Chaos in Tejas

Lorenz Attractor (butterfly)

Common sense inclines, on the one hand, to assert that every event is caused by some preceding events, so that every can be explained... On the other hand... common sense attributes to mature and sane persons... the ability to choose freely between alternative possibilities of acting" Sir Karl Popper - The Open Universe: An Argument for Indeterminism.
Imagine that I have been assigned the chore of measuring the length of the coastline of Texas. I can look at Texas on Google Maps and use the 100 mile long ruler (lower right corner scale) to guesstimate the length as 360 miles, give or take 10 miles. I have left out some bays and inlets though and so I switch to a smaller scale so I can measure more of the bends and bulges - I had assumed before that bends and bulges 'averaged out' before.

I'm not with the Band

I then set my ruler at 1 foot and set about measuring again. I am taking into account every feature of the coastline that is at least 1 foot long. I now end up with 500 miles of coast from 50000000 individual measurements where before I only needed 3.5 yardsticks. Now I realize that my rule of thumb - that the bulges and bends cancelled each other out - was completely wrong headed.

I get out my scanning electron microscope and discover that the coastline now measures 8,000,000,000 miles since I have captured the roughness of grains of sand, individual crystals of grains of sand, individual atoms in the lattice of the crystals in the grains of sand, dunes made of sand etc. Now I realize that I have fallen into the rabbit hole.

I also note that while measuring, the coastline of Texas really never changed it's nature. It was a series of rough quantum measures (the quanta being the fixed length of my ruler) that surprisingly looked similar at every scale of magnification. At the scale of the Planck length, the smallest physical dimension in nature (0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 meters) I am measuring individual quarks (a type of hadron which is in turn a type of fermion) and reducing the scale of measure stops. At this point the length of the coast of Texas is approximately equal to the length of all the coastlines of all the oceans. The measure of this sameness is pointed to by the Feigenbaum number, 4.669, a foundational measure of chaos and complexity.

Each diminution of my measuring rod can be called a fractional dimension. Each 'screen shot' or image  - at all scales of magnification is self similar. The mathematician Benoit Mandelbrot calls such a 'shape' a fractal. Further, Mandelbrot corrected His Holiness Sir Karl Popper with a philosophical crane instead of his indeterminism sky hook.

Here is the crane - a device which 'lifts' structural layers of physical meaning (the stuff of science) up from lower levels of complexity. Skyhooks merely hang from the sky by magic and flim-flammery, or as Sir Karl would say, a sky-hook is an instantiation of a reification fallacy which has no falsifiable notions built-in by careful description of underlying assumptions. In "making laws is like making sausages", sausage conflates to law-making - but laws are not made of pork. Sausages are also not made of stuff called 'policy' or 'statute'.

Only later in the century could Popper's indeterminism claim be falsified. First Ilya Prigogine, Nobel Laureate in Chemistry re-wrote the Second Law of Thermodynamics to include not only systems in equilibrium - all energy uniformly distributed throughout the closed system - but instead a heterogeneous pudding of near equilibrium entropy1 and pockets of entropy2 (extropy or entropy-lite) far from thermodynamic equilibrium.

Chaotic effects, including non-linear effects like the feedback noise from a microphone, pervade our reality. A mike held up to a loudspeaker that amplifies any small random noise (a snap, crackle or pop...) in a particular frequency, such that the noise on the mike, feeds the speaker. The speaker then amplifies that frequency which the mike picks up along with the original snap, crackle, pop. Which is further amplified by the speaker. Which is picked up by the mike. Et Cetera.

Prigogine noted that this is deterministic chaos. In other words there is only a narrow range of frequencies that are amplified and of all those frequencies there is an even narrower range of frequencies that bootstrap non-linear feedback. This is a apt model for life on earth - my meat is a dissipative structure far from thermodynamic equilibrium (DSFFTE). The thermonuclear star nearest earth adds energy to the Mother's energy. A dense iron core and a hot turbulent mantle of viscous goo add to the elan vital of biota. Our minds are grown from the radiated heat of second generation stars (there are nothing but hydrogen and helium in first generation of stars formed in the early universe).



The second surprising finding, from the study of predator/prey ratios in animal populations. Imagine a savannah populated by lions and gazelles. During a rainy season more gazelles are born and supported to sexual maturity on more abundant grazing. Then, having more prey, the lion population increases.

The open system of two populations interacting with the cosmic environment now has very large and unstable sets of predator and prey populations. These entire populations are incredibly sensitive to small fluctuations in rainfall. Not only yearly rain fall totals but even week to week or day to day weather change can upset this balance. In the Lorenz Butterfly above, each 'plane' or wing can be 'flipped' into the other plane by the strange attractor at work in the fabric of space time.

In the lions vs. gazelle grudge match, 'daily rainfall' has become an unusually fertile source of population crash .OR. the equally likely plausible world where both populations continue to surge. Rainfall has become a 'strange attractor' or a very small system input with monumental leverage on outcomes. We can postulate a network of causation but unless we are looking for the strange attractor (daily rainfall) we lose the signal to the noise.

What you see on the left is a rotating image of the Mandelbrot set (the blue thingy) and the logistics map (the red spider webby thing). This a mathematical solution to Popper's indeterminacy arguement. Notice that each branch of the logistics map is a locus of indeterminacy while the rest of the paths are deterministic. Punctuated deterministic chaos.

Chaos is a polysemous word - a word with many disjoint meanings (like paper in "read the paper" or "she works at the paper" or "let's paper over the differences"). There are two almost completely incompatible definitions for chaos. Epistemic ignorance on stilts.

1. Complete disorder and confusion. ("The rock concert was epic chaos")
2. Behavior so unpredictable as to appear random, owing to great sensitivity to small changes in conditions (mathematical representation of non-linear systems).

Confusing the two is a category error. The same category error people commit when describing their sky father as 'infinite' and then bouncing on the trampoline of conflation and folding the Flying Spaghetti Monster's accomplishments into a rhetorical infinite. Specifically, a prosaic mathematical fail - MATH does not trade in dictionary definitions but has a rigid framework symbolic meaning. In mathematics infinity is a contrivance like the square root of -4 (defined as 2i, an imaginary numbers). Once again 'imaginary' is polysemous with a prosaic or commonplace meaning and a rigorously defined mathematical meaning.

When I snark that economics is not a science I am referring to a framework bereft of chaos theory. Paco himself lS a strange attractor and therefore worthy of our respect.