5.3 Complexity

So what's the connection between entropy and complexity? Well however you conceive of the term complexity it intuitively includes concepts like 'special', 'rare', 'not uniform' in comparison to less complex being 'uninteresting', 'common', 'uniform'. In the program “Life” commented by David Attenborough he frequently uses the words “remarkable”, “amazing” and “incredible” to describe unusually special features or characteristics of living creatures suggesting their complexity. There is one area of mathematics which is all about such terms.. probability.

The occurrence of rare events is measured by the ratio of the number of them divided by the total number of possibilities and the answer is always between 0 and 1. Its the chance of such an event occurring based on the assumption that all possible events have the same chance of happening. 0 = impossible, 1 = certain. We talk of 'events' not objects, such as in the occurrence of 3 aces in a poker hand. In probability speak the occurrence implies both the acts of shuffling (randomizing) and fairly dealing the hand! If the deck were not shuffled or the deal unfair there would be a bias toward certain combinations which would significantly reduce the accuracy of the probability calculation. As the number of possibilities increases so does the 'improbability' of the event occurring, ie the number of possibilities is a measure of improbability. Which logically suggests improbability as a measure of complexity!

That statement is significant because from the work of Ludwig Boltzman and Gibbs came an equation for entropy (s) which related it directly to the logarithm of the number of possible states (called micro states - W) that particular class or type of arrangement (called macro states) can have. ie inverse of probability or improbability (by convention it is called thermodynamic probability)! Max Plank had the equation [ s = kLog W ] inscribed on Botzman's gravestone. Its true significance is sadly underestimated and not understood as it should be. This equation puts a numerical scale on the axis of complexity! It reveals the vital link between entropy, complexity and probability. The real answer to Mark Riddleys problem.

Have a nice day..