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| ====== The Encyclopedia of Mathematical and Philosophical Proofs ====== | |
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| Welcome to our comprehensive collection of mathematical and philosophical proofs. 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. | |
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| ===== About This Project ===== | |
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| 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 | |
| |
| ===== Getting Started ===== | |
| |
| ==== For Readers ==== | |
| * [[wiki:syntax|DokuWiki syntax guide]] | |
| * [[wiki:math|Mathematical expression formatting]] | |
| * [[wiki:navigation|How to navigate this encyclopedia]] | |
| * [[wiki:difficulty|Understanding our difficulty ratings]] | |
| * [[wiki:glossary|Glossary of terms]] | |
| |
| ==== For Contributors ==== | |
| * [[wiki:proofstyle|Proof formatting standards]] | |
| * [[wiki:contribute|How to contribute new proofs]] | |
| * [[wiki:review|Our peer review process]] | |
| * [[wiki:citation|Citation guidelines]] | |
| * [[wiki:editing|Tips for clear mathematical writing]] | |
| |
| ===== Mathematical Proofs ===== | |
| |
| ==== Foundations of Mathematics ==== | |
| * [[math:foundations:logic|Logic and Set Theory]] | |
| * [[math:foundations:logic:incompleteness|Gödel's Incompleteness Theorems]] | |
| * [[math:foundations:logic:compactness|Compactness Theorem]] | |
| * [[math:foundations:logic:completeness|Completeness Theorem]] | |
| * [[math:foundations:logic:lowenheim-skolem|Löwenheim-Skolem Theorems]] | |
| * [[math:foundations:logic:zfc|ZFC Axioms and Consequences]] | |
| * [[math:foundations:logic:choice|Axiom of Choice and Equivalents]] | |
| * [[math:foundations:logic:constructivism|Constructive Logic]] | |
| * [[math:foundations:numbers|Number Systems]] | |
| * [[math:foundations:numbers:peano|Peano Axioms]] | |
| * [[math:foundations:numbers:integers|Construction of Integers]] | |
| * [[math:foundations:numbers:rationals|Construction of Rational Numbers]] | |
| * [[math:foundations:numbers:reals|Construction of Real Numbers]] | |
| * [[math:foundations:numbers:complex|Complex Numbers]] | |
| * [[math:foundations:numbers:cardinals|Cardinal Numbers]] | |
| * [[math:foundations:numbers:ordinals|Ordinal Numbers]] | |
| |
| ==== Number Theory ==== | |
| * [[math:number:elementary|Elementary Number Theory]] | |
| * [[math:number:elementary:division|Division Algorithm]] | |
| * [[math:number:elementary:euclidean|Euclidean Algorithm]] | |
| * [[math:number:elementary:fundamental|Fundamental Theorem of Arithmetic]] | |
| * [[math:number:elementary:fermat-little|Fermat's Little Theorem]] | |
| * [[math:number:elementary:euler|Euler's Theorem]] | |
| * [[math:number:elementary:chinese|Chinese Remainder Theorem]] | |
| * [[math:number:analytic|Analytic Number Theory]] | |
| * [[math:number:analytic:pnt|Prime Number Theorem]] | |
| * [[math:number:analytic:dirichlet|Dirichlet's Theorem on Primes in Arithmetic Progressions]] | |
| * [[math:number:algebraic|Algebraic Number Theory]] | |
| * [[math:number:algebraic:quadratic|Quadratic Reciprocity Law]] | |
| * [[math:number:algebraic:units|Dirichlet's Unit Theorem]] | |
| * [[math:number:algebraic:class|Class Number Formula]] | |
| |
| ==== Algebra ==== | |
| * [[math:algebra:group|Group Theory]] | |
| * [[math:algebra:group:lagrange|Lagrange's Theorem]] | |
| * [[math:algebra:group:sylow|Sylow Theorems]] | |
| * [[math:algebra:group:isomorphism|Isomorphism Theorems]] | |
