Self-Dissimilarity

30 January 2011

Sunday


In A Note on Fractals and Banach-Tarski Extraction I suggested that there is a sense in which the mathematical methodology of Banach-Tarski extraction is the antithesis of a fractal, but I have realized that there is another kind of antithesis to a fractal, and this realization came to me by a rather circuitous chain of reasoning. The short version is this: fractals are self-similarity iterated to infinity, therefore the antithesis of a fractal is self-dissimilarity iterated to infinity. But it took me some time to get that far, so let me back up a bit and give a more detailed account of how my mind got from point A to point B.

My post Popular Revolt in the Arab World was tweeted by Ashwin Parameswaran, who added the cryptic (to me) comment: “The resilience-stability tradeoff in political regimes.” Mr. Parameswaran, a former banker, has a blog, Macroeconomic Resilience, which I of course took a look at, and found it to be of the greatest interest. Mr. Parameswaran describes his blog as, “Mostly about markets and macroeconomies as complex adaptive systems,” and in some of the posts he has written he draws heavily from ecology. This was of great interest to me given my recent formulation of Integral Ecology. His applications of ecological theory to economic theory are certainly in the spirit of a generalized conception of ecology that is universally applicable in formulating our conception of the world.

In his post The Resilience Stability Tradeoff: Drawing Analogies between River Flood Management and Macroeconomic Management, Mr. Parameswaran wrote:

“[The 2008 Bihar flood] was a disaster caused by the loss of system resilience, highlighted by the inability of the system to ‘withstand even modest adverse shocks’ after prolonged periods of stability… With the passage of time, a progressively greater degree of resources were required to maintain system stability and the eventual failure was a catastrophic one rather than a moderate one… stabilisation transformed the system into a state where eventually even minor and frequently observed disturbances would trigger a catastrophic outcome.”

As soon as I read this, while simultaneously acknowledging the truth of this as far as it extends, I immediately began thinking of counter-examples, of instances when this does not seem to be the case, instances when stability takes the form of a stable pattern that repeats over time (like a fractal in time). Do we perhaps think of stability a little too much as a form of Platonic eternity, in which the emphasis falls on the timeless, the unchanging, and invariant, and time is the mere moving image of eternity?

But what is stability? If an ecological system swings like a pendulum between too many predators and not enough prey, and too much prey and not enough predators, that is to say, if an ecosystem swings between feast and famine, is this a stable equilibrium, or is it an unstable system? One way to distinguish the two would be to say that if the change in the ecosystem escalates catastrophically resulting in damage to or destruction of that ecosystem, then this is not a system in equilibrium. On relatively short time scales this works out rather well, but if we think of longer time scales in which ecological succession, including stages of nudation, are common parts of the ecosystem, we have to ask in what sense the “destruction” of an ecosystem is a catastrophic failure and in what sense it is part of a larger ecological structure. When we consider the even longer time scales of evolutionary biology or geomorphology, in which the fungibility of the biome and the fungibility of the landscape mean that seas disappear and in time become mountaintops, then creation and destruction of entire ecosystems are part of the ongoing drama of life.

I have had similar thoughts in relation to punctuated equilibrium. The first time I ever heard about punctuated equilibrium I immediately said to myself, “Well, it all depends on how much of a period of time you take. What from a close-up perspective seems atomized and discontinuous may from a distance appear smooth and continuous. What appears as an exception to a rule on a small, local scale of time, in a larger, more comprehensive scale of time is the confirmation of a pattern.” I kept this idea of punctuated equilibrium in my thoughts, and when I went for a hike some years later on Mount Hood I saw a concrete illustration of in the White River Gorge, which runs down the east face of Mount Hood. I sat down in on a rock in the White River Gorge, thinking of punctuated equilibrium and the history of life, and I read the whole of Shelley’s (unfinished) poem The Triumph of Life. It was a memorable experience, and it stays with me to this day.

What about the White River Gorge put me in mind of punctuated equilibrium? The White River, fed by melting snow from the mountain, runs through a gorge with steep, slanted slopes of rounded, rolling rocks. It is quite difficult in places to walk over these rocks. The slopes seem to be just a little steeper than the angle of repose, so that if you wait and watch, you will eventually see rocks tumble down the slope. Most of these rocks are fairly small, but if you wait long enough you will eventually seek larger rocks roll down the slope. Is this a gradual process or a punctuated process? For each rock that rolls down the face of the slope, it is a punctuated event, but if you set up a camera to take one picture per day, and then made a film of several years worth of the White River Gorge, it would look like very slow and gradual change.

The White River Gorge on the east slope of Mount Hood.

