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A Rather Simple Fact Seemingly Overlooked by Science 


     It is quite simple to prove that absolutely all internally consistent explanations of absolutely any conceivable collection of facts is perfectly represented by the following rather concise mathematical expression:


$$\left \{\sum _i\vec{\alpha_i}\cdot\vec{\nabla_i}+\sum_{i\neq j}\beta_{ij}\delta(\vec x_i-\vec x_j) \right \}\Psi = \frac{\partial \;}{\partial t}\Psi=im\Psi$$


     The issue standing behind that expression is the fact that our knowledge, from which we deduce our explanations, can not consist of an infinite body of facts. This is an issue modern scientists seldom have interest in considering (see Zeno's paradox). The analysis presented here can be seen as effectively uncovering some deep significant implications of the fact that human knowledge is built on a finite basis.


     The same issue can be applied to human communication. Note that the entire collection of 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 understandings of the relevant experiences) the number of concepts they use to think about the issue is finite (it may be 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 total collection of all libraries, museums and other intellectual properties together with an inventory log of that collection.)


     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 [math](x_1,x_2,\cdots,x_n)[/math]. 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 [math](x_1,x_2,\cdots,x_n)[/math] you can use that data to reconstruct the meaning of each and every [math]x_i[/math] as that is exactly the 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 by the notation [math](x_1,x_2,\cdots,x_n)[/math]).


     But back to my book. The above circumstance leads to one very basic an undeniable fact. If one has solved the problem (that is, created a consistent mental model of their beliefs) then they can express those beliefs in a very simple form: [math]P(x_1,x_2,\cdots,x_n)[/math] which can be defined to be the probability that they believe the specific circumstance represented by [math](x_1,x_2,\cdots,x_n)[/math] is true. In essence, if they had an opinion as to the truth of the represented circumstances, [math]P(x_1,x_2,\cdots,x_n)[/math] could be thought of as representing their explanation of the relevant circumstance [math](x_1,x_2,\cdots,x_n)[/math].


     It is at this point that a single, most significant, observation can be made.  Those labels, [math]x_i[/math], are absolutely arbitrary. If any specific number is added to each and every numerical label [math]x_i[/math] 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) so long as that explanation is internally consistent; 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 [math]P(x_1+a,x_2+a,\cdots,x_n+a)[/math] with respect to a. 


     If [math]P(x_1,x_2,\cdots,x_n)[/math] were indeed a mathematical expression the above derivative would lead directly to the constraint that $$\sum_{i=1}^n\frac{\partial\;}{\partial x_i}P(x_1,x_2,\cdots,x_n)\equiv 0.$$ (That result arises from the fact that [math]\frac{\partial x_i}{\partial a}=1[/math] in absolutely all cases.) However, it should be evident to anyone trained in mathematics that the expression defined above above still 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 which must be handled before even continuing this deduction. First, the numerical labels [math]x_i[/math] are not variables (they are fixed numerical labels) and second, the actual number "n" of concepts labeled by those [math]x_i[/math] 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 rest of my deduction is devoted to handling all the issues related to transforming the above derivative representation into a valid mathematical function. To begin with, any attempt to handle the two issues just brought up 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 has to be erroneous. 


     There is another very important aspect of the above representation. Though the number of experiences standing behind the proposed expression [math]P(x_1,x_2,\cdots,x_n)[/math] 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 [math](x_1,x_2,\cdots,x_n)[/math]. 


     Observing that the expression, [math]\sum_{i=1}^n\frac{\partial\;}{\partial x_i}P(x_1,x_2,\cdots,x_n)\equiv 0,[/math] is exactly what physicists call an expression of shift symmetry (essentially the the origin of a coordinate system used to represent a system is of no consequence) I first suggest changing [math]x_i[/math] from "a number" to a point in a Euclidean coordinate system. Essentially this changes the representation from a list of numerical labels numbers to a pattern of points in a geometry. This can be seen as creating an analogous representation of the relevant facts as collections of printed words or perhaps geometric objects. Note again that this move was only to make "shift symmetry" a reasonable conceptual aspect of the representation. It does not convert the expression [math]P(x_1,x_2,\cdots,x_n)[/math] into a bonafide mathematical function. 


