Labeling meanings with numbers!

[Doctordick]

If you don't like to read long involved posts with subtle logic, don't bother reading this: you'll probably be severely disappointed. :hihi:

 

The Turtle has posted an interesting thread called "Katabatak Math-An Exploration In Pure Number Theory". In it he brings up the idea of attaching meaning to the numbers (colors, musical notes, ...) yielding some very interesting patterns. But he does not mention the possibility of considering the converse. That would be attaching numbers to meanings and then looking at internal relationships implied by mathematical processes. I have personally looked at that issue and found the results quite enlightening.

 

The first thing we need is a collection of meaningful symbols to which we can attach these numbers. A good name for a collection of meaningful symbols is the word "information". So, what I am proposing is attaching numbers to information. Now this really isn't a very alien concept at all; it's done with computers on a daily basis. :D What I would really like to do is find something of interest I could do with this information by applying mathematical ideas to these numerical "labels".

 

Before going any farther, I would like to separate the problem into two different issues. Suppose I could answer any questions about the information so labeled. It seems to me that I could define that as knowing the information: i.e., if I "knew" it all, I could recover any of it with little more than knowing the pattern to be located. Now this is a very uninteresting thing to do. In fact, that's what computers do all the time. On the other hand, "understanding" the information is quite a different issue. Now that would be an interesting computer operation and one I don't think has been seriously considered.

 

So how does one know when they understand something? Well, you could say they get a little light that goes on in their head and they then "know" they understand it, but I think you would be kind of gullible if that were all you wanted to ask of them. I personally would require an explanation before I would accept anyone's claim to understand. That being the case, let me give you my definition of "an explanation".

If the information is understood, then questions about the information can be answered given only limited or incomplete knowledge of the underlying information: i.e., limited subsets of the information. What I am saying is that understanding implies it is possible to predict expectations for information not known. The explanation itself constitutes a method which provides one with those rational expectations for unknown information consistent with what is known.

 

Thus I come to define "An explanation", from the abstract perspective, to be a method of obtaining expectations from given known information. If you have any arguments with that definition, it seems to me that you need to show me either a method of obtaining those expectations which can not be conceived to be an explanation or an explanation which provides no method of obtaining expectations. If you cannot show one of those circumstance, than you should agree that it is a usable definition of an explanation consistent with the common meaning of the term.

The above is taken directly from an earlier post on another thread.

 

Still looking for serious complaints.

 

Have fun -- Dick

 

Knowledge is Power

and the most common abuse of that power is to use it to hide stupidity

[Turtle]

___This sounds intriguing. So do I understand that an "explanation" is expressed as an algorithm/formula/equation which operates on the information?

___If that is right, now how does one load the information? By the same algorithms as explain it, or by others? How do you establish first principles?

___This looks like some kinda fun. :hihi:

[Turtle]

___On further reflection, I realized that the numerology I took the K function from is a meaning/number system. I found it arbitrary & just went for the number patterns. In the Katabatak thread is a key:

http://hypography.com/forums/attachment.php?attachmentid=70

___I actually edited it down from a more expansive key:

http://hypography.com/gallery/showimage.php?i=443&c=3

___Here in the fuller key, I have left meanings intact. These aren't numerology's meanings, rather my own that I attempted to arrive at in a similar manner as you suggest DoctorDick.

___I did have this expanded key up for awhile, but deleted it during a bout of bad attitude. You have improved my attitude, so now it is back. Looking forward to more. :hihi:

[Eclogite]

DD, I think this could rapidly metamorphose into a purely semantic discussion, but.....

You are using 'explanation' in a significantly different way from the one I would favour. I am certainly not saying that your way is wrong - after all you are the one defining it - but it jars with my interpretation of the word and its usage.

To me an explanation is an accounting of a process; a description of how and why things are as they are. In short, it deals, as directly and clearly as possible with what is. Your definition seems to address what will or might be. Now these two are related, but they are also different.

That difference, and the jarring effect of trying to fit your 'explanation' into a hole in my linguistic symbology, that does not want to accept it, are making it difficult for me to focus on where you want to take this idea.

[Doctordick]

To me an explanation is an accounting of a process; a description of how and why things are as they are. In short, it deals, as directly and clearly as possible with what is.

