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math-debrief
  • Math Debrief
  • Math: TIMELINE
  • 100-fundamentals
    • debrief-name: math section-code: 000 section-name: general section-desc: Elementary topics pervasive
      • About Mathematics
      • abstraction-in-math
      • About Math
      • Axiom schema
      • Basic concepts in math
      • Collections
      • Elementary concepts in objects
      • Elements of mathematics
      • math-as-a-language
      • Mathematical structures
      • List of mathematics-based methods
      • Mathematics and Reality
      • Mathematics: General
      • Controversial mathematics
      • the-elements-of-math
      • What is mathematics
    • The foundation of mathematics
      • Mathematical foundations
      • Foundations of Mathematics
      • Axiomatization of mathematics
      • Foundational crisis of mathematics
      • Foundations
      • Hilbert's problems
      • impl-of-math-in-set-theory
      • Gödel's Incompleteness Theorem
      • Theorems in the foundations of mathematics
      • The list of FOM candidates
      • Logicism
    • Philosophy of mathematics
      • Constructive mathematics
      • Constructive mathematics
      • Metamathematics
      • Philosophy of mathematics
      • Schools of mathematics
    • terms
      • Terms
      • Arithmetic
      • Axiom
      • The Axiomatic Method
      • discrete-math
      • 201 Discrete mathematics
      • Euclidean space
      • Formal system
      • Function
      • Generalization
      • Geometry
      • Higher-order
      • Impredicativity
      • Level of measurement
      • Mathematical definition
      • FAQ
      • Mathematical function
      • Mathematical induction
      • Mathematical object
      • Mathematical object
      • Equivalent definitions of mathematical structures
      • Mathematics
      • Mathematical model
      • mathematical-notation
      • Mathematical pages
      • Mathematical terminology
      • Mathematical adjective
      • Numbers
      • plane
      • Primer: Set Theory
      • Mathematical primitive
      • Set
      • Space
      • theory
      • Variable
  • 200 Set and Set theory
    • Sets: Hierarchy
    • set.TERMS
    • SETS › TOPICS
    • 201 Set concepts
      • Mathematical collections
      • The notion of sets
      • Specification of sets
    • Set cardinality
      • Cardinality of the continuum
      • Cardinality
      • Set Cardinality
      • cardinality2
      • Set cardinality
    • Set operations
      • Disjoint sets
      • Overlapping sets
      • Product
      • set-interactions
      • Set qualities
      • Set relations
    • Set properties
      • Basic set properties
      • Set properties
    • Set relations
      • Basic set relations
      • Disjoint sets
      • Inclusion relation
      • Membership Relation
      • Set membership
    • Summary
      • Set FAQ
      • Sets: Summary
    • Set theories
      • Axiomatic set theory
      • Set Theories
      • Naive Set Theory
      • Morse-Kelley set theory
      • von Neumann-Bernays-Gödel Set Theory
      • Quine's New Foundations (NF)
      • Cantor's set theory
      • Zermelo-Fraenkel set theory
    • Axioms of set theory
      • axiom-of-choice
      • The Axiom of Extensionality
      • Axiom of infinity
      • axiom-of-pairing
      • Axiom of powerset
      • Axiom of Regularity
      • Axiom of replacement
      • Axiom of union
      • Axiom of well-ordering
      • axiom-schema-of-comprehension
      • Axiom Schema of Specification
      • Axioms of set theories
      • List of axioms in set theory
      • ZFC Axioms
    • Sets: Terms
      • Bell Number
      • Cardinal number
      • Class (set theory)
      • Closure
      • empty-set
      • Extended set operations
      • Extensions by definitions
      • Family of sets
      • Fundamental sets
      • fundamental-sets2
      • Georg Cantor
      • History of set theory
      • Implementation of mathematics in set theory
      • Indexed family of sets
      • Extensional and intensional definitions
      • Involution
      • list-of-axioms-of-set-theory
      • Implementation of mathematics in set theory
      • Set membership
      • Naive Set Theory
      • Number of relations
      • empty-relation
      • Set Partitioning
      • Powerset
      • Russell's paradox
      • Set-builder notation
      • Set equivalence
      • Set Notation in latex
      • Set notation
      • Set partition
      • Intensional and extensional set specification
      • Set notation
      • Basic concepts in set theory
      • set-theory
      • Set Types
      • set
      • subset
      • Transfinite number
      • Tuples
      • ur-elements
  • Relations
    • basic-concepts
      • algebraic-axioms
      • Elements of a relation
      • Types of Relations
      • Named Relations
      • Relation theory
      • Relations
      • Types of relations
    • Relations
      • Definitions
      • Reflexivity
      • Symmetry
      • Transitivity
    • relation-properties
      • Uniqueness properties of relations
    • Types of relations
      • Transitivity
      • Binary Relation
      • Congruence relation
      • Connex relation
      • axioms-sets-zfc
      • Endorelation
      • Equivalence relation
      • Euclidean
      • Finitary relation
      • Heterogeneous relation
      • Homogeneous relation
      • Transitivity
      • Partial equivalence relation
      • Transitivity
      • Transitivity
      • Reflexive relation
      • Reflexivity
      • Index of relations
      • Serial relation
      • Symmetry
      • Transitivity
      • Ternary relation
      • Trichotomy
