Number

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

Definitions of specific numbers, like "2"

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

I use "two" and "2" completely interchangeably.

It is tempting to say something like "the definition of "2" is: the symbol succeeding "1" in the counting algorithm." That is not correct, because when I say something like "those are two cows," I do not mean "those are (the symbol succeeding "1" in the counting algorithm) cows." Rather, I am identifying the quantity of cows with a quantity of symbols.

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.