Primitive notion: Difference between revisions

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: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 expression 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,...

In [[axiomatic set theory]] the fundamental concept of ''set'' is an example of a primitive notion. As [[Mary Tiles]] wrote:

:[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."

When an [[axiomatic system]] begins with its [[axiom]]s, the primitive notions may not be forgottenexplicitly stated. Susan Haak (1978) wrote, "A set of axioms is sometimes said to give an implicit definition of its primitive terms."

'''Examples'''. In:

* [[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.

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

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

* [[Euclidean geometry]], under

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