section_path: Properties/In axiomatic systems
section_name: In axiomatic systems
paragraph_text: In axiomatic systems
  
  The concept of line is often considered in geometry as a primitive notion in axiomatic systems, meaning it is not being defined by other concepts. In those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the properties of lines are dictated by the axioms which they must satisfy.
  
  In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance, it is possible to provide a description or mental image of a primitive notion, to give a foundation to build the notion on which would formally be based on the (unstated) axioms. Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation. These are not true definitions, and could not be used in formal proofs of statements. The "definition" of line in Euclid's Elements falls into this category. Even in the case where a specific geometry is being considered (for example, Euclidean geometry), there is no generally accepted agreement among authors as to what an informal description of a line should be when the subject is not being treated formally.