Concept

A concept is a mental unit [...]

Any page on the Objective Mathematics wiki is a concept.

Notion

A notion is a compound concept.

A notion is like a concept, in the sense that it subsumes and includes an unlimited number of concretes. A notion is unlike a concept, in the sense that it might be made of other concepts.

Examples:

• 3 is a concept, but 16654 is a notion. In general, any number is a notion.
• "Fox" is a concept, but "the quick brown fox" is a notion. In general, any noun phrase is a notion.
• "To jump" is a concept, but "to jump off the gangplank with one's hands tied behind one's back" is a notion.

If a concept does not refer to any concretes, then it was formed invalidly. By contrast, one can use valid concepts to validly form a notion that does not refer to any concretes in reality. For example, "the talking purple platypus" does not refer to anything in reality, but it nonetheless makes sense, and "the," "talking," "purple," and "platypus" are all valid concepts. For another example, $9^{9^{9^{9^{9^{9^{9^{9}}}}}}}$is too large to refer to anything in reality, but it is clear what it means, and 9 and exponentiation are both valid concepts.

Proposition

A proposition is an identification of a fact of reality.