Beyond Chance and Credence

In the postscripts to “A Subjectivist’s Guide to Objective Chance,” David Lewis (1986: 118) wrote: “To the question how chance can be reconciled with determinism, ...my answer is: it can’t be done.” But a number of philosophers have tried, in the past thirty years or so, to show that Lewis was mistaken. Motives for doing so are not hard to find. Classical gambling systems such as coin flips and dice rolls, to which we ascribe objective probabilities, are arguably deterministic in their dynamics, or nearly so; and one of the two huge areas of physics in which objective probabilities play a crucial role, statistical mechanics, is based on fully deterministic underlying dynamics. Finding a clear account of such probabilities that makes them objective, yet compatible with determinism, has been a chief goal of philosophy of probability, with novel proposals having been defended by David Albert (2000), Barry Loewer (2001), Michael Strevens (2003), Aidan Lyon (2010), Jenann Ismael (2009, 2011), Jonathan Cohen and Craig Callender (2009), and myself (Hoefer 2019), among others.Beyond Chance and Credence (hereafter, BCC) develops and defends an approach to objective probability that Myrvold first presented in a 2012 article: epistemic chance. It is a notion that blends physical objectivity with epistemic considerations about the credences of cognitively limited agents such as human beings. But the name, and this description, may seem to suggest a fifty-fifty mix of objectivity and subjectivity, and that is very far from the truth. Epistemic chances are essentially determined by the physical dynamics, and hence are as objective as anyone should wish. In BCC, Myrvold has given us the best account yet of how objective probabilities can exist for the behaviors of (at least many of the) physical systems whose dynamics is deterministic. But BCC offers much more than that as well.BCC begins with a historical overview of the introduction of probabilistic considerations into physics, and the resultant problem of how to understand such probabilities when applied to deterministic systems. Myrvold is far more careful to get the history right than most philosophers of physics (or probability) tend to be, giving the reader a first taste of the rigor and clarity that will be maintained throughout the book. Chapter 2 introduces the notions of chance and of credence (agents’ subjective probabilities), offering a more sophisticated introduction to the latter than many readers will ever have seen. With these concepts in hand, BCC returns to the historical narrative begun in chapter 1, and to how the ideas of treating probabilities as either purely epistemic, or as frequency-based, arose in the nineteenth and twentieth centuries. Chapter 3 explains why the probabilities we want to have for deterministic dynamical systems can neither be understood along classical lines (based on equiprobable cases or principles of indifference), nor as purely subjective credences, nor along frequentist lines. This chapter’s arguments are far more compelling than the laundry list of familiar objections that one typically finds in discussions of early approaches to probability.Chapters 4 and 5 are the conceptual core of BCC, introducing the reader to the fundamental ideas of epistemic chance (hereafter, EC). Chapter 4 introduces a mathematical example of a deterministic dynamical system, one that Myrvold then uses in several subsequent chapters to illustrate key features of EC: the “parabola gadget.” This is a system in which a point-sized ball moves according to a set of rules that make the ball follow a series of vertical and horizontal movements (see fig. 1).A single iteration of the dynamics takes the ball from some starting point on the diagonal and moves it vertically until it meets the parabola, and then horizontally until the diagonal is met once more. This system amplifies any uncertainty regarding the initial position of the ball fairly quickly, so in case one has such uncertainty, one wants to use probabilistic considerations to figure out how likely the ball is to be in a given interval of the x axis after n iterations of the gadget’s operation. Equal-sized intervals of x turn out not to be equiprobable as locations where the ball may be found, after the gadget has run for a while—not by a long way! Figure 2 shows the kind of frequency distributions that running the gadget tends to produce after many iterations, almost no matter where the ball starts from.1As Myrvold shows, there is an invariant distribution of the parabola gadget dynamics, U-shaped as seen roughly in figure 2, and precisely in figure 3 below, toward which almost any initial probability distribution over the initial x-condition evolves fairly rapidly as n increases. For this reason, the gadget can be said to be a dynamical system for which epistemic chances exist—chances of the ball being found in a given interval of x-value at the end of an iteration, if the gadget has been running for a while.The invariance of the distribution in figure 3 is the key to understanding what an epistemic chance is. Consider a rational agent who does not know the precise initial condition with which a gadget has been set in motion. Perhaps the agent knows only that it was very near to 0.6; perhaps she knows that it was between 0.2 and 0.8, and more likely near the middle than the extremes of these values, but nothing else; perhaps she knows nothing at all about where it started. Her credences would be representable by something like a sharply peaked Gaussian centered on x=0.6, or perhaps a narrow, flat-topped bar centered on x=0.6, in the first case; a gently peaked hump-shaped curve from 0.2 to 0.8, perhaps, in the second case; and a flat plateau from 0 to 1 in the third. Suppose that the