Read for Meaning

When you read an English statement, such as the one which you are presently reading, I'm sure that you read the statement which you are reading with the purpose of ascertaining the meaning of the statement. I'm sure that you are not just reading English words, but that you are "hearing" the meaning of these words.

Likewise, when you read a mathematical statement, a statement expressed in mathematical shorthand, in mathematical notation, I hope that you read the statement which you are reading for meaning.

For instance, when you read:

	(*) A = { a in B | P(a) },

I hope that you read this for what it means:

	(*) A is the set of all elements of the given set, B, for which the given property, P, is satisfied.

NOTE: If either one of the two objects, the set, B, and the property, P, is not given, then the above statement, (*), is "meaningless". This will be demonstrated if you are asking yourself:

	(**) What is this set, "B", about which statement (*) is talking?
	(***) What is this property, "P", about which statement, (*), is talking?

Until both of these questions, (**) and (***), are appropriately answered, then you have no hope of understanding what is the set, A, that is being defined by statement, (*).

Until you know exactly which set, "B",is being talked about in statement, (*), you won't be able to know what is this set, A, that is being defined by this statement, (*).

Until you know exactly which property, "P", is being talked about in statement, (*), you won't be able to know what is this set, A, that is being defined by this statement, (*).

Once you know exactly which set, "B", is being talked about in statement, (*), and which property, "P", is being talked about in statement, (*), then you know exactly which set, A, is being defined by this statement, (*). After all, this is precisely what statement (*) tells you in such a "meaningful" context (i.e. in such a context where statement (*) has a meaning). Namely, it tells you, in such a context, precisely which set, A, is being defined by it.

In such a "meaningful" context, statement (*) tells you which objects in the "universe" are in the set, A. Namely, those objects which are not only elements of the given set, B, but also satisfy the given property, P.

In such a "meaningful" context, statement (*) tells you which objects in the "universe" are not in the set A. Namely, those objects which are either not in the given set, B, or, which, although they are in the given set, B, they do not satisfy the given property, P.

In such a "meaningful" context, statement (*) gives you precise knowledge of the set, A. It tells you "what's in" and "what's out" of the set A.