more complex than one that’s either random or homogeneous. The correlation condition thus fails to hold. A successful measure of complexity, then, should account for why there seems to be a “sweet spot” in between maximal and minimal Shannon entropy where the complexity of associated systems seems to peak, as well as give an account of how in general we should go about representing systems in a way that lets us appropriately judge their Shannon entropy.
Finally, fractal dimension suffered from one very large problem: it seems difficult to say how we can apply it to judgments of complexity that track characteristics other than spatial shape. Fractal dimension does a good job of explaining what we mean when we judge that a piece of broccoli is more complex than a marble (the broccoli’s fractal dimension is higher), but it’s hard to see how it can account for our judgment that a supercomputer is more complex than a hammer, or that a human is more complex than a chair, or that the global climate system on Earth is more complex than the global climate system on Mars. A good measure of complexity will either expand the fractal dimension measure to make sense of non-geometric complexity, or will show why geometric complexity is just a special case of a more general notion.
2.1 Dynamical Complexity
With a more concrete goal at which to aim, then, let’s see what we can do. In this section, I will attempt to synthesize the insights in the different measures of complexity discussed above under a single banner—the banner of dynamical complexity. This is a novel account of complexity which will (I hope) allow us to make sense of both our intuitive judgments about complexity and open the door to making those judgments somewhat more precise. Ultimately, remember, our goal is to give a concept which will allow us to reliably differentiate between complex systems and simple systems such that we can (roughly) differentiate complex systems