***3) IN a nutshell, which differential equation is Dick saying gives solutions which match other laws of physics ... or you can say,"Go see his paper again" if you don't want to type it here. ***
It is equation (1.27) at
***4) In your descriptions, a few sentences appear to need amendment in order to say something.
a) "He has proved that any set of numbers whatsoever must conform to a particular differential equation ..." -- is the word "conform" here used in a familiar way?***
I think so.
***I am not familiar with it.***
Oh I think you are, Mike. I used the word in the ordinary sense of the dictionary definition: "2. to be in accord or agreement: as, the novel conformed to my notion of a good story" (Webster)
***What could persuade you to divulge what it means for a set of numbers to "conform" to a differential equation?***
Well, first you need to ask, which you did, and secondly, I need to be able to explain it, which I will herewith attempt to do.
A set of numbers is said to "conform" to a differential equation if and only if by substituting members and subsets of that set of numbers for the variables in the equation in a manner consistent with the definitions given for those variables, the equation is seen to be true, i.e. the two sides of the equation will be equal.
Now, the thing that makes it more complicated than, say an algebraic equation is that instead of merely substituting numbers directly for variables, there are functions involved as variables. This means that, for example, subsets of the set of numbers might be ranges, domains, coefficients, parameters, etc. for functions that might be known or unknown. It can, and in Dick's case does, get much more complex than that.
So, to make a single sentence that would describe how an arbitrary set of numbers must conform to equation (1.27), it would have to be very long and convoluted and I would think be unintelligible as a result.
But let me give it a try anyway:
In any old big set of numbers, if there are any patterns whatsoever among these numbers such that there is some kind of rule that allows one to predict with a certain probability that a particular pattern might occur, and considering any collection of such patterns and rules with the associated probabilities for their individual occurences to be represented by numeric values which indicate the density of these probabilities as distributed over the collection, then any function which can possibly describe that probability density distribution must make equation (1.27) true in the ordinary sense of functions making differential equations true.
That's the best I can do before breakfast.
***I'd be grateful.***
***5) "God had no choice.." ...too general, if you'll forgive me for saying so.***
I forgive you.
***Sounds like a little more rhetoric...and unnecessary to support the math thesis.***
Yes, it does sound that way. Please forgive me.
And, yes, it is unnecessary to support the math thesis.
The reason I threw it in, aside from not yet having used up the allotted word count, was to indicate one of the profound philosophical implications of Dick's discovery.
What I meant by that remark is that no matter how omnipotent God might be, if he/she creates a universe which has enough order in it that it may be described in human-type communication, then it must necessarily obey the same laws of physics that operate here in our universe. That ought to tell us something, shouldn't it?