Natural number

A natural number or whole quantity is the concept which measures multitude.

Natural number is a second-level concept. To arrive at this concept, one must first arrive at some of the first-level concepts which it subsumes, like 1, 2, 3, 4, etc., as well as some of the related concepts which it doesn't subsume, like 1/2, 7/12, 3.14, etc.

It is intuitively clear what 1, 2, 3, 4, etc. refer to in reality. 2 refers to sets of two entities. 3 refers to sets of three entities, etc. (see §Examples if confused). However, this will not suffice as a definition, because it is self-referential. The definitions of those concepts are surprisingly sophisticated, and will be developed below.

Successorship

For one word B to be a successor to another word A means that in some context, one is supposed to say B after one says A. [TODO this definition is not general enough]

Examples

In a context where one is singing a scale in the solfège system (do, re, mi, fa, so, la, ti, do), the successor to "fa" is "so".

In a context where one is counting some multitude, the successor to "one hundred seventeen" is "one hundred eighteen."

In a context where one is dealing with the Greek alphabet, the successor to ζ is η.

Counting

Counting is a method to measure multitudes by means of perceptual symbols. [TODO I don't think that this definition fits all my examples.]

Examples

An ordered set of things can be counted by pointing at each of them, one after another, and saying a particular (natural) number out loud each time you do so. The numbers which are said should follow the rules that the first number said is "one", and the number stated after one states "X" is "Y", where Y=X+1.

Things can be counted in a similar way by using numbers in base 2, base 16, strings of letters of the alphabet, etc.

Tally marks are a nice way to count things in some situations.

A very primitive man might make use of a unary counting procedure, where he counts things by scratching marks on a clay tablet. This would certainly be much less practical than modern ways of counting, but it's better than nothing: it still records some information and has some use.

Small multitudes can be counted on one's fingers.

Some things can be counted using negative numbers. For example consider the following procedure, where the sifference between the number of things present in two groups is said out loud (and where the first row happens first, then the second row, etc):

Contents of group 1	Contents of group 2	The thing to say while counting
♣♣♣♣	♦	"three"
♣♣♣	♣♦	"one"
♣♣	♣♣♦	"negative one"
♣	♣♣♣♦	"negative three"
	♣♣♣♣♦	"negative five"

cool, right?

1, 2, 3, 4, etc.

We may now finally give definitions of the individual numbers

1, or one, is the concept of a unit.

2, or two, is the successor to 1.

3, or three, is the successor to 2.

4, or four, is the successor to 3.

Etc.

I will now explain why I deem successorship to be the essential characteristic of these numbers.

Examples

• 

  1 cat.

• 

  4 engines.

NNs

This is the easiest step. Natural number, $\mathbb {N}$, is just the concept [TODO]

Addition

To add two natural numbers $m$ and $n$ is to regard the multitudes to which they refer as a single set. The number measuring the resulting multitude may be denoted as $n+m$.

This is another way of thinking about a disjoint union of sets.

Multiplication

To multiply two natural numbers $m$ and $n$ is to add $m$ to itself $n$ times. The number measuring the resulting multitude may be denoted as $nm$ or $n\times m$. One always obtains the same quantity by adding $n$ to itself $m$ times, or in other words, it is always the case that $nm=mn$.

This is another way of thinking about a cartesian product of sets.

Primes

A natural number is prime if it cannot be factored nontrivially.