| * [[math:algebra:group:classification|Classification of Finite Simple Groups]] | |
| * [[math:algebra:ring|Ring Theory]] | |
| * [[math:algebra:ring:ideals|Ideal Theory]] | |
| * [[math:algebra:ring:ufd|Unique Factorization Domains]] | |
| * [[math:algebra:ring:pid|Principal Ideal Domains]] | |
| * [[math:algebra:field|Field Theory]] | |
| * [[math:algebra:field:extension|Field Extensions]] | |
| * [[math:algebra:field:splitting|Splitting Fields]] | |
| * [[math:algebra:field:algebraic-closure|Algebraic Closure]] | |
| * [[math:algebra:galois|Galois Theory]] | |
| * [[math:algebra:galois:fundamental|Fundamental Theorem of Galois Theory]] | |
| * [[math:algebra:galois:solvability|Solvability by Radicals]] | |
| * [[math:algebra:galois:quintic|Unsolvability of the Quintic]] | |
| * [[math:algebra:linear|Linear Algebra]] | |
| * [[math:algebra:linear:determinants|Determinants]] | |
| * [[math:algebra:linear:eigenvalues|Eigenvalues and Eigenvectors]] | |
| * [[math:algebra:linear:diagonalization|Diagonalization]] | |
| * [[math:algebra:linear:jordan|Jordan Normal Form]] | |
| * [[math:algebra:linear:spectral|Spectral Theorem]] | |
| |
| ==== Analysis ==== | |
| * [[math:analysis:calculus|Calculus]] | |
| * [[math:analysis:calculus:limit|Limit Theorems]] | |
| * [[math:analysis:calculus:continuity|Continuity]] | |
| * [[math:analysis:calculus:differentiation|Differentiation Rules]] | |
| * [[math:analysis:calculus:mean-value|Mean Value Theorem]] | |
| * [[math:analysis:calculus:integration|Integration Techniques]] | |
| * [[math:analysis:calculus:fundamental|Fundamental Theorem of Calculus]] | |
| * [[math:analysis:real|Real Analysis]] | |
| * [[math:analysis:real:sequences|Convergence of Sequences]] | |
| * [[math:analysis:real:series|Convergence of Series]] | |
| * [[math:analysis:real:continuity|Continuity Properties]] | |
| * [[math:analysis:real:differentiation|Differentiability Theorems]] | |
| * [[math:analysis:real:riemann|Riemann Integration]] | |
| * [[math:analysis:real:lebesgue|Lebesgue Integration]] | |
| * [[math:analysis:complex|Complex Analysis]] | |
| * [[math:analysis:complex:cauchy|Cauchy's Integral Theorem]] | |
| * [[math:analysis:complex:residue|Residue Theorem]] | |
| * [[math:analysis:complex:laurent|Laurent Series]] | |
| * [[math:analysis:complex:liouville|Liouville's Theorem]] | |
| * [[math:analysis:complex:maximum|Maximum Modulus Principle]] | |
| * [[math:analysis:functional|Functional Analysis]] | |
| * [[math:analysis:functional:baire|Baire Category Theorem]] | |
| * [[math:analysis:functional:banach-steinhaus|Uniform Boundedness Principle]] | |
| * [[math:analysis:functional:open-mapping|Open Mapping Theorem]] | |
| * [[math:analysis:functional:closed-graph|Closed Graph Theorem]] | |
| * [[math:analysis:functional:hahn-banach|Hahn-Banach Theorem]] | |
| * [[math:analysis:functional:spectral|Spectral Theory]] | |
| |
| ==== Geometry and Topology ==== | |
| * [[math:geometry:euclidean|Euclidean Geometry]] | |
| * [[math:geometry:euclidean:pythagoras|Pythagorean Theorem]] | |
| * [[math:geometry:euclidean:congruence|Triangle Congruence Theorems]] | |
| * [[math:geometry:euclidean:similarity|Similar Triangles]] | |
| * [[math:geometry:euclidean:circles|Circle Theorems]] | |
| * [[math:geometry:non-euclidean|Non-Euclidean Geometry]] | |
| * [[math:geometry:non-euclidean:hyperbolic|Hyperbolic Geometry]] | |
| * [[math:geometry:non-euclidean:elliptic|Elliptic Geometry]] | |
| * [[math:geometry:differential|Differential Geometry]] | |
| * [[math:geometry:differential:curves|Theory of Curves]] | |