      • Universal relation
      • Well-foundedness
    • terms
      • Relations
      • Binary relation
      • Relations
      • _finitary-rel
      • Relations: Overview
      • Relations
      • Index of relations
      • Binary relations
      • Composition of relations
      • Equivalence class
      • Notation
      • Relation
      • Relations
      • Sets: Summary
      • Aggregation: Sets, Relations, Functions
  • Order theory
    • Order theory
    • List of order structures in mathematics
    • List of order theory topics
    • Order theory
      • Hasse diagram
      • Order theory
      • ordered-set
      • Partial order
      • Partially ordered set
      • Total order
  • Function Theory
    • Function Theory: GLOSSARY
    • Function Theory: HIERARCHY
    • Function Theory: LINKS
    • Function Theory: TERMS
    • Function Theory: TOPIC
    • Function Theory: WIKI
    • _articles
      • about-functions
      • Function
      • Formal definition
      • Definition
      • constant
      • Introduction
      • Types of functions
      • Functions: Summary of Notations
      • Functions: Overview
      • Properties of functions
      • Function properties
      • Functions: Summary
      • Function
    • Abjections
      • Bijective function
      • Function (abjections)
      • Injective function
      • Surjective function
    • topics
      • Function: TERMS
      • Codomain
      • Composition of functions
      • Currying
      • Division of functions
      • Domain
      • Function fixed points
      • Function cardinality
      • Function definition
      • Elements of a function
      • Function in mathematics
      • Function notion
      • Function operations
      • Function properties
      • Functional statements
      • Functions in programing languages
      • Image and Preimage
      • Image
      • Inverse function
      • Notion of functions
      • Number and types of functions between two sets
      • Operation
      • Range
      • Successor function
      • Time complexity classes
  • debrief-name: math section-code: 280 section-name: domain-theory section-desc:
    • Domain theory: LINKS
    • Domain theory
  • Logic
    • Logic: CHRONO TERMS
    • Logic: CLUSTERS
    • lo.GLOSSARY
    • Logic: Wiki links
    • 305-basic-concepts
      • Introduction to Logic
      • Argumentation
      • Logic: Basic terminology
      • Logic: Terminology
      • Truth function
      • Truth function
    • README
      • Mathematical Logic
      • Types of Logic
      • BHK interpretation
      • FOL
      • Index of Logic Forms
      • History of logic
      • Logic Indices
      • Interpretation of symbols in logic and math
      • logic-systems
      • Mathematical Logic: People and Events
      • Index of logical fallacies
      • Logical symbols
      • Mathematical conjecture
      • Mathematical induction
      • Mathematical lemma
      • Mathematical Logic
      • Mathematical proof
      • Mathematical theorem
      • Mathematical theory
      • Monotonicity of entailment
      • Satisfiability Modulo Theories
      • Sequent Calculus
      • Sequent
      • Tableaux
      • Truth tables
    • 360-propositional-logic
      • Propositional Logic
      • Propositional Logic
    • 370-predicate-logic
      • Predicate Logic
      • First-order logic
      • Predicate calculus
      • Examples of predicate formulae
    • 380-proof-theory
      • Argument-deduction-proof distinctions
      • Direct proof
      • Mathematical induction
      • Mathematical induction
      • Mathematical proof
      • Natural deduction
      • Natural deduction
      • Proof by induction
      • Proof by induction
      • proof-calculus
      • Proof Theory
      • Structural induction
      • System L
      • Proof theory
    • Logic: Indices
      • Gödel's Incompleteness Theorem
      • The History of Mathematical Logic
      • forallx
      • Logic for CS
      • Lectures in Logic and Set Theory
      • _logicomix
    • Logic
      • Logical connectives
      • Logical equivalence
    • Rules of Inference
      • WIKI
      • Conjunction elimination
      • Conjunction introduction
      • Cut rule
      • Disjunction elimination
      • Disjunction introduction
      • Disjunctive syllogism
      • Exportation
      • implication-elimination
      • implication-introduction
      • Rules of Inference: Index
      • Rules of inference
      • Rules of Inference for Natural Deduction
      • Logical Inference
      • Reiteration
      • Rule of inference
      • Structural rules
      • substitution
    • Logic
      • The principle of bivalence
      • The principle of explosion
      • The Law of Identity (ID)
      • Laws of thought
      • Properties of logic systems
      • List of laws in logic
      • The law of non-contradiction
    • Logic
      • Logic systems: LINKS
      • Logic system
      • logic-systems
      • logic-typ
      • logics-by-purpose
      • _logics
      • Affine logic
      • Algebraic logic
      • Bunched logic
      • Classical logic
      • Traditional first-order logic
      • Hoare logic
      • Linear logic
      • Modal logic
      • Non-monotonic logic
      • Syntax
      • Predicate logic
      • Propositional Logic
      • Relevance logic
      • Separation logic
      • Substructural logics
      • Syllogistic logic
    • Logic: Sections: Elementary
    • Logic: Topics
      • Pages in Logic
      • Logic ❱ Terms ❱ List
      • Logic ❱ Terms ❱ Definitions
      • Absoluteness
      • Assumption
      • Automated theorem proving
      • Canonical normal form
      • Categorical proposition
      • Classical linear logic
