Primitive notion: Difference between revisions

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* [[Euclidean geometry]]: Under [[Hilbert's axiom system]] the primitive notions are ''point, line, plane, congruence, betweeness'', 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 these notions: For the class-calculus ([[set theory]]) he used [[relation (mathematics)|relation]]s, taking [[set membership]] as a primitive notion. To establish sets he also requires [[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==