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Mathematical object

Definition

A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a symbol, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even formal theories are considered as mathematical objects in proof theory.

Related concepts

Abstract and concrete Abstract concept Abstract logic Abstraction (mathematics)Ackermann set theory Aleph number Alfred North Whitehead Algebraic logic Alphabet (formal languages)Analytic philosophy Ancient Greek philosophy Andrey Markov (Soviet mathematician)Anna Sfard Anne Sjerp Troelstra Applied mathematics Argument Arithmetic Arity Atomic formula Atomic model (mathematical logic)Atomic sentence Authority Automata theory Automated theorem proving Axiom Axiom of choice Axiom of infinity Axiom schema Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bertrand Russell Bijection Binary function Binary operation Biology Bonnie Gold Boolean algebra Boolean algebras canonically defined Boolean function Branches of science Brouwer–Hilbert controversy CZF Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Chaos theory Chess Church encoding Church–Turing thesis Class (set theory)Classical logic Codomain Combinatorics Compactness theorem Complement (set theory)Complete theory Completeness (logic)Complex numbers Computability theory Computable function Computable set Computably enumerable set Concrete category Concrete object Conservative extension Consistency Constructible universe Construction of the real numbers Constructive analysis Constructive set theory Constructivism (philosophy of mathematics)Contemporary philosophy Continuum hypothesis Contradiction Convenient fiction Countable set David Hilbert Decidability (logic)Decision problem Deductive system Diagram (mathematical logic)Differential geometry Domain of a function Electron Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Entity Enumeration Epistemology Equiconsistency Equivalence relation Errett Bishop Euclid's Elements Euclidean geometry Exceptional object Existence Existential quantification Existential quantifier Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Fictionalism Finitary relation Finite-valued logic Finite model theory Finite set Finitism First-order logic Fixed-point logic Forcing (mathematics)Formal Logic Formal grammar Formal language Formal proof Formal semantics (logic)Formal system Formalism (philosophy of mathematics)Formation rule Formulas Foundations of geometry Foundations of mathematics Four-dimensional Free logic Free variables and bound variables Freeman Dyson Function (mathematics)Functional predicate Fuzzy set General relativity General set theory Glossary of mathematical symbols Gottlob Frege Graph of a function Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hartry Field Henri Poincaré Hereditary set Hermann Weyl Higher-order logic Hilary Putnam Hilbert's axioms Hilbert's program Hilbert space Hilbert system History of logic History of mathematical logic Idealism Identity (philosophy)Image (mathematics)Impossible object Inaccessible cardinal Independence (mathematical logic)Inference Infinite-valued logic Infinite set Information theory Inhabited set Injective function Integers Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Intuitionism Isomorphism John P. Burgess Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel L. E. J. Brouwer Lambda calculus Language of mathematics Large cardinal Lemma (mathematics)Lindström's theorem List of Hilbert systems List of axioms List of first-order theories List of formal systems List of mathematical objects List of mathematical shapes List of mathematical theories List of set identities and relations List of shapes List of statements independent of ZFC List of surfaces List of two-dimensional geometric shapes Logic Logical axioms Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical truth Logicism Ludo Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Material conditional Mathematical and theoretical biology Mathematical knot Mathematical logic Mathematical physics Mathematical structure Mathematics Metalanguage Metaphysical naturalism Metaphysical realism Metaphysics Minimal axioms for Boolean algebra Model complete theory Model theory Modern logic Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory NP (complexity)Naive set theory Natural deduction Natural numbers Natural science Nature (philosophy)Negation Nelson Goodman New Foundations Nikolai Aleksandrovich Shanin Nominalism Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Number Ontological commitment Ontology Open formula Operation (mathematics)Ordinal analysis Ordinal number Otávio Bueno Oxford English Dictionary Oxford University Press P (complexity)P versus NP problem Paradoxes of set theory Partition of a set Paul Benacerraf Paul Bernays Paul Halmos Peano axioms Philip J. Davis Philosophy of mathematics Physical object Physical objects Physical world Physics Planet Plato Platonic realm Platonism Power set Predicate (mathematical logic)Predicate logic Predicate variable Prediction Premise Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Proof (mathematics)Proof by contradiction Proof of impossibility Proof theory Property (philosophy)Proposition Propositional calculus Propositional formula Propositional variable Pure mathematics Quantifier (logic)Quantifier rank Quantum mechanics Quine–Putnam indispensability argument Raphael Sanzio Rational numbers Real analysis Real numbers Reality Recursion Recursive set Relation (mathematics)Relation (philosophy)Reuben Hersh Reverse mathematics Robinson arithmetic Roger Penrose Rule of inference Russell's paradox Satisfiability Saturated model School of thought Schröder–Bernstein theorem Science Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequent calculus Set (mathematics)Set theory Shape Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Stanford Encyclopedia of Philosophy Statement (logic)Stewart Shapiro Strength (mathematical logic)String (formal languages)Structuralism (philosophy of mathematics)Structure (mathematical logic)Substitution (logic)Substructure (mathematics)Supertask Surjective function Syllogism Symbol (formal)Syntax (logic)T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautology (logic)Term (logic)Term logic Tesseract The Foundations of Arithmetic The School of Athens The Unreasonable Effectiveness of Mathematics in the Natural Sciences Theorem Theories of truth Theory (mathematical logic)Theory of forms Thought Three-valued logic Timeline of mathematical logic Transfer principle Transitive set Trefoil knot True arithmetic Truth predicate Truth table Truth value Turing machine Type (model theory)Type theory Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Urelement Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Well-formed formula Willard Quine Zermelo–Fraenkel set theory

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