      • Consequence
      • Decidability
      • Deduction systems
      • deduction-theorem
      • Deductive reasoning
      • Diagonal lemma
      • Fallacy
      • Fitch notation
      • Formal language
      • formal-system
      • Formalism
      • Formula
      • functionally complete
      • Hilbert system
      • Hoare logic
      • horn-clause
      • Mathematical induction
      • Induction
      • Inductive Reasoning
      • Intuitionistic logic
      • Intuitionistic logic
      • Intuitionistic logic
      • Judgement
      • Judgments
      • Linear logic
      • Logic in computer science
      • Logic
      • Logical connective
      • Logical consequence
      • Logical constant
      • Logical form
      • axioms-sets
      • Logical reasoning
      • Ludics
      • Non-logical symbol
      • Predicate
      • Premise
      • Quantification
      • Realizability
      • Boolean satisfiability problem
      • DPLL algorithm
      • Satisfiability
      • Semantics of logic
      • Skolemization
      • SAT and SMT
      • Syntax
      • Tautology
      • Term
      • Unification
      • Validity
  • 510 Lambda Calculi
    • Lambda Calculus: GLOSSARY
    • Lambda calculi: LINKS
    • Lambda Calculus: OUTLINE
    • Lambda Calculus: Basic concepts
      • Introduction
      • Lambda expressions
      • Free variables
    • Lambda Calculi
      • Lambda calculus: LINKS
      • Lambda calculus combinators in Haskell
      • Lambda calculus: Combinators
      • Combinators
      • combos-all.js
      • combos-bird.js
      • combos-birds-list.js
      • combos-birds.js
      • Fixed-point combinator
      • Fixpoint operator
      • Lambda calculus: Fixpoint
    • combinatory-logic
      • algebraic-structures
      • Combinatory logic
      • Combinatory logic
      • relation-classification
      • 04-definition
    • Lambda calculus encoding schemes
      • bohm-berarducci-encoding
      • Index of Church encodings
      • Church encodings
      • Church Numerals
      • Encoding data structures
      • Encoding schemes in lambda calculi
      • Lambda encoding
      • Mogensen-Scott encoding
      • Parigot encoding
      • encodings
        • Encoding data structures
        • Encoding of Data Types in the λ-calculus
        • church-booleans
        • Church data structures
        • Church encoding
        • Church Numerals: Church encoding of natural numbers
        • Lambda Calculus: Church encoding
        • Lambda Calculus: Church encoding
        • church-numerals
        • Lambda Calculus: Church encoding: Numerals
        • Church pair
        • Pair
        • Lambda Calculus: Church encoding
        • Alternative encodings
        • Encoding schemes
        • Encoding schemes
        • Encodings in Untyped Lambda Calculus
        • Lambda calculus
        • Scott encoding
        • Lambda calculus: Scott encoding
    • lambda-calculus-evaluation
      • Call-by-name
      • Call-by-need
      • Call-by-value
    • lambda-calculus-forms
      • Beta normal form
      • Lambda terms
      • Fixity of lambda-terms
    • lambda-calculus-reductions
      • Alpha conversion
      • Beta reduction
      • Delta reduction
      • Eta conversion
      • Eta conversion
      • Lambda calculus: η-conversion
    • lambda-calculus
      • Alonzo Church
      • Inference rules for lambda calculus
      • Lambda Calculus: Introduction
      • Lambda abstraction
      • Lambda application
      • Lambda Calculus: Definition
      • About λ-calculus
      • Type inference
      • Lambda Calculus
      • Lambda Calculus: Introduction
      • Introduction to λ-calculus
      • Lambda calculus
      • Definition of Lambda Calculus
      • Functions in lambda calculus
      • History of Lambda Calculus
      • Using the Lambda Calculus
      • Name capturing
      • Variable occurrences
      • Variables
    • Lambda Calculus
      • Church-Rosser theorem
      • Curry's paradox
      • De Bruijn index
      • de Bruijn notation
      • Deductive lambda calculus
      • Kleene-Rosser paradox
      • Aspects of the lambda calculus
      • Function Refactoring
      • Lambda lifting
      • Let expression
      • Reduction strategy
      • Substitution
    • typed-lambda-calculi
      • Lambda Cube
      • Simply typed lambda calculus
      • System F
      • Typed lambda calculi
  • Type theory
    • Type Theory: GLOSSARY
    • Type theorists
    • Type Theory: SUMMARY
    • TERMS: Type Theory
      • Types
      • History of type theory
      • History of Type Theory
    • curry-howard-correspondence
      • The Curry-Howard Correspondence in Haskell
      • Curry-Howard correspondence
      • Curry-Howard correspondence
      • Curry-Howard correspondence
      • Curry-Howard-Lambek correspondence - HaskellWiki
    • dependent-types
      • Dependent type
      • Dependent type
    • Hindley-Milner type-system
      • Hindley-Milner type system
      • Monomorphism vs polymorphism
      • Let-polymorphism
      • The Hindley-Milner type system
      • Algorithm W in Haskell
      • Hindley-Milner Type Inference: W Algorithm
      • hindley–milner-type-system
      • Hindley-Milner type system
      • HM inference examples
      • HM in ML
      • Type Inference
    • Homotopy type theory
      • Homotopy type theory
      • Univalent Type theory as the foundations of mathematics
    • Intuitionistic type theory
      • Inductive definition
      • Inductive type
      • Intuitionistic type theory
    • Type Theory
      • TTTools
      • Coinduction
      • Impredicativity
      • Lean
      • Subsumption
    • Type Theory : Topics
