More Televised Mathematics: Chaos

Just as I was planning both to blog less, and to watch less television, BBC4 is provoking me by broadcasting yet more Mathematics.

There was a programme about chaos on BBC4 8-9 pm on Tuesday 14th October.

I much preferred that program to the previous night’s broadcast on Mathematics. The central idea of chaos was clearly explained, with a careful distinction between chaos and complexity, though they did miss an opportunity by not giving a simple arithmetical example, such as the one in my own essay on the subject.

My only serious doubts concerned the remarks about climate and the economy in the latter part of the programme. I thought that in both cases they were wrong to state categorically that those are examples of chaos. There is reason to suspect that they may both be chaotic, but I think it is still an open question whether they are.

They appeared to reason that because weather is chaotic and climate describes weather, climate must also be chaotic. I do not think that argument conclusive. Climate is a sort of statistical averaging of weather and the averaging might eliminate the chaos. Of course it might not, but I’d prefer not to assume climate chaotic without other evidence.

Some of the discussion of climate, with mention of tipping points and irreversible of changes, reminded me of catastrophe theory, something we used to talk about a lot in the 1970’s. I remember I once made a catastrophe machine from two pieces of cardboard and an elastic band. I wonder if I still have the instructions? A picture of a similar machine is shown here. But I found the computer animation hard to follow and recommend readers to make a model for themselves.

In the case of the economy, they seemed to assume that it must be chaotic because it is not predictable, but as I understand it, unpredictability alone does not constitute chaos.

Chaotic systems are ones that are deterministic and unpredictable. Typically they are in principle predictable, and in practice their behaviour can be predicted with moderate accuracy for a short time, but accumulation of errors limits that time span.

The problem with weather forecasting is that the underlying theory is such that errors accumulate and eventually overwhelm our calculations. The problem with economic forecasting is that there is no plausible theory capable even in principle of making precise predictions; calculations cannot be overwhelmed by accumulating errors because there are no plausible calculations to be overwhelmed. In my own essay referred to earlier, I quote Herbert Spencer’s description of what form a theory of society would take if there were one, and it is clear that we have no such theory.

It is true that social and economic trends sometimes resemble the behaviour of chaotic systems, but I hesitate to state categorically that they are chaotic. They might be, but they might be unpredictable because they are not deterministic, or because they are too complicated for us to understand.

Another possibility is that prediction of the behaviour of society is vitiated by feedback loops whereby human beliefs about society affect the development of the society. In that case it might be only members of the society who are unable to predict its behaviour. It might still be possible for reliable predictions to be made by inhabitants of another planet observing us without our knowledge.