Metaphysics and the Philosophy of Science ENION PROBABILITY ANALYSIS

A broad class of compositional theories dissolve the aggregation problem by taking a statistical approach: they divide the system into individuals such as molecules or organisms (which I call “enions”), they assign probability distributions to the behavior of the enions, and they derive the behavior of the whole by aggregating the relevant probabilities. I call this strategy enion probability analysis, or EPA for short (Strevens 2003).

Enion probability analysis cannot be used to model just any property of a complex system: it is limited to tracking aggregate properties of populations. You can use it (when conditions are favorable) to model changes in the number of tortoises in an ecosystem, for example, or the number of male tortoises in the system between one and two years of age, but not to follow the day-to-day movements of a particular tortoise. (For the latter purpose, I know of no compositional theory that does not suffer from the aggregation problem.) This is a relatively small subset of the properties of a system, but it is an important one, containing the things we need to predict and explain in order to undertake many of the projects falling within the ambit of statistical physics, evolutionary biology, ecology, economics, and so on. Consequently, EPA has a significant role to play in accounting for the predictive and explanatory success of the compositional sciences of complex systems, and a philosophical understanding of EPA has significant lessons to teach about the ontologies of successful compositional theories—and thus, given my explanationist approach to these issues, about the ontology of the world.

In the boreal forest of Canada, snowshoe hares eat underbrush and Canadian lynxes eat snowshoe hares. The populations of the two species cycle regularly: every eight to eleven years, the hare population booms (for reasons about which there is still considerable argument), and the lynx population, sustained by regular helpings of lapin a la mode, booms soon after. Then the hare population crashes and the lynx population follows on its heels.

Ecologists wishing to understand this famous predator/prey population cycle build mathematical models of the system. Although the models themselves tend to have a deterministic mathematics, their structure can be understood as rooted in stochastic foundations, as it was by Alfred Lotka, who gave his name to these “Lotka-Volterra” models (Lotka 1925). This is the statistical method that I am calling enion probability analysis; after describing its application to lynxes and hares, I will examine its presumptive ontology.

A Lotka-Volterra model tracks populations, attempting to predict the size of future populations—say, the numbers of lynxes and hares in a month’s time—from present populations. Some sophistications are possible: weather patterns or the luxuriance of the vegetation can be incorporated into the model, or subpopulations can be tracked (such as sexually mature female hares). But let me put all of this aside to focus on the basics.

Assume that the model has only two variables, representing the number of lynxes and the number of hares. What is wanted is a set of equations that relates the values of the variables at one time to their values at a later time.

It might seem that a complexity explosion is imminent. What happens to an individual hare depends on small wrinkles in local matters of fact. Forage under the old pine this morning and you will be lynx food. Head instead for the spruces, and you will live to graze another day. Tracking the fates of individual hares requires, then, an accurate record of particular movements and the principles that drive them. That, surely, must add up—when hundreds or thousands of hares and many lynxes are involved—into something very complex, certainly more complex than a set of simple equations relating only two variables. Or so it would appear.

Lotka’s approach gets around these difficulties using the following recipe. To the members of each population or sub-population in the ecosystem, assign probabilities for the kinds of outcomes that make a difference to populations: birth, death, perhaps migration. Assume that these “enion probabilities” are independent of one another. Use the law of large numbers—assuming here that populations are reasonably large—to derive frequencies for various events equal to the enion probabilities. If, for example, there is a 0.05 probability that any given hare is killed by a lynx over the course of a month, assume that one-twentieth of the hare population is lost to lynx predation every month. You now have a table of per capita rates: the rate of hare reproduction (per hare), the rate of hare predation, the rate of lynx reproduction, the rate of lynx death, and so on.

Some of these rates depend on other variables. The rate of hare predation depends, for example, on the population of lynxes: more lynxes means proportionally more hares served up for dinner. These dependences typically (though not invariably) end up in the model. It is crucial, then, that they bring with them into the model’s equations only quantities that the model is constructed to represent. In the simple case at hand, the rates should depend only on the total number of hares or the total number of lynxes in the system, or both, or neither. (If the rates as a matter of fact depended on subpopulations, the model would have to track those subpopulations—young female hares, for example—thereby becoming more complex.) The rates will satisfy this independence requirement just in case the enion probabilities satisfy the requirement. The probability of a hare’s being killed by a lynx over the course of a month should, for example, depend only on the number of lynxes (and perhaps the number of hares).

Let me suppose that the necessary enion probabilities, and so the rates, are fully determined by the physical facts. (Strevens [2003] tries to specify exactly what fundamental states of affairs fix the facts about the probabilities.) Then, with the probabilities’ existence secured, it is straightforward to write down two equations representing, respectively, the rate of change of the hare and lynx populations. The equation for the hare population might, for example, set the change in the population equal to the rate of hare reproduction (multiplied by the current hare population) less the rate of predation (multiplied by the current hare and lynx populations) less the rate of hare death from other causes (multiplied by the current hare population). The lynx equation will do something similar. Together, these equations make up a Lotka-Volterra model, which can be simulated or solved and then either tested against observed population changes or used to explain those changes.

That is one way—the EPA way—to model a complex system, deftly avoiding the problems posed by the complexity explosion.

The enion probability analysis of the lynx/hare system appears at first to have a wedding cake ontology: it proceeds by dividing the system into enions, individual hares and lynxes, that are spatiotemporally discrete. A closer look shows, however, that although the enions are indispensable, the most important elements of EPA— not the enions themselves, but the probabilities that describe the enions’ behavior— are individuated rather differently.