      • Type Theory : Terms
      • Recursion types
      • Recursive data type
      • Subtyping
      • Type Class
      • Type Equivalence
      • Type Inference
      • Type rule
      • Type system
      • Variance
    • type-theories
      • Calculus of Constructions
      • Constructive type theory
      • ramified-type-theory
      • simple-type-theory
      • Substructural type systems
    • type-theory-general
      • Linear types
      • History of Type Theory
      • Type Theory
      • Overview
      • Type Theory
  • Abstract Algebra
    • 410-group-theory
      • Abelian group
    • algebras
      • Associative Algebra
      • Field
      • Group-like algebraic structures
      • group
      • Lattice
      • Magma
      • monoid
      • Overview of Algebras
      • Quasigroup
      • Rack and quandle
      • Ring
      • Semigroup
      • Algebra of sets
      • Setoid
    • boolean-algebra
      • Boolean algebra
      • Axioms in Boolean Algebra
      • Boolean algebra
      • Boolean Algebra Laws
      • Boolean Algebra Laws
      • Two-element Boolean algebra
      • Boolean algebra
      • Boolean domain
    • terms
      • Algebra
      • Axioms of abstract algebra
      • Algebraic notation for algebraic data types
      • Algebraic structure
      • Algebraic structure
      • Field of sets
      • Homomorphism
      • Isomorphism
      • Algebraic structures
      • Mathematical structure
      • Polynomials
      • Relation algebra
  • Category Theory
    • CT GLOSSARY
    • Category Theory: OUTLINE
    • CT SUMMARY
    • A First Introduction to Categories (2009)
      • Sets, maps, composition
      • 02-history
      • axioms-logic
      • Bijection of functions
      • Commutative diagram
      • Directed graph
      • CT prerequisites
      • String diagram
      • Transitive closure
    • Category Theory Fundamentals
      • Introduction
      • Interpretation
      • Fundamental concepts
      • Category theory
      • Category
      • Category Theory: Definitions
    • Key concepts
      • Duality
      • Functor
      • Homeset
      • Initial Object
      • Morphism
      • Natural transformation
      • Object
      • Terminal Object
    • Categorical constructions
      • Categorical constructions
      • Coproduct
      • Diagram
      • Product
      • Universal construction
    • Types of categories
      • Concrete category
      • Discrete category
      • Functor category
      • Groupoid
      • Hask
      • Kleisli category
      • Locally small category
      • Monoid
      • monoidal-categories.md
      • Index of named categories
      • Opposite category
      • Ordered category
      • Set category
      • Small category
      • Subcategory
    • Types of Functors
      • Adjoint functor
      • relation-arity
      • Endofunctor
      • Faithful functor
      • Forgetful functor
      • Hom functor
      • Identity functor
      • Inverse functor
      • Monad
      • Powerset functor
    • Types of Morphisms
      • Anamorphism
      • Automorphism
      • Catamorphism
      • Endomorphism
      • Epimorphism
      • Homomorphism
      • Hylomorphism
      • Idempotent morphism
      • Identity morphism
      • Inverse morphism
      • Isomorphism
      • Metamorphism
      • monomorphism
      • Natural isomorphism
      • Split morphism
    • 20-advanced-concepts
      • Coalgebra
      • (Co)Inductive types
      • Recursion Schemes
    • Category Theory
      • Category Theory: TERMS
      • Algebraic Data Types
      • Category Theory
      • Category
      • Coproduct
      • Function type
      • Functoriality
      • Initial Object
      • Limits and Colimits
      • Natural Transformation
      • 5. Products
      • Terminal Object
    • Category Theory :: Contents
      • CT :: Links
      • Category Theory :: Terms
      • Category :: Definition
      • F-Algebra
      • Functor
      • Initial object
      • Monoid
      • Natural Transformation
      • Number of morphisms
      • Terminal object
      • Transitive closure
      • Types of morphisms
      • Categories by cardinality
      • Types of functors
  • Number Theory
    • Invariance and Monovariance Principle
    • 615-arithmetic
      • Addition
      • Aliquot sum
      • Arithmetic function
      • Laws
      • Arithmetic operations
      • Index of arithmetic operations
      • Arithmetic operations
      • Arithmetic
      • Divisibility rules
      • Divisibility
      • division
      • Divisor Function
      • Divisor Summatory Function
      • Divisor
      • Euclidean division
      • Hyperoperations
      • hyperops
      • Modular arithmetic
      • Multiplication
      • Number Theory: primer in numbers
      • Percentage
      • Rules of Divisibility
      • Subtraction
    • The fundamental sets of numbers
      • Algebraic numbers
      • Complex numbers
      • Fractions
      • Fundamental number sets
      • Imaginary numbers
      • Integers
      • Irrational numbers
      • Natural number
      • Rational numbers
      • Real numbers
      • Transcendental numbers
      • Ulam's spiral
      • The whole numbers
    • COUNTING THEORY
      • Counting Theory
      • counting
      • Fundamental Counting Rules
    • 630-combinatorics
      • Combinatorics
      • Combinations
      • Combinatorics
      • Counting theory
      • Counting theory
      • Enumerative combinatorics
      • Partition
      • Pascals triangle
      • Permutations
      • Twelvefold way
    • Probability theory
      • Statistics › Probability theory: Glossary
      • Statistics › Probability theory › Topics
      • Statistics › Probability theory › Wiki Links
