Primitive notion

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 to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms.^[1] 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 (per the regress problem).

For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them,^[2] their interplay is ruled (in Hilbert's axiom system) by axioms like "For every two points there exists a line that contains them both".^[3]

^ More generally, in a formal system, rules restrict the use of primitive notions. See e.g. MU puzzle for a non-logical formal system.
^ Euclid (300 B.C.) still gave definitions in his Elements, like "A line is breadthless length".
^ This axiom can be formalized in predicate logic as "∀x₁,x₂∈P. ∃y∈L. C(y,x₁) ∧ C(y,x₂)", where P, L, and C denotes the set of points, of lines, and the "contains" relation, respectively.

[1] More generally, in a formal system, rules restrict the use of primitive notions. See e.g. MU puzzle for a non-logical formal system.

[2] Euclid (300 B.C.) still gave definitions in his Elements, like "A line is breadthless length".

[3] This axiom can be formalized in predicate logic as "∀x₁,x₂∈P. ∃y∈L. C(y,x₁) ∧ C(y,x₂)", where P, L, and C denotes the set of points, of lines, and the "contains" relation, respectively.

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