Concept

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A concept is a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.

A concept is a mental unit. A concept refers to an unlimited number of concretes.

With the exception of proper names, there is basically a one-to-one correspondence between the concepts that a man uses and the words that he uses. There are some minor exceptions to that rule. For example, in my mind, I think that "dimensional analysis" is treated as a single concept, rather than as a type of analysis which is about dimensions (though the name is apt). Conversely, in my mind the different words "squadron" and "battalion" denote the exact same concept (though I am sure that they denote different concepts for people in the military).

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 a basic identification of a fact of reality. It is the form in which we make conceptual identifications.