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Form and Shape
Logic and Order
Logic and Order
A History Lesson
Date last modified
2015 Easter Day
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Mathematical Models of the generation of Physical Form and Shape

While morphology sounds like a discussion of form and shape, as a science it is mostly concerned with the creation of mathematical models that describe the process that results in a particular form or shape, sometimes called morphogenesis. Alan Turning published "The Chemical Basis of Morphogenesis" in 1952 which very literally showed how the leopard got its spots. In general terms, the form and shape of everything in the physical world is a direct result of the process that created that thing. To understand the thing, we need to understand the creation process. Understanding, for humans, typically comes from analogy or modeling. How is one thing like another, or how can the process be modeled with a mathematical equation. The essence of modern science is to emulate the success that physicists have achieved in creating mathematical models of the fundamental processes of reality.

A Culture of Change and Growth

Before the current era (BCE) the concepts of change and growth were generally unknown to men. The world and all that was contained within it had been given to men by creatures beyond our understanding. Men lived through cycles, endless repeating the old forms. From time to time one particularly great man could transcend the endlessly repeating cycle, but that was the exception, not the rule. And it was nearly always men that attained this breakthrough. After the destruction of the Jewish temple in 70 CE, the Christian religion started to preach about a arc of history with a known beginning and end; but it was not until the middle of the 19 century that men of learning generally adopted the concept of change in the human condition. The new field of science (nee natural philosophy) began to spread widely a new attitude that men could learn of their own observation how the world, and all that was in it, was ordered. The great success of the physical sciences, in particular, were considered by men (like John Stuart Mill in "A system of Logic") to be the crown jewel of sciences, one that all other science, including the political sciences, could aspire. The universe, and all that was contained therein, was just a great clockwork ordered by the laws of physics. Man might not yet have attained all the knowledge to predict the arc of History, but Karl Marx was not alone in thinking that was a definite possibility. Plank, Einstein and Heisenberg destroyed that certainty during the first two dozen years of the 20th century.

Confusion of model and reality

The success of modeling in physics has lead many physicists to confuse the model with reality. This sense of reality has been generated by the proofing process for a new theory. For example the electron theories of Paul Dirac resulted in a prediction of the existence of the anti-electron, or positron. The prediction of hitherto unknown species of particles is taken as proof positive of the correctness of the model. The scientists are correct in their belief that the model gains more credibility as a result of testing it against novel cases which the model correctly describes. But most scientists go beyond that proofing process to a declaration that the model is the reality. This view is personified in a tee shirt that I was given that says "And god said {the maxwell equations} and then there was light." The tee shirt is only funny because it exposes a prejudice in favor of the model. As Steven Hawking has said "God Created the Integers. Everything else is the creation of man." Which probably should be said as, God created countable things, man created the mathematical models.

The unnatural success of mathematical models

Several examples exist of mathematical models that describe order in the physical world. Peter S. Stevens in "Patterns in Nature" gave many examples, including river meanders. Meanders are a great example of the multiplicity of mathematical models of physical reality. There are at least three completely independent models of meanders where the only thing in common between the models is their result:

  • The mechanical model described in Albert Einstein "Essays in Science" Dover ISBN-13: 978-0486470115.
  • The uniform expenditure of Energy model where the scour of the river against the bank makes the bends smooth and uniform with no sharp changes in direction. The smoothness minimized further erosion so that the rivers does as little work going around the bends and possible. It seems like the river intends to give up its energy in as uniform a manner as possible. In straight shallow stretches it is by ripples int he surface. In the deep and quite pools, it twists itself into a bend.
  • The probability model was developed by Hermann von Schellig of General Electric. It posits a general rule that any line of fixed length that stretches between two fixed points will follow a meander with high probability.
Luna Leopold has a good description of meanders at this site This shows that models of (1)bank erosion, (2)uniform expenditure of energy and (3)probability all come up with the same result. Even quantum mechanics started life with two very distinct mathematical models (Schrödinger's wave equation and Heisenberg's matrix equations), which also came up with the same results. In his book "Conscious Mind" David Chalmers lists 5 levels of interpretation of the meanings of these models of the quantum. In the end he points out that none of these interpretations are entirely satisfactory. Here is a case where mathematical models just can't cope with the reality of our physical world.

I suspect that other mathematical models of physical processes are equally likely to be duplicated if we searched hard enough. What those equivalent mathematical models demonstrate, is that any given model of reality, is never reality, no matter that each model might predict all sorts of new and undiscovered facts, they are, in the end, just models, not reality. Some Physicists have a very hard time with that last statement, given the huge investment in human time and energy that has gone into their creation.

The Rashomon Effect

In the 1950 movie "Rashomon" four participants in a brutal crime all give quite different versions of the events that happened. Similar conceits were used in the "Alexandra Quartet" by Lawrence Durrell and "One Hundred Dollar Misunderstanding" by Robert Grover, both written shortly later. It seems that the differing views of reality by different observers is part of the new reality of physics as well. In Measurements on the reality of the wavefunction the authors demonstrate that either (1) there is no objective reality at all, or (2) the probabilities derived from the wavefunction are the only realities.

Condensed Matter (Solid State aka Squalid State) Physics

Classical physics deals with condensed states. Balls of matter that fall after being shot from a cannon, or even a large collection of electrons that can make a visible dot of color on a television screen. This is the physics of Galileo or Newton. It is the clockwork physics of the world before Max Plank. It worked well until physicists tried to apply the classical rules to the very smallest particles of matter or energy. Starting with the photoelectric effect, the classical rules failed miserably. But it is only aggregates of matter that we can measure unambiguously. While the rules of quantum mechanics are very good and giving answers, the only answers they give us are probabilities.