Jonathan Lawhead

“ After all, at least some of our judgments of complexity seem (at least at first glance) to have very little to do with shape: when I say (for instance) that the global economy is more complex today than it was 300 years ago, it doesn’t look like I’m making a claim about the shape of any particular object. ”

“ If all systems are complicated, then, what makes one system a complex system, and another a simple system? This is not an easy question to answer, and an entirely new academic field—complex systems theory—has grown up around attempts to do so. Despite the centrality of the concept, there’s no agreed-upon definition of complexity in the complex systems theory literature. ”

“ What’s the problem with calling a purely random message more complex? To see this point, let’s consider a more real-world example. If we want Shannon entropy to work as a straight-forward measure for complexity, it needs to be the case that there's a tight correlation between an increase (or decrease) in Shannon entropy and an increase (or decrease) in complexity. That is: we need it to be the case that complexity is proportional to Shannon entropy: call this the correlation condition. ”

“ Fractal-like systems (or, at least the fractal-like systems we’re interested in here) show interesting details at multiple levels of magnification, and the interesting details present at each level are self-similar.
With this intuitive picture on the table, let’s spend a moment looking at the more formal definition of fractals given in mathematics. Notice that we’ve been calling physical systems “fractal-like” all along here—that’s because nothing in nature is actually a fractal, in just the same sense that nothing in nature is actually a circle. ”

“ Just as in Chapter One, my focus here will be primarily on exploring the actual practice of contemporary, working science: I’m interested in what climate scientists, economists, and statistical physicists (as well as others working in the branches of science primarily concerned with predicting the behavior of complex systems) can learn from one another, rather than in giving a priori pronouncements on the structure of these branches of science. ”

“ Before we can actually engage with complex systems theories (and bring those theories to bear in exploring the foundations of climate science) , we’ll need to articulate what exactly makes a system complex, and examine the structure of complex systems theories generally. ”

“ A particular string of noises coming out of my cat are (in general) far more likely than any particular string of noises that comes out of my roommate; my roommate’s speech shows a good deal more variation between messages, and between pieces of a given message. A sentence uttered by him has far higher Shannon entropy than a series of meows from my cat. So far, then, this seems like a pretty good candidate for what our intuitive sense of complexity might be tracking: information about complex systems has far more Shannon entropy than information about simple systems. ”

“ There are no simplistic sciences: all science is complicated in the sense of being difficult, multi-faceted, and messy. Similarly, there are no simplistic systems in nature; no matter how we choose to carve up the world, the result is a set of systems that are decidedly complicated (and thank goodness for this: the world would be incredibly boring otherwise!) . This point should be clear from our discussion in Chapter One. ”

“ Moreover, it turns out that the more rugged a coastline is, the higher fractal dimension it has: coasts with outlines that are very complex have higher fractal dimension than coasts that are relatively simple and smooth.
The most serious problem with using fractal dimension as a general measure of complexity is that it seems to chiefly be quantifying a fact about how complex an object’s spatial configuration is: the statistical self-similarity that both broccoli and coastlines show is a self-similarity of shape. ”

“ However, it seems like we also want to say that my brain was more complex when it was capable of supporting both life and significant information processing than it was after it was randomized—we want to say that normal living human systems are more complex than corpses. But now we've got a problem: in randomizing the state of my brain, we've increased the Shannon entropy of the associated message encoding its state. A decrease in complexity here is associated with an increase in Shannon entropy. ”

“ Intuitively, my roommate’s vocalizations are the more complex.
We can also think of “surprise” as tracking something about how much I miss if I fail to hear part of a message. In messages that are more surprising (in this sense) , missing just a small amount of data can make the message very difficult to interpret, as anyone who has ever said expressed incredulity with “What?!” can attest; when a message is received and interpreted as being highly surprising, we understand that just having misheard a word or two could have given us the wrong impression, and request verification. ”

