Science Forums 2

Doctordick · 11-08-2007

Quote:

Originally Posted by Buffy

Go back and read your last three posts.
Think about revising what you appear to define the word "example" as.

I have done that and I think I understand what you are trying to say; however, I think much of what you are thinking is beside the point.

Quote:

Originally Posted by Buffy

You are indeed starting with a purely mathematical argument which is fine, but if you want to apply it to "what we know of reality" then mappings--even abstract meta-mappings--to "reality"--or meta-realities, which I know is *really* the point--have to be drawn and justified.

The problem is that such mappings or even abstract meta-mappings need to rest upon the behavior of solutions to that equation. Without those solutions, I am merely talking about the nature of representation itself. I am sure you have heard of the Ancient tablets with the cuneiform texts. There are apparently two versions referred to as linear A and linear B. One has been translated and, in spite of diligent effort, the other has not. We have a collection of symbols which are not defined and the information content just isn't sufficient to decipher them. Perhaps someone in the future will make a lucky guess and find some pattern within the known data which makes sense and then his (or her) start will lead to other rational interpretations. But meanwhile, it is merely a problem to be analyzed.

Normally, the very first step in analyzing such a problem is to label the various symbols under examination so that they can be easily referred to. The significant fact of interest to me is that the actual label used in such a circumstance is totally immaterial and that fact itself has consequences.

Quote:

Originally Posted by Buffy

The crux of what Q and I both--I think, and maybe even Anissi--are trying to figure out is whether or not your equation maps onto anything at all, and whether that mapping is valid, even if it is pure logic and math. You have yet to address Q's question of why partial differentials are required, and my ongoing unstated question remains "why must it be zero?"

I had thought I had answered both those questions a number of times. I will try to explain it in another way.

I have set things up so that “what is to be explained” is a set of numbers (those arbitrary numerical labels) which are displayed as a set of points in the $(x,\tau,t)$ plane. I have defined "an explanation" to be a method of obtaining expectations from given known information. Your expectation (which are actually a specific subset of of those numerical labels) can be represented as a probability (a number bounded by zero and one) with which you expect a given specific subset.

Thus it is that the solution (an explanation; that method of obtaining expectations which is in fact the epistemological construct to be discovered) can be seen as a mathematical function: the argument of the function consists of the set of specific labels and the output of the function is the probability that set will appear in a future $(x,\tau)$ plane (the future being defined to be “not part of the given known information”).

Since I want to omit not a single possibility for that “mathematical function”, I represent the “method” via a scalar product of an abstract vector function $\vec{\psi}$ which can represent any possible transformation from one set of numbers to another set of numbers (any set of information into a second set of information) and where the transformation into a number bounded by zero and one is an easy thing to define.

Now, the important thing in all this is that the given known information is actually the things being labeled, not the labels themselves. Since it can be represented as a mathematical function of those numerical labels, we know certain symmetries must exist in that representation. In particular we know shift symmetry must be a property of the representation. If we have discovered a solution, (a method of obtaining expectations from given known information) that method must yield the same result if we were to change all of our numerical labels by simply adding some number a to each and every numerical reference label (the three axes of our representation can be considered independent sets here). The point being that the output of the function is a function of the things being labeled and not actually of the labels themselves. By making the representation a function of the labels, we have introduced the shift symmetry.

To put it another way,

$\vec{\psi}(x_1+a,\tau_1,x_2+a,\tau_2,\cdots, x_n+a,\tau_n,t)=\vec{\psi}(x_1,\tau_1,x_2,\tau_2,\cdots, x_n,\tau_n,t)$

or

$\vec{\psi}(x_1+a,\tau_1,x_2+a,\tau_2,\cdots, x_n+a,\tau_n,t)-\vec{\psi}(x_1,\tau_1,x_2,\tau_2,\cdots, x_n,\tau_n,t)=0.$

Since a can be any number, it should be obvious that

$\frac{d}{da}\vec{\psi} = \lim_{\Delta a \rightarrow 0}\frac{ \vec{\psi}(x_1+a+\Delta a,\tau_1,x_2+a+\Delta a,\tau_2,\cdots, x_n+a+\Delta a,\tau_n,t)-\vec{\psi}(x_1+a,\tau_1,x_2+a,\tau_2,\cdots, x_n+a,\tau_n,t)}{\Delta a}=0.$

