Because It Works. That’s Why!Review of The Concept of Probability in Statistical Physics by Y.M. Guttmann Originally published at Books & Culture
Richard Feynman once remarked that unless one is able to make one’s ideas understandable to college freshmen, one doesn’t really understand them. On the other hand, when asked by a reporter to explain why he was awarded the Nobel Prize, Feynman remarked, “Listen buddy, if I could explain it in fifty words or less, it wouldn’t be worth a Nobel Prize.”
There are two truths here: (1) Important ideas can be made accessible without dumbing them down; (2) The details of a scientific theory are important and typically inaccessible except to individuals with the requisite training. The hallmark of good science writing — the Feynman Test, let’s call it — is the ability to heed both of these truths at the same time.
Yair Guttmann’s project centers on a question of fundamental interest, growing out of the increasing use of probabilistic reasoning in physics over the last 150 years: Why do statistical methods work so well at predicting the behavior of huge numbers of particles enclosed in containers even though the physics that describes the individual behaviors of those particles is purely deterministic? Hence, it is an ideal subject for a book that seeks to make important scientific and philosophical issues accessible without garbling essential technical details. Alas, Guttmann fails the Feynman Test, but his failure is instructive.
Guttmann observes that a “philosophical temperament” is needed for a book such as this “on the nature of probabilities in statistical mechanics. . . . For most physicists, the topic is too ‘academic.'” Guttmann himself champions what he calls the “pragmatist” approach to statistical mechanics (statistical mechanics explains the macroscopic properties of a physical system by a probabilistic description of its constituent particles). Essentially, the pragmatist approach says that we use probabilities in statistical mechanics because they work — they give us a successful theory. As Guttmann puts it in chapter 5:
Pragmatists believe that we have a total freedom to choose the concepts that appear in our scientific theories. We do not have to justify the initial choice of concepts at all [here statistical/probabilistic concepts]. All we need is to make sure that the theories we construct will imply correct predictions.
Well, I’m certainly sympathetic to pragmatism in science. “The scientific method, as far as it is a method,” the Nobel laureate Percy Bridgman said, “is nothing more than doing one’s damndest with one’s mind, no holds barred.” This account of pragmatism is perfectly fine diet for scientists, whose livelihood depends on getting results. But it’s rather thin gruel for philosophers, whose livelihood depends on analyzing conceptual difficulties. Here, then, is Guttmann’s challenge.
He does a fine job (once one wades through the technical morass and finds the relevant sections) of showing that an “objectivist” approach to statistical mechanics cannot succeed. That is, he shows that trying to locate probabilities in the purely physical properties of a particle system won’t deliver a satisfactory account of why probabilities successfully characterize such a system. But when he considers non-objectivist, or what we might call conceptually based alternatives (the Bayesian or “subjectivist” approach, the “ergodic” approach, and the pragmatist approach), he is less successful at distinguishing these and making a case for his own preferred pragmatist approach.
The problem with subjectivist approaches is that strictly speaking their only criterion for assigning probabilities is internal coherence (probabilities have to be assigned so that a betting scheme based on them cannot always be a loser for one party who bets — heads I win, tails you lose is, for instance, not allowed). But there is a physical tie-in with statistical mechanics, and thus something more than coherence is required.
Guttmann also considers but finds wanting the ergodic approach to statistical mechanics. The ergodic approach treats probabilities as relative waiting times during which a particle stays in a given portion of phase space (that is, the hypothetical space that describes all the potential states of a system). Thus for a particle that spends half its time in a given portion of phase space, the corresponding probability of that portion of phase space is 1/2. Although the ergodic approach offers a powerful way of understanding statistical mechanics and is a fertile area of mathematical research, it falls short as a justification for applying probabilities to particle systems where the individual particles are controlled by deterministic physical laws. Many assumptions with no physical justification need to be incorporated (e.g., assumptions about the limiting behavior of particle trajectories and about the type of flows in phase space).
