Number

A number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified.

This is also basically the definition of "numeral" and "counting."

Numbers work by identifying a quantity of things with a quantity of symbols. That is, the quantity of these dots {*,*} is the same as the quantity of these symbols {"1", "2"}.

Definitions of specific numbers, like "2"

[TODO unify this page with the natural number page.]

The definition of the noun "2" is: the numeral succeeding "1" in the counting algorithm.

The definition of the adjective "2" is: the quantity of numerals up to and including "2," in the counting algorithm.

I use "two" and "2" interchangeably.

The two definitions above can be generalized to arbitrary natural numbers. Indeed, we define "3" as the numeral succeeding "2" in the counting algorithm, "844" as the numeral succeeding "843" in the counting algorithm, etc.

Is a fraction a number?

Yes.

To identify a quantity as $p/q$ is to say that it consists of $p$ pieces of quantity $1/q$. It is not fundamentally different from identifying a quantity as $p$; it's just that the unit is different. In the fraction, the unit is being regarded as a $q$th of another unit.

Is a negative a number?

Yes.

We can consider a sequence that goes backwards; 1, 0, -1, -2, -3, and so forth. Then we can identify a quantity like -5 by putting it in correspondence with a position in that sequence.

Is $\pi$ a number?

No.

π is the ratio between the circumference and the diameter of a circle. Unlike standard mathematics, by "circle," OM means real, finite, physical circles, with thickness and height and bumps and other non-uniformities. Observe the following facts about OM's concept of π, which are not true of the standard concept:

1. Since π is the result of a continuous measurement, it is not a specific value, but is rather a range of possible values. For example, one circle might have a circumference and diameter whose ratio is found to be 3.1 ± 0.1.
2. π is not the same range for every circle. For one circle, we might measure π = 3.1 ± 0.1; for another we might measure π = 3.15 ± 0.02; for yet another, we might measure π = 3.14159 ± 0.00005.

OK I'm failing to differentiate between two things. On the one hand, there's the actual ratio of the actual circumference and diameter of the actual circle. On the other hand, there's the thing we measure....

To treat something as a circle is to say like "all these ways in which I might measure its circumference, they're the same."

Is ${\sqrt {2}}$ a number?

No, I don't think so.

First of all we must settle on a definition of ${\sqrt {2}}$. There are two potential definitions that I can think of

1. (The usual definition) ${\sqrt {2}}$ is the number that squares to 2.
2. ${\sqrt {2}}$ is the side length of a square of area 2.

If definition 2 is correct, then ${\sqrt {2}}$ is not a number, for basically the same reason that π is not a number. It seems that ${\sqrt {2}}$ defines a range of numbers (and one which, to boot, differs between concrete situations) rather than a specific number.

If definition 1 is correct, then what is the fixed sequence of symbols which we are using to identify ${\sqrt {2}}$? It's certainly not fractions. I don't think can be anything else, either.

Is a number just "an identification of a quantity"?

No.

My sofa has some length. In saying that, I have identified a quantity, but I haven't given a number.

Is an element of the cyclic group $\mathbb {Z} _{n}$ a number?

Yes, I think so.

For example, an amount of time could be identified by saying something like: after that much time has passed, it will be 4:00.

We are identifying the amount of time by drawing a comparison between it and a "sequence" of fixed symbols appearing on a clock. This sequence is $\mathbb {Z} _{12}$, or $\mathbb {Z} _{12}\times \mathbb {Z} _{60}$, or $\mathbb {Z} _{12}\times \mathbb {Z} _{60}\times \mathbb {Z} _{60}$, depending on the clock.