Corpus Demonstratum: Collection of Proofs and Arguments

Welcome to our comprehensive collection of mathematical proofs and philosophical arguments. This wiki aims to be a definitive resource for students, scholars, and enthusiasts seeking rigorous demonstrations across the spectrum of mathematical disciplines and philosophical traditions.

This encyclopedia serves multiple purposes:

• To collect and preserve important proofs in accessible formats
• To provide clear explanations with appropriate levels of rigor
• To highlight connections between different areas of mathematics and philosophy
• To trace the historical development of key theorems and arguments
• To serve as an educational resource for learners at various levels

• Mathematical expression formatting guide
• Mathematical proof template
• Philosophical argument template

• Logic and Set Theory
  • Gödel's Incompleteness Theorems
  • Compactness Theorem
  • Completeness Theorem
  • ZFC Axioms and Consequences
  • Axiom of Choice and Equivalents
• Number Systems
  • Peano Axioms
  • Construction of Integers
  • Construction of Rational Numbers
  • Construction of Real Numbers
  • Complex Numbers

• Elementary Number Theory
  • Division Algorithm
  • Euclidean Algorithm
  • Fundamental Theorem of Arithmetic
  • Fermat's Little Theorem
  • Euler's Theorem
• Analytic Number Theory
  • Prime Number Theorem
  • Dirichlet's Theorem on Primes

• Group Theory
  • Lagrange's Theorem
  • Sylow Theorems
  • Isomorphism Theorems
• Ring Theory
  • Ideal Theory
  • Unique Factorization Domains
• Field Theory
  • Field Extensions
  • Splitting Fields
• Galois Theory
  • Fundamental Theorem of Galois Theory
  • Solvability by Radicals
• Linear Algebra
  • Determinants
  • Eigenvalues and Eigenvectors
  • Diagonalization

• Calculus
  • Limit Theorems
  • Mean Value Theorem
  • Fundamental Theorem of Calculus
• Real Analysis
  • Convergence of Sequences
  • Convergence of Series
  • Riemann Integration
  • Lebesgue Integration
• Complex Analysis
  • Cauchy's Integral Theorem
  • Residue Theorem
  • Laurent Series
• Functional Analysis
  • Baire Category Theorem
  • Hahn-Banach Theorem
  • Spectral Theory

• Euclidean Geometry
  • Pythagorean Theorem
  • Triangle Congruence Theorems
  • Similar Triangles
• Differential Geometry
  • Theory of Curves
  • Theory of Surfaces
  • Manifolds
• Point-Set Topology
  • Metric Spaces
  • Continuous Functions
  • Compactness
• Algebraic Topology
  • Fundamental Group
  • Homology Groups
  • Fixed Point Theorems

• Probability Theory
  • Kolmogorov Axioms
  • Probability Distributions
  • Limit Theorems
• Differential Equations
  • First-Order ODEs
  • Linear ODEs
  • Partial Differential Equations

• Fermat's Last Theorem
• Four Color Theorem
• Poincaré Conjecture
• Riemann Hypothesis
• P vs NP Problem

• Ontological Arguments
  • Anselm's Ontological Argument
  • Descartes' Ontological Argument
  • Modal Ontological Arguments
• Causation
  • Cosmological Arguments
  • Kalam Cosmological Argument
  • Infinite Regress Arguments
• Philosophy of Mind
  • Arguments for Dualism
  • Arguments for Physicalism
  • Knowledge Argument (Mary's Room)

• Nature of Knowledge
  • Knowledge as Justified True Belief
  • Gettier Problems
  • Reliabilist Theories
• Skeptical Arguments
  • Brain in a Vat
  • Descartes' Evil Demon
  • Dream Argument
• Problem of Induction
  • Hume's Problem of Induction
  • Goodman's New Riddle of Induction

• Normative Ethics
  • Utilitarian Arguments
  • Kantian Arguments
  • Virtue Ethics
• Metaethics
  • Moral Realism
  • Error Theory
  • Emotivism
• Applied Ethics
  • Trolley Problem
  • Value of Life Arguments

• Formal Logic
  • Propositional Logic
  • Predicate Logic
  • Modal Logic
• Informal Logic and Fallacies
  • Deductive Fallacies
  • Inductive Fallacies
• Theories of Meaning
  • Reference Theories
  • Use Theories
  • Verification Principle

• Demarcation Problem
  • Popper's Falsificationism
  • Kuhn's Paradigm Shifts
• Scientific Explanation
  • Deductive-Nomological Model
  • Causal Accounts
• Scientific Realism vs. Anti-Realism
  • No Miracles Argument
  • Pessimistic Induction

We welcome contributions from mathematicians, philosophers, educators, and enthusiasts. To contribute:

• Follow our standards: Use the proper templates for consistency
• Use proper notation: Ensure mathematical expressions use LaTeX via the MathJax plugin
• Provide references: Include sources for all theorems, proofs, and arguments
• Consider accessibility: Write clear explanations that build on prerequisites

For questions or assistance with this wiki, please contact ayman@aymanlafaz.com.