Integer

An integer is a concept measuring the difference between two multitudes, considered as two members of an ordered set. In Objective Mathematics, the concept of integer is often denoted by a symbol, $\mathbb {Z}$.

Differences

The difference between two quantities A and B, is the quantity which would have to be added to the lesser quantity in order to make it equal to the greater quantity.

In the case where there is no difference between A and B, we say that the difference between them is zero, or 0.

Examples

The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

The difference between a pile consisting of 5 apples, and a pile consisting of 3 apples, is 2 apples.

The difference between 3 and 5 is 2.

Non-examples

"A difference between me and my friend is that I like chocolate ice cream, but he doesn't." This is a perfectly valid use of the concept of difference, but it's not a difference of quantities.

Signed differences

A difference is something that is identified with respect to two quantities. For a difference, the order of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.

A signed difference, which Objective Mathematics sometimes calls a sifference, is a concept much like a difference, except that it keeps track of the order the two quantities under consideration. Let A and B denote two quantities, where A is greater than or equal to B. The sifference between A and B is the difference between A and B; the sifference between B and A is the difference between A and B, but with a slight asterisk to remind us about the order.

Describing things like I have, in the English language, may give the reader a slightly incorrect idea, because "and" is often considered to be symmetrical. Indeed, "Bob and Jane" usually means the same thing as "Jane and Bob." In our context, however, it is very important to distinguish between the two noun phrases. This emphasis on the order of "and" is not completely foreign to English, however: authors know that at times, the subtle change in emphasis between "Bob and Jane" and "Jane and Bob" matters.

Symbolically, we write the sifference between $X$ and $Y$ as $X-Y$, and we write the sifference between $Y$ and $X$ as $Y-X$.

Examples

The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

in fact there is a symmetry between them.