Notes on Strevens, Bigger than Chaos
I've been asked to review Michael Strevens's new book, Tychomancy. This motivated me to have another look at his earlier book Bigger than Chaos.
The aim of Bigger than Chaos is to explain how apparently chaotic interactions in highly complex systems often give rise to simple large-scale regularities, such as the laws of thermodynamics, the stability of predator/prey population levels, or the economic cycle. The basic explanatory strategy, which Strevens calls enion probability analysis (EPA), consists in aggregating the probabilistic dynamics for the individual components of a complex system into a probabilistic dynamics for macro-level features of the system.
The book is not an easy read. Strevens introduces a lot of new jargon, and carefully works through abstract mathematical results to the point where one has almost forgotten the big picture. What's more, in some respects the big picture still remains a little elusive, at least for me.
Here is a somewhat more detailed outline of the plot.
Strevens's topic are complex systems consisting of a large number of fairly independent but interacting parts, such as the animals in an ecosystem or the gas molecules in a box. Strevens calls these parts, whose behaviour determines the behaviour of the system, enions (a made-up word, in case you wondered).
At each point in time, the precise state of a complex system is given by a large number of microvariables specifying the state of each individual enion. By contrast, macrovariables such as temperature or pressure or rabbit population give coarse-grained, statistical information about the system as a whole. The puzzling fact Strevens wants to explain is that the evolution of macrovariables over time often obeys mathematically simple laws.
This is puzzling because the dynamics of the system is determined by a large number of interactions between its many enions. One might therefore have expected the evolution of macrovariables to likewise depend on specific information about the system's microstate. But often it doesn't. Somehow the details cancel out. According to Strevens, we can see how that happens by paying close attention to the microlevel dynamics of individual enions. In fact, he argues that we can derive the simple macrolevel behaviour of complex systems from general microlevel facts about the behaviour of their enions.
The key assumption in this derivation is that the relevant microlevel facts include probabilistic facts of a certain kind. For example, if we know that (1) each rabbit in an ecosystem has a five percent chance of dying within a month, (2) no new rabbits are born, and (3) the deaths of individual rabbits are stochastically independent, then we can infer that (4) the overall rabbit population is likely to decline by about 5 percent each month. The inference from (1)--(3) to (4) is an example of EPA. (1) and (3) are the lower-level probabilistic facts.
More generally, suppose for each enion e in a system, there is a well-defined probability P(y_e/X) that the enion will evolve into microstate y given that the entire system is in macrostate X. Strevens calls P(y_e/X) the enion probability for y_e given X. Suppose these probabilities are stochastically independent for different enions, so that, for example, P(y_e & y_f / X) = P(y_e/X)P(y_f/X), where y_f is the event of some other enion f evolving into microstate y. We can then easily aggregate all the individual probabilities into a probability that the entire system ends up in microstate (y_e & y_f &...) given that it is presently in macrostate X. Now each macrostate Y corresponds to a set of microstates. So by summing the probability of all Y-type microstates we get a macrolevel probability that the system will go from state X to state Y. This is the macrolevel law. In the first place, it is a probabilistic law, but often the laws of large numbers can be used to turn it into an approximate non-probabilistic law, as in the ecosystem example above.
Accounts along these lines are popular in statistical mechanics, population dynamics, and other areas of science. The most obvious and most widely discussed philosophical question they raise concerns the interpretation of the enion probabilities: in what sense does each rabbit in an ecosystem have a 5 percent chance of dying within a month? In what sense does each gas molecule have an equal chance of being in different subregions of a box? Oddly, Strevens sets this question aside. He argues that the metaphysics of enion probabilities doesn't really matter; rather, what matters are certain structural features of enion probabilities that are required for EPA to work.
Return to the probability that a given rabbit dies within the course of a month. Intuitively, this probability seems to depend on the precise microstate of the system -- for example, on the rabbit's proximity to hungry foxes. But then it is hard to see how aggregating a lot of microstate-sensitive enion probabilities could give rise to macrolevel probabilities that are no longer sensitive to microlevel information. According to Strevens, enion probabilities should therefore be independent of microlevel details.
