 The Alternating Harmonic Series and its Inverse Analog
  85. 1.6.7 Transformations and Representation Theory
  86. 1.6.8 Matrix Determinants and the Infinite Symmetry Group
  87. 1.6.9 The Infinite Sphere of Tangent Spaces
  88. 1.6.10 Summary
  89. 1.7 Prime Numbers and the Geometry of the Minimal Element

  90. 1.7.1 Prime Numbers and the Geometry of the Minimal Element
  91. 1.7.2 The Geometric Primes and the Higher-Ordered Forms
  92. 1.7.3 Prime Number Theory and the Geometry of Groups
  93. 1.7.4 Primality
  94. 1.7.5 Discussion and Summary
  95. 1.8 The Flowing Point and the Theory of Light

  96. 1.8.1 The Flowing Point and the Theory of Light
  97. 1.8.2 The Flowing Point of Axial Light
  98. 1.8.3 The Quantum Flowing Point
  99. 1.8.4 Relativistic Mechanics and the Unified Field
  100. 1.8.5 The Unit Invariant Mass and the Flowing Point Ideals
  101. 1.8.6 The Theory of Light
  102. 1.8.7 Summary
  103. Conclusion
  104. References

This edition

Arith­moí

A Study of Math­e­mat­i­cal Co­he­sive­ness in Physics and Meta­physics

Volume I, Part 1:

Math­e­mat­i­cal Foun­da­tions and the Sin­gu­lar­i­ty

Jason Glowney

Athene Noctua Press

Boul­der, Col­orado USA

2023

This book is ded­i­cat­ed to all of those who have fought, and con­tin­ue to fight, to bring hu­man­i­ty out of in­tel­lec­tu­al dark­ness and into the light.

To my wife Emily and my sons Max­im­i­lien and Aidan.

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