Part I – Representing the Situation:
There exist four concepts in the English language which need to be carefully examined in order to comprehend how they influence the fundamental aspects of all scientific studies. Those four words are “information”, “understanding”, “explanation” and “communication”. The central issue of any scientific endeavor is to understand the problem confronting one. One can refer to what is to be understood as “information” and the purpose of understanding is to provide an “explanation”. Finally, without communication, understanding is to a great extent rather worthless. In my definition of “communication” I include what one could call self communication which essentially amounts to consciously thinking about the issue to be explained.
Since my purpose here is to create a mathematical representation of all four concepts, the first step is to come up with a mathematical representation of “information” which makes no constraints whatsoever on any aspects of communicating that information. I will use an approach somewhat related to the concepts behind the the idea of TCP/IP internet packets. Internet packets essentially amount to collections of numerical labels for particular elements essential to the intended communication, the actual meaning of those labels is established by the design of the TCP/IP packets. I will use a collection of such numerical labels to denote what I call specific circumstances. The idea here is that any communications can be represented by a collection of such “circumstances”: i.e., any communication can be seen as a collection of meaningful elements. If every meaningful element is labelled with a specific numerical label, then the communication can be see as a collection of such numerical labels.
I intentionally omit any information as to what each of those labels actually mean as coming up with a meaning is very much a central aspect of the problem to be represented here. Learning the meaning of the communication symbols is the first step of solving any problem and is a process involving a great many assumptions to be avoided. That is, if we assign meaning to those labels, we are presuming what they represent is known. My intention is to include all possible interpretations of those given numerical labels and a presumption that we know their meanings amounts to a total failure to include absolutely all possibilities. .
That brings up a second extremely important issue. It is a very common presumption that there exists but one interpretation of of what “understanding the represented information” means. Understanding the represented information is ordinarily interpreted to mean that the examiner knows exactly what was intended by the creator of that message (whoever it was that assigned those labels). The fact that a serious unwarranted assumption has been made there should be recognized by the reader.
There clearly exists no way of proving that one's “understanding” of a specific message is actually what the sender intended (if there were, secret codes could not exist). The only process available to clarify misunderstandings involves further communication. That is exactly the process used by teachers to determine if the communication of a subject is understood by their students. It is by the means of questions and answers that the character of “an understanding” is determined. Again we are confronted with the problem of representing all possible questions via an acceptable mathematical notation. Once again, Alan Turing and computer technology comes to our rescue.
Today it has been shown quite clearly that any question on any subject may be represented by a finite collection of true/false questions expressed by means of exactly the same language used to present the original "information" which we are interested in understanding. It follows that the desired questions can be represented by exactly the same kind of circumstances used to represent the information to be explained: i.e., collections of circumstances represented by specified sets of numerical labels.
Essentially, what I have just pointed out is the fact that one's “understanding” of a given collection of circumstances can be represented by assigning true/false values to a collection of numerical labels written as [math](x_1, x_2,\cdots, x_n)[/math]. This brings up another subtle issue. As any teacher well knows, the answers can be a function of the context the student has in mind; essentially, the same question can be asked multiple times and the answer may vary. Since I wish to be very careful to include all possibilities I will include allowing fractional answers between zero (for false) and one (for true) which are to indicate the probability the understanding will yield a true answer for the circumstance represented by a specific set of numerical labels.
Now the first issue to be clarified regarding my analysis is the fact that everyone makes the erroneous assumption that there exists but one “understanding” of any given set of information (often referred to as the “correct” understanding). It should be clear to the reader that it is impossible to prove a specific understanding is correct. So long as the volume of information on which the understanding is based is less than all possible information, the possibility exists that there exists a piece of information not yet examined which will invalidate that understanding.
From the above, I will assert that there exists a function [math]P(x_1, x_2,\cdots, x_n)[/math] which expresses exactly what the probability which a specific understanding gives to the truth of the circumstance represented by [math](x_1, x_2,\cdots, x_n)[/math]: i.e., the probability it could be a member of the body of information to be explained. It should be recognized here that (at this point in the discussion) everything is finite. The number of elements in a “circumstance” is finite, the number of known circumstances making up the information of interest is finite and the number of known cases for values of “P” is finite: i.e., the best we can possibly do is to create a table expressing values of “P” for specific circumstances [math](x_1, x_2,\cdots, x_n)[/math].
The issue now is the idea that every possible understanding of any possible collection of information can be represented by a particular table of “P”. There are two issues of significance here. Is there indeed a one to one mapping between all possible understandings and all possible such tables of “P”? Clearly if two thinking entities produce exactly the same table of answers for “P”, there exists no evidence that their understanding of the subject differs in any way. On the other hand, one different answer is sufficient to imply the understanding of the two entities is different in some way. This suggests that, the moment a different value of [math]P(x_1, x_2,\cdots, x_n)[/math] is found, that table of "P" can be taken to indicate a possibly different understanding.
Thus all possible understandings of any possible information can be represented by the given functional notation,
P(x_1, x_2,\cdots, x_n)
which, at this point, is represented as a finite table of entries.
Does anyone find any part of that unintelligible?
Thank you -- Dick