Teleology is the principle that final causes exist in nature, that is, that there is something or someone that has established a purpose or goal to which a system such as the universe develops in time. Teleology makes an implicit appeal to some absolute primary observer, which is why it finds favour with religious people.

Teleology is regarded in science as an unscientific principle because either it is stated in a contextually incomplete form designed to lead to an acceptance that there is an ultimate primary observer (the designer of the universe), or else it is stated with explicit reference to a metaphysical primary observer such as the God concept.

Chaos

As science and mathematics develop, there arises, surprisingly frequently, popular interest in various new or rediscovered ideas. One of these is chaos theory. It is of interest to non-scientists because their lives can be affected by chance events, the consequences of even the smallest change can be magnified over time to have devastating unforeseen consequences. Chaos theory is a mathematical discussion of such phenomena.

Dynamical systems exhibiting chaotic behaviour occur in both classical and quantum theory. It is classical chaos that is generally referred to in popular debate, because that is where it seems most surprising to the non-mathematician. After all, quantum mechanics is well known for its random outcomes, whilst standard classical mechanics was, until relatively recently, always thought of as fully deterministic. In fact, chaos and strict determinism are not inconsistent. Far from it: chaotic phenomena are most spectacular when discussed within a strictly deterministic context.

We said above that Newtonian mechanics is based on a process time perspective: given an initial state ?_{0} of an SUO, the equations of motion can be used to determine ?_{t}, the final state at a later time t > 0. That is a strictly deterministic picture.

Now suppose we alter the initial conditions, from ?_{0} to say ?0, reminiscent of Reichenbach’s mark method discussed above. Then the equations of motion applied to ?0 will evolve it to a unique final state ?*. The question is, what is the relationship between the ordered pairs (?_{0}, ?_{t}) and (?0, ?*)?

To quantify this, we need some way of describing differences between states. A typical method would be to use the properties of real numbers as follows. Given the two initial states ?_{0} and ?0, construct in some agreed way a positive number d(?_{0},?0), with the property that d(?_{0},?_{0}) = 0 and d(?_{0},?0) > 0 if ?0 = ?_{0}. This will serve as a definition of ‘distance’ between the states.

The question then is, what is the relationship between d(?_{0}, ?0) and d(?_{t}, ?*(t)) for any t greater than zero? This question can be discussed in several ways. We give now a heuristic (hand waving) account of one, involving the so-called Lyapunov exponent.

A Lyapunov exponent X is a specific measure of sensitivity to initial conditions that gives a measure of the timescale over which chaos sets in. It can be estimated for various realistic SUOs. For such systems, there is an approximate relationship of the form

for large enough time t and small enough d(?_{0}, ?0). The exponent X may itself depend on the initial states. If the maximal Lyapunov exponent is positive, then the system is taken to display chaos: given d(?_{0}, ?0) > 0, then d(?_{t}, ?*) will eventually exceed a given bound. For example, if we want to ensure d(?_{t}, ?_{г}*) < S, where S is positive, then if (9.1) is taken to be exact, then by a time T = X^{_1} ln {S/d(?_{0}, ?0)} we have reached the limit.

The point here is that if an SUO demonstrates chaos, then there will be a characteristic timescale beyond which predictions become meaningless. An essential feature here is that chaos cannot be attributed to an SUO alone: a deterministic SUO evolves without any concern for predictability. Chaos is a discussion of a relationship between an SUO, time, and the observer.