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On The Issue Of “Hypothetical” Versus “Real”!


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#1 Doctordick

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Posted 31 March 2013 - 02:37 AM

This is an excerpt from a private conversation between Anssi and myself. It brings up an issue Anssi seems to have overlooked and I thought others might also have overlooked the same issue.

I am afraid you are missing a very important issue. My presentation is supposed to be absolutely general: i.e., applicable to all conceivable possibilities. There exists a possibility which your comment explicitly omits.

I'm just saying that there's a risk that anyone who pretty much sees everything as hypothetical already, might read your text and decide you don't understand that issue since you are only referring to a sub-set of defined things as hypothetical.

You are clearly omitting the possibility that some of the things you are trying to explain are “real”.

In fact, there is a big difference between “real” and “hypothetical” which must be kept in mind. Since my analysis is to be applicable to all possible explanations, one must include the possibility of “real” elements in the circumstances to be explained.

Actually, when it comes down to the difference between “real” and “hypothetical” those “real” elements possess a property the “hypothetical” elements do not. That property is the simple fact that all possible explanations of a given set of circumstances must explain the “real” elements. On the other hand, specific “hypothetical” elements may or may not exist in some other explanation.

If you go back to my original preface, you will find the line,

This paper is a presentation of my analysis of the question, "can one make any constraints on the range of possible explanations without making any constraints whatsoever on the assumptions embedded in that explanation?”

It should be clear that I have made no assertion that the explanation lacks assumptions. That issue must be left totally open. Embedded in that thought is the idea that only what is “real” actually needs explanation. The problem of identifying what is or is not real is, in fact, very much a side issue. My initial definition of a circumstance makes the assumption that what is being explained is real: i.e., the initial definition of a circumstance consists of what is to be explained and if those elements must be in all explanations then they certainly must be “real”.

The problem of identifying what is and what is not real is simply laid aside as undecidable: i.e., without making any constraints whatsoever on the assumptions embedded in the explanation.

In fact, when it comes down to the possibility of a true valid explanation, only those “real” elements need be explained. An issue of deep and significant importance.

Have fun -- Dick

Edited by Doctordick, 07 April 2013 - 03:44 PM.


#2 AnssiH

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Posted 31 March 2013 - 10:57 AM

In fact, there is a big difference between “real” and “hypothetical” which must be kept in mind. Since my analysis is to be applicable to all possible explanations, one must include the possibility of “real” elements in the circumstances to be explained.

Actually, when it comes down to the difference between “real” and “hypothetical” those “real” elements possess a property the “hypothetical” elements do not. That property is the simple fact that all possible explanations of a given set of circumstances must explain the “real” elements. On the other hand, specific “hypothetical” elements may or may not exist in some other explanation.


Ah, yeah I remember this issue. I think it would be important to make that issue clear at the moment you bring up the concept of hypothetical elements (basically explain exactly what you explain in the latter quoted paragraph), as that is a very relevant bit of the definition of what you mean by hypothetical elements. Otherwise I think you run the risk having people mis-interpret your intention about hypothetical elements. Maybe I'm wrong though, maybe it's not a big issue, all I can go with is that I remember having some troubles with it. It was one of those things that I decided would make sense only once I get further along, and so it was.

-Anssi

#3 Doctordick

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Posted 29 November 2014 - 05:02 PM

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



#4 Rade

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Posted 29 November 2014 - 10:56 PM

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...

.

But, a scientific explanation of reality is not constructed by deduced concepts, it is constructed by defined concepts with such definitons taking two forms (1) theoritical definitions and (2) operational definitions. Therefore not only must each concept have a unique label, but associated with each label must be a unique theoritical and operational definition. Given this fact it is clear that your notation cannot 'represent absolutely' any circumstance for the reason that each numerical reference label can have an infinite hypothetical number of theoritical and operational definitions associated with it.

#5 AnssiH

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Posted 30 November 2014 - 06:06 AM

But, a scientific explanation of reality is not constructed by deduced concepts, it is constructed by defined concepts with such definitons taking two forms (1) theoritical definitions and (2) operational definitions. Therefore not only must each concept have a unique label, but associated with each label must be a unique theoritical and operational definition. Given this fact it is clear that your notation cannot 'represent absolutely' any circumstance for the reason that each numerical reference label can have an infinite hypothetical number of theoritical and operational definitions associated with it.


I think you are here also not seeing the forest from the trees. What he is saying is very simple; one cannot generate an explanation, that would demand a circumstance to be represented with an infinite number of elements.

Of course there cannot be, how could you represent a circumstance if it required an infinite number of elements to be actually laid down? It would be impossible by definition to ever lay down those elements.

Referring to a concept of infinity is completely different thing, than actually creating individual references to each element in an infinite set. For that same reason you just draw the symbol of infinite, rather than start writing down a number. Surely you see the difference?

#6 Rade

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Posted 01 December 2014 - 06:01 AM

What he is saying is very simple; one cannot generate an explanation, that would demand a circumstance to be represented with an infinite number of elements.