Yes indeedy do! That is certainly the first requirement of an explanation. If the "explanation" is not one hundred percent consistent with "what is" (i.e., the information being explained) then it's a bogus explanation from the get go. But there is more to "an explanation" than that. If your explanation has nothing to say about what is to be expected it is a rather worthless construct. As the older generation used to tell me, "what is, is what is, and don't worry about it!" That "deals, as directly and clearly as possible with what is" but I doubt many people would regard it as an explanation. :eek2:

Your definition seems to address what will or might be. Now these two are related, but they are also different. That difference, and the jarring effect of trying to fit your 'explanation' into a hole in my linguistic symbology, that does not want to accept it, are making it difficult for me to focus on where you want to take this idea.

I think that what you are missing is the necessity of that prediction of expectations. Even in a case where there is nothing to predict, (i.e., the story which constitutes the explanation is completely finished at its conclusion) the construct itself unfolds through the issue of prediction. Point by point, the explanation sets up a process for establishing expectations for the next step until the finished state is established.

 

Secondly, I would say that you are jumping the gun by worrying about where I want to take this idea. I think the "jarring effect of trying to fit [my] 'explanation' into a hole in [your] linguistic symbolism is a consequence of the common perception that explanations are "correct": i.e., a failure to take into account the fact that all explanations of anything must be taken as tentative and possibly erroneous. I have found in my life that everybody is willing to admit that they could be wrong; but never about the case under discussion. :eek_big:

 

And, where do I want to take it? Well, I think the idea of an explanation is a rather basic concept (a first principal thing if you would be so kind as to consider it such). What can we talk about without the concept of an explanation? If we are going to talk, we are going to use words and how do I know what you are saying if I don't know what the words mean? That is the first explanation required. It is entirely possible you are speaking in a secrete code and I am missing the entire import. Probably not; but certainly possible and any errors in communication essentially arise from exactly the same fundamental phenomena (your meanings for words and my meanings may not be exactly the same).

 

That's why I like mathematics. Mathematicians have labored for centuries trying to make sure their systems are internally self consistent. As a consequence, it really makes no difference if I have misunderstood their intended meaning. It is the internal consistency of the defined procedures which establish the system (secrete code or not). That is, my construct (since it obeys the same internally self consistent collection of rules) is an analog of his and we know we will arrive at the same conclusions. (Or one could face the real issue and admit that, logically speaking, mathematics is pretty dammed simple to understand.) In short, we can communicate.

 

But back to the issue of assigning numbers to meanings. Turtle, you are assigning numbers to meanings and, as you mention, the assignment is arbitrary. In order to make an assignment such as you propose, the meanings being referred to have to be understood (and meaningful, so to speak). In essence, you have already made assignments of the connections and relationships between these many "defined entities" you are assigning numbers to. But, understanding a language is a complex thing. It is very hard to keep in mind all the various usages of these entities and thus quite difficult to be absolutely sure that the inferred connections and relationships are indeed internally consistent.

 

On the other hand, mathematical relationships are pretty well guaranteed to be to be internally consistent so I can be rather caviler in any manipulations based on those mathematical operations. What draws me to the issue is the fascinating things which happen when you don't set it up any internal connections or relationships (outside those mathematical operations already well defined): i.e., I presume there is no information outside the language usage itself (which is actually quite a reasonable proposition). In many respects, I am looking at the problem of finding the possible internally consistent connections and relationships which can be inferred from a finite set symbols, signs, concepts ... (what ever one wishes to call these meaningful things). In many respects this is the very essence of the problem of learning a language.

 

As an aside, notice that I said, "there is no information outside the language usage itself" (the "language" being this collection of "things" which can be referred to). If there is no information outside the collection, we can refer to the collection as "a universe unto itself" (a mathematician might refer to it as a "closed" universe). On the surface it sounds like I am asking the impossible: find all the possible internally self consistent connections and relationships which can be inferred from an undefined collection of supposedly meaningful references. But it certainly isn't impossible, millions of babies come into the world (a closed universe of experiences) every year and within a few short years, they manage to infer, for themselves, whole collections of internally self consistent connections and relationships between symbols, signs and concepts which were unknown to them only a matter of months before.