Notice that, in the above, the arguments of $\vec{\psi}$ are themselves shown as functions of the shift parameter. Any rudimentary knowledge of partial differentiation should include the fact that the above has to require that

$\frac{d}{da}\vec{\psi}=\sum_{i=1}^n \frac{\partial}{\partial x_i}\frac{\partial x_i}{\partial a}\vec{\psi}=0$

and, adding the fact that $\frac{\partial}{\partial a}x_i = 1$ , one is clearly led to the conclusion that

$\sum_{i=1}^n \frac{\partial}{\partial x_i}\vec{\psi}=0.$

You might be bothered by the fact that the $x_i$ are discreet numbers and the partials above treat them as continuous variables. This shouldn't be bothersome as no new information (which would add possible new values) can violate any of the steps given. It follows that, in the continuous limit, we should expect the result to be valid. Plus that, it provides about the simplest possible extension of information into unknown regions. Lastly, the above can be done in each of the orthogonal axes of our representation. As to the question, why are these differentials required? I don't see that they are except for the fact that they lead to a convenient way of expressing the constraints required by the existence of that shift symmetry.

Quote:

Originally Posted by Buffy

If this is all just an exercise in throwing around arbitrary formulas, then the answer "why not?" is perfectly valid, but if you're (eventually) go on to the next step of saying "this formula can be made a *valid assumption* in constructing internally consistent worldviews" then you need to start mapping the pieces of your formula on to the next step. If you don't its just a meaning-free arbitrary formula!

I would agree with that 100%. At the moment, it is absolutely nothing more than a meaning-free formula. I don't know that I would use the word “arbitrary” as it is somewhat strongly dependent upon the definitions I have already put forth.

Quote:

Originally Posted by Buffy

To apply this issue onto your thesis: your formula may or may not be consistent with any particular worldview, but it all depends on the $vec{psi}$ it exists within!

I won't comment on this because I do not understand what you had in mind when you wrote it. My thesis is that there always exist an interpretation of the facts known to me which satisfies my representation. In no way does that make it consistent with “any particular world-view”. It can be no more than an interpretation of what I know of a specific world-view and it rests upon a paradigm which insures that as more information becomes available to me, the new information will not lead to a violation of that interpretation.

Quote:

Originally Posted by Buffy

So please *do* think about the examples above and try to work with them: Maybe Q's example is too limited (although quite frankly your response to it was tremendously useful!), but anything you can do to provide analogies will assist us meatheads in "understanding what you're talking about."

The problem here is that any problem which can be solved at all requires such a large base of information that that the actual process required is far beyond anyones physical capability (at least on a conscious level). And I certainly don't regard anyone here as a “meathead”. I am very willing to work on trying to communicate. I think you may find my following response to Q enlightening. If not, I apologize.

Qfwfq, I may have misjudged you. You may be trying to resolve a problem which I have not yet mentioned. I have avoided the issue because, once it is broached, most everyone refuses to take me seriously. I will try and clarify exactly what I have in mind.

Quote:

Originally Posted by Qfwfq

Specify what the arbitrary labels are and how the mapping may be chosen. Show me what is consequential, I need to catch up with Anssi before his math skills improve enough to get him even further ahead.

The arbitrary labels are “arbitrary labels” and the mapping may be chosen anyway at all. Essentially, nothing is consequential; the consequences are entirely in your epistemological construct (your interpretation of what is going on). It makes utterly no difference what that epistemological construct is, if it is internally consistent, the temporal behavior of the ontological elements can be interpreted such that my equation is a valid constraint.

I suspect you may have noticed that there need be no underlying rules to reality at all: i.e., essentially embedded in my work is the question “what would reality appear to look like if it were absolutely and totally random?”. Most everyone is quite confident that “it certainly wouldn't appear as it does”. I do not know “what it would look like”; however, I do know that any explanation you could develop for it could be interpreted in a way which would guarantee the ontological elements would obey my equation. That leads to the conundrum, “how could I possibly achieve an explanation of a totally random reality. Actually, the answer is quite simple.

