8.1 Who framed the number sequence?
- Jessica Rabbit explains -
The power of Jessica Rabbit.As verified by professor Lambeau in Good Will Hunting, there's something deeply sensual about the number sequence (1, 2, 3, 4, 5...etc). There's also something deeply sensual about Jessica Rabbit. So... perhaps the two have something in common.
A girl like Jessica Rabbit.It all started with Pythagoras who is not only celebrated for discovering that the hypotenuse is equal to the square root of the sum of the squares of the two other sides of a triangle, but also for founding the religion of the Pythagoreans, with at the heart the belief that the universe can be represented by numbers. Like this:
After all, there's an amount of anything, and only so many ways things can relate to each other. The number sequence and the mathematics (which is the study of patterns) that was derived from the number sequence consider the properties of numbers before they are connected to a unit. The properties of a number (like 6) remain after it is connected to a unit (like apples or star ships or jokes). Since 6 is 2 plus 4, 6 apples is 2 apples plus 4 apples, 6 star ships is 2 star ships plus 4 star ships, and 6 jokes is 2 jokes plus 4 jokes. Numbers can be studied without a practical application, in a way a lot like Jessica's sensuality can be studied without sticking this sensuality to an actual image.
And that has a very peculiar effect. Most of us have seen a car. Most of us have seen a hundred cars parked on a lot. But not many of us have seen a million cars at once. Everybody has seen an apple. But not many of us have seen a billion apples. Not many of us have seen a billion of anything. Well, perhaps atoms. When we look up to the stars we can see billions and billions of particles at once, but still this is only a minute, almost negligible fraction of the amount of particles that exist.
It seems that big numbers are less likely to have a practical application. But mathematics considers numbers before their application. That means that big numbers are as real and proper as littler numbers. To math, 2 is as down right normal as 2,000,000,000,000,000, or an even larger number such as the googol, which is 10100 (and which is also the entropy of the total universe, hence the name of the famous search engine), or the googolplex, which is 10googol (as well as the nickname of the Google buildings - funny!). And it goes on. How about googolplexgoogolplex! Or googolplex raised to the power of googolplex raised to the power of googolplex raised to the power of googolplex...! Pfew, a number is never so large that it can't be even larger, that 1 can't be added to it, or that it can't be multiplied by 2 or 100 or by itself, or by the square of itself.
In other words, the number sequence runs ad infinitum, and the numbers that have bearing on reality are an infinitesimal small part of the whole sequence. But even though the number sequence runs towards infinity, infinity is not a point on the number sequence. After all, from every point on the number sequence (that means every number) we can go to a larger number, or a smaller one. Not with infinity. Two times infinity is still infinity. Infinity divided by 2 is still infinity. In fact, infinity divided by any point on the number sequence, from 1 to googolplex raised to the power of googolplex raised to ... etc, will still yield infinity. That means that infinity is equally far removed from any point on the number sequence: infinitely far!
Good, you'll say, because infinity does not occur in nature. If we just consider the first big chunk of the number sequence, we're in the clear and numbers may represent the universe. Sorry, no cigar...
The number 2 can only exist when 1 exists. Just like helium can only exist when hydrogen exists. All numbers are tied into each other. All numbers can only exist when all other numbers exist; every number derives its existence from the entire number sequence. The number sequence is infinite, and so there's traces of infinity in every number...
Defining numbers.The number sequence works because of some very rigid rules. Suppose there was some hidden worm hole that could take us from, say, 4 to a number far up the slope of the sequence that has the same properties of 4. That would cause the entire sequence to collapse. Every number must have an individual spot, where only that number may live. Math studies patterns, including patterns in the number sequence and over the millenia it has become evident that all numbers are somehow connected to all other numbers, like the cells of a large organism. And cells like that must have a common ground that makes them relate so vigorously. What is the common ground of numbers? The very heart of the number sequence? It's the fact that they are all segments of an infinite system. Infinity is the fundamental identity of the sequence, and the elements that make it up.
In order for the number sequence to work the following must be true for every number 'n': Whatever is smaller than n is not equal to n, and whatever is larger than n is also not equal to n. Numbers smaller than n is any number from zero to n, but numbers larger than n is any number from n to infinity! That means that in the definition of every number the mystery of infinity is lurking. There's infinity in every number, and since infinity does not occur in nature, numbers the way we know them do not apply to nature. And the reason that we see 2 apples in a basket, 5 birds in the sky and 12 cars parked in the lot is that Jessica's voluptuary bouquet of pixels is so attractive that we forgive and forget that she's not really real. We love her. We wouldn't know how to live without her. And to rescue her we dismiss the definition of numbers that identifies the number as profiling against the entire sequence, and seek to define a number by means of its nearest neighbors:
When we say 2 we mean exactly 2, not 2,00001 or 2,0000000000000001 or 2 with a novemdecillion zeroes and then a 1. When we say 2 we mean a 2 with an infinite amount of zeroes following it. Any amount of zeroes that is not infinite would allow a 1 to be added to it, making the 2 not exactly so. Just like the number pi needs infinite detail to be fully described by numbers, so does any number. Even round numbers. Every number.
There's something very unreal about numbers. But there's also something very real about them. Numbers help us to stilyze reality into a tangible shape that we can push and punch and move around. Numbers help us to define what needs to be debated, they help us built the world we live in. And although they work on a principle that depends on something that they themselves can not reach or fathom, they imitate the universe so well that a glimpse of that which they can not contain shines dimly from their ranks. It has been overly reported that the randomness which lies at the bottom of all reality actually managed to creep into the number sequence, like Jessica's very real sensuality has crept into her flat and frozen pose. And as a rose by any other name would smell as sweet, the number sequence is saturated with a perfume carrying the fragrance of nature, real nature, like a heart beating in Jessica's paper chest, pumping paper blood to her paper promontories.
Say goodbye to Jessica Rabbit and all her beauty and go to the next chapter:
There's something about the Number Sequence.