In Praise of Chaos and its Ordered Understanding

If you think you know what chaos theory is, keep reading. If you think it has anything to do with butterflies, keep reading. If you think it is anything to do with dinosaurs, keep reading. If you think it is a type of randomness, keep reading. If you think you know what a fractal is, keep reading. If you have never heard of Mandelbrot and Julia, or of non-linearity, if you think continuously discontinuous is something Donald Rumsfeld gets laughed at for saying, keep reading. And above all, if you think Chaos Theory has anything to tell us about human relations, the future or the meaning of life, you should very definitely keep reading.

Insofar as the Hickory Wind is about anything much beyond my ramblings, it is about truth- the nature of truth and the ways in which we deceive ourselves and others. There are many scientific theories that are used symbolically by writers, journalists, culturalists and ‘ordinary people’, as though they were directly applicable to and descriptive of other areas of activity or thought, or just to the world in general. This usually comes from ignorance- a failure to understand the theory itself, the limits of its reference and the nature of scientific, or of rational thought. False analogy is an error so common in the reasoning even of people who imagine they know how to think that it is scarcely worth pointing it out most of the time, but certain errors of the type outlined influence the way of interpreting the world of very large numbers of people, and should be challenged, identified and corrected.

Gödel’s incompleteness theorem, the general theory of relativity, Heisenberg’s uncertainty principle, Darwinian evolution are some of the main tools by which people who do not understand them try to make sense of the world about them. They are, in effect, using words as totems to give the world a shape they can be happy with. They are not using the ideas themselves, since they have no inkling of what they really mean and they do not realize that they are quite useless when applied outside their areas of reference. I intend to write about all of these in time, but today it is the turn of Chaos Theory.

Chaos Theory is essentially the mathematics of dynamical systems that are sensitive to initial conditions. Specifically it deals with systems of functions in which an arbitrarily small change in a variable produces a change in the result that cannot be held within a determined limit. This is what is meant by continuously discontinuous; it happens at every point. It doesn’t mean that the result is not defined by the starting conditions; the result is completely defined, but changing them very slightly can cause a very large change in the result. Many physical systems can only be modelled by such mathematical systems, the weather being the best known example, and since the initial variables in a physical system cannot be precisely determined, the results are necessarily unreliable over time or space.

The beauty of mathematical chaos does not lie in anything it supposedly tells us about the uncertainty of nature and the lack of determinism in human life- such interpretations are utter nonsense. Chaos is an old word and an old idea; mathematical chaos is a precisely defined concept which only tells us anything about mathematics. And the beauty is in the expression of the ideas behind it, the transformations that can be done with it, and what it reveals about the abstract world of numbers. It also helps us to create some lovely images, but the beauty I refer to is different. It is an aesthetic response, felt directly in the soul, uplifting and immensely satisfying, as the contemplation of any kind of physical beauty may be. Intellectual beauty is no different.

Non-linear dynamics can describe, often with great simplicity, systems that until recently were inaccessible to analysis, many believed to be irreducibly complex. It is simplicity and elegance which are the key to mathematical beauty, but also the efficiency and precision of that elegance, without which you merely have a drawing by Van Gogh, not a sunflower. The psychobabble and cultural studies mob prattle about chaos, but in order to learn anything about the world you have to create a model for the system, and do the mathematics. That’s the hard bit, and it’s much easier just to waffle. After all, the people who listen to you won’t notice.

My purpose in this and, I hope, subsequent posts, is not to tell you you're thick, but to suggest that there is much more to understand than you might have realized, and that it can be understood (try the links). And above all, perhaps, to dispel the idea that these theories, theorems, tools, methods are in any sense philosophy. It is this misunderstanding that causes the most trouble.