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With a deep passion for learning and knowledge, Chris is an avid enthusiast of mathematics, computer science, physics, philosophy of mind, and economics. If you have any questions or comments about what you read on this website, feel free to email him at: NoSpam@ChristopherLind.Com (Replace NoSpam with Chris) Article: Patterns In Mathematics: Randomness, Order, and Information 5-7-2005

## Patterns In Mathematics: Randomness, Order, and Information

What is randomness?
What does it mean when someone says that a number or event is random? Intuitively, a non random event is one that follows a pattern.

In theory, tossing a coin into the air is supposed to be a random event.

For example, which of the following is the result of tossing a coin into the air 20 times and writing a 1 when the coin lands heads and a 0 when the coin lands tails:

01110100001010001101

or

10101010101010101010

It is intuitively obvious that the first series of 1's and 0's was a result of my tossing a coin into the air 20 times, but not the second. Why? Because the first result DOES NOT follow a definite pattern, whereas the second one does. In other words, we percieve the first string of 1's and 0's to be random BECAUSE it does not follow a specific pattern.

As another example, imagine you have a friend go into another room and toss a coin 20 times in the air. When your friend returns from the room, they tell you that the result of their tossing a coin 20 times into the air was 10101010101010101010. Would you believe them? Probably not. Why, because this series of 1's and 0's too closely matches an obvious pattern.

The amazing thing is, even though it is intuitively obvious that the first series of 1's and 0's is random, according to classical probability, BOTH of these results have EQUAL liklihood of occuring. According to probability theory, There are 220 different permuations of a coin tossed 20 times into the air. Each of the above sequences of 1's and 0's represent just one of these 220 permutations. So the likelihood of the first result occuring is 1/220 and the likelihood of the second event occuring is also 1/220.

So probability alone (or the unliklihood of an event occurring) CANNOT be a measure of randomness. If this were not the case then you would have NO reason to believe that your friend had lied when they told you that the result of there 20 coin tosses was 10101010101010101010.

Where these 2 sequences of 1's and 0's DO DIFFER is that the second one can be produced / generated by an algorithm, whereas the first one cannot.

An algorithm is a step by step list of instructions. (For more information on algorithms, and how they relate to computers, click HERE)

An algorithm to print the sequence 10101010101010101010 is:

Write "10" ten times.

An algorithm written in Visual Basic to produce the sequence 10101010101010101010 is:

dim i as integer
for i = 1 to 10
print "10"
next i

On the other hand, an algorithm to produce the sequence 01110100001010001101 is Write "01110100001010001101".

An algorithm written in Visual Basic to produce the sequence 10101010101010101010 is:

print "01110100001010001101"

Notice what happens when we extend the two sequences of numbers from 20 to 40 digits:

The algorithm to produce 1010101010101010101010101010101010101010

is: print "10" 20 times,

whereas the algorithm to produce the sequence 0111010000101000110101100111110101100000

is: print "0111010000101000110101100111110101100000"

If you think about it, you will see that as the random number gets larger and larger, the size of the algorithm whiech produces it must grow proportionately larger and larger, whereas the algorithm to produce the non-random number 1010101010... will stay roughly about the same size, regardless of the size number it produces.

I first learned of this definition of randomness off of IBM researcher Gregory Chaitin's website. He is the co-discoverer of this concept

## Another way of stating the above is that "algorithms CREATE ORDER or PATTERNS". Whenever you see a pattern, there is an underlying algorithm that has caused it.

In mathematics, algorithms are called functions and in physics, algorithms are called natural laws. A function is an algorithm that relates one or more input values with a single output value.

For example:
C = A * B is a function.
This function relates any two numbers A and B with a single product C.
It is also an algorithm. There is a step by step list of instructions you must follow in order to arrive at the answer C.

If you were given: C = 5 * 6, and did not know your multiplication tables, you would have to follow the instruction to "ADD 5 to itself 6 times".
Furthermore, addition itself can be broken down into an algorithm.

