Sunday, May 13, 2007

Mathematical Politics: Part 6

Mathematical IntractabilityRandomness is a complexity upper limit – size for size nothing can be more complex than a random distribution generated by, say, the tosses of a coin. A sufficiently large random distribution configurationally embeds everything there possibly could be. And yet in spite of this complexity, it is a paradox that at the statistical level randomness is very predictable: for example, the frequency of sixes thrown by a die during a thousand throws can be predicted with high probability. In this sense randomness is as predictable as those relatively simple highly organized physical systems like a pendulum or the orbit of a comet. But in between these two extremes of simplicity and complexity there is vast domain of patterning that is termed, perhaps rather inappropriately, “chaotic”. Chaotic patterns are both organised and complex. It is this realm that is not easy to mathematicise.

We know of general mathematical schemes that generate chaos (like for example the method of generating the Mandelbrot set), but given any particular chaotic pattern finding a simple generating system is far from easy. Chaotic configurations are too complex for us to easily read out directly from them any simple mathematical scheme that might underlie them. But at the same time chaotic configurations are not complex enough to exhaustively yield to statistical description.

The very simplicity of mathematical objects ensures that they are in relatively short supply. Human mathematics is necessarily a construction kit of relatively few symbolic parts, relations and operations, and therefore relative to the vast domain of possibility, there can’t be many ways of building mathematical constructions. Ergo, this limited world of simple mathematics has no chance of covering the whole domain of possibility. The only way mathematics can deal with the world of general chaos is to either simply store data about it in compressed format or to use algorithmic schemes with very long computation times. Thus it seems that out there, there is a vast domain of pattern and object that cannot be directly or easily treated using statistics or simple analytical mathematics.

And here is the rub. For not only do humans beings naturally inhabit this mathematically intractable world but their behavior is capable of spanning the whole spectrum of complexity – from relatively simple periodic behaviour like worker-a-day routines, to random behaviour that allows operational theorists to make statistical predictions about traffic flow, through all the possibilities in between. This is Super Complexity. When you think you have mathematicised human behaviour it will come up with some anomaly....

To be continued.....

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