Saturday, 20 April 2013

The Problem of Points - Symbols

Christian Kramp invented the usage of the symbol ! to represent factorial.  The idea of $n(n-1)(n-2)..2.1$ as an interesting product which represents all the ways that n unique and completely distinct objects can be permuted dates to at least the 12th century.  To show this is the case imagine the following thought experiment.

Imagine that a man walks into one of the world's most sumptuous brothels.  His reward was to have a sexual encounter with every available woman in the brothel.  He enters a room filled with seven of the most desirable women he has ever seen.  One has jet black hair, is nearly six feet tall and has a full and firm ass.  Two is smaller with short black hair and a wicked twinkle in here eye.  Three is central Asian looking, small perfectly formed breasts.  Four is from south east Asia, an imperturbable and fiercely independent soul.  Five is from central Europe, innocently cheeky.  Six is from the Caribbean  tall and rangy.  He knows that he will fuck them all.  He picks number one, goes with her, comes back, picks number three this time.  Repeats until he is done.  This is a permutation.  A single permutation.  How many ways could he have chosen?  The answer is 6!, which   is 720.  He could have had 720 chronologically different sexual experiences.

Imagine you are him.  Imagine his decision moments.  His first decision was to pick the tall jet black haired girl.  This was a so-called free choice.  A choice of one in six.  Immediately start imagining a total of six parallel worlds.  In each world, he started with one or another of the six girls.  OK.  Focus on his reality again.  In the real world, he now has to chose which one of the five remaining girls to fuck.  Now, in each of the 6 parallel worlds, he'll similarly need to make a 1 in 5 choice.  So for each and every one of the 6 imagined worlds, a further multiplication of 5 occurs.  There are now 6 times  5 or thirty worlds which capture all the possibilities he has to chose from.  Repeat and you will find 720 different worlds.  In making his particular ordering choice,  he in essence picked one of 720 different worlds.

Without knowing the man, his sexual preferences, etc, if we had to guess his selection we could imagine each of the 720 worlds was equally likely to be chosen.  That is, you might say that the likelihood of choosing the fucking order of <tall-dark then short dark then central Asian then SE Asian then central European and finally Caribbean> was $\frac{1}{6!}$. 

Insofar as every object is uniquely distinguishable from every other object, then this idea of permutations is quite core to the most basic choosing activity - their ordering.  Visually you can see this as lining them up in a particular way, touching them in a particular, order, placing them sequentially on a particular chronology.

Imagine you got to know that some kind of selection process has 1,000,000 different distinct possibilities, the final result being a collection of 6 uniquely distinguishable objects.  Well, if you didn't actually care about the ordering your set of possibilities would be reduced down to $\frac{1000000}{6!}$.  Imagine you needed to pick 6 women to come on a journey with you.  Someone informs you that the selection process imposed on you allowed you to pick a particular order-important set of 6 women with equal chances $\frac{1}{1000000}$.  But you know that it doesn't matter to you what order they got picked.  They're coming with you all the same.  In this case, you'd divide your million possibilities by 6!.  Thus $\frac{6!}{1000000}$ is the likelihood of picking those 6 women (or indeed any 6 women).  Here you're using the permutation formula to make order irrelevant.  By knowing the set of choices implied in a strict ordering, you can use that knowledge to throw away the importance of ordering.

Klamp chose the word 'factorial' because it sounded more French than its rivals.