“ In the case of circles, we know exactly what it means to say that there are no circles in nature: no natural systems exist which are precisely isomorphic to the equation that describes a geometric circle: things (e.g. basketball hoops) are circular, but on close enough examination they turn out to be rough and bumpy in a way that a mathematical circle is not. ”

“ You can probably guess by now that the answer is ‘no, not entirely.’ There are certainly cases where fractal dimension accords very nicely with what we mean by ‘complex:’ it excels, for instance, at tracking the rugged complexity of coastlines. Coasts—which were among Mandelbrot’s original paradigm cases of fractal-like objects—are statistically self-similar in much the same way that broccoli is. Viewed from high above, coastlines look jagged and irregular. ”

“ For a more formal definition of fractal dimensionality (especially in the case of Pythagoras Tree-like figures) , see Lofstedt (2008) . ↑ In the illustration here, we had to build in the presence of “copies” by hand, since a featureless line (or square or cube) has no self-similarity at all. That’s OK: the action of bisecting the figure is, in a sense, a purely abstract operation: we’re not changing anything about the topology of the figures in question by supposing that they’re being altered in this way. ”

“ More specifically, I’ll argue that the parallels between climate science and other complex systems sciences—particularly economics—have been largely overlooked, and that this oversight is primarily a result of the tradition of dividing the sciences into physical and social sciences. ”

“ This concept, which I will call dynamical complexity, is presented as a physical interpretation of some very recent mathematical advancements in the field of information theory. The central problem that shall occupy us in the next chapter, then, is how to transform a discussion of complexity that seems to work very well for things like messages into an account that works well for things like climate systems. My hope is that dynamical complexity offers this bridge. ”

“ In such systems, the whole is more than the sum of the parts, not in an ultimate, metaphysical sense, but in the important pragmatic sense that, given the properties of the parts and the laws of their inter-action, it is not a trivial matter to infer the properties of the whole. In the face of complexity, an in-principle reductionist may be at the same time a pragmatic holist... ”

“ This isn't going to do it, then: while size certainly matters somewhat, the mereological size measure fails to capture the sense in which it matters. Bigger doesn't always mean more complex, even if we can solve the all-important problem of defining what "bigger" even means.
In the case of Strevens’ proposal, we might well be suspicious of what Wikipedia editors would recognize as “weasel words” in the definition: a complex system is one that is made up of many parts that are somewhat independent of one another, and yet interact strongly. ”

“ Finally, consider our broccoli-like fractal: the Pythagoras Tree. The Pythagoras Tree, as you can easily confirm, has a fractal dimension of 2: at each step n in the generation, there are 2n copies of the figure present: 1 on the zeroth iteration, 2 after a single iteration, 4 after two iterations, 8 after three, 16 after four, and so on. Additionally, each iteration produces figures that are smaller by a factor of √2/2. ”

“ The Shannon entropy has thus increased. ↑ To see this point, think of two pieces of DNA—one of which codes for a normal organism (say, a human being) and one of similar length, but which consists only in cytosine-guanine pairs. Each DNA string can be encoded as a message consisting entirely of the letters A, C, G, and T. The piece of DNA that codes for a functional organism will be associated with a message with far higher Shannon entropy than the piece of DNA associated with a message that consists entirely of the string ‘CG’ repeated many times. ”

“ Even supposing that we can easily (and uncontroversially) find a way to represent the physical state of my brain as something like a message, [13] it seems clear that we can construct a case where measuring Shannon entropy isn't going to give us a reliable guide to complexity. ”

“ The same is true of fractals; if we continue to subdivide the broccoli stalk discussed above, eventually we’ll reach a point where the self-similarity breaks down—we can’t carry on getting smaller and smaller smooth green stems and round green bristles forever. Moreover, the kind of similarity that we see at each level of magnification is only approximate: each of the lobes looks a lot like the original piece of broccoli, but the resemblance isn’t perfect—it’s just pretty close. ”