     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 whatsoever on the possible 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. 

Edited by Doctordick
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  • 4 months later...

I am quite surprised by the fact that this thread (which was posted over a year ago) has been viewed over a hundred times without a single response. It seems to imply that none of those viewers found even a single issue which they found to be erroneous. I would certainly like to comprehend why that is the case.


Thanks Dick 

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  • 1 month later...

Perhaps the following is a little more comprehensible. Maybe someone out there might comprehend where I am coming from. I try to  put things as simply as possible but I seldom comment on what made me think the way I do.


In many respects, modern physics is a religion, not a science! I went into physics for the very simple reason that math and physics were the only fields where my teachers made a real effort to logically back up their explanations. Now mathematics is a logical construct based entirely on internal definitions thus, though mathematical arguments are used through out physics, math really has to do with logic and not reality. So physics it was which I wanted to understand.


Now physicists always made significant efforts to explain things to me but there were incidents where my questions were answered with "you will have to talk to someone with a more advanced education because I do not know the answer to that question! That is exactly what brought me to graduate school. Now graduate school was a great disappointment to me as I have never met a faculty member who had any interest in understanding reality; their interest seemed rather to be recognized for their authoritative knowledge. (A very strong symptom of religion.)


As a graduate student I had a bad habit when it came to reading journals. Whenever my class work required reading a referenced article, I would read the referenced journal volume cover to cover. When one does that, one gets a very different impression from what one gets if they read only the referenced articles. If one reads only the referenced articles one gets the impression that they are dealing with intelligent people. If one reads the entire journal one gets the impression that most of the publications are not worth the paper they are printed on. Common journal articles are seldom well thought out. 


As a result, I had little interest in publishing. I decided that I would make no effort to publish unless I had something serious to bring up; understanding the universe was my real interest.  Thus I have only three official publications to my name. My thesis, and two publications related to my first employment after receiving my Ph.D in January 1971. (I ended up earning my living outside physics, thinking about physics issues on my own time.)


I tried to publish on a number of occasions but was totally blocked by the "peer review process". In 1982, I tried to publish what I felt was a rather important insight but was rejected by every physics journal I sent it to. I don't think it ever saw a referee as the response was too quick to allow anyone to read the thing. Their responses were almost identical: they said what I had written was philosophy and not physics and was of no interest to physicists. 


I took it to my thesis advisor for help in getting it published but he refused to even look at it with the comment: "Stafford, no one will ever read your stuff because you haven't paid your dues!" He made it quite clear that not publishing was an unforgivable slight to the academy.


It is funny but, because of the physics response, a couple of years later I sent it to several philosophy journals and received the response: "this is mathematics and not philosophy and is of no interest to philosophers." I showed it to some mathematicians I knew and obtained the response, "there is no new math here, this is physics and would be of no interest to mathematicians." So I have apparently discovered something of no interest to any scientific field. I will present another version of my thoughts which might be comprehensible to someone here.


I contend that there exists a specific mathematical relationship that must be a valid constraint on any self consistent explanation of anything. (That opening equation on this thread.)

There exists only one universal constraint which applies to absolutely all theories. That would be the fact that any acceptable theory must be internally self consistent. The only way to examine the consequences of that constraint is to first establish an absolutely general notation capable of representing absolutely any collection of facts. 


That notation must be capable of representing the language necessary to express any possible explanation, without knowing what that language might be. That fact brings up two very important points that are explicitly under focus in my analysis. 


Before one can even discuss a theory, one must first learn the language necessary to express that theory. The intellectual process of learning that language is identical to the intellectual process of learning the facts to be explained. If the notation must be capable of representing any language, without knowing what that language might be, the underlying information required to comprehend the language must be included in the complete body of facts to be explained: i.e., the notation itself can make absolutely no presumptions whatsoever.  