| * [[math:geometry:differential:surfaces|Theory of Surfaces]] | |
| * [[math:geometry:differential:manifolds|Manifolds]] | |
| * [[math:geometry:differential:gauss-bonnet|Gauss-Bonnet Theorem]] | |
| * [[math:topology:point-set|Point-Set Topology]] | |
| * [[math:topology:point-set:metric-spaces|Metric Spaces]] | |
| * [[math:topology:point-set:continuity|Continuous Functions]] | |
| * [[math:topology:point-set:compactness|Compactness]] | |
| * [[math:topology:point-set:connectedness|Connectedness]] | |
| * [[math:topology:algebraic|Algebraic Topology]] | |
| * [[math:topology:algebraic:fundamental-group|Fundamental Group]] | |
| * [[math:topology:algebraic:covering-spaces|Covering Spaces]] | |
| * [[math:topology:algebraic:homology|Homology Groups]] | |
| * [[math:topology:algebraic:cohomology|Cohomology Theories]] | |
| * [[math:topology:algebraic:fixed-point|Fixed Point Theorems]] | |
| |
| ==== Applied Mathematics ==== | |
| * [[math:applied:probability|Probability Theory]] | |
| * [[math:applied:probability:axioms|Kolmogorov Axioms]] | |
| * [[math:applied:probability:conditional|Conditional Probability]] | |
| * [[math:applied:probability:distributions|Probability Distributions]] | |
| * [[math:applied:probability:limit-theorems|Limit Theorems]] | |
| * [[math:applied:statistics|Statistics]] | |
| * [[math:applied:statistics:estimation|Parameter Estimation]] | |
| * [[math:applied:statistics:hypothesis|Hypothesis Testing]] | |
| * [[math:applied:statistics:regression|Regression Analysis]] | |
| * [[math:applied:differential-equations|Differential Equations]] | |
| * [[math:applied:differential-equations:first-order|First-Order ODEs]] | |
| * [[math:applied:differential-equations:linear|Linear ODEs]] | |
| * [[math:applied:differential-equations:systems|Systems of ODEs]] | |
| * [[math:applied:differential-equations:pde|Partial Differential Equations]] | |
| * [[math:applied:numerical|Numerical Analysis]] | |
| * [[math:applied:numerical:interpolation|Interpolation]] | |
| * [[math:applied:numerical:integration|Numerical Integration]] | |
| * [[math:applied:numerical:optimization|Optimization Methods]] | |
| |
| ==== Famous Theorems and Problems ==== | |
| * [[math:famous:fermat|Fermat's Last Theorem]] | |
| * [[math:famous:four-color|Four Color Theorem]] | |
| * [[math:famous:poincare|Poincaré Conjecture]] | |
| * [[math:famous:riemann|Riemann Hypothesis]] | |
| * [[math:famous:p-np|P vs NP Problem]] | |
| * [[math:famous:goldbach|Goldbach Conjecture]] | |
| * [[math:famous:twin-prime|Twin Prime Conjecture]] | |
| |
| ===== Philosophical Arguments and Proofs ===== | |
| |
| ==== Metaphysics ==== | |
| * [[philosophy:metaphysics:ontology|Ontological Arguments]] | |
| * [[philosophy:metaphysics:ontology:anselm|Anselm's Ontological Argument]] | |
| * [[philosophy:metaphysics:ontology:descartes|Descartes' Ontological Argument]] | |
| * [[philosophy:metaphysics:ontology:modal|Modal Ontological Arguments]] | |
| * [[philosophy:metaphysics:ontology:plantinga|Plantinga's Ontological Argument]] | |
| * [[philosophy:metaphysics:causation|Causation]] | |
| * [[philosophy:metaphysics:causation:cosmological|Cosmological Arguments]] | |
| * [[philosophy:metaphysics:causation:kalam|Kalam Cosmological Argument]] | |
| * [[philosophy:metaphysics:causation:leibniz|Leibniz's Cosmological Argument]] | |
| * [[philosophy:metaphysics:causation:regress|Infinite Regress Arguments]] | |
| * [[philosophy:metaphysics:mind|Philosophy of Mind]] | |
| * [[philosophy:metaphysics:mind:dualism|Arguments for Dualism]] | |