      • Conditional Probability
      • Distribution
      • Probability theory
      • Probability
    • Number theory
      • euclids-lemma
      • gcd-lcm
      • Induction
      • Infinity
      • Numbers and numerals with interesting properties
      • Lagrange's four-square theorem
      • Matrix
      • Matrix
      • List of Number Systems
      • Number Theory
      • Number Theory with Glenn Olsen
      • Number
      • Arithmetic
      • Numbers
      • numeral-prefixes
      • Numeral system
      • Numeral
      • Ordinal numbers
      • Parity
      • Peano axioms
      • Polynomial
      • Polynomial
      • Positional notation
      • Probability
      • Symbol
      • Well Ordering Principle
    • topics
      • Coprimality
      • Facorization of composite numbers
      • Fundamental Theorem of Arithmetic
      • Prime factorisation
      • Prime number
      • Prime numbers
  • Theory of computation
    • Theory of computation: Abbreviations
    • Theory of computation: CHRONOLOGICAL TOPICS
    • Theory of computation: GLOSSARY
    • Theory of Computation: HIERARCHY
    • Theory of computation: LINKS
    • Theory of computation: TERMS
    • Theory of computation: TOPICS
    • Theory of computation: WIKI
    • Theory of Computation
      • _toc-more
      • Theory of Computation
    • 610-automata-theory
      • Abstract machine
      • Automata Theory
      • Automaton
      • Edit distance
      • Finite-state Machine
      • Automata Theory: WIKI
    • Formal systems
      • Abstract interpretation
      • Alphabet
      • Binary combinatory logic
      • Chomsky hierarchy
      • Epsilon calculus
      • Formal language
      • Iota and Jot
      • Regular expression
      • Regular Language
      • SKI combinator calculus
    • 621-grammar
      • Backus-Naur Form (BNF)
      • Context-free grammar
      • Context-sensitive grammar
      • Extended Backus–Naur Form (EBNF)
      • Regular Language
      • Terminal and nonterminal symbols
    • 622-syntax
      • Syntax
    • 624-semantics
      • Axiomatic semantics
      • Denotational Semantics: Summary
      • Denotational Semantics
      • Denotational Semantics
      • Denotational semantics
      • Formal semantics
      • Operational semantics
      • Semantics in CS
      • Semantics
    • 630-computability-theory
      • Computability (recursion) theory: TERMS
      • Computability (recursion) theory: TOPICS
      • Effective Computability
      • Church Thesis
      • Church-Turing Thesis
      • Computability theory
      • Computability
      • Computable function
      • Entscheidungsproblem
      • Halting problem
      • Machine that always halts
      • McCarthy Formalism
      • Super-recursive algorithm
      • Recursion theory
    • 632-recursive-function-theory
      • Recursion Theory
      • Ackermann function
      • General recursive function
      • Minimization operator
      • Partial functions
      • Recursion Function Theory
      • Sudan function
    • 634-primitive-recursive-functions
      • Primitive Recursive Function
      • Initial functions
      • The list of primitive recursive functions
      • Primitive combination
      • Primitive composition
      • Primitive recursion
      • Successor function
    • 640-models-of-computation
      • Models of computation: Summaries
      • Model of computation
    • 680-complexity-theory
      • Algorithmic Complexity
      • Complexity Theory
  • debrief-name: math section-code: 900 section-name: aggregations section-desc: Aggregations, indices,
    • Index of closures
    • List of mathematical entities
    • List of mathematical objects
    • Enumeration of mathematical structures
    • Math : Axioms as Formulae
    • 950-math-areas
      • Areas of mathematics
      • Areas of mathematics
    • 970-links
      • check
      • Math: Links
      • Math Debrief: Links
      • Math Primer: LINKS
      • Links
      • Math: LINKS: ncatlab
      • Math: LINKS
      • WIKI
      • WIKI
      • WIKI_ALL
      • Math: Wiki lists
      • Glossary of areas of mathematics
      • WIKI_collections
      • Mathematics for Computer Science
      • Mathematics Classification
      • math
      • Resources
      • Math on YouTubel Video Playlists
      • wiki resources
    • 980-hierarchy
      • HIERAR
      • Math: Hierarchy
      • Math HIERARCHY
      • classification
        • Mathematics
        • https://ncatlab.org/nlab/all_pages https://ncatlab.org/nlab/all_pages/reference https://ncatlab.org/
        • Math Classification and Topical Pages
        • Areas of mathematics
        • Areas of mathematics
        • Math Classification: CCS
        • Math hierarchy
        • Computational mathematics
        • Taxonomy: Mathematics
        • Areas of mathematics
        • Mathematics Subject Classification
        • Math fields
        • math-topics
        • Mathematics Subject Classification – MSC
        • MSC Classification Codes
        • mss-top-levels-filenames
        • MSC classification: Top Levels
        • Math classification
    • 990-appendix
      • Math glossary at ENCYCLOPÆDIA BRITANNICA
      • Bibliography
      • Math: Abbreviations
      • math.GLOSSARY
    • Math : Canon
      • Main branches of mathematics
      • Enumeration: Math paradigms
      • enum-math-symbols
      • List of mathematical theories
      • enum-algebras
        • Group-like algebraic structures
        • Group
        • Groupoid
        • magma
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  1. 100-fundamentals
  2. debrief-name: math section-code: 000 section-name: general section-desc: Elementary topics pervasive

