Being able to tell the future has always been valuable, whether it’s by astrologers, clairvoyants or entrail readers there has never been any shortage of people willing to give it a go. Of course there has been some success, most notably in our understanding the motion of the planets. Even before Newton there was a model that could roughly predict where all the planets were likely to be at any given time, but since Newton our ability to do so has improved vastly.
All sciences have at some stage wanted to replicate the sort of sheer predictive power of physics, a goal that is certainly worthy. The problem is that in general they are studying much harder problems. This is not a new observation of mine, but it is interesting to reflect that physics, while frequently regarded as a very difficult subject, in reality solves easy problems. Now to solve those problems at times involves some fairly sophisticated mathematics but ultimately the problems themselves are simple. Many other fields, and I am particularly thinking of economics, as that’s a lot of what I’ve been reading about recently, intrinsically tackles problems that are complex.
Firstly to make it clear I need to define what simple is. Simple is a system where the behaviour is governed by a small number of interactions. “Governed” meaning that while there may be other effects one far and away dominates. Thus we can get accurate orbits for Venus around the Sun or any other planet for that matter. Even though there are many bodies in the solar sytem which all have a gravitational effect on Venus the one that really matters is the Sun. The behaviour is governed by the gravity interaction with the Sun. Of course that is not the only gravitational interaction, but it gets us most of the way there and we can look at the others as merely small perturbations. Thus the solar system is “simple”, we can build a good model by modelling each planet orbiting the sun separately and then performing a few perturbations for planet-planet interactions and get everything just about perfect.
Traditionally physics attempted to work out the rules governing interactions of particles (forces of nature), and made its most successful predictions on the basis of these types of simple systems or by breaking problems into a series of very simple systems. For many things this is all that was needed to make some fairly spectacular predictions.
In my view the overwhelming success in physics has lead to a general belief in this sort of reductive method being able to reveal the full nature of the universe. Unfortunately things aren’t that simple. It’s well known that while we can model the interaction of two objects under the influence of gravity, there is no integrable solution to the three body problem, or indeed in general for the n-body problem. Similarly in Quantum mechanics, which it is possible to solve the Schrödinger equation to get a solution for the Hydrogen atom, such an analytic solution is impossible for larger atoms (eg Helium) because of the inherent many-body interactions.
Of course the non-existence of a nice integrable solution is not the end of the matter. There is any number of either asymptotic expansions or approximate solutions, and in this age there is always simulation by throwing buckets of computing power at it. This is fine for low and moderate complexity systems, but with number of interactions growing as the square of the number of constituent elements, the problem rapidly grows out of control.
The realisation is there is a fundamental bounding to the types of problems we can solve. What can we and what can’t we predict about the future? The ultra determinist line, summed up by Laplace:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
is in fact wrong. The problem is incalculable.
That’s not to say that physics has had no success treating problems of many interacting particles. Statistical Mechanics and Thermodynamics are both subjects of great importance and value, not to mention great success. However they are also not without assumptions and attendant limitations, with problems being derived for equilibrium states and the problem of non-equilibrium being barely dealt with at all.
In a great many other subjects, complex problems of this sort which are fundamentally incalculable are the fundamental problems of the subject. For example when we consider the problems of macro economics and a free market economy, it is the economic interactions between the individuals which drive the dynamics of the system. We buy, sell and earn mostly or wholly from private entities, who in turn do the same. If in a single day an average an individual has interactions, with thousands of different entities. The shops available that you could potentially buy from but decline to, are still form part of your decision even if you don’t actually exchange anything with them. With individuals in their tens of millions we have billions of interactions which determine the evolution of the system, even if all our individuals are automatons strictly following rules.
Such as system is massively nonlinear, and any sort precise evolution of it in the sense of being able to predict with reasonably accuracy the position of say even some broad aggregate like GDP, fundamentally impossible. It is, in short, a complex system.
Ultimately what I want to think about in this series of posts is a number of different areas where people make predictions, or alternatively try to make policy and what we can hope to achieve. Hopefully I wish to think about things such as the efficient markets hypothesis and trading, forecasting economic time series, weather forecasting and many other fun and interesting topics.