Problems with Probability
One of the often used justifications for inductive reasoning, especially in science, is probability theory. If all observed swans in the past have been white, then there is a high probability that the next observed swan will be white. David Hume quite rightly criticised this reasoning in Enquiries Concerning Human Understanding, but today I'd like to dwell on the issues surrounding probability theory. I can't see how probability theories can provide any justification for inductive reasoning. Standard approaches to probability such as Bayesian Probability require the prior probability of a phenomenon to be known. Clearly, this prior probability is impossible to deduce for anything other than artificial constructs such as games of chance, and as such, is of little use in science and the external world. Put simply, how can we know the actual probability of something before we have experienced it? Probability derived from observed frequency is equally problematic. Frequency probability is calculated by dividing the number of observed events (nx) by the number of trials (nt) so that as the number of trials becomes ever larger, the expected probability approaches the real probability. Thus:
P(x) = lim(nt -> ∞) nx / nt
It becomes immediately apparent that frequency probability has significant limitations when P(x) is close to 0 or 1. The expected probability will always lag the actual probability of a phenomenon, underestimating certainly when close to 0 and over estimating certainty when close to 1, which is precisely the domain where science is most interested. The story of Russell’s Turkey provides a vivid example of the ineffectiveness of frequency probability calculations, as even after 364 observations, on Christmas Eve the expected probability that the turkey will be fed at 9 am the following morning is 1. Perhaps a more fundamental flaw in justifications from probability based on the degree to which one could rationally believe a claim, and experience in general, is this. The calculated probability or expected likelihood of an event is contingent upon the observer’s knowledge of the phenomenon, however the observer’s knowledge is independent of the underlying nature of the phenomenon. There is a disjoint between expected and actual probabilities that cannot be determined a priori. As Hume (EHU V.I) noted, the particular causes by which all natural operations are performed, never appear to the senses. Because we cannot observe causes, only their effects, we have no way of knowing the nature of the phenomenon that causes those effects. Time for a thought experiment. Consider an example of an alien coin. Unlike coins made by man, this one is controlled by a different natural force, so that its characteristics are either random, consistent, or drawn to equilibrium. The characteristics are fixed and determined at the time of minting but are completely unobservable a priori. After flipping the coin and landing 10 heads in a row (the urge to imprint faces on coins seems truly universal), it remains impossible to accurately estimate the probability of the next toss also being heads. If the coin is random, then probability is independent and 0.5. If the coin is consistent, then the probability of another head is 1, while if the coin is drawn to equilibrium (say, has a memory), the probability is 0.000488 (1 / 211). As this example shows, inductive reasoning requires a uniformity of nature to be effective but that uniformity cannot be assumed and cannot be observed a priori. So, can probability ever be useful? Like induction, it seems to work most of time but can never be proved or be found conclusive before the fact, except for artificial games of chance. Personally, I get very concerned when people start using probability based on historical data to 'demonstrate' how some phenomena (say the share market) will or wont behave in a certain way.