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==Examples==
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics:
* [[Set theory]],: theThe 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'', page 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:The [[empty set]] is a primitive notion. To assert that it exists would be an implicit [[axiom]].
* [[Peano arithmetic]],: theThe [[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.{{Citation needed|date = January 2016}}
* [[Axiomatic system]]s,: theThe 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]],: underUnder [[Hilbert's axiom system]] the primitive notions are ''point, line, plane, congruence, betweeness'', and ''incidence''.
* [[Euclidean geometry]],: underUnder [[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).
==See also==
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