| * [[philosophy:metaphysics:mind:physicalism|Arguments for Physicalism]] | |
| * [[philosophy:metaphysics:mind:consciousness|Explanatory Gap Arguments]] | |
| * [[philosophy:metaphysics:mind:qualia|Knowledge Argument (Mary's Room)]] | |
| * [[philosophy:metaphysics:mind:zombies|Zombie Arguments]] | |
| |
| ==== Epistemology ==== | |
| * [[philosophy:epistemology:knowledge|Nature of Knowledge]] | |
| * [[philosophy:epistemology:knowledge:justified-true-belief|Knowledge as Justified True Belief]] | |
| * [[philosophy:epistemology:knowledge:gettier|Gettier Problems]] | |
| * [[philosophy:epistemology:knowledge:reliabilism|Reliabilist Theories]] | |
| * [[philosophy:epistemology:knowledge:coherentism|Coherentist Theories]] | |
| * [[philosophy:epistemology:knowledge:foundationalism|Foundationalist Theories]] | |
| * [[philosophy:epistemology:skepticism|Skeptical Arguments]] | |
| * [[philosophy:epistemology:skepticism:brain-in-vat|Brain in a Vat]] | |
| * [[philosophy:epistemology:skepticism:evil-demon|Descartes' Evil Demon]] | |
| * [[philosophy:epistemology:skepticism:dream|Dream Argument]] | |
| * [[philosophy:epistemology:skepticism:responses|Responses to Skepticism]] | |
| * [[philosophy:epistemology:induction|Problem of Induction]] | |
| * [[philosophy:epistemology:induction:hume|Hume's Problem of Induction]] | |
| * [[philosophy:epistemology:induction:goodman|Goodman's New Riddle of Induction]] | |
| |
| ==== Ethics ==== | |
| * [[philosophy:ethics:normative|Normative Ethics]] | |
| * [[philosophy:ethics:normative:utilitarianism|Utilitarian Arguments]] | |
| * [[philosophy:ethics:normative:deontology|Kantian Arguments]] | |
| * [[philosophy:ethics:normative:virtue|Virtue Ethics]] | |
| * [[philosophy:ethics:normative:natural-law|Natural Law Theory]] | |
| * [[philosophy:ethics:metaethics|Metaethics]] | |
| * [[philosophy:ethics:metaethics:moral-realism|Moral Realism]] | |
| * [[philosophy:ethics:metaethics:error-theory|Error Theory]] | |
| * [[philosophy:ethics:metaethics:emotivism|Emotivism]] | |
| * [[philosophy:ethics:metaethics:prescriptivism|Prescriptivism]] | |
| * [[philosophy:ethics:applied|Applied Ethics]] | |
| * [[philosophy:ethics:applied:trolley|Trolley Problem]] | |
| * [[philosophy:ethics:applied:value|Value of Life Arguments]] | |
| * [[philosophy:ethics:applied:consequentialism|Implications of Consequentialism]] | |
| |
| ==== Logic and Language ==== | |
| * [[philosophy:logic:formal|Formal Logic]] | |
| * [[philosophy:logic:formal:propositional|Propositional Logic]] | |
| * [[philosophy:logic:formal:predicate|Predicate Logic]] | |
| * [[philosophy:logic:formal:modal|Modal Logic]] | |
| * [[philosophy:logic:formal:non-classical|Non-Classical Logics]] | |
| * [[philosophy:logic:informal|Informal Logic and Fallacies]] | |
| * [[philosophy:logic:informal:deductive|Deductive Fallacies]] | |
| * [[philosophy:logic:informal:inductive|Inductive Fallacies]] | |
| * [[philosophy:logic:informal:relevance|Relevance Fallacies]] | |
| * [[philosophy:language:meaning|Theories of Meaning]] | |
| * [[philosophy:language:meaning:reference|Reference Theories]] | |
| * [[philosophy:language:meaning:use|Use Theories]] | |
| * [[philosophy:language:meaning:verification|Verification Principle]] | |
| * [[philosophy:language:meaning:grice|Gricean Pragmatics]] | |
| |
| ==== Philosophy of Science ==== | |
| * [[philosophy:science:demarcation|Demarcation Problem]] | |
| * [[philosophy:science:demarcation:popper|Popper's Falsificationism]] | |
| * [[philosophy:science:demarcation:kuhn|Kuhn's Paradigm Shifts]] | |
| * [[philosophy:science:explanation|Scientific Explanation]] | |
