|
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.
|
|