Primitive notion: Difference between revisions

Revision as of 17:49, 10 November 2022

Revision as of 13:38, 24 March 2023

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{{short description|Concept that is not defined in terms of previously defined concepts}}

In [[mathematics]], [[logic]], [[Philosophy|philosophy]], and [[formal system]]s, a '''primitive notion''' is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to [[Intuition (knowledge)|intuition]] and everyday experience. In an [[axiomatic theory]], relations between primitive notions are restricted by [[axiom]]s.<ref>More generally, in a formal system, rules restrict the use of primitive notions. See e.g. [[MU puzzle]] for a non-logical formal system.</ref> Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of [[infinite regress]] (per the [[regress problem]]).

For example, in contemporary geometry, ''point'', ''line'', and ''contains'' are some primitive notions. Instead of attempting to define them,<ref>[[Euclid]] (300 B.C.) still gave definitions in his ''[[Euclid's Elements|Elements]]'', like "A line is breadthless length".</ref> their interplay is ruled (in [[Hilbert's axiom system]]) by axioms like "For every two points there exists a line that contains them both".<ref>This axiom can be formalized in [[predicate logic]] as "[[universal quantifier|∀]]''x''₁,''x''₂[[Set membership|∈]]''P''. [[existential quantifier|∃]]''y''∈''L''. ''C''(''y'',''x''₁) [[logical conjunction|∧]] ''C''(''y'',''x''₂)", where ''P'', ''L'', and ''C'' denotes the set of points, of lines, and the "contains" relation, respectively.</ref>