Infinite Monkeys: The Limits of Probability Theory

Article of the Week from Fixing the Economist

by Philip Pilkington

No one could be more frank, more painstaking, more free from
subjective bias or parti pris than Professor Tinbergen. There is
no one, therefore, so far as human qualities go, whom it would be
safer to trust with black magic. That there is anyone I would
trust with it at the present stage or that this brand of statistical
alchemy is ripe to become a branch of science, I am not yet
persuaded. But Newton, Boyle and Locke all played with
alchemy. So let him continue.

— John Maynard Keynes speaking of one of the pioneers of econometrics

Recently the blogger Lord Keynes over at the always excellent Social Democracy for the 21st Century blog has been doing some posts about probability theory (examples here, here and here). There is a famous theorem in statistics called the ‘Infinite Monkey Theorem’ which states that as n approaches infinity the probability Xn approaches zero. We’ll get back to this theorem later but let me lay down a theorem of my own — one which I hope that Lord Keynes and others interested in this topic pay attention to: as a person’s wonderment at probability theory approaches infinity the probability that they will get anything of worth done approaches zero.

I say this because the issues surrounding probability theory and economic theory, while they are so profound as to be possibly the most important part of economic theory, are nevertheless a rather basic and simple metaphysical puzzle that can be laid out in clear terms and thus answered in clear terms. Once you understand this puzzle you “get it” and the deeper you dig into probability theory, the more this puzzle will repeat itself. Indeed, this puzzle has already been laid out and answered by the Post-Keynesian economist Paul Davidson in his writings on ergodicity in the social sciences.

Everything comes down to this: do you believe that the future mirrors the past? Or, put another way: do you believe in deterministic laws that govern the universe and can be understood by human beings? Everyone can answer that question themselves. It is a metaphysical question that borders on the theological — and it really goes all the way back to the debate between Erasmus and More on the existence of free will. Again, everything comes down to this. No matter how deep you dig into probability theory you will not find any other question and so you can save yourself a great deal of time by pondering this question in pure form rather than getting caught up in the nuances of probability theory.

(Okay, I’m not saying just avoid the whole of probability theory. It is indeed interesting. But I would advise against becoming obsessed, as I know how obsessive the structure of theory is. It could potentially — and I mean this literally — drive a person mad because it takes the form of an infinite series of questions which never provide any answers, simply because the meta-question is the one I just laid out above.)

But back to the monkey theorem for a moment. The theorem states that if a monkey sits at a typewriter hitting random keys it will eventually come up with the collected works of Shakespeare. When expressed in mathematical form the theorem makes sense, but when stated anecdotally and “brought back down to earth”, as in the monkey example, it makes no sense at all. First of all, a monkey’s behavior does not generate random walks. What the anecdote does is replace a truly random-generating machine with a monkey, but the analogy is incorrect because a monkey would not behave in a properly random manner. Secondly, infinity does not exist in our oh-so finite reality. It is either a figment of our imagination or it is some divine space inhabited by a higher power.

This is important because we should remember what we are really dealing with when we are dealing with such theorems. People often mistake these theorems as saying something tangible about our lived reality. But often this is simply not the case. What these theorems do is absorb and disguise debates that used to take place among metaphysicians and theologians and then pretend as if certain questions which philosophers have been tackling since time immemorial have been answered. One is tempted to put this down to the hubristic tendencies in the scientific method itself.

Whatever the reason, however, the lesson for the Sciences of Man should be clear: these questions have not been answered; methods derived from disciplines like probability theory (for example: econometrics) probably have a very limited reach; and you’re not going discover the secrets of the universe by becoming an expert in these disciplines, although society may bestow you with a role similar to that occupied by a priest or a haruspex in times past. If this is indeed your goal well, as Keynes said all those years ago, Newton, Locke and Boyle played at alchemy, so by all means continue.