Hi guys,
Sorry, but I have to relate a funny (to me anyway). My wife and I went to a Chinese buffet for lunch and got fortune cookies. Mine was, "The simplest and most necessary truths are the last to be believed". I found it rather appropriate, considering what I am trying to show here and the number of times people have baulked in the past.
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Originally Posted by CraigD
Doctordick, by “labeling meaning with numbers,” are you referring to a general technique of which Godel numbering is a specific example? |
In a word, no! I am merely stepping off in a strange direction not logically examined (to my knowledge) by anyone. To rephrase some of the steps in my abstract definition of an explanation, any explanation consists of two very different things. The first is, there has to be something which is to be explained. If my definition is to be abstract (applicable to anything which might be explained) I certainly cannot tell you what I am explaining. What I am saying is that I can attach the label "information" to whatever it happens to be: i.e., I think we can stretch the common meaning of "information" sufficiently to cover anything which might be "explained".
I think it is a fact that, when we go to explain something, we use a language to refer to the various portions of that information we are trying to explain. When we do that, we are essentially labeling those various portions so that we can refer to them in our explanation. The underlying problem in such a proposition is one of defining the labels themselves (how does one come to know the meanings of those labels). Clearly we are essentially working with a set of specific identifying labels and assigning meaning to the labels themselves is of fundamental importance.
Now the set of languages we can use to create these labels is quite diverse. In fact, if we allow the use of secrete codes (or Jargon), one could say the number of ways of specifying these labels approaches infinity. Since I don't know what language this explanation is going to be in (this is an abstract thing and cannot be a function of language), let me just refer to any specific label as "label i" where i is some number. Or hey, why not just use the number i itself to label these specific various portions (the elements) of whatever it is that is to be explained.
This next step is where everybody really goes ballistic. Instead of expressing the
B's with a list of these numbers, which label the elements of
B, instead, suppose we express the
B with a set of points on the real axis. After all, isn't the real axis little more than a way of expressing the entire set of real numbers, the abstract space with in which these numerical labels lie? And secondly, it is convenient to visual impressions. Notice that turtle uses this same linear lay out in his
discussion of the K function.
We do have one subtle difficulty with such a representation (as mere points on a line). Since
B was defined to be some collection of elements of
A (the real thing to be explained), we need to allow for the possibility that a specific element of
A could occur multiple times in
B. If we try to express
B with a set of points on the real axis, the fact of these multiple occurrences will disappear from sight: the real axis simply cannot express such a fact. If we want to express
B as a set of points, we need to use a two dimensional real space so that we can separate those multiple occurrences. I will refer to the original set of numbers (those numerical labels of the elements of
B) as a set of x values, having attached to each of them a second numerical label which will be used to provide a displacement orthogonal to x. Please make a mental note of the fact that the orthogonal displacement are a compete and utter fabrication of my mind introduced for the sole purpose of eliminating those multiple occurrences. This fact is a serious issue and will dealt with at a later date.
However, in the meantime every specific
B is conceptually represented by a set of points in a real (x,tau) plane. (I use the Greek tau to represent this orthogonal displacement for reasons which will become evident later.)
Since
B was introduced in order to represent changes in
C (which I eluded to as a collection of
B's)
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Originally Posted by Doctordick
Now certainly some elements of A need to be known (which we could call the basis of our explanation) and our model of the circumstance must include changes in the collection of known elements. These changes must consist of elements of A. What else can they consist of if we are talking about a closed universe? So I will simply refer to a change in the base information as B and the base information (the sum total of all changes from our opening position of zilch) as C.
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the set
C is very definitely a finite collection of sets
B. That fact means the sets
B can be ordered (note that the or. That being the case, I will attach a third number which I will designate as t to every
B which goes to make up
C. Now that I have that designation, I can see (or visualize) [b]C[b] as a collection of (x, tau) planes labeled by the set of numbers called t. That designations should be quite clear; it is certainly consistent with the idea that our knowledge about
A increases with "time".
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Originally Posted by Doctordick
Since the whole issue was begun with the prospect of assigning numbers to the elements of B, the question answer problem has now been reduced to finding a mathematical function where "validity" = Probability (B). If I knew exactly what that function was, I could use it to explain A. If you refuse to accept that notion, I think I could at least say I understood A as I would know my expectations exactly and I could not possibly be surprised in any new information.
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Now that I have told you how those numbers are to be assigned (and please note that the method is absolutely and completely general, thus allowing for any language conceivable: i.e., do it anyway you like), the problem of finding an explanation has been reduced to discovering the function P((x1,[tau]1), (x2,[tau]2), ... (xn,[tau]n), t) which yields the correct probability for any specific
B. A rather simple concept considering the territory it covers.
I started this all by suggesting it would be attaching numbers to meanings and then looking at internal relationships [color=-red]implied by mathematical processes[/color]. Now that we have attached those numbers, let's look at some implied relationships.
The first process of interest to anybody should be addition, about the simplest operation available. For the fun of it, let's look at what happens when we add some number "a" to all the numbers xi. P((x1,[tau]1), (x2,[tau]2), ... (xn,[tau]n), t) then becomes P((x1+a,[tau]1), (x2+a,[tau]2), ... (xn+a,[tau]n), t). The funny thing about that operation is that, if it applies to every number in every possible
B sub t, then both functions have to yield exactly the same result (remember the result is, by definition, the probability of that particular
B being correct). The operation of adding a to every number amounts to expressing
B in another language (where the translation is "add a number" to every numerical label). What is important here is the fact that the act does not shuffle the
B's being referred to and we are talking about the probability of the
B and not the probability of the outcome of a particular numerical labeling procedure.
We have the astounding outcome that, if that function P does indeed give us the correct probability of finding a particular
B it must also be true that
P((x1+a+b,[tau]1), (x2+a+b,[tau]2), ... (xn+a+b,[tau]n), t) – P((x1+a,[tau]1), (x2+a,[tau]2), ... (xn+a,[tau]n), t) =0
As the relationship must be valid for all a and b (even as the limit of either a or b goes to zero by the way), we know that we can divide by b and the expression still evaluates to zero. Look at that carefully; that's the definition of a derivative! We arrive at one and only one conclusion: the derivative of P((x1+a,[tau]1), (x2+a,[tau]2), ... (xn+a,[tau]n), t) with respect to a is zero.
Well, let's look at another simple variation of the above relationship. Let's us make a change of variables, setting zi=xi+a. If you know anything about partial differentiation, you should know that the derivative with respect to a is identical to the sum over all i of the partial of zi with respect to a times the partial of P with respect to zi. Since the partial of zi with respect to a is exactly one, we have the ultimate conclusion
[See the attached thumbnail]
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Originally Posted by Turtle
In my intuitive generalist view, the Universe IS the simplest expression of itself; we simply do not apprehend most of what is going on with our limited senses.  |
I wouldn't argue with that at all except that I might change apprehend to comprehend.
Have fun -- Dick
Knowledge is Power
and the most common abuse of that power is to use it to hide stupidity
