Philosophy and Probability

by Timothy Childers

Probability is increasingly important for our understanding of the world. What is probability? How do we model it, and how do we use it? Timothy Childers presents a lively introduction to the foundations of probability and to philosophical issues it raises. He keeps technicalities to a minimum, and assumes no prior knowledge of the subject. He explains the main interpretations of probability-frequentist, propensity, classical, Bayesian, and objective Bayesian-and uses stimulatingexamples to bring the subject to life. All students of philosophy will benefit from an understanding of probability, and this is the book to provide it.

Genres: Philosophy

213 pages, ebook

First published July 15, 2012

Reviews

[William Bies · Rating 4 out of 5]

In contrast to Donald Gillies, in the present related work (outwardly bearing a similar-looking title) the author Timothy Childers thinks for himself and is willing to confront reality.

The author leads off in chapter one with a sketch of elementary notions of frequentism with simple illustrations that may help the beginner to understand what is meant – in contrast to what is the norm these days in mathematics, where he will in the main be left to his own devices (sink or swim). The frequentist interpretation remains true to the eighteenth-century origins of probability theory and the view that chance ought to reflect a ratio of positive to negative outcomes in the limit as the number of trials diverges, appropriately averaged over an ensemble of cases. For instance, when tossing an unbiased coin, the proportion of heads to total number of tosses and the ratio of heads to tails ought to converge to ½ resp. 1 as long as no undue influence prevail over the selection of trials from the universal population. These ideas would once have been deemed axiomatic but controversy among experts centers on the question as to whether real-life conditions can indeed be chosen in such a way as to ensure power-law or faster fall-off in the tails of the empirical probability distribution function. True enough, according to the law of large numbers and its corollary, the central limit theorem. Probability theory, however, concerns the real not the ideal world. Therefore, we have to contend with two further issues: first, whether a meaningful algorithm can be constructed by which, in practice, to form an infinite collective of events and second, whether it be possible to select infinite subsequences from this collective according to a procedure specified in advance that would guarantee sufficient objectivity for the real-world cases to exhibit convergence properties analogous to the ideal. Wald criticizes the first, von Mises the second.

What does the present author have to add? Analysis of the four operations that can be performed on collectives (viz., selection, mixing, partition and combination) and of traditional objections to von Mises’ interpretation. Naturally, one expects that unbiased procedures should respect independence of outcome of measurement on method used, but in practice it is not at all clear that such independence will be true. For instance, could an outside agent have intervened in the 2020 presidential election in order to determine its outcome? At this level of reality, it becomes by no means evident that naïve expectations must bear out in practice. Here, one appreciates why probability theory is no domain for simpletons.

Chapter two: as before, several further homely examples preceding a concise discussion of propensity – what doesn’t Gillies say? Childers manages to say far more in a brief discussion than does Gillies in his one-chapter general survey and chapter-length exposition of his particular operationalist version (Gillies just fails to motivate his operationalism very well). Propensity as a disposition, the problem of assigning meaning to a single-case probability, indeterminism and existence of a reference class, connection with von Mises, worked numerical examples on Humphrey’s paradox etc. Nevertheless, Childers mostly delimits the scope of his ambition to raising awareness of issues to be pursued in the literature and fails to integrate them into a coherent perspective on the field of probability as a whole.

A clear exposition of fair bets and Dutch book arguments, Bayesianism, disconfirmation versus falsification, Duhem-Quine problem with worked numerical example, probabilities from likelihood versus from utility. The problem however, is that Childers fails to do anything interesting with these concepts – a drawback common to most thought these days, by the way. The treatments of subjective versus objective approaches in chapter four along with classical and logical interpretations in chapter five suffer from the same. It is useless to mock up sections on symmetry, principle of indifference, paradoxes, rule of succession, Carnap and logical foundations when there is nothing to teach.

Therefore, in this reviewer’s opinion, the sole chapter worth reading in the sense of putting in much effort to understand is the last, chapter six on the maximum entropy principle and its justification (there follows an appendix which any who know mathematics will deem pointless). The main point is to justify Claude Shannon’s formula for information in terms of the probabilities of possible states and to investigate where probabilities considered as numerical come from – a rather mysterious and favorite topic! The latter leads to consideration of language dependence and ontological status of information.

Childers' fine if informal text would repay further study for one transitioning into the field (and perhaps serve as a reference). Childers versus Gillies: which is the more philosophical? Seemingly Childers: yes; why? Gillies an ideologist more than a scientist. Real analysis entails caring enough and being willing to work hard enough to get to the truth of the matter. What John the Baptist, referring to Jesus, calls the ‘baptism of the Holy Spirit and with fire’ (Matthew 3:11).

Heart of the matter: what makes life interesting is a proper balance between the exploration of original ideas (in youth) and an appreciation of the contours of the whole (in advancing years). Western societies seemingly began to lose this concordance sometime during the aftermath of the second world war. Why? The Cold War placed a premium upon development and fielding of weaponry in order to counter the Soviet bloc, all of which can proceed quite well in the absence of the latter – indeed, the more furiously one thrusts oneself into research on applications, the less serious reflection upon the whole and the meaning of human existence itself becomes. In the limit, the latter half – what makes one human at all – evaporates and one is left with the shell of a human being, reduced to resource exploitation maximization.

[Brandon · Rating 4 out of 5]

This book is both difficult and interesting. It does what any introductory survey should do - instill a hunger for more learning. I am particularly interested in pretty much anything Bayesian and the logical interpretation, with the maximum entropy principle as a particularly intriguing idea.

My philosophical background is lacking, so when the author discussed Humean Supervenience and David Lewis's metaphysics, I was largely lost. This, and other sections like the logical interpretation, are where I think more length for sufficient explanations would have been appreciated. I know it's an introductory survey filled with recommended references, but some things are introduced with pretty much no explanation other than a completely arcane sentence or two.

There are some strange curiosities here. In the Maximum Entropy section, the author refers back to 1.2.6. Section 1.2.6 does not exist. In the Appendix, there is a short section defining a field. This is section A.2.1. The author then says, in A.2.1, that a formal definition of a field can be found in A.2.1. Wow. Amazing.

As said before, I was particularly interested in the Bayesian account and the logical account, despite barely understanding any of the logical interpretation. I also was sympathetic towards the propensity interpretation, but can certainly see the problems with it. It's hard to escape determinism under that framework. The relative frequency interpretation is what I learned in the probability class, so that section was easy to understand but it is a weak account of probability.

I think this book would be of interest to anyone interested in probability who has sufficient preparation in philosophy and mathematics, but that person probably shouldn't expect to actually learn the field from this independent source. The author beats the reader over the head with the idea that he should read further afterwards, and that is totally okay. In that, the author totally did his job. It's more of a guidebook for independent study than a source to be used in independent study.

I also get the sense that this book might be a strong method for cohesion after reading more broadly. Read a couple different accounts, then return here to see how they all run together. I'll probably end up doing that in a few years once I'm a bit more acquainted with the field.

[Richie · Rating 3 out of 5]

Took much more from SEP article on the topic, but perhaps that is because this book is more advanced.

The presentation of the logical interpretation was the clearest that I have encountered.