Primitive notion: Difference between revisions

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In [[mathematics]], [[logic]], and [[formal system]]s, a '''primitive notion''' is an undefineda concept. In particular, a primitive notionthat is not defined in terms of previously defined concepts,. butIt is onlyoften motivated informally, usually by an appeal to [[Intuition (knowledge)|intuition]] and everyday experience. In an [[axiomatic theory]], orrelations otherbetween primitive notions are restricted by [[formal systemaxiom]]s.<ref>More generally, the role ofin a primitiveformal notionsystem, isrules analogousrestrict tothe thatuse of primitive notions. See e.g. [[axiomMU puzzle]]. Infor axiomatica theories,non-logical theformal primitivesystem.</ref> notionsSome areauthors sometimes saidrefer to bethe latter as "defineddefining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of [[infinite regress]].

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>

==Details==