I wasn’t the only one to have this reaction to punctuated equilibrium. I remember one day reading an issue of Scientific American in the Newberg library (I was there on my lunch break; this had to be in the mid-1980s as that was the timeframe when I was driving to Newberg every day) there was a story on evolutionary theory in which the author stated something to the effect that what to the paleontologist appears as a rapid if not sudden change may appear to the biologist as a smooth and incremental transition. The punctuated equilibrium-oriented paleontologist and the gradualist biologist may simply be speaking at cross purposes.

The library in Newberg, Oregon, where I frequently spent my lunch breaks while my job at the time involved a layover in the town.

After I wrote the above paragraph I did a little research and I found the article “The Evolution of Darwinism” in the June 1985 issue of Scientific American. Here is the paragraph that remained in my mind (more or less) all these years:

“The dispute with the punctualists loses some of its focus when one recognizes that it is partly an artifact of a radical difference in time scales: the time scale of the paleontologists who propose the theory of punctuated equilibrium and that of the geneticists who were instrumental in formulating the synthetic theory. Since successive layers in geologic strata may have been laid down tens of thousands of years apart, morphological changes that developed over thousands of generations may make an abrupt appearance in the fossil record. In contrast, geneticists refer to changes that require 200 generations or more as gradual, since they exceed the time span of all experiments except those on microorganisms. In speaking on the one hand of sudden change and on the other of gradual evolution, the punctualists and the gradualists are in many cases talking about the same thing.”

Much later, only recently, I came across the scientific language used to express this distinction, and it is the language of symmetry. Richard Feynman, in his short book The Character of Physical Law, has a chapter on symmetry in physical theory, and here he observes that if you construct an apparatus for a scientific experiment in one place, and then do the same thing in another place and get the same result, this is spatial symmetry, sometimes called translation in space. Similarly, we can conduct a scientific experiment at two different times and this is temporal symmetry, or translation in time. Then he considers the possibility of constructing an experimental apparatus at one scale, conducting an experiment, and then trying the same apparatus and experiment conducted at different scales, either much larger or much smaller. This is translation in scale, and it demonstrate symmetry or asymmetry of scale. Many scientific experiments that can be translated in space or time cannot be translated in scale. Step back for a moment and this becomes obvious: macroscopic observations that, say, conform to Newton’s laws of motion, do not conform to the laws of quantum physics which operate on the scale of atomic and subatomic particles, and vice versa.

One way to define a fractal is that it is a structure that retains its properties under magnification. Fractals possess self-similarity, or symmetry, across translation in scale iterated to infinity. We can see from this that symmetry is the more comprehensive concept, and we can only define self-similarity (symmetry) or self-dissimilarity (asymmetry) in the context of symmetry construed quite broadly, which might be applied to translation in scale, but it might also be applied to translation in space or time.

In any case, the sense of symmetry sought in contemporary physical theory can be formulated in a much more intuitive sense, as well as a much more traditional sense, by understanding the it is a search for constants, or what would have once been called universal truths. This idiom of universal truths has fallen into disfavor (it has the flavor of theology about it), but if you are looking for a property that remains invariant under translations in space, translations in time, or translations in scale, with each of these symmetries you demonstrate you approach more closely to something that is universally true (whether or not you choose to call it that). This is a sense of “science” that would have been immediately recognized by Plato or Aristotle, before any distinction between science, philosophy, and theology was even imagined. For Plato, knowledge, in contradistinction to mere opinion, was about changeless, eternal truths, and in so far as physicists today are seeking symmetry in physical theory, they are seeking that within the changing appearances of the world that does not change.

It is the natural response of the philosophical mind, when presented with a putative universal truth to immediately seek counter-examples. Universal generalization (the “universal affirmative” in traditional logic) is given the lie by a single counter-example. That doesn’t mean that we have to necessarily give up on the universal generalization, though we do have to modify it, at very least acknowledging an exception. Much of science is also built on this instinctive contrarian response, and in so far as contemporary science deals more with statistical regularities than universal truths, we can except a law with the occasional exception, as long as we can account for the exception with some other law.

In seeking counter-examples to proposed universal laws, we are seeking asymmetry, that is to say, we are seeking self-dissimilarity within the world. While Feynman said in the same book of his mentioned above the human beings have a natural response to symmetry, that we intuitively appreciate and enjoy symmetry, it is no less human to exercise skepticism by calling symmetry into question, and perhaps even appreciating asymmetry. Ruskin’s famous essay on Gothic architecture makes much of what he calls the “wildness” of Gothic art, and anyone who has visited both ancient Greek temples and European Gothic cathedrals knows well the difference between Hellenistic rationalism framed in terms of eternal verities and universal truths on the one hand, and on the other hand Gothic irregularity and the insistence upon the particular and the exception, which is, in a sense, a form of empiricism.

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