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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 : 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 x$2$ + y$2$ = 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. Once we have a few

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