Primitive notion: Difference between revisions

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

In [[mathematics]], [[logic]], [[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]] andor everydaytaken experienceto be [[self-evident]]. 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 (geometry)|point]]'', ''line'', and ''contains'' are some primitive notions.

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==

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* Arithmetic of [[real number]]s: Typically, primitive notions are: real number, two [[binary operation]]s: [[addition]] and [[multiplication]], numbers 0 and 1, ordering <.

* [[Axiomatic system]]s: The primitive notions will depend upon the set of axioms chosen for the system. [[Alessandro Padoa]] discussed this selection at the [[International Congress of Philosophy]] in Paris in 1900.<ref>[[Alessandro Padoa]] (1900) "Logical introduction to any deductive theory" in [[Jean van Heijenoort]] (1967) ''A Source Book in Mathematical Logic, 1879–1931'', [[Harvard University Press]] 118–23</ref> The notions themselves may not necessarily need to be stated; Susan Haack (1978) writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms."<ref>{{citation|first=Susan|last=Haack|year=1978|title=Philosophy of Logics|page=245|publisher=[[Cambridge University Press]]|isbn=9780521293297}}</ref>

* [[Euclidean geometry]]: Under [[Hilbert's axiom system]] the primitive notions are ''point, line, plane, congruence, betweenessbetweenness '', and ''incidence''.

* [[Euclidean geometry]]: Under [[Foundations of geometry#Pasch and Peano|Peano's axiom system]] the primitive notions are ''point, segment'', and ''motion''.