 

I have already said that I am interested in "explanations" because I think that is the central issue of understanding (my first principle so to speak). As I have mentioned in my first post above,

The first thing we need is a collection of meaningful symbols to which we can attach these numbers. A good name for a collection of meaningful symbols is the word "information". So, what I am proposing is attaching numbers to information.

Now turtle, you make a habit of listing out numbers and their assignments; but the sheer volume of information being talked about here precludes the step of writing down symbols for the basic elements of this information. It is much easier just to plant the concepts of doing so in your head. For this reason, I will simply refer to the information to be explained as A, the set of supposedly meaningful "things": i.e., the meaningful things themselves are "the elements of A. So our purpose is to explain A (a pretty simple statement).

 

However, we have a slight problem there. As I explained to Eclogite, one of the essential properties of an explanation is that it must make predictions: i.e., it must predict things about A before they are known. This implies that, for the purpose of analysis, A (our universe) cannot be thought of as "known". Now certainly some elements of A need to be known (which we could call the basis of our explanation) and our model of the circumstance must include changes in the collection of known elements. These changes must consist of elements of A. What else can they consist of if we are talking about a closed universe? So I will simply refer to a change in the base information as B and the base information (the sum total of all changes from our opening position of zilch) as C.

 

At this point I have some meaningful entities to which I can attach numbers: the elements of C or "the elements of the collection of B's which go to make up C".

 

The second element of "an explanation" must be that prediction. As laid out above, all of the information on which the explanation is based is contained in the set C. Again, as I explained to Eclogite, even if the explanation concerns itself only with C making no predictions concerning unknown portions of A the construct itself still unfolds through the issue of predictive steps. In essence, what is desired is a prediction of the next set B which is to be added to the known collection C. Think of it as an answer to the question, "Ok, what's next?" Obviously the answer needs to be a collection of elements of A (or a new set B) as there is nothing else available (that's the definition of a "closed universe").

 

Designing an abstract model of a question is probably the hard part of the whole thing. Sure, if the set A constitutes everything, both the question and the answer are comprised of elements of A. However, the problem is that both questions and answers can be quite complex and may involve strings of sets B. We can get around the answer complexity by requiring a "twenty question" format (think of it as a multi-billion question game). In that case the answers become a simple binary yes/no but the questions can still be complex.

 

I hold that the most complex questions which can be asked can be seen as collections of changes in information (think of it as a sequence of concepts represented by sets B). In that case, the question can be reduced to a yes/no answer on each hypothetical set B. Another way to view the same issue is to see the answer as a collection of possible sets B with the question being, "can this collection of possible sets actually be expected as real changes in information?" Can they actually be possible valid sets?

 

That reduces the question answer problem down to, "given a particular arbitrary set B, can that set be accepted as a real possibility?" The answer is clearly either yes or no or something in between (a probability: a number bounded by zero and one). Since the whole issue was begun with the prospect of assigning numbers to the elements of B, the question answer problem has now been reduced to finding a mathematical function where "validity" = Probability (B). If I knew exactly what that function was, I could use it to explain A. If you refuse to accept that notion, I think I could at least say I understood A as I would know my expectations exactly and I could not possibly be surprised in any new information.

 

Believe it or not this is no more than setting the stage; but, before I go on I would like a little feed back on the clarity of what I have said. I admit it is a rather abstract opening but one shouldn't be afraid of abstraction. Abstraction is a powerful tool as it frees one to rationally examine things they do not understand. And I certainly don't understand exactly what this set A is! All I really know is that I have to come up with an explanation of it and, to do that, I need a road map to organize my thoughts. :eek:

 

Have fun -- Dick

 

Knowledge is Power

and the most common abuse of that power is to use it to hide stupidity

[CraigD]

Doctordick, by “labeling meaning with numbers,” are you referring to a general technique of which Godel numbering is a specific example?

[Turtle]

___I suspected Kurt's Hammer might apply. From the article Craig linked to :

 

"The Gödel numbering is not unique. The general idea is to map formulas onto natural numbers. An alternative Gödel numbering could be to consider each of the symbols of Step 1 to be mapped (through, say, a mapping h) to a digit of a base-22 numeral system, so a formula consisting of a string of n symbols s1s2s3...sn would be mapped to the number."

