for instance, would give rise to systems that might be fruitfully studied by psychologists but not by biologists. This is a problem for someone who wants to embrace position in a strict hierarchy as a measure of complexity: there may be no strict hierarchy to which we can appeal. Dynamical complexity cheerfully acknowledges this fact, and judges complexity on a case-by-case basis, rather than trying to pronounce on the relative complexity of all biological systems, or all psychological systems.
What aspects of fractal dimensionality does dynamical complexity incorporate? To begin, it might help to recall why fractal dimensionality by itself doesn’t work as a definition of complexity. Most importantly, recall that fractal dimensionality is a static notion—a fact about the shape of an object—not a dynamical one. We’re interested in systems, though, not static objects—science deals with how systems change over time. On the face of it, fractal dimensionality doesn’t have the resources to deal with this: it’s a geometrical concept properly applied to shapes. Suppose, however, that think not about the geometry of a system, but about the geometry of the space representing the system. Perhaps we can at least recover self-similarity and see how complexity is a fractal-like concept.
Start with the normal configuration space we’ve been dealing with all along. From the perspective of fundamental physics, each point in the space represents an important or interesting distinction: fundamental physics is a bit-map from point-to-point. When we compress the configuration space for treatment by a special science, though, not all point differences remain relevant—part of what it means to apply a particular special science is to treat some distinctions made by physics as irrelevant given a certain set of goals. This is what is meant by thinking of the special sciences as coarse-grainings of fundamental physics.