agent is asked to make bets on which decile the ball will be found in after five hundred iterations of the gadget dynamics. Crucially, if any of these three agents update their credences about the ball’s location by starting from their initial credence distribution and evolving it forward under the dynamics, the updated credence distribution will look, at a coarse-grained level, very, very similar to the shape of figure 3. All three agents’ credences about (for example) how likely the ball is to be found between x=0.3 and 0.4 will, to a high degree of approximation, be the same. The dynamics of the parabola gadget, in other words, imposes “the right” credences on agents who have almost any reasonable initial credence distribution over possible initial x-values and who update those credences in light of the gadget’s known dynamics. These “right” credences that such rational agents end up closely approximating are precisely the epistemic chances (ECs) for the x-values of balls in a long-running parabola gadget.The features described above are not, fortunately, restricted to parabola gadgets. As Myrvold demonstrates in chapter 8, equilibration in thermodynamical systems can be understood as a dynamical process that maps very diverse (but not unreasonable) credences about the initial microstate of the system into agreement as to the probabilities of coarse-grained final states of the system. More generally, statistical mechanical probabilities can be understood as epistemic chances arising from the deterministic dynamics (Newtonian, for classical Statistical Mechanics [SM]). With the tool of ECs—and a long career’s worth of accumulated wisdom concerning the philosophy of SM—Myrvold achieves wonderful results in chapter 8. Myrvold shows why we do not need to invoke a “Past Hypothesis” to let us explain why systems around us evolve to higher entropy conditions rather than lower.2 More generally, chapter 8 offers satisfying solutions to many of the main philosophical problems regarding classical SM. (On the way, chapters 6 and 7 give clear, compact, and elegant introductions to thermodynamics and classical SM, respectively; these are didactic gems in their own right, and an invaluable resource for future philosophy of physics.)Coming back to epistemic chances in general, a few further remarks will help to situate Myrvold’s approach compared to others, and to highlight its strengths and limitations. As Myrvold stresses, ECs have an epistemic side to them, which is ineliminable and not problematic. I would stress that their objectivity is what makes them well suited to play the role of objective chances—an objectivity that is exactly as strong as the objectivity of the dynamics of the systems to which they attach. Myrvold, who is (unlike me) willing to contemplate that there might be brute/primitive objective chances in the world (perhaps associated to brute probabilistic physical laws), does not want to consider his ECs to be objective chances. Nevertheless, within their domains, they can serve all the purposes that objective chances are taken to serve. One of the nice points of Myrvold’s account is his quick but plausible demonstration that ECs satisfy a close analogue of Lewis’s Principal Principle.The strengths of the EC approach come, however, with a corresponding limitation: the account only extends to chances regarding the behaviors of systems that are governed by the right sort of dynamics. Myrvold does not try to delimit the scope of the approach, other than by discussing it in connection with SM (both classical and quantum), and in connection with standard Quantum Mechanics itself, about which more in a moment. One may wonder: does it cover the whole range of classical gambling systems? It seems very likely that roulette wheels have a dynamics with the requisite features, and probably dice rolling also. About coin flips I am not so sure, since it is not clear that the dynamics will substantially wash out bumpy initial credences enough to make the entire class of reasonable initial credences converge to near 50 percent credence in H and T. And about chances in card games—for example, the chance of drawing to an inside straight in draw poker—it seems to me unlikely that the wide variety of deck-shuffling methods found in the real world can all be treated as kinds of dynamical systems with ECs arising from the dynamics. Looking beyond gambling systems, many other kinds of probabilities that we treat as at least quasi-objective, such as probabilities of being involved in a car accident, or of suffering a cancerous mutation in a lung cell, or recombination probabilities in genetics, are probably not amenable to treatment as ECs. So, while EC does cover many of the most central and important probabilities found in physics and in a wide class of dynamical systems, it is not apt for use in every context. By contrast, Humean Best Systems approaches to chance (e.g., Cohen and Callender 2009; Loewer 2001; Hoefer 2019), by making chances supervene on actual events themselves rather than on dynamical systems having certain mathematical features, are arguably better able to encompass the full range of contexts in which we postulate objective probabilities in science and in daily life.3The past few decades have seen an explosion of new work in the philosophy of probability, and Beyond Chance and Credence is without doubt one of the very best contributions to the field. Myrvold has crafted a novel, elegant account of objective probabilities. It may have a restricted domain of applicability, but within that domain its achievements are remarkable

Credence · Economics · Epistemology · Positive economics · Statistics · Mathematics · Philosophy · Philosophy and History of Science · Psychology

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