The standard coordinate system. N-dimensional points are represented as ordered n-tuples, with each element representing a distance along one of n orthogonal axes.

For example, the 2 dimensional cartesian system (the cartesian plane) represents points as (x, y), where x is the distance along the x axis, and y is the distance along the y axis. The distance between 2 points in a cartesian coordinate system is sqrt ((a1 - b1)^2 + (a2 - b2)^2 + ... + (an - bn)^2) (if you can fly -- it's sum (abs(a_i-b_i)) if you have to take a cab), where a is one point and b is the other. ak is the kth element of a.

The Cartesian coordinate System was created by a sickly French man named Rene Descartes. The actual concept was first considered when he was laying on a bed one day. As he stared up at his tiled ceiling he noticed two beams, one horizontal and one vertical. While waiting for his medication he noticed a small fly flying about his room. As it passed a tile he though how wonderful it would be to be able to assign an actual number value to the point where the fly landed. This eventually led to the Cartesian coordinate System.

All the crowned heads of Europe gasped in awe.

Descartes went on to demonstrate his "coordinate system" (as he called it) for the delectation and amusement of heads of state and cafe society in all the great cities of Europe. Wearing a snappy boater and accompanied by his beautiful blonde assistant Zimroel, Decartes danced from strength to strength, from triumph to triumph, for the final forty years of his life. His passing was greeted with inconsolable grief, and forty millions of livres were subscribed by an anguished public to build an awe-inspiring brass monument to the man, a monument now found in the Tuillieries in Paris.

Although I often laugh at the notion of Western Culture, the notion that a few hundred years of history in Europe are what seperates us from Hominids, I do have to give credit for Descartes for coming up with this coordinate system. Whenever, for reasons both practical and speculative, I need a concept of an unchanging system of reference, the first thing I envision is Descartes simple two axis system. Thinking of everything around me as being underlined by graph paper is how I try to begin my logical thought.

When I am driving around the streets of Portland, and I make a wrong turn, and then a few more wrong turns, where do I go back to find my bearings? Imagining that the city around me is laid out on the perfect right angles of Descartes. It is one of the most practical conceptual tools that I actually inherited from the grandest culture ever.

That being said, it should also be said, IIRC, that Descartes himself didn't believe in Cartesian space. He didn't belive that there was an absolute space underlying everything around us. Because, of course, only things with substance, or material, can have extension. This extension creates the illusion that there is a space underlying everything, but since space is not material, it can therefore not have extension. Thus, no absolute space or perfect vacuums can exist.

I won't even get into the Preexisting harmony of all monads