by: Paul Martin
In Chapter 1 of his paper, Dick defines a completely unspecified set of numbers, a collection of subsets of those numbers, and a finite subset of that collection with cardinality n. He then poses the question, "Could there be a function that can in all cases predict the makeup of the nth subset given a knowledge of the makeup of the other n-1 subsets?", (equation (1.2)), and proceeds to demonstrate logically that the answer to the question is "no".
Next, he poses the question, "Could there be a function that can predict the probability of the nth subset having a particular makeup, given a knowledge of the makeup of the other n-1 subsets?"
To answer this question, he assumes that such a function exists (equation (1.3)). This assumption raises two new questions. 1) Can we prove the existence of such a function? and 2) What, if anything, can we deduce about the nature of this function?
Then, by applying standard definitions from previous mathematics, (equation (1.4)), he defines yet another function which can be interpreted as producing what we might call probability density (equation (1.7)).
Finally, he proceeds, by strictly logical deductive arguments, to show that if such a function exists, it must obey a particular differential equation (various versions of his paper have labeled this equation as (1.27) and (1.29))
Dick chose notation in making his definitions that just happened to coincide with the notation used in modern physics to describe the Schroedinger and Dirac equations. That choice makes some of the important interpretations of Dick's results immediately evident.
If we view our access to the "universe" or "reality", (whatever they might be) as being a set of information available to "our senses", (whatever they might be) then that accessible set of information can be considered to be, or converted to, a set of numbers. Dick's differential equation applies to this set of numbers. Furthermore, the solutions to his differential equation, can be interpreted to mean that there are certain constraints imposed on any "universe" or "reality" that we can perceive via "our senses". The meanings of the terms in quotes, of course, have nothing to do with Dick's result or its derivation.
In Chapters 2-5, Dick has developed solutions to his differential equation which he shows to be equivalent to most of physics as it has been discovered so far by other methods. Armed with the differential equation and it's solutions, Dick has developed a completely general way of modeling the information available to us about our universe. This method involves plotting the numbers in the various subsets on a three-dimensional space, and parameterizing the subsets with a variable called 'time'.
By applying the solutions to his differential equation to this method of plotting, or displaying, the information, he shows that this model is consistent with the typical model that each of us develops naturally in our subconscious minds as a result of living in this world, and it is consistent with the model developed by conventional physics. N.B. This "Model" is not part of his formalism.
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