Monday, December 26, 2005

Geometry and the Philosophy of Mathematics.

I blogged this some time ago, but wish to again, rather than as a record of history, as a thought provoker. It was some time ago that I first pondered the nature of the system which underlies our whole scientific endeavour, that of course being the language of mathematics. I am speaking as it were of the philosophy and underlying principles, rather than the semiotics, i.e. Arabic or Indian numeral systems. There is of course the contention that for Homo sapiens Base 10 is ‘natural’ given we have 10 digits on our hands. This then gives us a real reason for base 10, but are we really to presume that maths exists solely because of our hands, or is it more likely that there exists a philosophy/science that exists in the very fabric of the universe. Our digits as it were simply allowed us a ‘key’ to access a comprehendible aspect, which in turn lead us to a greater understanding of the language itself. It could of course be argued that the first concept of math we experienced was Base 2, while in the womb; asleep and awake, noise and quiet, light and dark, and on our subsequent birth, day and night, warm and cold. The list is exhaustive, but it would seem that our early experience indoctrinates us to a binary concept. Let us then consider binary

0 1

Within this system lies the whole mathematical system, it appears everywhere in science, thought, action, philosophy, mysticism. Zero has a close companion that being ∞ as when a number is divided by either the result is the same as the dividing factor. In the wanderings of the minds of men, and probably in any creature that has a conscious mind, it is the pondering of the infinite/nothing principle that leads man to define, and the concept of one is, not to state the obvious, the first conception, the first ‘defined’. In the same way that night leads into day, the indefinable where are sight is limited, night, leads into day, where we think we see more clearly, and things are more clearly defined. In pondering the movement of night into day, we may consider 0 → 1 , and thus with the drawing in of night its counterpoint, 1 → 0. With this ‘movement’ we are presented with a linear positive progression, but also crucially a movement in the negative, -1, through the same field.

-1 0 1

Thus in pondering the procession of night into day and again its recession we are presented with what would appear as the archetypal dialectic, where 0 is pondered as undefined, then gaining definition with 1, but then producing -1 as it were through a reflection in 0. From thinking upon night and day as 0 and 1, we achieve day and night as -1 0 1. From this movement we receive a further movement and thus concept of 2, as from -1 to 1 is the ‘distance’ which covers that interval, thus, by extension, we achieve the base 2 progression.

-16 -8 -4 -2 -1 0 1 2 4 8 16

Although most commonly binary calculation occur in the positive field its very concept and progression can be seen to have its roots in the above dialectic progression. Although 0 and 1 maybe be seen as the prototypical dialectic the progression -1 0 1, seems to encapsulate dialectical thought more completely as it possesses thesis, antithesis and synthesis. Hidden within this system are the numbers present in the common decimal (base 10) system, both by movement between the numbers present but also when the binary system is considered, i.e. between 2 and -1 is a progression of 3 increments, and 11 in binary is 3 by the decimal. Thus the numbers seen to be missing from the base 2 progression outlined above are present as abstracts if not delineated in symbol. As is often shown in nature the geometric progression above is prevalent. This progession is as it were through one plane, one dimension, what if we extend it to two dimension, or at least two of the possibilities.

16
8
4
2
1
-16 -8 -4 -2 -1 0 1 2 4 8 16
-1
-2
-4
-8
-16

Here the progression takes place in two dimensions, through four vectors. There are however immediate implications when one considers the inherent geometric progression where 1 becomes two becomes 4 etc.. If we consider the decimal system and its progression interesting patterns emerge. This and further manipulations is best represented by the following numerical/geometric diagram

8 7 6 5 4 3 2 1 0 9 0 1 2 3 4 5 6 7 8
7 6 5 4 3 2 1 0 9 8 9 0 1 2 3 4 5 6 7
6 5 4 3 2 1 0 9 8 7 8 9 0 1 2 3 4 5 6
5 4 3 2 1 0 9 8 7 6 7 8 9 0 1 2 3 4 5
4 3 2 1 0 9 8 7 6 5 6 7 8 9 0 1 2 3 4
3 2 1 0 9 8 7 6 5 4 5 6 7 8 9 0 1 2 3
2 1 0 9 8 7 6 5 4 3 4 5 6 7 8 9 0 1 2
1 0 9 8 7 6 5 4 3 2 3 4 5 6 7 8 9 0 1
0 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 0
9 8 7 6 5 4 3 2 1 ∞ 1 2 3 4 5 6 7 8 9
0 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 0
1 0 9 8 7 6 5 4 3 2 3 4 5 6 7 8 9 0 1
2 1 0 9 8 7 6 5 4 3 4 5 6 7 8 9 0 1 2
3 2 1 0 9 8 7 6 5 4 5 6 7 8 9 0 1 2 3
4 3 2 1 0 9 8 7 6 5 6 7 8 9 0 1 2 3 4
5 4 3 2 1 0 9 8 7 6 7 8 9 0 1 2 3 4 5
6 5 4 3 2 1 0 9 8 7 8 9 0 1 2 3 4 5 6
7 6 5 4 3 2 1 0 9 8 9 0 1 2 3 4 5 6 7
8 7 6 5 4 3 2 1 0 9 0 1 2 3 4 5 6 7 8

This ‘number square’ represents one possibility. If the numbers where multiplied, squared, cubed, in fact any variation, then the fractal/vortex created would subsequently be different in actuality but common in form. Of course this is through four vectors, but potentially there could be as many vectors as could be imagined. The basic vortex concept is shown in simplistic form below.
This then is imagined as the ‘mechanics’ or philosophy that lies behind mathematics or geometry, and that the system known to H. sapiens has nothing to do with systematic addition of one item to another, but is ‘sewn’ into the very fabric of the universe, seen and unseen, our desire to quantify has merely touched upon its surface, most people never progressing beyond 1-10, looking at their fingers, objects visible in the world, and believing that to be the genesis of mathematics and geometry

http://www.what-means.com/encyclopedia/Complete_graph

1 comment:

Steven Tucker said...

I've got to tell you, that was WAY over my head - but very informative.