A second significant fact under focus is that all explantations must be based on a finite volume of information. If the information required to learn such an explanation together with the required language were infinite, it could not be learned and thus could not be understood. Thus it is that the only underlying requirement is that the proposed notation can represent absolutely any finite collection of information. 


The entire collection of information can be divided into two categories. I will refer to the two different categories with the labels "elements" and "facts".  Elements will consist of the collection of all information which can each be represented by means of a single concept significant to the theory of interest. Note that this is exactly the roll played by words in the required language but that it is not necessarily  constrained to actual words. There are often concepts embedded in a theory which require more than one word to express; however they may nonetheless be represented by a definition which is the specific representation of an elemental idea embedded in the represented information. 


Facts will consist of any information representable by a structured collection of those elements. You should comprehend that the definitions of the elements constitute "facts". You should also comprehend that, if there exist facts which can not be expressed via a structured collection of elements, the language designed to express the explanation is insufficient to its purpose. In essence this is no more than an expression of the fact that, If a theory can not be expressed through use of the fundamental concepts of the language, it can not be explained and is thus beyond representation.  


Given the above one can examine the problem of representing a specific theory.  The first point is that one cannot represent a theory without knowing that theory. It follows that, if one knew the specific theory to be represented, the required elements (together with the specific facts constituting the definitions of those elements) could be listed and numerically labeled. Once that list has been created, any specific elemental concept required to explain that theory may then be represented by a specific number which I will represent by "x": i.e., "x" is no more than that elements numerical label in the created list. 


One can mentally picture that as essentially a complete list of all entries in every relevant dictionary in existence at the time the specific theory is understood. That would essentially be the entire set of possible concepts which could be used to clarify the theory of interest. Again, It must be understood that the definitions are relevant facts and not elemental information. 


The next step is to use those numerical labels to represent any factual information presumed by the theory. The represented facts can be seen as a collection of statements in the relevant language specified by means of the numerical labels "x". The actual statements of interest can be single sentences, whole contents of books or even perhaps an entire library or collections of libraries. The single most important point here is that the collection of known facts is finite. The only issue of significance is that each and every fact can be specifically represented by the notation (x1,x2,…,xn): i.e., an ordered list of elements. (Assertions are expressed with words.)


Of particular interest to this analysis is the fact that the truth of the relevant statement can not be a function of the order with which the facts are considered and must be represented by the statement itself. If a different result is obtained via an alternate order of consideration of facts, the system is, by definition, "internally inconsistent".


A second issue of interest arises if one opens up the set of possible representations. If one allows all possibilities which are representable by the notation (x1,x2,…,xn), constrained only by the specific definitions of the numerical label "x", two important categories are added to the representation. These additions can be seen as no more than new expressions in the relevant language which are not part of the factual information presumed by the theory. 


The complete collection of expressions now being represented can be divided into three categories: a finite set which represents the facts on which the theory is based, a second set which represent assertions which could  be valid under the represented theory and a third set which constitute a collection of assertions which are either meaningless garbage or totally inconsistent with the relevant theory. In essence the third category constitutes expressions which can not possibly be valid.


Probabilities can be assigned to each of those three categories consistent with the theory being represented. First, those which constitute the facts on which the theory is based must be true. Second, those which represent assertions which are meaningless or totally inconsistent with the relevant theory must be false. And third, those which represent assertions which could possibly be valid under the theory (issues predicted by the theory but not yet confirmed) can be seen as having a probability bounded by zero and one. 


The complete set of probabilities just brought up can be represented (within the proposed representation) by the expression P(x1,x2,…,xn), where "P" represents the probability that the represented assertion is true. It follows that the entire collection of probabilities constitutes a complete representation of the specific theory of interest. Under this definition people who have different mental concepts of a theory would be considering a different theory: i.e., under my definitions, the theory is specifically represented by this set of probabilities.  


