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|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==

[[Alfred Tarski]] explained the role of primitive notions as follows:<ref>[[Alfred Tarski]] (1946) ''Introduction to Logic and the Methodology of the Deductive Sciences'', p. 118, [[Oxford University Press]].</ref>

:When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously. The sentence which determines the meaning of a term in this way is called a DEFINITION,...

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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>{{Citationcite thesis needed|date title=Mechanising JanuaryHilbert'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 2016}}</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''.

* [[Philosophy of mathematics]]: [[Bertrand Russell]] considered the "indefinables of mathematics" to build the case for [[logicism]] in his book ''[[The Principles of Mathematics]]'' (1903).

==Russell's primitives==

In his book on [[philosophy of mathematics]], ''[[The Principles of Mathematics]]'' [[Bertrand Russell]] used the following notions: for class-calculus ([[set theory]]), he used [[relation (mathematics)|relation]]s, taking [[set membership]] as a primitive notion. To establish sets, he also establishes [[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==

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*[[Foundations of geometry]]

*[[Foundations of mathematics]]

*[[Logical atomism]]

*[[Logical constant]]

*[[Mathematical logic]]

*[[Notion (philosophy)]]

*[[Object theory]]

*[[Natural semantic metalanguage]]

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[[Category:Set theory]]

[[Category:Concepts in logic]]

[[Category:Mathematical concepts]]