Editing Primitive notion (section)

==Examples==
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics:
* [[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 thesis | 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, betweenness '', and ''incidence''.
* [[Euclidean geometry]]: Under [[Foundations of geometry#Pasch and Peano|Peano's axiom system]] the primitive notions are ''point, segment'', and ''motion''.