To make this plausible, Strevens suggests that enion probabilities belong to event types. The probability that a particular rabbit will die within a month is sensitive to its proximity to hungry foxes, but the type-level probability that rabbits die within a month, Strevens claims, is not sensitive to the information that rabbit so-and-so is close to a hungry fox.
(I didn't really understand why this is supposed to be so, nor exactly why enion probabilities must be independent of microlevel details. I'm also confused about the nature of the algebra over which probabilities are defined: according to Strevens, one can meaningfully speak of the probability of an outcome type conditional on information about token events; so the algebra must somehow involve both types and tokens. But let's move on with the plot.)
If the microdynamical laws governing the dynamics of individual enions are deterministic, the enion probabilities for future states must then derive from a probability measure over the present microstate. Again, Strevens does not pause to ask what this measure is supposed to represent. He seems to assume that it is simply given as part of the system under investigation.
Now it turns out that if the microdynamics is of the right type, then the enion probabilities are largely insensitive to the precise distribution over "initial conditions". These results, known as "the method of arbitrary functions", are reviewed in chapter 2 (which goes on for over 100 pages). The basic observation is the following.
Consider a mechanism that produces certain outcomes depending on its initial state. Suppose the possible initial states can be partitioned into many small, contiguous regions each of which has the same ratio of subregions leading to the different outcomes. Mechanisms of this kind Strevens calls microconstant with respect to the relevant outcomes. As von Kries, Poincare, Hopf and others pointed out, if a mechanism is microconstant, then any reasonably smooth probability measure over initical conditions translates (by the dynamics of the mechanism) into approximately the same probability measure over outcomes.
Exactly why this is relevant to EPA remains a little unclear. Strevens explicitly rejects the idea that enion probabilities could be defined by the method of arbitrary functions. Rather, he assumes that for each enion in a system there is some true probability measure over initial conditions, determining the corresponding enion probability. What's good about a microconstant enion dynamics is then that we don't have to pay attention to (or know) the individual probabilities over initical conditions: as long as they are all sufficiently smooth, the enion dynamics alone determines the approximate probability of outcomes.
Microconstancy also helps with another problem. Aggregating individual enion probabilities is straightforward only if the probabilities are stochastically independent. That rabbit 1 dies within a month should be independent of whether rabbit 2 dies within a month. How can this be true, given the frequent causal interactions between enions in a system?
Again, a result due to Hopf points the way towards a solution. Hopf showed that if two experiments have a microconstant mechanism, and the probability distribution over their initial conditions are smooth, then the outcome probabilities are independent provided that the distribution over the joint initial conditions is also reasonably smooth. This is compatible with strong correlations in the distributions over initial conditions. Thus microconstant processes have the power to break down correlations: even if the initial states are strongly correlated, the outcomes are often independent.
In chapter 3 (which also has over 100 pages) Strevens provides further insight into this "shuffling power" of microconstant processes by looking at experiments with causal coupling and at sequences of experiments in which the outcome of one experiment is the input to the next. In both cases, Strevens explains that if the individual mechanisms have certain properties (that go beyond mere microconstancy) then the outcomes will be independent. These observations were new to me, and are perhaps the most important technical contribution of Strevens's book.
As usual in the literature on arbitrary functions, the discussion in chapters 2 and 3 remains at a rather abstract level, dealing with highly simplified and idealized scenarios. To his credit, Strevens spends quite some time in chapter 4 arguing that the dynamics of gas molecules and rabbits satisfies the relevant conditions of microconstancy. Nevertheless, the promised derivations remain sketchy (especially in the case of the rabbits). Readers expecting new insights into, say, the foundations of thermodynamics will find little here that goes beyond standard results of ergodic theory, except for all the new jargon (which might explain the the scathing reviews on amazon).
