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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* [[Set theory]]: The concept of the [[Set (mathematics)|set]] is an example of a primitive notion. As [[Mary Tiles]] writes:<ref>[[Mary Tiles]] (2004) ''The Philosophy of Set Theory'', p. 99</ref> [The] 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive, undefined, term. As evidence, she quotes [[Felix Hausdorff]]: "A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit."

* [[Naive set theory]]: The [[empty set]] is a primitive notion. To assert that it exists would be an implicit [[axiom]].

* [[Peano arithmetic]]: The [[successor function]] and the number [[zero]] are primitive notions. Since Peano arithmetic is useful in regards to properties of the numbers, the objects that the primitive notions represent may not strictly matter.<ref>{{cite documentthesis | title=Mechanising Hilbert's Foundations of Geometry in Isabelle (see ref 16, re: Hilbert's take) | author= Phil Scott|type=Master's thesis|publisher=University of Edinburgh | date=2008| citeseerx=10.1.1.218.9262 }}</ref>

* 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''.

==Russell's primitives==

In his book on [[philosophy of mathematics]], ''[[The Principles of Mathematics]]'' [[Bertrand Russell]] used thesethe following notions: For thefor class-calculus ([[set theory]]), he used [[relation (mathematics)|relation]]s, taking [[set membership]] as a primitive notion. To establish sets, he also requiresestablishes [[propositional function]]s as primitive, as well as the phrase "such that" as used in [[set builder notation]]. (pp 18,9) Regarding relations, Russell takes as primitive notions the [[converse relation]] and [[complementary relation]] of a given ''xRy''. Furthermore, logical products of relations and [[relative product]]s of relations are primitive. (p 25) As for denotation of objects by description, Russell acknowledges that a primitive notion is involved. (p 27) The thesis of Russell’s book is "Pure mathematics uses only a few notions, and these are logical constants." (p xxi)

==See also==