 

___Sounds a lot like Katabataks to me. :eek2: I have to also say, that the principle of synergy (or emergence) seems absent, if not contrary to the very idea of prediction DoctorDick is seeking. In my intuitive generalist view, the Universe IS the simplest expression of itself; we simply do not apprehend most of what is going on with our limited senses. :eek_big:

[Doctordick]

Hi guys,

 

Sorry, but I have to relate a funny (to me anyway). My wife and I went to a Chinese buffet for lunch and got fortune cookies. Mine was, "The simplest and most necessary truths are the last to be believed". I found it rather appropriate, considering what I am trying to show here and the number of times people have baulked in the past.

Doctordick, by “labeling meaning with numbers,” are you referring to a general technique of which Godel numbering is a specific example?

In a word, no! I am merely stepping off in a strange direction not logically examined (to my knowledge) by anyone. To rephrase some of the steps in my abstract definition of an explanation, any explanation consists of two very different things. The first is, there has to be something which is to be explained. If my definition is to be abstract (applicable to anything which might be explained) I certainly cannot tell you what I am explaining. What I am saying is that I can attach the label "information" to whatever it happens to be: i.e., I think we can stretch the common meaning of "information" sufficiently to cover anything which might be "explained".

 

I think it is a fact that, when we go to explain something, we use a language to refer to the various portions of that information we are trying to explain. When we do that, we are essentially labeling those various portions so that we can refer to them in our explanation. The underlying problem in such a proposition is one of defining the labels themselves (how does one come to know the meanings of those labels). Clearly we are essentially working with a set of specific identifying labels and assigning meaning to the labels themselves is of fundamental importance.

 

Now the set of languages we can use to create these labels is quite diverse. In fact, if we allow the use of secrete codes (or Jargon), one could say the number of ways of specifying these labels approaches infinity. Since I don't know what language this explanation is going to be in (this is an abstract thing and cannot be a function of language), let me just refer to any specific label as "label i" where i is some number. Or hey, why not just use the number i itself to label these specific various portions (the elements) of whatever it is that is to be explained.

 

This next step is where everybody really goes ballistic. Instead of expressing the B's with a list of these numbers, which label the elements of B, instead, suppose we express the B with a set of points on the real axis. After all, isn't the real axis little more than a way of expressing the entire set of real numbers, the abstract space with in which these numerical labels lie? And secondly, it is convenient to visual impressions. Notice that turtle uses this same linear lay out in his discussion of the K function.

 

We do have one subtle difficulty with such a representation (as mere points on a line). Since B was defined to be some collection of elements of A (the real thing to be explained), we need to allow for the possibility that a specific element of A could occur multiple times in B. If we try to express B with a set of points on the real axis, the fact of these multiple occurrences will disappear from sight: the real axis simply cannot express such a fact. If we want to express B as a set of points, we need to use a two dimensional real space so that we can separate those multiple occurrences. I will refer to the original set of numbers (those numerical labels of the elements of B) as a set of x values, having attached to each of them a second numerical label which will be used to provide a displacement orthogonal to x. Please make a mental note of the fact that the orthogonal displacement are a compete and utter fabrication of my mind introduced for the sole purpose of eliminating those multiple occurrences. This fact is a serious issue and will dealt with at a later date.

 

However, in the meantime every specific B is conceptually represented by a set of points in a real (x,tau) plane. (I use the Greek tau to represent this orthogonal displacement for reasons which will become evident later.)

 

Since B was introduced in order to represent changes in C (which I eluded to as a collection of B's)

Now certainly some elements of A need to be known (which we could call the basis of our explanation) and our model of the circumstance must include changes in the collection of known elements. These changes must consist of elements of A. What else can they consist of if we are talking about a closed universe? So I will simply refer to a change in the base information as B and the base information (the sum total of all changes from our opening position of zilch) as C.

the set C is very definitely a finite collection of sets B. That fact means the sets B can be ordered (note that the or. That being the case, I will attach a third number which I will designate as t to every B which goes to make up C. Now that I have that designation, I can see (or visualize) C as a collection of (x, tau) planes labeled by the set of numbers called t. That designations should be quite clear; it is certainly consistent with the idea that our knowledge about A increases with "time". :eek_big:

Since the whole issue was begun with the prospect of assigning numbers to the elements of B, the question answer problem has now been reduced to finding a mathematical function where "validity" = Probability (B). If I knew exactly what that function was, I could use it to explain A. If you refuse to accept that notion, I think I could at least say I understood A as I would know my expectations exactly and I could not possibly be surprised in any new information.

