Сompositionality and complexity

Choose a class of systems: halide molecules, coronaviruses, predator/prey ecosystems. A compositional theory of such a class divides the systems into parts, assigns the parts properties in virtue of which they behave in certain ways, and then aggregates the behavior of the parts to derive, for predictive or explanatory purposes, the behavior of the whole—for any system in the class.

Newtonian gravitational theory, for example, divides the world into objects, assigns those objects masses in virtue of which they exert and experience gravitational force, and then by way of a principle of aggregation—the rule that individual forces are to be summed as vectors—makes predictions about the movements of a system of such objects.

In principle, Newtonian theory can predict the behavior of any system in its scope; there lies its compositional strength. In practice, great difficulties may emerge in the aggregation. Given only three gravitating bodies, the mathematics of even approximate derivations can, if the bodies are of similar mass, become complex, in part because such systems may be highly sensitive to initial conditions. For greater numbers of bodies, combinatorics piles on difficulty: a small change here can make for moderate changes in many other places, which in turn change things further here—and so on. This is a complexity explosion; it limits the usefulness of even so practically important a compositional theory as Newtonian gravitation.

What you might call the aggregation problem arises again and again at many different levels in the sciences. The quantum chemistry of large atoms is difficult enough; that of large molecules is more challenging still. Modeling the complex genetic networks at work in embryological development is fiendishly hard. Predicting many of the significant consequences of interacting human minds—housing bubble collapses, Hollywood megahits, popular revolutions—is quite beyond us. Sometimes, however, compositional theories of complex systems lie within our grasp: we can build successful models in statistical physics, evolutionary biology, and computational psychology. What many of these models have in common is their being non-spatiotemporal in some aspect of their underlying ontology: they decompose systems into parts or properties, some of which are not spatiotemporally discrete units or anything close. The need to solve the problem of aggregation, then, imposes on higher-l evel sciences ontologies that depart at least in part from the wedding cake ideal.

Is this because theories with a wedding cake ontology are especially prone to the aggregation catastrophe? Is there something about decomposing a system into spa- tiotemporal parts that renders the calculus of aggregation especially intractable? Perhaps not. It might simply be that the great majority of lower-level ontologies are unsuited to tractable aggregation, and that since spatiotemporal cohesion is a strong constraint and so wedding cake ontologies are only a small proportion of the possible ontologies, we ought not to have expected them to be particularly successful.

Since the spatiotemporal constraint is chosen not at random but because it simplifies the organization of science, that is a somewhat weak explanation of the aggregation problem. I think we can do better: the nature of the wedding cake ontology to some extent does explain why it, in particular, suffers from the aggregation problem.

A rule of aggregation pulls together the connections between the parts into which a system is decomposed. In a compositional theory with a wedding cake ontology, then, the aggregation rule will calculate the net effect of the relevant connections— causal connections, let me suppose—between its spatiotemporal parts. The aggregation problem tends to arise for wedding cake theories because of the combinatorial complexity and sensitivity of these relations.

I call the relations between spatiotemporal parts sensitive because their effect on the aggregate is sensitive to small changes in the state of the part. I do not mean “sensitive” in the chaotic sense, but in a much weaker sense: the effect on the aggregate is not wholly independent of the small changes, or in other words, small changes have some effect on the aggregate. For example, although the gravitational force exerted by an object does not depend chaotically on the object’s position, it does depend on exact position: move the object slightly and the force exerted at any point changes slightly, and so the aggregate force exerted on an object at that point changes slightly. The same is true for mass: a slight change in mass means a slight change in force, rather than none whatsoever.

The relations between spatiotemporal parts are combinatorially complex because, as you increase the number of objects, the number of relations to keep track of increases. I do not mean that it increases exponentially, or even non-linearly—just that there is an increase.

These two properties tend to lead, because of increasingly intricate whorls of dependence, to complexity explosions: as in the case of gravitation, small alterations in one place proliferate quickly to many other places, where they further nudge conditions at the original locus of change. The web of mutual influence becomes a hopeless tangle.

For this reason, wedding cake ontologies typically prove to be unsuitable ground for compositional theories of complex systems. What theories do better?

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