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.

The three concepts of number, quantity, and numeral are closely related. They conceptualize the same thing, but from three different perspectives. Number is epistemological. Quantity is metaphysical. Numeral is symbolic. Note that the latter two are referred to in the definition of number. Those concepts are more basic; quantity is an irreducible primary, and numerals are some symbols that man created. Number is an integration of the two.

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.

[TODO maybe I should delete "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.

Basic arithmetic operations

[TODO unify with the several other articles, especially multitude.]

[TODO quantity assumes the unit perspective...]

Succession

The successor to a numeral is the numeral which comes after it in the fixed sequence of numerals.

Successorship is the function computing successor. There is a very simple algorithm computing successorship, which is learned by schoolchildren. [TODO copy paste the part where I wrote it?]

Counting

Counting is the process which produces a number by means of repeated application of the successorship function.

Addition

The sum or disjoint union of two quantities of existents is the quantity of those existents considered together as a single group. (See also the "disjoint union" thing.)

Addition is the function which takes two numbers as input, and outputs the number identifying the sum of their corresponding quantities.

Subtraction

The difference of two quantities is the quantity which would have to be added to the lesser in order to make it equal to the greater. (Note: this sense of the word is used derivatively. It should not be confused with that primary and axiomatic concept of "difference," which stands in opposition to the axiomatic concept of "similarity.")

Subtraction is the function which takes two numbers as input, and produces the number identifying the difference between their corresponding quantities.

Multiplication

see page multiplication.

Before we can give a definition of multiplication, we must say a bit more about how numbers work.

In the identification of a quantity as a number (in the "numbering" of a quantity), a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x meters" or "there are x cows" or "there are x Hertz" etc.

[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Actually I am thinking now that I will probably bite the bullet and say that a ratio is not a number. Hmm I guess I'll come back to this later. OK it's later now. Ratios ARE numbers. And ratios actually do come with units. Like I am driving at 30 MPH, that's expressing a ratio but it has units. Even if I take something with the same units, like the ratio of two measurements of length, maybe I shouldn't omit that fact from the notation. Like maybe we should write 3.14 m/m instead of just 3.14. The former retains some information about where it came from, the latter doesn't; it's just a numeral and that's it.]

Multiplication (of A and B) is an algorithm which produces the number identifying a quantity in units of v, where B identifies the number of vs making up one u, and A is the quantity of us.

Division

[TODO]

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.