In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously defined concepts. It is often motivated informally, usually by an appeal tointuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress. For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them, their interplay is ruled by axioms like "For every two points there existsa line that contains them both".
Details
explained the role of primitive notions as follows: An inevitable regress to primitive notions in the theory of knowledge was explained by Gilbert de B. Robinson:
Examples
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics:
Set theory: The concept of the set is an example of a primitive notion. As Mary Tiles writes: '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.
Axiomatic systems: 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. The notions themselves may not necessarily need to be stated; Susan Haack writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms."
Euclidean geometry: Under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness, and incidence.
