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Rather than trying to go for a solid demarcation between complex and simple systems immediately, it might be easier to start by comparing systems. Here are some comparisons that seem intuitively true[1]: a dog’s brain is more complex than an ant’s brain, and a human’s brain is more complex still. The Earth’s ecosystem is complex, and rapidly became significantly more complex during and after the Cambrian explosion 550 million years ago. The Internet as it exists today is more complex than ARPANET—the Internet’s progenitor—was when it was first constructed. A Mozart violin concerto is more complex than a folk tune like “Twinkle, Twinkle, Little Star.” The shape of Ireland’s coastline is more complex than the shape described by the equation x2 + y2 = 1. The economy of the United States in 2011 is more complex than the economy of pre-Industrial Europe. All these cases are (hopefully) relatively uncontroversial. What quantity is actually being tracked here, though? Is it the same quantity in all these cases? That is, is the sense in which a human brain is more complex than an ant brain the same sense in which a Mozart concerto is more complex than a folk tune? One way or another, what’s the significance of the answer to that question—if there’s an analogous sense of complexity behind all these cases (and I shall argue that there is, at least in most cases), what does that mean for the practice of science? What can we learn by looking at disparate examples of complex systems? Let’s look at a few different ways that we might try to make this notion more precise. We’ll start with the most naïve and intuitive paths, and work our way up from there[2]. Once we have a few

  1. I'm going to rely quite heavily on our intuitive judgments of complexity in this chapter; in particular, I'll argue that some of the definitions we consider later on are insufficient because they fail to accord with our intuitive judgments about what counts as a complex system. Since constructing a more rigorous definition is precisely what we're trying to do here, this doesn't seem like much of a problem. We've got to start somewhere.
  2. For an even more exhaustive survey of different attempts to quantify “complexity” in the existing literature, see Chapter 7 of Mitchell (2009). We will not survey every such proposal here, but rather will focus our attention on a few of the leading contenders—both the most intuitive proposals and the proposals that seem to have gotten the most mileage—before offering a novel account of complexity that attempts to synthesize these contenders.