Saturday, April 20, 2013

On complexity

From "A Different Universe" by Robert B. Laughlin.

… Seeing structures like these for the first time causes even a hard boiled reductionist to pause and wonder whether they might be caused by some agency other than elementary quantum mechanic It is one thing to explain ordered crystals of atoms with simple microcroscopic rules, but quite another to do so with complex lifelike structures and shapes, especially when one cannot deduce from first principles that these shapes should emerge. But this common and perfectly reasonable viewpoint is exactly backward. In a world with huge numbers of parts the unusual thing is not complexity but its absence. Simplicity in physics is an emergent phenomenon, not a mathematically self-evident state from which any deviation is a worrisome anomaly.

It is somewhat easier to explain and defend this assertion if you substitute the word random for complex. Thus you roll a die and the number three comes up at random. This statement means that YOl did not know ahead of time which face would come up, that it is something unpredictable, and that the degree of unpredictability is measured by the number of possible outcomes, in this case six. Then is nothing random about the number three itself once it has been selected. It makes no sense for any particular die face to be "random.' Similarly, it makes no sense for an isolated shape to be “complex” Only the selection of one shape out of many, a physical process, can be complex. When we say a shape is complex we really mean that the physical process by which it forrned is unstable and with a slight nudge could have generated one of many different shapes. Similarly, we say a shape is simple if it is guaranteed to be forrned by a physical process the same way every time, even when nudged fairly violently.

Once you understand that simplicity in nature is the exception. rather than the rule, it becomes easy to imagine that lifelike patterns might emerge if the microscopic circumstances were suitable. It is not possible to prove that they emerge, but it is possible to prove that their emergence is reasonable and does not violate common sense. |

One does so by means of complexity theory, a branch of mathemathics borne in the 1970s that subsumes the topics of chaos, fractals, and cellular automata. The strategy of complexity theory is to so simplify and abstract the equations of motion of matter that they can be solved reliably by computer. This abstraction, however, is a pact with devil, since the resulting equations so grotesquely distort things that you no longer have a faithful representation of nature. The value of complexity theory is thus limited to showing that emergence of complex patterns is reasonable. It cannot supply predictive models of any natural phenomenon, and it is certainly not a fundamentally new way of thinking. |

A simple example of such a model is the mountain range fractal. A computerized map grid is refined again and again, each time assigning a fictitious height to the new grid point that is the average of eights of the adjacent old ones plus a random increment that becomes smaller and smaller as the refinement proceeds. The heights generated simulate the appearance of real mountain ranges so effectively that they are often used in movies to generate backdrops, …

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