"Number" is at the foundation of Mathematics. It's what we use to represent things we count. The original idea covers the natural numbers (=1,2,3,...), and then extends to zero, negative numbers, and off into lots of complication.

There has been some research that suggests that humans have an innate number sense. The research involved presenting babies with situations which obeyed preservation of number, and comparing their reaction to when they saw situations which violated this.

For example, they would be shown a screen, and shown that nothing was behind it. Then, they would see one hand put a doll behind the screen. When the screen was removed, there would be two dolls there. In that case, even babies less than 1 year old acted more surprised than when there was the expected number of dolls there. (The way you tell reactions of someone who can't talk is by looking at their eyes - babies look at unexpected things longer than they look at expected things.

It makes sense that there would be at least a basic genetically programmed ability to use numbers - like language, numbers are useful for many social interactions.

A number is an idea, nothing more than the result of axiomatic systems such as ZFC and Peano's Axioms

Philosophy students like to say things like "Would two mean anything if there weren't two objects for two to refer to." This is foolishness... Numbers need have no basis in the physical realm, they are ideas.

To state it simply, math is truth, truth that transcends what is observable. Math when done in an axiomatic sense has no "belief system." It is true.

# A set theoretic derivation of the numbers...

let 0=null

Now, define the successor of a number as the successor ordinal of that number... that is, let x+1=p(x)=x U {x}

So, we have...

0=null
1={null}={0}
2={null,{null}}={0,1}
3={null,{null},{null,{null}}}={0,1,2}
...

Now that we have the natural numbers, we have define that rationals...

for example, we can define 1/2 as the set of all ordered pairs {(1,2),(2,4),(3,6),(4,8),...}

Then, a given real number can be defined as the set of all Cauchy sequences convergent to that number.

Further, imaginary numbers can be defined as ordered pairs... for x=a+ib, let x=(a,b) where a and b are sets of Cauchy sequences convergent to a and b.<./p>

# Further Notes

One good treatment of this construction is Enderton's Elements of Set Theory, published in 1977.

Instead of constructing the reals using Cauchy Sequences, it is also possible to use Dedekind Cuts, but I have always found that derivation to be more painful.

In 1908, Zermelo proposed to define the integers as:
0=null
1={null}
2={{null}}
...
von Neumann proposed the definition given above which has become standard because of the property that for all y<x,yεx. This leads to some nice things when doing arithmatic.

I can't believe there's nothing like this already, considering all the many varied number nodes there are. As you can imagine, it's a big PITA hunting down all these numbers, so keep in mind that this is a work in progress, and if you know of a number node that's not on here, msg me and I'll add it on. Enjoy.

##### Other Relevant Number Nodes

In theatrical music, a number is a particular piece of music from a larger work such as an opera, cantata, oratorio, musical, or set of incidental music. In colloquial speech, the term number is most often used to refer to a particular song from such a work, but can also be applied to the music played between songs.

This usage comes from the fact that musical segments in a theatrical music score are numbered. This is for convenience and easy reference in rehearsal. (Segments within numbers are lettered.) The numbers were designed to provide a common reference for all the musicians, actors, and crew involved. This solved the difficulty where an "Act I, Scene 5" reference for the actors would leave the pit orchestra out of the loop, a "Semele's da capo aria" would leave the crew out, and "The beginning of the countersubject of the fugato section, please" would leave both the singers and the crew clueless.1

A common system in modern scores is to give major songs or beginnings of scenes their own numbers such as "3" and subsequent related sections or instrumental interludes number-letters such as "3A". An example is this excerpt from the table of contents of the 1956 Candide Broadway score:

```1    Ensemble: The Best of All Possible Worlds (Pangloss, Candide, Cunegonde, Chorus)
2    Duet: Oh, Happy We (Candide, Cunegonde)
2A   Chorus and Instrumental: Wedding Procession, Chorale and Battle Scene
3    Instrumental: Candide Begins His Travels
3A   Song: It Must Be So (Candide)
3B   Instrumental: Candide Continues His Travels
```
In a work that is sung throughout, such as an opera or an oratorio, number-letters are generally not required. For example, this excerpt from Handel's Messiah:
```1              Sinfonia (Overture)
2    Recit.    Comfort ye my people
3    Air       Ev'ry valley shall be exalted
4    Chorus    And the glory of the Lord
```

Small bit of trivia: the famous "Hallelujah Chorus" from Messiah is No. 44.

