User:Boris Tsirelson/Sandbox1

Defined or undefined

Facts are formulated via notions.

In the non-axiomatic approach, notions are nodes of a network whose connections are definitions. If some notions are forgotten they probably can be restored from the others.

Searching Google for "define:line" we get "a length without breadth or thickness". Similarly we find definitions for breadth, thickness and so on, recursively. Doing so we would get a large subnetwork; here is its small fragment:

• Line: a length without breadth or thickness
  • Length: linear extent in space
    • Linear: along a line↑
    • Extent: a range of locations
      • Location: point or extent↑ in space
    • Space: unlimited expanse in which everything is located↑
  • Breadth: the extent↑ from side to side
    • Side: a surface forming part of the outside of an object
      • Surface: the extended↑ two-dimensional outer boundary of a three-dimensional object
  • Thickness: the dimension through an object as opposed to its length↑ or width
    • Width: the extent↑ from side↑ to side↑

(Up arrows mean: see above.) We observe that

• circularity appears routinely; for example: line→length→linear→line;
• the definition of a single notion involves recursively a large number of other, quite remote notions.

Such system of notions is unsuitable for a mathematical theory.

In the non-axiomatic approach, notions are a tower of defined notions, grounded on the basis of more fundamental notions called undefined primitives. If all defined notions are forgotten they surely can be restored from the undefined primitives. The undefined primitives are sparse and simple, not to be forgotten.