It has become quite obvious to me that practically no one seems to comprehend what I have put forth in my book. I recently attended a Sigma Xi conference in Glendale Arizona where I spoke to several people about my logical presentation. From their reactions, I think I have a somewhat better comprehension of the difficulty they perceive. The opening chapters of the book seems to emphasize the wrong issues.
The two first chapters seem to overcomplicate a rather simple issue. I suggest one might consider the following post to be a simpler replacement of those opening issues.
The underlying issue I was presenting was the fact that our knowledge, from which we deduce our beliefs, constitutes a finite body of facts. This is an issue modern scientists usually have little interest in thinking about. See Zeno's paradox. My analysis can be seen as effectively uncovering some deep implications of the fact that our knowledge is built on a finite basis.
The same issue can be applied to human communication. Note that all languages spoken by mankind to express their beliefs is also a construct based on a finite number of concepts. What is important here is that the number of languages spoken by mankind is greater than one. This fact also has implications far beyond the common perception.
Normal scientific analysis of any problem invariably ignores some issues of learning (and understanding) the language in which the problem is expressed. Large collections of concepts are presumed to be understood by intuition or implicit meanings. One should comprehend that they can not even begin to discuss the logic of such an analysis with a new born baby. In fact I suspect a newborn can not even have any meaningful thoughts before some concepts have been created to identify their experiences. Any concepts we use to understand any problem had to be mentally constructed. The fact that multiple languages exist implies that the creation of those concepts arise from early experiences and that the representation itself is, to some degree, an arbitrary construct.
The central issue of my deduction is the fact that once one has come up with a theoretical explanation of some phenomena (that is, created a mental model of their experiences) the number of concepts they use to think is finite (maybe quite large but must nonetheless be finite). It follows that, being a finite collection, a list of the relevant concepts can be created. (Think about the creation of libraries, museums and other intellectual properties together with an inventory log.)
Once one has that inventory log, numerical labels may be given each and every log entry. Using those numerical labels, absolutely every conceivable circumstance which can be discussed may be represented by the notation $(x_1,x_2,\cdots,x_n)$. Note that learning a language is exactly the process of establishing the meaning of such a collection from your experiences expressed with specific collections of such circumstances: i.e., if you have at your disposal all of the circumstances you have experienced expressed in the form $(x_1,x_2,\cdots,x_n)$ you can use that data to reconstruct the meaning of each $x_i$ as that is actually the central issue of learning itself.
I would like to point out that, just because people think they are speaking the same language does not mean their concepts are semantically identical. Each of them possess what they think is the meaning of each specified concept. What is important here is that "what they think those meanings are" was deduced from their experiences with communications; i.e., what they know is the sum total of their experiences (that finite body of facts referred to above).
But back to my book. The above circumstance leads to one very basic an undeniable fact. If one has solved the problem (created a mental model of their beliefs) then they can express those beliefs in a very simple form: $P(x_1,x_2,\cdots,x_n)$ which can be defined to be the probability that they believe the specific circumstance represented by $(x_1,x_2,\cdots,x_n)$ is true. In essence, if they had an opinion as to the truth of the represented circumstance, $P(x_1,x_2,\cdots,x_n)$ could be thought of as representing their explanation of the relevant circumstance $(x_1,x_2,\cdots,x_n)$.
It is at this point that a single, most significant, observation can be made. Those labels, $x_i$, are absolutely arbitrary. If any specific number is added to each and every numerical label $x_i$ in the entire defined log, nothing changes in the patterns of experiences from which the solution was deduced. In other words the following expression is absolutely valid for any possible solution representing any possible explanation (what is ordinarily referred to as one's belief in the nature of reality itself); i.e.,
$\lim_{\Delta a \rightarrow 0}\frac{P(x_1+a+\Delta a,x_2+a+\Delta a,x_n+a+\Delta a)-P(x_1+a,x_2+a,x_n+a)}{\Delta a}\equiv 0.$
What is important here is that, if this were a mathematical expression, it would be exactly the definition of the derivative of $P(x_1+a,x_2+a,\cdots,x_n+a)$ with respect to a.
If $P(x_1,x_2,\cdots,x_n)$ were a mathematical expression the above derivative would lead directly to the constraint that $$\sum_{i=1}^n\frac{\partial\;}{\partial x_i}P(x_1,x_2,\cdots,x_n)\equiv 0.$$ However, it should be evident to anyone trained in mathematics that the expression defined above above does not satisfy the definition of a mathematical expression for a number of reasons.
The reader should comprehend that there are two very significant issues before even continuing the deduction. First, the numerical labels $x_i$ are not variables (they are fixed numerical labels) and second, the actual number of concepts labeled by those $x_i$ required to represent a specific circumstance of interest is not fixed in any way. (Consider representing a description of some circumstance in some language; the number of words required to express that circumstance can not be a fixed number for all possible circumstances.)
The remainder of chapter 2 is devoted to handling all the issues related to transforming the above derivative representation into a valid mathematical function. Any attempt to handle the two issues above will bring up additional issues which must be handled very carefully. The single most important point in that extension of the analysis is making sure that no possible explanation is omitted in the final representation: i.e., if there exist explanations which can not be represented by the transformed representation the representation is erroneous.
There is another important aspect of such a representation. Though the number of experiences standing behind the proposed expression $P(x_1,x_2,\cdots,x_n)$ is finite, the number of possibilities to be represented by the explanation must be infinite (the probability of truth must be representable for all conceivable circumstances $(x_1,x_2,\cdots,x_n)$.
I take care of the first issue by changing the representation from a list of numerical labels to a representation by patterns of points in a geometry. This would be something quite analogous to representation via printed words or geometric objects representing the relevant represented concepts. I handle the second issue is by introducing the concept of hypothetical objects, a very common idea in any scientific explanation of most anything.
At this point another very serious issue arises. If the geometric representation is to represent all possible collections of concepts, that geometry must be Euclidean. This is required by the fact that all "non Euclidean" geometries introduce constraints defining relationships between the represented variables. Only Euclidean geometry makes absolutely no constraints on the relationships between the represented variables. This is an issue many theorists omit from their consideration.
I look forward to any issues which the reader considers to be errors in this presentation.
