Concept

[TODO: this page is under construction]

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

This whole article is largely a rehash of ITOE, but with a few things added to it like notions and propositions.

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 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).

In German, the one-to-one correspondence between concepts and words is closer to being true, because they have more compound words than English (e.g "stop sign" is "Stoppschild"). In Chinese, the one-to-one correspondence between concepts and characters is far from being true, because often a single Chinese character is ambiguous until combined with other characters. This fact has been recognized by Chinese, which has a word ("词语") that effectively means: a grouping of one or more characters that stands for a single concept. "词语" gets translated into English as "word."

Most pages on the Objective Mathematics wiki are concepts.

Unit

[TODO must write this part]

Measurement

Measurement is "the identification of a quantitative relationship, by means of a standard that serves as a unit."

Definition

A definition is "a statement that identifies the nature of a concept’s units."

Any definition provides a genus, and some differentia. The genus tells us in what broad category of our mind we should look to find the concept. The differentia tell us how to distinguish the concept from the other concepts in that broad category.

For examples of definitions, look at the first sentence of any page on the Objective Mathematics wiki.

Standard math's idea of definitions

In standard mathematics, the definition of a term is considered to be all there is to the term.

For example, take the standard math definition of a group:

Definition. A group $(G,\mu ,i,e)$ is a set $G$, a map $\mu :G\times G\rightarrow G$, a bijection $i:G\rightarrow G$, and an element $e\in G$, satisfying the following axioms

1. Associativity: $\mu (x,\mu (y,z))=\mu (\mu (x,y),z)$ for all $x,y,z\in G$.
2. Identity: $\mu (e,x)=x$ for all $x\in G$.
3. Inverse: $\mu (x,i(x))=e=\mu (i(x),x)$ for all $x\in G$.

In standard math, a group is nothing less and nothing more than what these definitions say it is. A group has exactly those properties which follow deductively from its definition, and no other properties.

In contrast, the properties of objects in real life are identified rather than deduced from axioms. Consequently, there is always the possibility that a real-life object has properties which have not yet been identified---that some totally novel property of the object could be discovered. However, anything that is discovered about (the standard math notion of) a group, though it may have been previously unknown, is not novel in the same way: it is necessarily just a deductive consequence of the group axioms.

For the concepts of standard mathematics, there are no borderline cases or grey areas. Given some mathematical structure (in standard mathematics), either it is a group, or it isn't a group. In real life, though, there are borderline cases. There are objects whose measurements don't clearly fall inside the range of a given concept, but also don't clearly fall outside the range of that concept. For example, is a feminine hermaphrodite a woman? I imagine that there is not always a clear answer to that question.

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.

I reserve the right to be sloppy about the distinction between notion and concept, in situations where I think it doesn't matter.

Proposition

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