Jonathan Lawhead

“ While there is no single value for effective complexity, we can perfectly coherently talk about summing all the useful ways given our goals and values. The value of this sum will change as we make new scientific discoveries—as we discover new patterns in the world that are worth paying attention to—but this again just serves to emphasize the point from Chapter One: the world is messy, and science is hard. Complexity theory is part of the scientific project, and so inherits all the difficulties and messiness from the rest of the project. ”

“ Dynamical complexity, on the other hand, tracks facts about how systems change over time, and (moreover) embraces the notion that change over time can be tracked in numerous different ways, even for a single system. ”

“ We’ll introduce the basics of climatology and atmospheric science, and examine the difficulties involved in creating a working model of the Earth’s climate. From there, we will consider the particular challenges that climate science faces, given that it explicitly deals with a system of high dynamical complexity, and think about and how have those challenges been met in different fields. We’ll examine why it is that scientists care about dynamical complexity, and what can be learned by assessing the dynamical complexity of a given system. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ I shall argue that the markedly high degree of dynamical complexity in the global climate system is best dealt with by strongly interdisciplinary scientific inquiry, and that a failure to recognize the role that dynamical complexity plays in shaping the practices of some branches of science is what has led to most of the general criticism faced by climate science. ”

“ The central thrust of McAllister’s criticism is that it is difficult to assign a determinate value to the effective complexity of any physical system, as that system might contain a myriad of patterns, and thus fail to be best represented by any single ensemble. The question of what effective complexity we assign a system will depend on what string we choose to represent the system. ”

“ This is a consequence of the fact that (as we saw in Chapter One) some special science compressions are more lossy (in the sense of discarding more information, or coarse-graining more heavily) than others: biology is, in general, a more lossy encoding scheme than is organic chemistry. This is (again) a feature rather than a bug: biology is lossy, but the information discarded by biologists is (ideally) information that’s irrelevant to the patterns with which biologists concern themselves. ”

“ This is why the distinction just made between special-scientific compression and useful special-scientific compression is essential—if we were to attend only to shifts that changed a system enough for a particular special science’s compression to fail entirely, then we wouldn’t be able to account for the uncontroversial reduction of complexity that coincides with the removal of the human from our kitchen-system. ”

“ The choices of a state-space also constitutes a choice of dynamics: for a system with a high degree of dynamical complexity, there are a large number of state spaces which evince not only interesting static detail, but interesting dynamical detail as well. ”

“ This chapter has consisted in a sustained description of the concept, and an argument for its role as a marker for the kind of complexity we’re after when we’re doing science. The insight at the heart of dynamical complexity is that complexity, at least as it concerns science, is a feature of active, changing, evolving systems. Previous attempts to define complexity have overlooked this fact to one degree or another, and have tried to account for complexity primarily in terms of facts about the static state of a system. ”

“ What, then, is the relationship between effective complexity and dynamical complexity? Moreover, if effective complexity is the right way to formalize the intuitions behind complexity, why is this the case? ”

“ We can think of the compression from physics to (say) chemistry, then, as resulting in a new configuration space for the same old system—one where points represent regions of the old space, and where every point represents a significant difference from this new (goal-relative) perspective, with the significance stemming from both the discovery of interesting new macrostates and interesting new dynamics. This operation can be iterated for some systems: biology can define a new configuration space that will consist of points representing regions of the original configuration space. ”

“ Dynamical complexity escapes from this problem by relaxing the pyramidal constraint on the relationship between the various branches of science. As I argued in Chapter One, the intersections between the domains of the various sciences are likely to be messy and complicated: while many psychological systems are in fact also biological systems, there may well be psychological systems which are not—the advent of sophisticated artificial intelligence, for instance, would give rise to systems that might be fruitfully studied by psychologists but not by biologists. ”

“ Gell-Mann argues that the best representative of a random string is the uniform ensemble—that is, the ensemble of strings that assigns all possible strings equal probability. This is supposed to resolve the compressibility issues in the traditional information-theoretic account of complexity. It’s easy to see why: suppose we want to print a random string of length n. Rather than printing n characters directly, Gell-Mann proposes that we instead write a program that prints a random character n times. ”

“ The regions of configuration space that evolve in ways that interest biologists are less precisely defined than the regions of configuration space that evolve in ways that interest chemists, but the biologists can take advantage of that fact to (in a sense) do more work with less information, but that work will only be useful in a relatively small number of systems—those with paths that remain in a particular region of configuration space during the time period of interest. ”

“ In this section, I will argue that dynamical complexity succeeds in cherry-picking the best features of the mereological size measure, the hierarchical position measure, the information-theoretic measure, and the fractal dimension measure, while avoiding the worst difficulties of each of them. ”

“ Because McAllister believes his observation that effective complexity contains an observer-relative element to be a fatal flaw in the concept, he doesn’t consider the possibility that we might obtain a more reliable value by summing over the effective complexity values for the system. ”

“ 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. ”

“ Before we close this theoretical discussion and move on to consider the special problems faced by climate science as a complex science, it’s worth briefly reviewing the attempted definitions of complexity we surveyed in Section 2.1 to see how dynamical complexity fares as a unifying definition of complexity. ”

“ First, notice that this account of dynamical complexity also gives us a neat way of formalizing the state of a system as a sort of message so that its Shannon entropy can be judged: the state of a system is represented by its position in configuration space, and facts about how the system changes over time are represented as patterns in how that system moves through configuration space. ”

“ Still, there’s a clear parallel to standard statistical self-similarity: fractal dimension for normal physical objects roughly quantifies how much interesting spatial detail persists between magnification operations, and how much magnification one must do move from one level of detail to another. ”

“ Moreover, since we’re tying dynamical complexity not to composition but behavior, we don’t need to answer the uncomfortable questions that dog the avid proponent of the mereological size measure—we don’t need to say, for instance, which method of counting parts is the right one. ”

“ This is an insight that is clearly incorporated into dynamical complexity: since dynamical complexity deals with the number of different ways of carving configuration space that yield informative time-evolution patterns for a given system, the presence of interacting constituent parts is indeed, on this view, a great contributor to complexity. ”

“ All these facts can easily be expressed numerically. The deeper conceptual problem with Shannon entropy remains, though: if the correlation condition fails (which it surely still does) , how can we account for the fact that there does seem to be some relationship between Shannon entropy and dynamical complexity? ”

“ For a particularly salient example, think of the search for extraterrestrial life: we need to both identify conditions that must obtain on extrasolar planets for life to plausibly have taken hold and, given that identification, try to predict what sort of life might thrive on one candidate planet or another. ↑ If this assertion seems suspect, consider the fact that patterns identified by economists (e.g. the projected price of fossil fuels vs. the projected price of cleaner alternative energies) are now helpful in predicting the evolution of the global climate. ”