The use of numbers gave rise to phrases such as "No. 6 from the top" or "Next number, please." Gradually, instead of just being the labels for the music, numbers became the music, so that conductors would say "What about we try your solo number again?" to an unprepared soprano.

Nowadays, the word number has gone past theatrical usage and made its way to the concert stage and the cabaret. One might conceivably ask the cabaret or club singer, "What numbers do you have in store for us today?" or compliment him or her by saying "Your rendition of that number was fantastic." In fact, this word is now in common usage as a general word for any kind of performed song.

And now you know why!

1 That said, when we (the pit) want to keep to ourselves and not have those pesky singers2 bother us, we still like to use phrases such as "the countersubject of the fugato section" or "the first theme in the recapitulation of the overture." Just for fun. ::evil grin::

2 Never "vocalists"—that would imply they know what they're doing. And certainly never "musician." ::shudder::

::laugh:: Okay okay, all in good fun. I sing, myself, and know what's like to be on the other side. And to be fair, there sometimes are good singers—good vocalists—on the stage. And when that happens, it's wonderful.

Sources:
Handel, G. F.; Charles Jennens; and Watkins Shaw (ed.). Messiah. Vocal Score. The New Novello Choral Edition. Novello Handel Edition. London: Novello & Co. Ltd., 1992.
Hutchins, Michael H. Candide / The 1958 Broadway Score. <http://www.geocities.com/bernsteincandide/58score.html>. Accessed 22 January 2005.

Num"ber (?), n. [OE. nombre, F. nombre, L. numerus; akin to Gr. that which is dealt out, fr. to deal out, distribute. See Numb, Nomad, and cf. Numerate, Numero, Numerous.]

1.

That which admits of being counted or reckoned; a unit, or an aggregate of units; a numerable aggregate or collection of individuals; an assemblage made up of distinct things expressible by figures.

2.

A collection of many individuals; a numerous assemblage; a multitude; many.

Ladies are always of great use to the party they espouse, and never fail to win over numbers. Addison.

3.

A numeral; a word or character denoting a number; as, to put a number on a door.

4.

Numerousness; multitude.

Number itself importeth not much in armies where the people are of weak courage. Bacon.

5.

The state or quality of being numerable or countable.

Of whom came nations, tribes, people, and kindreds out of number. 2 Esdras iii. 7.

6.

Quantity, regarded as made up of an aggregate of separate things.

7.

That which is regulated by count; poetic measure, as divisions of time or number of syllables; hence, poetry, verse; -- chiefly used in the plural.

I lisped in numbers, for the numbers came. Pope.

8. Gram.

The distinction of objects, as one, or more than one (in some languages, as one, or two, or more than two), expressed (usually) by a difference in the form of a word; thus, the singular number and the plural number are the names of the forms of a word indicating the objects denoted or referred to by the word as one, or as more than one.

9. Math.

The measure of the relation between quantities or things of the same kind; that abstract species of quantity which is capable of being expressed by figures; numerical value.

Abstract number, Abundant number, Cardinal number, etc. See under Abstract, Abundant, etc. -- In numbers, in numbered parts; as, a book published in numbers.

Num"ber, v. t. [imp. & p. p. Numbered (?); p. pr & vb. n. Numbering.] [OE. nombren, noumbren, F. nombrer, fr. L. numerare, numeratum. See Number, n.]

1.

To count; to reckon; to ascertain the units of; to enumerate.

If a man can number the dust of the earth, then shall thy seed also be numbered. Gen. xiii. 16.

2.

To reckon as one of a collection or multitude.

He was numbered with the transgressors. Is. liii. 12.

3.

To give or apply a number or numbers to; to assign the place of in a series by order of number; to designate the place of by a number or numeral; as, to number the houses in a street, or the apartments in a building.

4.

To amount; to equal in number; to contain; to consist of; as, the army numbers fifty thousand.

Thy tears can not number the dead. Campbell.

Numbering machine, a machine for printing consecutive numbers, as on railway tickets, bank bills, etc.

Syn. -- To count; enumerate; calculate; tell.

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