At this point I contend that I have developed "an absolutely general notation capable of representing any conceivable theory"


Using this notation, I can prove that there exists a specific mathematical relationship that must be a valid constraint on any self consistent explanation of anything. I will present here only the first step of that proof as the entire proof is too long to specify here.


There is a very interesting fact embedded in the representation just defined. The assignment of the numerical labels "x" is an absolutely arbitrary process. The actual numerical labeling is of no consequence. Consider two lists of exactly the same required elements numerically  labeled via a totally different procedure. Let "x" represent a numerical label of a concept in the first list and let "y" represent the numerical label of exactly the same concept in the second list.  In that specific case, the probability represented by P(x1,x2,…,xn) must be exactly the same as the probability represented by P(y1,y2,…,yn). 


At this point a very interesting consequence appears. If one examines the rather simple alteration where yi=xi+a, it must be absolutely true that the derivative of P(y1,y2,…,yn) with respect to "a" must vanish exactly by the very definition of a derivative. This would be a very significant observation were P(x1,x2,…,xn) a mathematical expression. Note that although It may look like a mathematical expression, in fact it certainly is not.


The expression P(x1,x2,…,xn) arose from a defined mechanism capable of representing "all possible theories" and it's creation must be carefully understood. It does not qualify as a mathematical expression for several reasons. First, if it were a mathematical expression, the number "n" would be the same for every assertion under consideration and, second, if it were a mathematical expression, the numerical labels "xi" would be variables. In this representation they are fixed constants and not variables. 


The idea of shift symmetry (the phenomena represented by  yi=xi+a)  together with the arbitrary nature of the labeling procedure should lead one to think about possible alterations in the representation which might transform it into a valid mathematical representation. 


If such a notational shift could be achieved it would open a massive collection of deductive possibilities. Performing that notational shift is in fac, the second step of my proof. Achieving that transformation without introducing a constraint which would invalidate the issue of representing "any conceivable theory" requires a complex analysis too involved to be presented here. I will nonetheless give a very rough sketch of my attack.  


The first possible step is to convert the expression, (x1,x2,…,xn), into a collection of points plotted on an "X" axis. This representation certainly satisfies the required shift symmetry and, in a certain sense, converts the "xi" into variables.  At the same time such a conversion creates a number of other serious problems which need to be handled. 


It should be clear that important information contained in the original representation is lost in the proposed conversion. The original definition of the representation made no constraint that an elemental concept (think "a word") could be used only once in a given assertion. If an elemental concept were used more than once, multiple entries in (x1,x2,…,xn) would plot to exactly the same point and the existence of multiple occurrences in the original notation would be lost information.


Another factor in the original representation which would be lost is the order of the elements in the expression (x1,x2,…,xn). The order in the proposed plot would end up being the actual value of the numerical label "xi" and would not represent the indicated subscript.


It turns out that repetitions, order and specific number of elements can all be handled by a rather simple change in the definition of the relevant concepts. Note that these concepts can be thought of as "elements" of the language. It turns out that the addition of "hypothetical" concepts can be used to restore the lost information brought up in the previous paragraphs. 


An example would be converting this graphical plot to a two dimensional plot by adding an additional hypothetical position information for every element, essentially converting each "xi" to a vector in that two dimensional space. This allows elements which would otherwise plot to exactly the same x coordinate to be separated in that hypothetical dimension. 


Similarly, the order problem may be solved by adding a hypothetical ordering parameter. And finally "n" can also be made to be the same in all expressions through the addition of a sufficient number of hypothetical elements.


The actual process required to achieve the desired result is far from trivial and extreme care must be taken to assure that the final representation can indeed represent absolutely all possible theories without making any constraints beyond internal consistency.


I would very much like to find someone in the world who would be willing to take the time to follow what I have to say.  If I have made an error in my deduction, I would very much like to have that error pointed out to me.


I would love to find someone interested in thinking -- Richard Stafford, Ph.D. in theoretical physics. 

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