Which leaves the pragmatist approach. As I indicated, I’m sympathetic to this approach. Indeed, I think it the best of the alternatives to the objectivist approaches that Guttmann considers. But Guttmann treats haphazardly what to my mind is the key question raised by the pragmatist approach. The pragmatist approach, when applied to statistical mechanics, enjoins us to use probabilities because they work — they enable us to construct a predictively accurate and scientifically fecund theory. But why should they do so? As we have seen, physicists are not inclined to pursue this question, “be cause discussing it is not likely to improve our ability to make better predictions, to discover new effects, or to explain phenomena that we do not yet under stand.” And that is why, Guttmann says, “it is the task of a philosopher” to answer this question.
Objectivist approaches fail because the underlying dynamics of individual particles, at least as far as statistical mechanics is concerned, is deterministic (bringing quantum mechanics into the mix doesn’t help here since we still have to deal with ensembles of particles and statistics gets applied to the ensembles). And as we’ve seen, subjective and ergodic approaches are also unsuccessful, either failing to connect with physical reality or adding what amounts to a frequentist superstructure with all the problems that raises. Why then do probabilities work? More precisely, why is the mechanics for ensembles of particles statistical?
It’s precisely at this point, just when he is homing in on the central question, that Guttmann seems repeatedly to lapse into the sorts of loose justifications that the founders and shapers of statistical mechanics employed to motivate their probabilities in the first place. Why, for instance, in an ensemble of particles does the future behavior of a particle become stochastically independent of its past behavior as the time separating past and future increases? As the time increases, the particle, being part of an ensemble, will undergo a lot of collisions with other particles. All those collisions will tend to wash out any influence the past behavior of the particle has on its (distant) future behavior. Justifications like this are intuitive and vague. But they can be given precise mathematical form, and such mathematical forms can in turn be used to construct a powerful physical theory, to wit, statistical mechanics.
The role of such loose justifications and why they could stimulate the development of statistical mechanics should have been topics of particular scrutiny in a book that finds a pragmatist approach as “the most convincing account of statistical mechanics.” Guttmann recounts such justifications, but does not adequately show how they underwrite the approach he advocates.
This failure at the heart of the book is written large in its conception and execution. There are some wonderfully lucid passages here which the educated lay reader would find particularly helpful in understanding the controversies surrounding the foundations of probability theory, especially as they relate to statistical mechanics. But these passages, though of general philosophical interest, are embedded in highly technical discussions aimed at specialists in probability theory, statistical mechanics, and the philosophical controversies connected with these fields.
One of the most illuminating passages in the book, for instance, is buried in Appendix II (the second of three), which is devoted specifically to the foundations of probability. It comes right after an all-too-succinct appendix devoted to measure theory and topology, topics about which most nonmathematicians will not have a clue. The material in the second appendix needed to be placed at the very beginning of the book to frame the ensuing discussion — indeed, the ensuing discussion presupposes it. (To his credit, Guttmann advises nonspecialists to consult the appendixes, but the first appendix is sufficiently offputting to intimidate readers from going further; what’s more, accessible material that is of central importance deserves more than an appendix).
Guttmann seems to have written primarily for fellow philosophers who specialize in the foundations of statistical mechanics, unaccountably throwing in an appendix here and a summary there for popular consumption. With more care, this book could have been made accessible to a general audience while still engaging the technical philosophical and mathematical concerns that Guttmann raises.
It’s evident, though, that care was not exercised. As someone who has published in the same series in which this volume appears, I can vouch that the problem would not have been with the copyeditors or the typesetters. The book is strewn with mathematical typos — nothing serious if you’re a re search mathematician and know what the relevant mathematical theory says. But for an amateur mathematician trying to work through the details, reading this book will be a nightmare. The chapters themselves are disconnected, and one has little sense of a unifying framework.
All in all, this is a disappointing book from an author who clearly could have done much better (there are flashes of insight and lucidity that convince me of this). The book seems to have been cobbled together, largely from technical papers addressed to experts in the field. Introductory and transitional material is minimal, and lacking is a gentle guiding hand into a highly technical field that nonetheless has wideranging philosophical import. I would love to see plans for a second edition placed in the hands of an exacting editor.
William A. Dembski is based at Baylor University’s newly founded Michael Polanyi Center for Complexity, Information, and Design. He is the author of The Design Inference (Cambridge Univ. Press) and Intelligent Design: The Bridge Between Science and Theology (InterVarsity).
Copyright © 2000 by the author or Christianity Today, Inc./Books & Culture Magazine, March/April 2000, Vol. 6, No. 2, Page 42