Now that I have told you how those numbers are to be assigned (and please note that the method is absolutely and completely general, thus allowing for any language conceivable: i.e., do it anyway you like), the problem of finding an explanation has been reduced to discovering the function P((x1,[tau]1), (x2,[tau]2), ... (xn,[tau]n), t) which yields the correct probability for any specific B. A rather simple concept considering the territory it covers.

 

I started this all by suggesting it would be attaching numbers to meanings and then looking at internal relationships implied by mathematical processes. Now that we have attached those numbers, let's look at some implied relationships.

 

The first process of interest to anybody should be addition, about the simplest operation available. For the fun of it, let's look at what happens when we add some number "a" to all the numbers xi. P((x1,[tau]1), (x2,[tau]2), ... (xn,[tau]n), t) then becomes P((x1+a,[tau]1), (x2+a,[tau]2), ... (xn+a,[tau]n), t). The funny thing about that operation is that, if it applies to every number in every possible B sub t, then both functions have to yield exactly the same result (remember the result is, by definition, the probability of that particular B being correct). The operation of adding a to every number amounts to expressing B in another language (where the translation is "add a number" to every numerical label). What is important here is the fact that the act does not shuffle the B's being referred to and we are talking about the probability of the B and not the probability of the outcome of a particular numerical labeling procedure.

 

We have the astounding outcome that, if that function P does indeed give us the correct probability of finding a particular B it must also be true that

 

P((x1+a+b,[tau]1), (x2+a+b,[tau]2), ... (xn+a+b,[tau]n), t) – P((x1+a,[tau]1), (x2+a,[tau]2), ... (xn+a,[tau]n), t) =0

 

As the relationship must be valid for all a and b (even as the limit of either a or b goes to zero by the way), we know that we can divide by b and the expression still evaluates to zero. Look at that carefully; that's the definition of a derivative! We arrive at one and only one conclusion: the derivative of P((x1+a,[tau]1), (x2+a,[tau]2), ... (xn+a,[tau]n), t) with respect to a is zero.

 

Well, let's look at another simple variation of the above relationship. Let's us make a change of variables, setting zi=xi+a. If you know anything about partial differentiation, you should know that the derivative with respect to a is identical to the sum over all i of the partial of zi with respect to a times the partial of P with respect to zi. Since the partial of zi with respect to a is exactly one, we have the ultimate conclusion

 

[see the attached thumbnail]

 

In my intuitive generalist view, the Universe IS the simplest expression of itself; we simply do not apprehend most of what is going on with our limited senses. :)

I wouldn't argue with that at all except that I might change apprehend to comprehend. :)

 

Have fun -- Dick

 

Knowledge is Power

and the most common abuse of that power is to use it to hide stupidity

[Doctordick]

Well, I am quite surprised at the complete lack of any response to my last post. It seems to me that the fact that any explanation of anything can be interpreted in such a manner as to be a solution of the equation attached to that post is absolutely astounding for many different reasons.

I started this all by suggesting it would be attaching numbers to meanings and then looking at internal relationships implied by mathematical processes. Now that we have attached those numbers, let's look at some implied relationships.

...

We arrive at one and only one conclusion: the derivative of P((x1+a,[tau]1), (x2+a,[tau]2), ... (xn+a,[tau]n), t) with respect to a is zero.

Either no one here found that astounding or no one here found my arguments that it is true valid. In either way, I would sure like to here some clarification of the issue. There is a lot more down that road yet to be seen.