Mathematics: General

Mathematical elements:

  • mathematical primitive

  • mathematical axiom

  • mathematical definition

  • mathematical conjecture

  • mathematical proof

  • mathematical theorem

  • mathematical lemma

  • mathematical theory

  • Mathematical method

    • Primitives

    • Axioms

    • Inference rules

    • Rigorous argumentation

  • Although there is written evidence of mathematical activity in Egypt as early as 3000 BCE, many scholars place the birth of mathematics in ancient Greece around the 6th century BCE, when deductive proof was first introduced. Aristotle credited Thales of Miletus with recognizing the importance of not just what we know but how we know it, and finding grounds for knowledge in the deductive method. Around 300 BCE, Euclid codified a deductive approach to geometry in his treatise, the "Elements". Through the centuries, Euclid's axiomatic approach was held as a paradigm of rigorous argumentation, not just in mathematics, but in philosophy and other sciences as well. The "Elements" takes the second place of the most printed book of all times. Until the XIX century it was used as a proper textbook for geometry classes across Europe. (at around that time, the other, the so-called non-Euclidean, geometries were established, spawning out of the problematic Euclid's 5th axioms about parallel lines).

Math*=-related

  • invented or discovered

  • abstraction, generalization

  • analysis, synthesis

  • searching for patterns

  • abstracting the details

  • aggregating chunks of knowledge

  • eventually giving rise to a mathematical field

  • integration into the entire body of human knowledge

  • Mathematics is about finding patterns and lifting them into abstraction in its ever going search for truth.

  • Mathematical fields and disciplines are formed through the work of researches that, nowadays, publish their findings in dedicated academic journals.

  • Mathematical knowledge is expressed beginning with conjectures, which are then explored in an effort to prove them. The worst kind of conjecture is the one that can neither be proven nor disproven (Goldblach's) because mathematicians hate the middle. Some conjectures are known as hypothesis (Riehmann's) to mix things up. Those conjectures shown to be true go on to become theorems. Conjectures that are proven become theorems. A set of theorems constitutes a theory. A small theorem used as a steppingstone when proving a larger theorem is called a lemma (from old norse 'Lemmi').