Consider the circumstance that is to be explaned is the idea called the 'number line' {1,2,3...infinity}. It is factually true that this circumstance must be represented with an infinite number of elements {1,2,3...infinity}. It is rather simple to then propose a set of statements to describe the set of facts associated with the idea of {1,2,3..infinity} that can be used to clarify the consequences of {1,2,3...infinity}, e.g., to explain why and how it is true.

Clearly the claim that no explanation can be given for a circumstance represented with an infinite number of elements is false. Why is this comment causing such grief ?

#7 Doctordick

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Posted 01 December 2014 - 03:31 PM

Consider the circumstance that is to be explaned is the idea called the 'number line' {1,2,3...infinity}. It is factually true that this circumstance must be represented with an infinite number of elements {1,2,3...infinity}. It is rather simple to then propose a set of statements to describe the set of facts associated with the idea of {1,2,3..infinity} that can be used to clarify the consequences of {1,2,3...infinity}, e.g., to explain why and how it is true.

Clearly the claim that no explanation can be given for a circumstance represented with an infinite number of elements is false. Why is this comment causing such grief ?

 

Your explanation of your ideas required a very small number of concepts

 

1: Consider

2: the

3: circumstance

4: that

5: is

  [math]\cdots[/math]

14: 'number line'

15: {

16: 1

17: 2

  [math]\cdots[/math]
 
It should be clear to you that the paragraph you presented did not require an infinite number of elements to represent it!
 
You assert that, "It is factually true that this circumstance must be represented with an infinite number of elements." If that were indeed the case, how in the world could you possibly represent it with such a small number of concepts? Even if you include all the books, movies and other means of communication, as necessary to clearly define the elements you actually used, the number would still be finite. 
 
It should also be somewhat clear to you that you cannot present any circumstance if the representation of that circumstance does require an infinite number of elements to be "actually laid down" in detail. 
 
You are clearly confusing the representation with what is being represented.  They are quite different issues.
 
Please put a little thought into your comments!
 
Have fun --Dick

Edited by Doctordick, 01 December 2014 - 03:36 PM.


#8 Rade

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Posted 01 December 2014 - 05:24 PM

It should be clear to you that the paragraph you presented did not require an infinite number of elements to represent it!

This is not correct. The paragraph I presented does indeed require an infinite number of elements to represent it...they are detailed in the representation I provided {1,2,3...infinity}. I do agree with you that more thought should have been put into understanding the difference between my representation and how it was represented.

#9 AnssiH

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Posted 02 December 2014 - 02:04 PM

This is not correct. The paragraph I presented does indeed require an infinite number of elements to represent it...they are detailed in the representation I provided {1,2,3...infinity}.


You still misunderstand what it means. The paragraph where you described a concept implying infinity, was in itself a representation of the concept you wanted to represent, and you accomplished that representation with a finite number of elements.

Representing a concept never requires infinite number of elements. Representing a concept of continuous line does not require an infinite number of elements; just just need to be able to define what you mean in some sense.

Actually laying out a continuous line would be impossible. Representation of it, is not.

Any explanation represents the concepts it defines, with a finite amount of elements. No explanation can actually lay out infinite number of elements to represent something. That's all that was said. Pretty simple, right?

#10 Rade

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Posted 03 December 2014 - 05:58 AM

1. Actually laying out a continuous line would be impossible. Representation of it, is not....

2. No explanation can actually lay out infinite number of elements to represent something. That's all that was said. Pretty simple, right?


1. Suppose a robot was tasked to walk from point a to point b and then back to point a, etc. continuously without stopping. Assume it has an endless power supply and a bag with an endless supply of numbers that it places (lays out) on the ground while it walks, one number every meter. The mechanical life of the robot is estimated to be 1000 years. Within the lifetime of any human it would be factually true to claim that the robot was actually laying out a continuous line. So, your claim #1 is false.

2. This is not correct. The concept of infinity can be explained by actually laying out an infinite number of elements to represent the concept, it just takes lots of time and conmittment to detail. Anyone reading this has the actual ability to begin a process of actually laying out an infinite number of elements to represent any concept they wish, the process will stop the day they pass away. As you read this a computer is in the process of acutally laying out the infinite numbers of pi, e.g., the computer is in the process of explaining pi. There is no philosophic requirement that all explanations must come to an end. Some explanations not only seem like they go on forever (the long winded diatribe), in theory, they can actually go on forever, if the person explaining has a greater lifetime than the person to whom something is being explained, or the example of the computer now in the process of explaining pi.

What this means is that you cannot define the concept of explanation by constraining it in such a way that it must, under all circumstances, actually contain a finite number of concepts to represent it. In other words, by rejecting such a definition of explanation one must reject the entire presentation of DD...pretty simple, right ?

Edited by Rade, 03 December 2014 - 06:38 AM.


#11 AnssiH

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Posted 03 December 2014 - 02:53 PM

Hehehe, I must admit I chuckled... :D Why do you even respond to anything when all you do is try so hard to avoid thinking about it...