Strevens seems to think that all that stands in the way of completing the derivation of macrolevel laws from microlevel facts is mathematical complexity and our ignorance about the relevant mechanisms. I am less optimistic.
I suspect that macrolevel laws generally can't be derived, even in principle, from general, law-like information about the microlevel. There are no precise micro-laws governing the behaviour of rabbits in an ecosystem, especially since ecosystems are never completely isolated. Similarly, albeit to a lesser extent, for gas molecules in a box.
The issue is related to the issue Strevens tried to set aside: the interpretation of the probability measures. Strevens seems to assume that we have a clear, determinate concept of objective probability that (among other things) gives non-trivial values to initial states of physical systems. The very same concept of probability then figures in macro-dynamical laws: P(Y/X) is the sum of all P((y_e & y_f & ...) / X) with (y_e & y_f & ...) in Y; P((y_e & y_f & ...) / X) is the product of P(y_e / X), P(y_f / X), etc.; and P(y_e / X) is determined by the distribution over initial conditions. It's the same probability measure throughout. As a consequence, in systems where the assumptions of independence and microconstancy are only approximately satisfied, Strevens argues that the simple models used in science are also only approximately true: their probabilistic claims do not quite match the probabilistic reality. (See the remarks on pp.273f.)
From this perspective, it is natural to think that the metaphysical interpretation of the probability measure is orthogonal to the explanation of macro-level simplicity. But that can't be quite right. Strevens's probabilities essentially apply to event types, and they apply not only to future states given present states, but also to present realizations of present macrostates. This rules out, for example, an interpretation as primitive propensities. It also seems to rule out straightforward epistemic interpretations, since epistemic probabilities presumably don't pertain to types.
Consider also Strevens's claim (on pp.147ff.) that causal independence is sufficient for stochastic independence. It is not at all obvious that this is true on, say, an epistemic reading of the probabilities.
In fact, I'm inclined to think the picture becomes clearer if we don't take the probabilities as primitively given.
Suppose the aim of macro-level models is not to accurately record primitive probabilistic facts, but to capture robust patterns in the relevant systems (as I argue in this paper). Strevens's puzzle still arises: why do complex system often display simple, robust macro-level regularities? In particular, why can we make fairly accurate predictions of macrovariables without knowing the exact present microstate? The mathematical results reviewed and extended by Strevens point towards an answer, but the answer is a little more indirect than on Strevens's account.
Consider the method of arbitrary functions applied to a simple wheel of fortune. The method shows that any smooth probability distribution over initial velocities yields approximately the same probabilities for red and black. This is useful even if we don't assume that there is some true objective probability distribution over initial velocities. It tells us that any sufficiently chaotic pattern of initial velocities determines the same kind of probabilistic pattern of outcomes. Thus if we have reason to think that the initial velocities are sufficiently chaotic at the micro-level, and that they are likely to be so under counterfactual circumstances, we have reason to think that the outcomes are well modelled by assigning fixed, IID probabilities to red and black.
Similarly, Strevens's results about colliding coin tosses show that even though the two processes strongly interact, any sufficiently chaotic pattern of initial conditions gives rise to an outcome pattern in which each outcome occurs approximately a quarter of the time. The outcomes are independent in the sense that the compound experiment is well modelled by a probability measure that assigns to compound outcomes the product of the values assigned to individual outcomes by adequate models of the individual processes.
The main advantage of this perspective is that it makes immediate sense of what probabilistic theories in science are up to. On Strevens's picture, we should be reluctant to trust probabilistic models of evolution or population dynamics, since we have little to no evidence that they accurately represent the ultimately low-level probabilistic facts. Another advantage is that we no longer have to assume that Strevens's unified role of physical probability has a realizer. Moreover, on the alternative perspective we can happily live with merely approximate results. A population model that treats the fate of individual rabbits as statistically independent can capture salient patterns even if on a lower level the required kind of statistical independence only holds to a certain approximation.