Metamathematics

Definition

Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.

Related concepts

Abductive reasoning Abstract algebra Alan Turing Alfred North Whitehead Alfred Tarski Algebra Algebraic geometry Algebraic number theory Algebraic topology Algorithm Alonzo Church Analytic geometry Analytic number theory Analytic philosophy Analytical philosophy Analytic–synthetic distinction Antecedent (logic)Antinomy Applied mathematics Argumentation theory Arithmetic Arithmetic geometry Axiom Axiomatic system Begriffsschrift Bertrand Russell Boeotia Calculus Calculus ratiocinator Cantor's theorem Category theory Church–Turing thesis Classical logic Combinatorics Commutative algebra Completeness (logic)Complex analysis Complex geometry Computational complexity theory Computational geometry Computational mathematics Computer algebra Computer science Consequent Consistency Contradiction Control theory Critical thinking David Hilbert Decidability (logic)Decision problem Deductive closure Deductive reasoning Definition Description Dichotomy Differential equation Differential geometry Differential topology Diophantine geometry Discrete geometry Discrete mathematics Douglas Hofstadter Effective method Effective procedure Effectively calculable Elementary algebra Emil Post Engineering mathematics Entscheidungsproblem Euclidean geometry Finite geometry First-order logic Formal proof Formal semantics (natural language)Formal system Formula Foundational crisis of mathematics Foundations of geometry Foundations of mathematics Functional analysis Future of mathematics Game theory Gauss General topology Geometric topology Geometry German language Glossary of mathematical symbols Gottfried Wilhelm Leibniz Gottlob Frege Graph theory Gregory Chaitin Group theory Gödel's completeness theorem Gödel's incompleteness theorem Gödel's incompleteness theorems Gödel, Escher, Bach Harmonic analysis Hilary Putnam Hilbert's program Hilbert's second problem Hilbert program History of logic History of mathematics Homological algebra Homotopy theory Hyperbolic geometry Hypercomplex analysis If and only if Independence (mathematical logic)Index of logic articles Inductive reasoning Inference Inference rule Informal logic Informal mathematics Information theory Interpretation (logic)Knot theory Kurt Gödel Lambda calculus Language Large numbers Linear algebra Linguistic entailment List of Boolean algebra topics List of fallacies List of films about mathematicians List of logic symbols List of logicians List of mathematical logic topics List of mathematicians List of paradoxes List of rules of inference List of set theory topics Lists of mathematics topics Logic Logic in computer science Logical consequence Logical form Logical truth Löwenheim–Skolem theorem Mathematical analysis Mathematical and theoretical biology Mathematical chemistry Mathematical economics Mathematical finance Mathematical geology Mathematical linguistics Mathematical logic Mathematical optimization Mathematical physics Mathematical psychology Mathematical rigour Mathematical sociology Mathematical statistics Mathematical theorem Mathematical theory Mathematicians Mathematics Mathematics and art Mathematics education Max Newman Measure (mathematics)Meta (prefix)Metalogic Metatheorem Metatheory Michael Dummett Modal logic Model theory Multilinear algebra Name Natural language Natural numbers Necessity and sufficiency Non-Euclidean geometry Non-classical logic Number Number theory Numerical analysis Operations research Order theory Outline of logic Paradox Paul Benacerraf Paul Cohen Philosophical logic Philosophy Philosophy of logic Philosophy of mathematics Predicate logic Premise Principia Mathematica Probability Probability theory Proof theory Proposition Real analysis Reason Recreational mathematics Recursion theory Recursive definition Reference Richard's paradox Russell's paradox Satisfiability Schema (logic)Semantic theory of truth Semantics of logic Set theory Soundness Statement (logic)Statistics Stephen Cole Kleene Stephen Kleene Substitution (logic)Syntax (logic)System of types Systems science T-schema Theorem Theory of computation Thoralf Skolem Thought Timeline of mathematics Topology Truth Turing machine Type theory Type–token distinction Universal algebra Use–mention distinction Validity (logic)Willard Quine Zermelo–Fraenkel set theory

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