Complexity, Reality, and Rubber Sheets
Consider a complex function of many time varying variables. Choose any two variables. Plot the value of the function over the ranges of the chosen two variables creating a three dimensional surface. Observe this surface over time. If there are a sufficient number of time varying variables and if a sufficient number of these variables are not simply correlated then the surface will appear chaotic.
In nearly all physically realizable systems there are constraints on the macroscopic rate of change from point to point in space. In other words, a surface representing real phenomena must be continuous.
In our example, the surface was not constrained to be continuous. Nonphysical chaotic systems (mathematical simulations) are not constrained. Their surface plot can have many areas and points of discontinuity.
One could consider complex functions as driver functions. These driver functions are, however, constrained in their real world behavior by characteristics of the representational media. Such constraints can be perceived as “forces”. Thus inertia acts like a force to limit the rate of change in the position of objects which have mass. Communication speed acts as a retarding force to limit the rate of change in mob behavior.
An analogy of such system constraints would be to conceive of our example function driven surface as connected to a two dimensional rubber sheet where each point in the rubber sheet is connected to the function surface by tiny rods. As the function attempts to drive these rods up and down their movement is constrained by the physical characteristics of the rubber sheet and by the strength of the tiny rods. The resulting surface of the rubber sheet is a constrained representation of the underlying function.
Note that, depending upon the characteristics of the rubber sheet and the connecting rods, it may be quite difficult to know the value of any particular point in the driving function. The transfer function between a point on the sheet’s surface and the driving function is an interaction of the sheet characteristics, the rod’s characteristics and not only the particular point value of the driving function but also of the values of the driving function at many points surrounding the particular point. Thus observing the rubber sheet provides limited understanding of the driving function.
As the number of variables involved in the driving function increases, our ability to derive understanding from observing the rubber sheet rapidly decreases. As the complexity of the rubber sheet’s characteristics and the characteristics of the connecting rods increases, our ability to derive understanding also rapidly decreases.
In nearly all physically realizable systems there are constraints on the macroscopic rate of change from point to point in space. In other words, a surface representing real phenomena must be continuous.
In our example, the surface was not constrained to be continuous. Nonphysical chaotic systems (mathematical simulations) are not constrained. Their surface plot can have many areas and points of discontinuity.
One could consider complex functions as driver functions. These driver functions are, however, constrained in their real world behavior by characteristics of the representational media. Such constraints can be perceived as “forces”. Thus inertia acts like a force to limit the rate of change in the position of objects which have mass. Communication speed acts as a retarding force to limit the rate of change in mob behavior.
An analogy of such system constraints would be to conceive of our example function driven surface as connected to a two dimensional rubber sheet where each point in the rubber sheet is connected to the function surface by tiny rods. As the function attempts to drive these rods up and down their movement is constrained by the physical characteristics of the rubber sheet and by the strength of the tiny rods. The resulting surface of the rubber sheet is a constrained representation of the underlying function.
Note that, depending upon the characteristics of the rubber sheet and the connecting rods, it may be quite difficult to know the value of any particular point in the driving function. The transfer function between a point on the sheet’s surface and the driving function is an interaction of the sheet characteristics, the rod’s characteristics and not only the particular point value of the driving function but also of the values of the driving function at many points surrounding the particular point. Thus observing the rubber sheet provides limited understanding of the driving function.
As the number of variables involved in the driving function increases, our ability to derive understanding from observing the rubber sheet rapidly decreases. As the complexity of the rubber sheet’s characteristics and the characteristics of the connecting rods increases, our ability to derive understanding also rapidly decreases.
Labels: complexity, experimental method, modeling, observation, reality, understanding
posted by PsySciGuy at 3:22 PM
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