It seems I have great difficulty clarifying the issue I have been trying to bring up since I first tried to present my thoughts some fifty years ago. Personally, it never even dawned on me that such an obvious constraint on the logical analysis of reality could be so totally ignored by the entire intellectual community. For over five thousand years, the existence of thousands of different languages created to express mankind's understanding of reality has been an issue totally ignored by everyone. Their utter refusal to deal with the issue that the total information available to build that understanding cannot be anything but finite is, to me, incomprehensible.

The issue I have been trying to discuss revolves around the actual problem confronting an entity which begins with "utterly no concept of reality whatsoever" (the obvious starting point for all intelligent analysts: i.e., what can be said about the facts available to that entity "without presuming any definitions of those facts at all". That has to be a valid representation of the universe upon which any entity must build their understanding.

As I point out in the opening chapter of my book, Sir Arthur Eddington had pointed out the vast number of unexamined assumptions embedded in any world view. He lays aside the issue as insoluble with the comment that one can not study signs and indications without identifying the signs and indications to be studied.

I avoid the issue of creating such definitions by referring to the relevant experiences by nothing except an arbitrary collection of numerical labels. (What that numerical label actually refers to is only available after the entity comes up with a specific explanation of those experiences and develops a language capable of representing his thoughts.)

There is only one fact of interest in what I am trying to present. That fact follows from my definition of "an explanation". Let me once again define "an explanation" to be "nothing more or less then a means of determining the truth of a collection of supposed facts". I propose that any such "fact" may be represented by a finite ordered sequence of some collection of numerical labels (each of which refers to specific undefined experience expressed in the created language).

In such a case any fact may be represented by the notation [math]P(x_1,x_2,\cdots,x_n)[/math], the probability that the indicated sequence of numerical indices, [math]x_1,x_2,\cdots,x_n)[/math], specifies a valid fact within the entities perception of reality). What is really interesting about that specific representation is the fact that the actual collection of numerical labels is absolutely and totally immaterial. That fact has subtle consequences not examined by any scientific inquiry of which I am aware.

Exactly the same result must be obtained from [math]P(y_1,y_2,\cdots,y_n)[/math] so long the experience referred to by the collection of indices, "[math]y_i[/math]" is exactly the same experience referred to by "[math]x_i[/math]". This leads to a rather astonishing relationship. For example, if one defines [math]y_i=x_i+c[/math] for all "i" then it becomes absolutely required that [math]P(x_1,x_2,\cdots,x_n) \equiv P(y_1,y_2,\cdots,y_n)[/math] no matter what explanation is being represented.

If you go one step further and examine two specific cases, one where [math]c=a[/math] and a second where [math]c=a+\Delta[/math], this relationship extends to the fact that one can assert that the expression

[math]\lim_{\Delta a\rightarrow 0}\frac{P(x_1+a+\Delta a ,x_2+a+\Delta a ,\cdots,x_n+a+\Delta a )-Px_1+a ,x_2+a ,\cdots,x_n+a )}{\Delta a}[/math]

must exactly vanish. (Note that the numerator is exactly zero and the denominator is never exactly zero.) Anyone who understands calculus will realize that this expression is, in fact, exactly the definition of the derivative of [math]P(y_1,y_2,\cdots,y_n)[/math] with respect to "a", that simple additive change in those numerical labels. One may thus infer that:

[math]\frac{d}{da}P(y_1,y_2.\cdots,y_n)=0.[/math]

That fact leads to another rather remarkable universal constraint on the expression [math]P(y_1,y_2,\cdots,y_n)[/math]. From the above expression the following expression

[math]\sum_{i=1}^{n}\left [\frac{\partial}{\partial y_i }P(y_1,y_2,\cdots,y_n)\frac{dy_i}{da} \right ][/math]

must also vanish. But, even more important is the fact that, in the two specific cases being discussed here, the expression [math]\frac{dy_i}{da}\equiv 1[/math]. That leads to an even simpler universal constraint on these references as defined:

[math]\sum_{i=1}^{n}\frac{\partial}{\partial y_i }P(y_1,y_2,\cdots,y_n) =0.[/math]

But I started with the fact that these numerical labels are absolutely immaterial. That certainly implies that exactly the same constraint must exist on the original "fact" representation [math]P(y_1,y_2,\cdots,y_n)[/math].

However, that conclusion may be carried quite a bit further. The only real constraint is the fact that the indices referred to by [math]x_i[/math] and [math]y_i [/math] must refer to the same specific concept. That issue lies entirely with the interpretation of the language represented by those indices. Since the actual indices used to represent the specific concepts are an entirely open issue it should be clear that, no matter what those indices actually refer to, there must always exist an interpretation of the entire collection which will make every [math]x_i[/math] and [math]y_i [/math] refer to the same concepts once the specific known facts are actually specified.

On the other hand, we are not talking about the actual experiences, we are instead talking about the explanation of those experiences, "a means of determining the truth of a collection of supposed facts"; i.e., development of the actual probabilities represented by [math]P(x_1,x_2,\cdots,x_n)[/math]. Once that result has been achieved, The actual numerical indices used to refer to underlying concepts is still a totally immaterial issue. The final result is that, so long as the concepts referred to by [math]x_i[/math] are the same concepts referred to by [math]y_i[/math] one can prove that

[math]\sum_{i=1}^{n}\frac{\partial}{\partial y_i }P(y_1,y_2,\cdots,y_n) =0.[/math]

There is another perspective which arises from the idea of the free nature of those numerical indexes used. One could just as well picture these indices as points in a graphical presentation. There are indeed some subtle difficulties with such a representation which I discuss in Chapter two of my book. The graphic representation leads directly to a great number of required relationship between "facts" which turn out to be, in many cases, exactly the relationships presumed by modern physics. Further extensions towards creation of physics concepts which are universally applicable to all possible representations of reality are discussed in Chapter three.

The remainder of my book is concerned with issues which must be valid under my representation.

Have fun ---Dick