In physics, natural laws are written down as mathematical functions, Such as Force = Mass * Acceleration, and therefore natural laws are also algorithms. The point is that algorithms, mathematical functions, and natural laws are one and the same thing.

Patterns and Natural Law philosophy
Reality has an underlying order to it. This order is created by natural laws (the "rules" of physics). These natural laws apply to everything and everyone in the universe, whether they are aware of these laws or not. Natural laws are discovered through the use of REASON. Reasoning can be used to discover natural laws precisely because reality has an underlying order to it. Natural laws are discovered by using your intelligence (ability to reason) to perceive and recognize patterns in nature. New mathematics is discovered by using your ability to reason to perceive and recognize patterns in numbers.

Furthermore, algorithms (natural laws and functions) COMPRESS information.
The string 1010101010101010101010101010101010101010
can be compressed down to:
print "10" 20 times

There is a saying that "Theories destroy facts". What this saying means is that before a law of nature is known, you simply have LOTS and LOTS of data or information. Before Isaac Newton discovered the laws of gravitation, people like Kepler had produced detailed charts of the motions of planets across the sky. You could use these charts to help determine where the position of a planet should be. When newton discovered his law, you no longer needed all of the facts from the chart, all of the information contained in the chart had been compressed(reduced in size) into a simpler formula(algorithm).

Understanding IS Data Compression
You understand something when you can create an algorithm which can produce the pattern, event, or phenomenon you are trying to explain.

However, an algorithm that is as large as the the phenomenon you are trying to explain does not explain anything at all.

print "10" 20 times
is an algorithm that is 19 characters long(including spaces). This algorithm explains the sequence of supposed coin tosses: 1010101010101010101010101010101010101010

An algorithm to explain nature reduces the information you see down to a smaller amount of information..."Theories(algorithms) Destroy Facts".

Phenomenon that CANNOT be explained in terms of a simpler algorithm, truly are RANDOM, in the since that they cannot be explained in terms of something simpler.

The size of an algorithm, and the size of the output it generates, can be measured in terms of INFORMATION. This brings up the question, What is information?

## Information and the Explanatory Filter

What is the difference between
01110100001010001101

and

10101010101010101010

They BOTH contain the SAME amount of information. The difference is that one of them can be generated by a simpler algorithm, the other cannot.

What is the difference between
"Hey fiddle diddle, the cat in the middle, the cow jumped over the moon" and
"sukachfulsad noa uhnah asiofjoihasd paisfdh aosidhjoaish roihas dfoiha"

They BOTH contain the SAME amount of information.
Also NEITHER of them can be generated by a simpler algorithm.
According to the definition of Randomness I gave at the start of this article, they are BOTH random!

Yet intuitively, it does not seem possible that the "Hey fiddle diddle" string could be random. There seems to be a major difference between these 2 strings. What is this difference?

Imagine you have a friend go into another room and toss 70 scrabble pieces into the air. When your friend returns from the room, they tell you that when the scrabble pieces landed, they landed in this order: "Hey fiddle diddle, the cat in the middle, the cow jumped over the moon" Would you believe it? Probably not. Anyone would believe that your friend had intentionally laid the scrabble pieces onto the floor. The "Hey fiddle diddle" string fits a specified pattern of information that most people are familiar with, the old Mother Goose nursery rhyme. On the other hand,it is possible that when your friend tosses the scrabble pieces into the air, the letters "dog" may appear. Why, because the word "DOG" only contains 3 letters. It does not contain enough information in it to rule out the possibility of chance turning up the letters "DOG" in succession.

Whenever an event or phenomenon is sufficiently large in size(contains enough information), Cannot be generated by a simpler algorithm, and contains fits a SPECIFIED pattern, it can be concluded that the phenomenon or event can be attributed to an intelligent agent. Information that meets all 3 of these criteria is known as Complex Specified Information.

The concept of Complex Specified Information, and the method to detect it, were discovered by mathematician William Dembski.
This is another link to info on Dembski's Work.