  • Abstraction is the process of extracting the essence from a concept by severing its dependences on the real world, then generalizing it softly until it bleeds utility.

  • Platonic view of math: mathematical objects have existence independent of the human mind. (Whose mind? Yours? Mine? "Collective mind"?)

  • The goal of science is to understand what is, by describing and understanding the universe. Math, on the other hand, seeks to understand what must be.


  • Mathematicians investigate mathematical concepts, try to formulate conjectures and establish their truth by rigorous deduction from a set of axioms and previously defined theorems, all within a particular formal system.

  • A formal science is a branch of knowledge concerned with the properties of formal systems based on a set of axioms (definitions and rules of inference). Unlike other science, formal science is not concerned with the validity of theories based on observations in the physical world.

  • Logical reasoning guides us in understanding and constructing proofs; it helpes us reason about formal mathematical objects (numbers, lists, trees, formulas, etc., even theorems themselves).

  • Definitions define the objects of discussion. For instance, a straight line (versus a curved one) might be defined as "a line which lies evenly with the points on itself", as in Euclid.

  • Axioms describe what we can do with the things we've defined. For example, the axiom of symmetry says that "If A=B, then B=A". In this example, you could see the axiom as something you can do ("you can switch the sides of an equation") or as defining what it means that two things are equal.

  • On top of this foundation, mathematics is built with logic. Given the definitions and axioms, certain conclusions follow as inescapable consequences. These conclusions we call theorems.

  • Mathematics is not static, and the axioms and definitions we use are not discovered (as in they were here all along waiting to be revealed) but invented (as in pulled out of ass).

  • As we seek deeper understanding, we often come to a point where we realize our earlier understanding was incomplete or even incorrect, and we seek to fix the foundations. This has occurred over and over again in the history of mathematics.

  • Mathematics is a quest for understanding what must be, but the very concepts we try to understand are not set in stone. The objects of mathematics are defined by people, and as we understand them better, the definitions and axioms we base our understanding on change.

  • Math is a quest of understanding what must be. The basis for this quest are the axioms and definitions. Definitions define the objects of discussion. Axioms describe what we assume those objects can do.

  • Another way that definitions are invented is when mathematicians want to generalize an idea to a more general situation. Another way to say this is that mathematicians are trying to somehow identify the intrinsics of an idea. This is a major theme of modern mathematics. "What does it mean to be a shape?", "What does it mean to multiply things?". These two questions lead to the complete reformulations of many branches of mathematics.

Until Bernhard Riemann, a shape was always visualized in the plane or in space (or perhaps a higher dimensional ℝⁿ). Pondering the question of what makes a shape a shape, Riemann concluded that the essential property of shapes is that, at any point on the shape, one can travel in a certain number of directions. The usual visualization of shapes in space was a crutch that distracted us from the intrinsic properties of that shape. These "many-fold quantities", as Riemann called them, now called manifolds, have become the basis for geometry.

By generalizing multiplication, group theory in abstract algebra emerged. A group is a collection of things you can multiply. It's immaterial whether these things are matrices, functions, numbers, shapes, symmetries, etc.; as long as they follow the group axioms, you will know how to multiply them.

These kinds of generalization often seem to further complicate something that is already complex. But abstracting away the details and focusing on the core ideas turns out to be very valuable. First, sometimes it makes it easier to prove results you care about. Second, by unifying very disparate ideas (such as matrix multiplication and rotations and normal multiplication), if you can prove a theorem about groups in general, than it applies to all of these very different situations.

The system of logic

  • Mathematical proofs are not eternal, existing independently of any other concept, shape or form. A proof is a logical consequence of a specific theory that was developed in a certain formal system of logic. The theory and the proofs derived are valid only with regards to some logic system.

For example, the classical logic is a formal logic system that has two truth values (true and false) and it holds that a proposition is either true or false, not both and not neither.

This and other behaviours are enforced by the axioms, one of which, called the law of excluded middle, states that either a proposition is true, or that its negation is true. Another axioms, called the principle of double negation, states that if a statement is true, then it is not the case that the statement is not true.

On the other hand, the intuitionistic logic, which is also a mainstream system of logic, doesn't accept neither of these two axioms that are fundamental in the classic logic.

Sand castle

  • Modern mathematics has the set-theoretic foundation (concretely ZF set theory), meaning the whole of mathematics could be derived from the set theory. In fact, it would be better to say that the mainstream mathematics is based on set theory because majority of mathematicians today agree with that. However, there are other groups of mathematicians that have different views.

  • Recently, the category theory has gained enough momentum to present itself as the alternative candidate for the foundations of mathematics. With the introduction of sets and soon after the related paradoxes, the type theory has started development, which makes another alternative for the foundations of math. The foundations of mathematics based on functions (lambda calculus and combinatory SK calculus) have been, and still are, investigated for this role as well.

