Tuesday, February 15, 2011

This misnomer of "self organizing" systems

The study of self organizing is extremely important in modern science, whether in evolutionary biology, economics, or even the physical sciences. I once went to a talk on the study of self-organizing systems in the context of crime syndicates in Los Angeles. There are even some interesting applications at the basic level of image segmentation: mathematicians working in nonlinear analysis are now able to devise computer algorithms to detect different parts of a photograph using evolutionary partial differential equations. There is a key idea encompassing all of these fields of research: complex systems can "organize themselves" not according to a preset plan, but according to certain rules, with certain types of organization being "selected" over others. For instance, in image segmentation, it would be pointless to try to give a computer all the necessary preset criteria for identifying a piece of an image in a photograph. (How do you know where a mountain ends and a sky begins? Try writing down the exact criteria just to see how impossible this is.) Instead, if you give a computer general rules (dictated by a PDE) by which to allow its image segmentation to evolve, eventually an appropriate segmentation will be selected.

So why do I say "self organizing" is a misnomer? The argument can be put pretty simply in a mathematical framework. A self organizing system is nothing more and nothing less than a dynamic system. Let x(t) by the state of a system at time t, with the state space given by H, and let f be a function. Then we can write x'(t) = f(x(t)) for given initial data x(0) in H, where x' is the time derivative of x. This defines a dynamic system. For discrete time, we can simply write x(n+1) = f(x(n)). The long-time behavior of a dynamic system is, in a completely literal sense, the result of evolution. The crucial point is this: this behavior is governed by the function f. It would be absurd mathematically and otherwise to say that x(t) determines itself. On the contrary, it is clearly determined by f, over which it has no control. (I could mention stochastic dynamic systems, but the same principles apply there as well; randomness does not mean the system is somehow freer.)

There is a wide range of phenomena that can be observed in a dynamic system. The simpler ones tend toward some fixed point (such as zero). Dynamic systems arising in engineering applications tend to be of this simpler type. Those arising in other fields exhibit much more complicated behavior, such as "dynamic equilibrium." Essentially, whatever the long-term behavior of the system is what has been "selected" by the dynamic system in question. Take, for instance, a row of numbers that evolve in the following way: each number is multiplied by its nearest neighbors at each time step. If zeros are present to begin with, eventually they will be selected and the system will tend to all zeros. On the other hand, even if the system begins with all positive numbers, certain trends will be noticeable: for instance, evens will tend to be selected over odds (since an even times anything is an even).

As stated above, the observable characteristics of the state x(t) is entirely determined by the function f. Therefore the function f is doing the "selecting." It seems unnatural and even misleading to talk as if the state were organizing itself. Perhaps when self organizing systems were first being studied, people were only used to thinking of deliberately constructed systems, with each component playing a role specified in advance. It is true that in a dynamic system, the resulting structure cannot be described as deliberately constructed, but selected. Yet even if it seems at first mysterious that a useful structure could emerge apart from deliberate construction, that does not mean such structures "arranged themselves." Quite the opposite is true; the real story both mathematically and scientifically is that such structures emerge due to no plan of their own.

There are certainly fundamental differences between a dynamic system, which undergoes a selection process, and something deliberately constructed. In fact the former is usually more robust: the state (or collection of states, rather) selected by the system will usually be unperturbed by slight modifications to the system. By contrast, a deliberately constructed system (such as a computer) is notoriously fragile, and can only be made robust by means of intentional redundancy (making back-ups, etc.). Learning to use dynamic systems is to our advantage in the long run, because of this robustness. Yet sadly it requires an epistemological sacrifice: the more complex a dynamic system is, the less we can actually determine about its precise behavior. It helps to concentrate on the questions we can answer and actually care about. For instance, it is possible to study the long-term stability of a system without knowing what happens at each moment in time (this is what I attempt to study in my research). In other words, it's very possible to get results without even approaching an exhaustive description of a system.

My point is this: both structures resulting from deliberate construction and those resulting from selection can be said to be products of "design." In fact, one can think of a dynamic system as getting more for less: simply establish a few preset rules, and you get structures that exhibit behavior suited to those rules. Deliberate design is just the opposite: you get less than you put in, since not only do you have to give each and every component its particular place, but the resulting structure is subject to unpredictable forces which you could not account for (natural wear and tear, so to speak). One might say that the former is simply a more clever form of design, but to be fair, it involves a lot of things being weeded out before the selection process is complete. I don't think there's anything other than prejudice to philosophically justify calling one "design" and the other not.

The words "self organizing" imply a sort of ontological independence that many people find philosophically appealing; in particular, the theory of evolution is common fodder for arguments against the existence of God. For this reason I couldn't help but write a little note explaining why this independence is rather illusory: everything in a dynamic system depends on the laws by which it is governed. So now I finish with a theological point: so what if God really is the law-giver, and the universe simply follows his laws? Is this an unjust way to "design" the universe? "But why all the waste?" you ask. Yet you yourself, being the product of evolution, your mind the result of this enormously complex process of selection, are in no position to ask such questions. Or, to put it another way, "Nay but, O man, who art thou that repliest against God? Shall the thing formed say to him that formed it, Why hast thou made me thus?" I don't presume to have proven some point here, but to merely illustrate a possibility and to raise a question. Is it really possible for us to know how we ought to have been "designed"? And thus is it really possible to know whether God made us by studying how it is we came into being?

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