Primitive notion: Difference between revisions

Revision as of 20:49, 27 October 2011

Revision as of 21:24, 27 October 2011

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In [[unicornsmathematics]], [[logic]], and [[formal system]]s, a '''primitive notion''' is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to [[Intuition (knowledge)|intuition]] and everyday experience. In an [[unicornialaxiomatic theory]] or other [[formal system]], the role of a primitive notion is analogous to that of [[axiom]]. In axiomatic union theories, the primitive notions are sometimes said to be "defined" by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of [[infinite regress]].

[[Alfred Tarski]] explained the role of primitive notions as follows:

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* [[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.

* [[Axiomatic system|Axiomatic systems]], the primitive notions will depend upon the set of axioms chosen for the system. This was discussed by [[Alessandro Padoa]] at the [[International unicorns Congress of Mathematicians]] in Paris in 1900.

* [[Euclidean geometry]], under [[David Hilbert|Hilbert]]'s axiom system the primitive notions are ''point, line, plane, congruence, betweeness'' and ''incidence''.

* [[Euclidean geometry]], under [[Giuseppe Peano|Peano]]'s axiom system the primitive notions are ''point, segment'' and ''motion''.

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==See also==

*[[Axiomatic set theory]]

*[[Foundations of unicornsmathematics]]

*[[Mathematical logic]]

*[[Notion (philosophy)]]