 

Have fun -- Dick

 

Knowledge is Power

and the most common abuse of that power is to use it to hide stupidity

[CraigD]

Well, I am quite surprised at the complete lack of any response to my last post. ... Either no one here found that astounding or no one here found my arguments that it is true valid. In either way, I would sure like to here some clarification of the issue. ...[/i]

My apologies, Doctordick. I’ve been worked to exhaustion the past couple of weeks, and haven’t been able to dedicate the time and thought necessary to draft a relevant reply to your previous posts. My attention span of late has been mostly appropriate to thinking about such things as shooting skydivers out of cannons, and other “applied” stuff.

 

Please have patience – I hope my leisure time will take a upward turn in the next few days, and I can give your work the attention it deserves.

[Erasmus00]

This sounds a great deal like the philosphical language that John Wilkins proposed in the 17th century. He tied meanings to characters in the hope of creating a language where any true sentence would be a guaranteed mathematical certainty. Look for the book An Essay Toward a Real Character and a Philosophical Language. It will probabl be very difficult to find outside a good university library, as its mostly old and forgotten. I think Neal Stephenson brought it back into vogue a bit with one of his novels.

-Will

[CraigD]

This sounds a great deal like the philosphical language that John Wilkins proposed in the 17th century. He tied meanings to characters in the hope of creating a language where any true sentence would be a guaranteed mathematical certainty. Look for the book An Essay Toward a Real Character and a Philosophical Language. It will probabl be very difficult to find outside a good university library, as its mostly old and forgotten. I think Neal Stephenson brought it back into vogue a bit with one of his novels.

-Will

Interesting observation (regardless or its pertinence to what DoctorDick is describing). If Stephenson’s being historically accurate in his descriptions of the philosophical language in the Baroque Trillogy (Quicksilver, The Confusion, The System of the World), it bears a startling resemblance to a Godel numbering – according to Stephenson’s description, each Real Character in it is assigned a prime number, allowing relationships between Characters to be represented by composite numbers. Although this differs substantially from Godel numbering, which is position sensitive, each character being represented by the exponent of a prime, the resemblance is still striking.

[Doctordick]

My apologies, Doctordick. I’ve been worked to exhaustion the past couple of weeks, and haven’t been able to dedicate the time and thought necessary to draft a relevant reply to your previous posts. My attention span of late has been mostly appropriate to thinking about such things as shooting skydivers out of cannons, and other “applied” stuff.

 

Please have patience – I hope my leisure time will take a upward turn in the next few days, and I can give your work the attention it deserves.

No apologies required. My wife and I are headed to China for a month and won't be back until October so I will be out of contact (unless I just happen to run across the appropriate facilities which I doubt) for a good while. I was just hoping for a sign that someone was putting some thought into what I was saying.

 

Have fun -- Dick

 

Knowledge is Power

and the most common abuse of that power is to use it to hide stupidity

[Doctordick]

Well, I have been back for several weeks now (been pretty busy with "Katrina" problems) but I haven't looked at this thread because no one had posted to it since I left. :) Since "Qfwfq" seems to have lost interest in in my comments (see the "Defining the nature of rational discussion!" thread on the "philosophy of science" division) I thought I would take a look at how this thread had been left. It seems I owe an answer to Erasmus00. :)

This sounds a great deal like the philosphical language that John Wilkins proposed in the 17th century. He tied meanings to characters in the hope of creating a language where any true sentence would be a guaranteed mathematical certainty

You are wrong, it is not at all similar to John Wilkins proposition. In fact, it is exactly the opposite. :) One of the problems with philosophy is that, as far as I am aware, all philosophers do exactly what you say John Wilkins proposed. They fundamentally waste their time trying to carefully tie meanings to characters in the hope of creating a language where any [seemingly] true sentence would be guaranteed to be valid. Think about it for a moment. Is not the idea of establishing the right meanings for the words (which are symbols by the way) a perfect description of the common approach of all philosophers trying to clarify what they mean? :)

 

Take a look at my opening post to "Defining the nature of rational discussion!" In that post I try to clarify the difference between two specific modes of thought which I refer to as "squirrel" thought and "logical" thought. So far I don't think anyone has comprehended why I make so much noise about the separation. The issue is actually quite simple: "logical" or "deductive" thought is so patently limited in scope as to be, for all practical purposes, worthless while "squirrel" or "inductive" (think intuitive or Zen) thought is clearly impossible to defend as valid. It is only with the combined power of both that we are able to come up with the powerful and quite defendable ideas upon which our modern scientific perspective is built. :)