Essentially the circumstance implies something very significant: your explanation of reality must actually be a data compression algorithm which, in reality, (to use the word in a slightly differently way) says nothing about reality other than “what is” is, “what is”. Most everyone immediately drops into the “that can't possibly be true” mode when they realize this and want to hear not another word of it. In their opinion, I must be nuts to even consider such a thing: i.e., to make a metaphor, it doesn't even mention the god of science “causality”. I suspect there is not a single professional scientist who would even dream of considering the possibility as his professional standing would immediately vanish were he to do so. The position of the scientific community is that “an absolutely random universe could not have any rules”. The really significant issue here is that I am not talking about reality, I am talking about your interpretation of reality (your explanation of your experiences), quite a different thing. The fact is that it is your interpretation of reality which must have rules and the ”what is”, is “what is” is the only interpretation which exists without rules.

Quote:

Originally Posted by Qfwfq

Logic does not depend on reality. Mathematics is a type of constructing on logic and certain definitions, in essence choices, and therefore does not depend on reality. Logic and mathematics are not a belief about reality. Whether or not the definitions are chosen to describe some aspect of (a given) reality (in a given universe) is of no whatsoever concern to a mathematician. Reality is of no influence on the validity of mathematics; any valid mathematical construct is valid "in" any reality or universe.

It is valid even if that universe is totally and absolutely random. Nevertheless, such an assertion quite often brings forth nothing except total rejection, quite analogous to that “I burned my math book because it had no mention of God.” Modern science is indeed a religion as they think they understand the universe and believe there cannot be another perspective as successful as theirs.

Just an interesting comment. Many years ago, when the Internet first became available, I ran across a very interesting web site. I have searched for it many times since but have never been able to find it again. I was a mathematical presentation which proved that, for any given pattern of numbers, there existed a number of numbers within which that pattern could not be avoided. This was a long time ago and I didn't really examine his proof carefully so I cannot stand behind it but I think a simple example of what he was talking about could be “two numbers the same distance apart”. If you try to construct such a set of integers where the pattern absolutely does not appear, you will always run into difficulties.

Start with $x_1$ and $x_2$ . Then no number which differs from either of them by $(x_1-x_2)$ or $(x_2-x_1)$ can be used. (In fact, no pairs which differ by those amounts can be included in the set; there may be subtleties to this extension which have far flung import.) There are still a lot of numbers available so we can add $x_3$ to our collection. Now no number which differs by $(x_1-x_2)$ , $(x_2-x_1)$ , $(x_1-x_3)$ , $(x_3-x_1)$ , $(x_2-x_3)$ or $(x_3-x_2)$ can be used. As I said, I do not remember the proof but it seems to me the result was a consequence of the fact that the number of numbers which could not be used expanded much more rapidly than the number of numbers in the collection (every time one number is added, the number of additional numbers to be excluded exceed the total number of numbers already in the set). As I said, I wish I could find that paper so I could examine it carefully (I haven't been able to prove it myself). It is nevertheless quite clear that, with a sufficiently large set of random numbers, the probability that any specific repeated patterns will be found becomes closer and closer to one.

This implies that, even in a totally random universe, patterns will exist in any collection of numerical labels given to those elemental entities if the number of elemental entities is sufficiently large. Considering that ”what is”, is “what is” table (that set of points in my $(x,\tau,t)$ space) must reflect some $10^{20}$ references for $10^7$ years means that “no recognizable patterns” is a very improbable possibility: i.e.. even a totally random universe will most probably have a great number of recognizable patterns even if no rules are given as to how the mapping is to be chosen. I think this issue will become much clearer if we can get to the solutions of that equation.

And finally, as to expectations, considering the volume of information represented by the past compared to that which constitutes the present, one should certainly understand that the probability that the present will make any significant contribution to any pattern based description of the past, has to be negligibly small: i.e., the best statement of your expectation should be “the patterns I see in the future should look a lot like the patterns I used to describe the past”. This implies a good data compression mechanism is most probability the best representation of the universe possible and is probably the rational which stands behind our view of reality. We notice patterns and mark them with labels and begin to deal, not with the patterns themselves but, with patterns in those names. That is almost the definition of a data compression mechanism.

Now, I have proved that certain constraints must be imposed upon such a solution and that those constraints are embedded in my “fundamental equation”. The issue is, have I sufficiently defended that position that you will take the trouble to look at solutions to that equation without scoffing at the applicability of my constraints?

Have fun -- Dick

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