The mathematics of chaos in economics and nature

The Mathematics of Chaos — a documentary that was aired on BBC Four not too long ago is an informative, enlightening but disturbing viewing experience. In brief, it contrasts the modern understanding of causality (and the mathematics this presupposes) with the more recent, perhaps in a sense “postmodern” understanding of causality (again with a concomitant mathematical model) and on the basis of this distinction, draws some interesting parallels between economics and natural events.

The reason why I said “in a sense” postmodern, is because there are so many ways to conceive of postmodernity (and postmodernism, which is not the same thing). The one I have in mind here is perhaps closer to poststructuralism, with its signal rejection of linear ways of thinking and binary logic (either/or) in favour of complex modes of thinking and paradoxical logic (such as both/and). From this perspective, one can characterise postmodernity as the culture or society of complexity at may different levels — including the cultural, economic, scientific and political.

The documentary I have referred to charts the scientific differences between a Newtonian universe, where linear mathematical thinking guarantees, with certainty, the ability to calculate the coordinates of a body in space at some future or past time, provided the requisite information about its present position is known, on the one hand, and a “chaotic”, or “complex” universe, where what may seem like a calculation unaffected by an infinitesimal deviation or modification of initial conditions, turns out to be hugely affected by it in the long run.

The 19th century mathematician Poincaré already worked out this “mathematics of chaos”, specifically with reference to the problem of two bodies in space, travelling parallel paths, where the “Newtonian” calculations customarily demonstrated that these bodies would stay on exactly the same course, but Poincaré, by contrast, showed that an apparently insignificant difference in trajectory under initial conditions would eventually cause these bodies — for example hot air balloons — to shoot off in totally different directions.

In other words, this mathematician already had an inkling of the complex interrelationships among events in the universe, something that was corroborated, decades later, by a (by now) well-known meteorologist, Edward Lorenz, in the early 1960s when he discovered that what seemed like an insignificantly small mathematical variation in the trajectory of a weather prediction, resulted in a completely different set of conditions. This phenomenon eventually became known as “the butterfly effect” (a phrase first used by science fiction writer Ray Bradbury) and denotes what is technically known as “the sensitivity involved in dependence on initial conditions” in chaos theory.

At this stage, intelligent readers will no doubt be able to guess where I am going with this, even if they haven’t seen the BBC Four documentary. To cut a long story short: in the documentary (which is almost an hour long, so I cannot repeat it here in all its detail) a persuasive contrast is drawn between the modern, “Newtonian” mindset, intent on “controlling” things at all costs (and believing it is possible, of course) whether it is in economics or as far as nature is concerned, and the “postmodern” insight into the kind of universe hinted at by Poincaré’s mathematics of chaos and Lorenz’s later rediscovery of this in meteorology.

Importantly, in this, very much contemporary understanding of things, neither economics nor natural events are, strictly speaking, controllable. The insightful discussion, in the documentary, of the 1929 Wall Street crash, as well as the events following the 1986 interlinking of all the economies in the world, makes it easier to understand why it is that, despite positivistic beliefs to the contrary, human beings just cannot predict, nor control, the long-term developments of complex systems like the global economy (which comprises a large number of interdependent, but slightly different, economies). Nor can one predict, with certainty, the development of global meteorological systems, which form a part of the complex global ecological system.

To mention but one, very relevant thing: In the presenter’s discussions with several scientists, the question of “tipping points” comes up — something that features in a lot of the speculations about climate change today. Al Gore is one of those who believe that, if humanity could decrease its carbon emissions significantly within the next 15 years, we could reverse the effects of having reached the “tipping point” in global warming — there is a “window period” for us to act and prevent a cataclysm.

However, one of the scientists interviewed in the documentary, Dr Lovelock, the “godfather of the ecological movement”, believes that this is “nonsense” because it rests on the fallacious belief that humans could reverse a process that has already gone a certain (irreversible) way. Instead of talking about a “tipping point”, he prefers the metaphor of a “slope” on which we find ourselves, and which is gradually getting steeper, until it reaches the point where we will simply “fall off”.

Part of his reasoning involves the fact that at least 60% of the sea ice at the North Pole has melted, and when the point is reached, possibly in the next 10 years or so, where it has all gone, the earth’s cooling system — the circulation of cold water by means of sea currents dependent on the difference between the cold north and the warm equator — will effectively shut down. Because Lovelock knows the mathematics of chaos regarding nature, he will not make a firm prediction, but he does venture to say, reasonably, that humans cannot put that ice, so vital for the well-being of the planet, back there, and hence we can reasonably expect the Earth’s average temperature to rise by about 5 degrees, if the ice does disappear. I’m not sure whether he was talking Fahrenheit or Celsius, but in either case, this is likely to give rise to weather conditions that are so far removed from anything we know that we can only speculate about them.

I have noticed that Gordon Brown is allegedly elated about the “historic agreement” reached at the G8 Summit concerning the cutting of emissions by large margins — so large that they would be 80% down by 2050 — but in light of the scientists’ comments in The Mathematics of Chaos, this seems to me to be almost totally insignificant. And yet, although it may indeed be the case that, whatever we do at this stage to lessen carbon emissions, it could already be far too late to avert a catastrophe, this is no reason NOT to try — but not by aiming for 2050. The time to do it is NOW, lest we wake up, one morning, to a world where climate conditions have become so turbulent that none of the economic activities humans engage in daily can continue unhindered. That would be a recipe for internecine strife, even war.