 

What I have discovered (and have been unable to communicate to anyone) is a very effective method of handling the very real difficulty that squirrel thought can not be defended as valid. The very first step (which is apparently very difficult for anyone to comprehend) is to fully recognize that language (and by language, I mean the act of tying meanings to words) is in itself a "squirrel" construct. Powerful as it may be, one can never be assured that the interpretation they put on a sentence is the one the speaker intended. Languages are inherently vague as one can only come to understand a language through induction and, as any decent philosopher will admit, inductive conclusions can not be deemed certain. This is the very crux of the difficulty avoided by all. :)

 

Clearly, denying one the right attach meanings eliminates language itself; but, without language, communication is impossible. This is a serious dilemma and we are forced to work with something we cannot prove is valid. However, as Popper has pointed out, the term reliable is another characteristic of communications which is of great value. We are quite lucky in that great thinkers have spent thousands of years fabricating a language which is just about as empty of vague definitions as is conceivable. That language is called mathematics! :) In fact, I personally define mathematics as the invention and study of internally consistent systems. As Feynman is noted for saying, "mathematics is the distilled essence of logic". Without mathematics, logical thought is constrained to roughly three or four steps (what we can keep in our conscious mind simultaneously); with mathematics, logical thought can be extended to relationships hard for the mind to comprehend. :)

 

So, although logic and mathematics are themselves squirrel constructs and thus inductive and unprovable, they are (as Popper would say) "reliable". That is, they are understood by many people and, of all the languages known to man, the most apt to achieve agreement. What is important here is that, even if the concept of some mathematical relationship in your head differs from the concept in my head, there always exists an isomorphism with a one to one mapping between the two. That is to say, there is sufficient agreement upon the definitions of basic entities and embedded procedures that we can have extreamly high confidence that, following some specific mathematical prescription where our starting entities were mapped to one another, our finishing entities will also map to one another: i.e., agreement on the consequences of our logic will very probably be achieved. If it cannot be achieved, there are clearly either entities or procedures thought to be part of mathematics which must be removed (the ones which lead to that disagreement). Note that this step is deductive and not inductive so its validity can be investigated. :)

 

The point here is that, although being a squirrel construct and thus possibly invalid, we have very strong evidence that it is a construct not yet proved invalid. (It is indeed the language Wilkins was trying to construct he just didn't think it was sufficient to his needs.) :) However, once one limits oneself to mathematics as the only internally consistent language available it be comes quite clear that omitting general squirrel constructs (induction) is so limiting that nothing can be accomplished. In fact, the situation is an absolute impasse so long as we try to use these tools as independent entities; which is exactly what every philosopher I have ever read does. :) All they do is stir the pot of those vaguely defined terms (presuming this or that is a valid concept) in the fond hope that something of use will float to the top. Well, valuable concepts do occasionally float to the top :ud: (that's the whole source of scientific progress) but the results must always be doubtful. :xx:

 

What all the philosophers seem to miss is that the error occurs not with the squirrel constructs themselves but with our assumption they are valid. It follows that the only solution is to move these constructs into an abstract form such that no meanings whatsoever are attached to the associated symbols. By this means, the squirrel constructs may be included in our analysis without worrying about their validity (it can be left to a later examination: i.e., by keeping the constructs abstract, we have overtly held on to the fact that we have not assumed they are valid). :D

 

What is important here is that serious logical analysis can be done from that basis without presuming any of these squirrel constructs are valid (except for mathematics which is being used only for communication purposes). It is the result of this logical analysis so constrained which I am trying to get under discussion but, so far, I can't even get anyone to consider the thing. :eek: As soon as I get even close to laying out the logic, everyone disappears into the wood work. :) Is there no one out there with any confidence in their ability to think at all? :) No one willing to give the slightest attention to what I am saying? :)

Please have patience – I hope my leisure time will take a upward turn in the next few days, and I can give your work the attention it deserves.

Again, I would really appreciate that. :)

 

Have fun -- Dick

 

Knowledge is Power

and the most common abuse of that power is to use it to hide stupidity