  • The foundation of mathematics itself also became the mathematical discipline to encompass the effort of searching for mathematical foundations. However, it came about late in the history of mathematics, when many mathematical disciplines were already formed.

Namely, in the XIX century, after many surprises that questioned stability of mathematics, German mathematician David Hilbert has set forward the motion for axiomatization of mathematics.

The goal was to put the entirety of math on a solid cornerstone upon which the rest of mathematics could be build by devising a strong theory with a carefully chosen set of axioms.

Any mathematical theory consists of a set of mathematical axioms, given without the proof. Besides the axioms, mathematical primitives are similarly privileged, require no proof as well.

  • A theory may start its development with a mathemtical primitive as its most central concept, with axioms describing its behavior (primitive as the central entity whose behavior is described by the axioms). Alternatively, a primitive itself may be introduced into a theory as one of the axioms.

The fact that primitives and axioms require no proof has to do with infinite regression (also nicknamed vicious circle). Any theory has to start somewhere, it introduces some concept, some term that wasn't previously defined. If it were to define that term, then all the terms used in a definition would aslo require a definition of their own. And so on and on, ad nauseam. This is the vicious circle of regression that would prevent any theory to even begin.

Therefore, it is allowed to "make a cut" and arbitrarily choose the starting concept for a theory. That concept is usually the most fundamental concept of the theory, either introduced as a primitive or explicitly given as an axiom. Then, a minimal set of axioms (compactness) is chosen in such a way that it prohibits deriving contradictory statements within this system (consistency). Also the axioms should allow deriving all the true statements within the system (completeness). No axiom should be chosen if it can be derived from other axioms.

One can choose to construct the new set theory, adding an axiom as a fix for every encountered problem.

The higher ground

Even if the foundation of mathematics is not unanimously agreed upon, there's no time to waste waiting for the consensus, so mathematicians proceed exploring their fields of interest anyway, forming new theories and deriving new proofs from them. This means that the proofs depend on the theory and if you accept the foundation of some theory, you can be certain that the proof will be correct within that theory. However, if the future shows that the foundation of that theory was wrong, there are always alternative to get on board with.

You can even work with respect to two opposing theories and then produce your work in both versions, which would be the case if you were to work within the systems of both classical and intuitionistic logic.

Whether one mathematician chooses to construct natural numbers in set-theoretic terms (zero as the empty set, {}, the successor as S(n) = n U {n}), and another using lambda calculus (zero as the lack of function application, λsz.z, the successor as λnfx.f(nfx)), the other mathematicians will happily welcome them both at the next level of abstraction, where natural numbers are concerned as the standalone mathematical objects anyway, abstracted away by removing them from their foundational background.

So, just get to the higher abstraction level somehow and see everyone in agreement once again, at least initially.


  • Mathematics tries to recognize patterns, and use them to formulate conjectures, which are resolved by means of mathematical proof. Those conjectures found to be true become theorems.

  • When a mathematical structure turns out to be a good model for some real world phenomena, then mathematical reasoning can provide further insight or predictions about the nature of that phenomena.

  • Math was developed from counting, calculating, measuring and the systematic study of shapes and motions of physical objects through the use of abstraction and logic,

  • Practical mathematics has been a human activity from as far back as written records exist.

  • The research required to solve mathematical problems can take years or even centuries of effort.

  • The rigorous method of math had first appeared in ancient Greeks: it was established by Euclid in his "Elements".

  • Since the pioneering work of Giuseppe Peano, David Hilbert and many others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction based on the initial set of axioms and definitions.

  • Mathematics was in slow development until the Renaissance, which was a period when innovations and new scientific discoveries led to an increase in math discoveries, a trend that continued to the present day.

  • Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory.

  • Pure mathematics ("math-pour-math-ism") is math for the sake of math, a pure exploration of math unburdened with practical application. However, an application is often discovered later.

  • Many areas of math began as inquiry into practical problems, but then the underlying rules and concepts were identified and extracted - abstracted into abstract structures. For example, geometry originated from practical calculation of distances and areas; algebra started from methods for solving problems in arithmetic.

  • Definitions of mathematics. Math has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions, all controversial.

  • Philosophy of mathematics aims to provide an account of the nature and methodology of mathematics and to understand the place of math in people's lives.

Refs

https://en.wikipedia.org/wiki/Definitions_of_mathematics https://en.wikipedia.org/wiki/Philosophy_of_mathematics https://www.youtube.com/playlist?list=PLFJr3pJl27pIp1EsDD2rYaTI7GxoXqrLs https://infinityplusonemath.wordpress.com/2017/06/17/what-is-math/ https://infinityplusonemath.wordpress.com/2017/06/21/where-do-axioms-come-from/

Astronomy-and-trigonometry https://www.britannica.com/science/geometry/Astronomy-and-trigonometry

mathematics https://www.britannica.com/science/mathematics

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