| * [[philosophy:science:explanation:deductive-nomological|Deductive-Nomological Model]] | |
| * [[philosophy:science:explanation:causal|Causal Accounts]] | |
| * [[philosophy:science:realism|Scientific Realism vs. Anti-Realism]] | |
| * [[philosophy:science:realism:no-miracles|No Miracles Argument]] | |
| * [[philosophy:science:realism:pessimistic-induction|Pessimistic Induction]] | |
| |
| ===== Interdisciplinary Topics ===== | |
| |
| ==== Mathematics and Philosophy Connections ==== | |
| * [[interdisciplinary:foundations|Philosophical Foundations of Mathematics]] | |
| * [[interdisciplinary:platonism|Mathematical Platonism]] | |
| * [[interdisciplinary:formalism|Formalism]] | |
| * [[interdisciplinary:intuitionism|Intuitionism]] | |
| * [[interdisciplinary:structuralism|Structuralism]] | |
| |
| ==== Mathematics and Science ==== | |
| * [[interdisciplinary:physics|Mathematical Physics]] | |
| * [[interdisciplinary:biology|Mathematical Biology]] | |
| * [[interdisciplinary:social|Mathematical Models in Social Sciences]] | |
| |
| ===== Resources ===== | |
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| ==== Teaching Resources ==== | |
| * [[resources:teaching:undergraduate|Undergraduate Mathematics]] | |
| * [[resources:teaching:graduate|Graduate Mathematics]] | |
| * [[resources:teaching:philosophy|Philosophy Courses]] | |
| |
| ==== Reference Materials ==== | |
| * [[resources:reference:notation|Mathematical Notation Guide]] | |
| * [[resources:reference:symbols|Symbol Reference]] | |
| * [[resources:reference:glossary|Glossary of Terms]] | |
| * [[resources:reference:bibliography|Comprehensive Bibliography]] | |
| |
| ===== Proof Templates ===== | |
| * [[templates:mathematical|Mathematical Proof Template]] | |
| * [[templates:philosophical|Philosophical Argument Template]] | |
| * [[templates:references|Reference and Citation Template]] | |
| * [[templates:theorem|Theorem Template]] | |
| * [[templates:lemma|Lemma Template]] | |
| |
| ===== Recent Additions ===== | |
| * //2025-03-15 - Wiki structure established// | |
| * //2025-03-15 - Initial proof templates created// | |
| * //2025-03-15 - Mathematical expression formatting guide added// | |
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| ===== How to Contribute ===== | |
| |
| We welcome contributions from mathematicians, philosophers, educators, and enthusiasts. To contribute: | |
| |
| * **Follow our standards**: Review our [[wiki:proofstyle|proof formatting standards]] | |
| * **Use proper notation**: Ensure mathematical expressions use LaTeX via the MathJax plugin | |
| * **Provide references**: Include sources for all theorems, proofs, and arguments | |
| * **Maintain organization**: Follow our established namespace structure | |
| * **Consider accessibility**: Write clear explanations that build on prerequisites | |
| * **Add value**: Provide historical context, alternative proofs, and applications | |
| |
| For substantial contributions, please: | |
| * Contact an editor to discuss the addition | |
| * Submit drafts for peer review | |
| * Be open to feedback and revisions | |
| |
| ===== Editorial Team ===== | |
| |
| This encyclopedia is maintained by an editorial team of mathematicians and philosophers. For questions, suggestions, or to join the editorial team, please contact the administrators. | |
| |
| ===== Contact ===== | |
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| For questions or assistance with this wiki, please contact [[mailto:ayman@aymanlafaz.com|ayman@aymanlafaz.com]]. | |
