In early November, I attended the 2014 Sigma Xi Annual Meeting & International Research Conference in Glendale, Arizona. I brought the issue I have been talking about on this forum to a number of attendees who seemed to show a little interest. For the most part their responses were essentially the same as those I have found from others; however, the person to person contact has given me a somewhat different view on the difficulties others perceive in my presentation. Below is a simplified presentation which might be seen as more easily understood.

Our scientific explanations of our experiences are actually little more than a self consistent expressions of our experiences convenient to predicting our expectations. The real issue of interest in my treatise is that those experiences themselves constitute a finite set. That simple fact has serious consequences. Furthermore, it is an absolute fact that any explanation of anything may be represented by a finite number of concepts. Learning those underlying concepts is one of the fundamental issues of our childhood experiences. The underlying beliefs acquired in childhood are so extensive as to be essentially beyond common logical analysis. On the other hand, the finite nature of those experiences yields a fundamental attack which should be examined by all serious scientists.

The number of concepts expressible via any given language may be quite large but it must, none the less be finite; even if you include every language spoken on earth. Consider all the documents on earth providing information about the meaning of any specific conceptual issue. That total number is not, and cannot be, infinite. The meaning of any specific language element is essentially a concept defined by some finite collection of relationships expressed within in the sum total of all communications.

The fact that a new concept can always be added leads to an infinity of concepts is a spurious argument in that, the moment one ceases to add concepts and begins to search for explanations of the known information, the number of concepts available is finite.

Another issue of great importance here is the fact that any explanation of reality is constructed through a finite number of those deduced concepts: i.e., if [math]x_i[/math] constitutes a numerical reference label to a specific concept, then the notation [math](x_1.x_2,\cdots,x_n)[/math] can be used to represent absolutely any circumstance of interest..

The final fact of interest is that an explanation of any collection of circumstances must provide the truth of a given circumstance. In fact the collection of probabilities [math]P(x_1.x_2,\cdots,x_n)[/math] (where P represents the probability the indicated circumstance is valid) can be seen as capable of representing any specific explanation. It is a further fact that order in the specific circumstances standing behind an explanation can not be an issue of significance. That fact is central to internal consistency itself: i.e., internal consistency constitutes the fact that the order with which specific circumstances are considered cannot influence the truth of that circumstance.

**The single most significant issue embedded in the above presentation is the fact that the actual numerical labels used to reference those specific concepts is absolutely arbitrary. **

That fact leads to an issue of far reaching significance. If each and every numerical label used to refer to those concepts are incremented by a single specific constant, there can be no change in the probability of any specified circumstance.

In essence, if one has the both the finite collection of circumstances [math](x_1,x_2,\cdots,x_n)[/math] and [math]P(x_1,x_2,\cdots,x_m)[/math], the entire collection of truths deduced from that specific explanation, the result can not be altered by adding a specific constant to every numerical label [math]x_i[/math]. That fact leads to the absolute necessity of the following expression:

$$\lim _{\Delta a \rightarrow 0} \frac{P(x_1+a+\Delta a,x_2+a+\Delta a,\cdots, x_n+a+\Delta a)-P(x_1+a,x_2+a,\cdots, x_n+a)}{\Delta a}=0$$

since [math]P(x_1+a+\Delta a,x_2+a+\Delta a,\cdots, x_n+a+\Delta a)-P(x_1+a,x_2+a,\cdots, x_n+a)[/math] is zero no matter what the value of a or [math]\Delta a[/math] might be. That incontestable result further insures that, if one defines [math]z_i=x_i+a[/math] one can immediately write down

$$\frac{d\;}{da}\sum_{i=1}^n\frac{\partial \;}{\partial z_i}P(z_1,z_2,\cdots,z_n)\frac{d\;}{da}=0$$

Since [math]\frac{\partial z_i}{\partial a} \equiv 1[/math] for all i, it must follow that one may look at the limit where a=0 (which requires all [math]z_i=x_i[/math]) and realize that

$$\sum_{i=1}^n\frac{\partial \;}{\partial x_i}P(x_1,x_2,\cdots,x_n)=0$$

That result is very interesting as mathematically it represents what is commonly referred to as "shift symmetry". If [math]x_i[/math] represented a position in a geometric representation, some very important deductions may be made; however, the representation given here can not be trivially converted to such a geometric representation. The given numerical labels are not mathematical variables; they are actually nothing more or less than simple numerical labels. On the other hand, it should be clear to the reader that any collection of such labels can be seen as representable as a pattern of points in a geometric representation. In fact, the points of ink on a paper, a normal printed document, may be seen as capable of representing any circumstance of interest (consider all the books in all the libraries of the world).

The underlying problem is that transforming the above representation into a mathematical function drops some significant information. Chapter two of my book (available in PDF format on Anssi's blog site - http://foundationsof...cs.blogspot.com) details the transformation to a valid graphic representation which includes all possible explanations of anything.

I hope the reader finds this a clearer presentation than what I had written earlier.

Have fun -- Dick