Defining the nature of rational discussion!

[Turtle]

I don't know if you are joking or what!

Have fun -- Dick

 

I never joke; but seriously Doc, I summarized honestly. Here is now what I think you want me to focus on:

...dismiss the idea that "knowing the information" could be conceived to be "an explanation...

 

...define[d] "an explanation" to be a method of obtaining expectations from given known information

 

___Ok. Now suppose a specific example of 'given known information' C. Let C be an apple. And since A is what C is made of, let A be chemical elements. The information change B of the information 'apple of chemical elements' is rotting apple. With your method of explaining how do I contruct my expectational model.

___In common terms you are 'explaining' to us an understanding of pattern you have experience of. Is this common 'explanation' congruent to your constructing a tautology on an analytical proposition?

___:) Having Fun.

Turtle

[questor]

DD, if we can somehow come to understand your tautological propositions, how will that be of help to us or science ?

[Doctordick]

DD, if we can somehow come to understand your tautological propositions, how will that be of help to us or science ?

The answer will become quite evident should you ever come to understand what I am trying to convey.

I never joke

Now I am sure the "never" is a bit of an exaggeration; (I've read a number of your posts) but I will take you at your word that you are trying to understand what I am saying. ;)

___Ok. Now suppose a specific example of 'given known information' C. Let C be an apple. And since A is what C is made of, let A be chemical elements. The information change B of the information 'apple of chemical elements' is rotting apple. With your method of explaining how do I contruct my expectational model.

I do not propose any method of explaining anything. That is the work of the person providing the explanation. All I am going to do is deduce the logical consequences which are required by my definition. The issue under discussion at this moment is no more than your willingness to accept my definition. From that perspective, what you have said above can only be taken as an example of something you don't believe is consistent with that definition. So let's look at it and try to resolve what you are trying to say in terms my defined components. ;)

 

If C, "the given known information" is to be "an apple" then we have to be very careful to express exactly what you mean when you call "what you know" "an apple". The known information is "what you know" and, by definition, you know nothing else (for a compartmentalized problem we can adjust this to "nothing else relevant to what is to be explained"). B is the name I use to specify the "change in that known information". Now, since you referred to the known information as an apple, we could interpret the "given known information" to be some specific apple (not an apple in general).

 

Thus the known information is, "what you know about this particular apple". And B is there to accommodate the possibility that "what you know about this particular apple" may change (perhaps it starts to smell different or its color changes). A is what you are trying to explain (what you would know about this apple if you were all knowing) but, since you are not "all knowing", there must be some aspects about this apple of which you are ignorant (for example, perhaps the exact point on the skin where rot will first be seen). The whole purpose of introducing B was to allow "what you know", or C, to change. And the elements of B must be elements of A if A is "everything you could possibly know" about the thing you are explaining.

 

However, "what you know about this particular apple" might be that you had dropped it and it had a bit of a bruise on one side (the consequence of acquiring a bit of information earlier: a B(t) in the assemblage of your current C, "what you know"). This might encourage you to believe that the exact point on the skin where rot will first be seen would be somewhere in that bruised area.

 

Now there is an expectation! Something you do not know but with which you have some expectations. And, if I were to ask you, why do you think the skin will first rot there, you will probably tell me, "because that's where the bruise is". That's your explanation. In simple terms, you have theorized that apples tend to rot when they are bruised. And you have a simple method of obtaining your expectations, the method is no more than looking at the apple and finding the bruise. So, "an explanation" is a method of obtaining expectations from known information.

 

Now, in your attempt to communicate to me the example problem you wished to discuss, you said "let A be chemical elements". This casts quite a different perspective on the makeup of C, "what you know", which could be quite confusing. Remember A was to be what is to be explained. This means that what is to be explained is "chemical elements" and what is known is "an apple". It becomes somewhat difficult to resolve exactly what you mean. About the only thing which comes to mind is that you are going to try to explain "chemical elements" based on nothing except your knowledge of "an apple" and I doubt that is what you had in mind.

 

Now I suppose you could have been thinking of "an apple" from the perspective of "its chemical elements" under the presumption that eventual collection of all information on that apple (the set of all possible B's would eventually include all of chemistry) but then, C, all the information relevant to the problem, could hardly be called "an apple". Of course, you could be a very narrow minded agricultural chemist who specialized in apples who might refer to "everything he knew" as "apples". In this case, his expectations might be quite complex and involved and his explanations (his theory: his actual method of coming up with those expectations) could be far more complex and involved than the example I gave above.

___In common terms you are 'explaining' to us an understanding of pattern you have experience of. Is this common 'explanation' congruent to your constructing a tautology on an analytical proposition?

If I understand what you are trying to say here, the answer would be yes. The analytical proposition is that "an explanation" is a method of obtaining expectations from known information. No more, no less! What I am concerned with is exactly what can be deduced from that proposition. I have experienced the construction of that deduction and would like to communicate the construct to you. :)

 

Have fun -- Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

[Turtle]

...Now I am sure the "never" is a bit of an exaggeration; (I've read a number of your posts) but I will take you at your word that you are trying to understand what I am saying. ;)...

 

I do not propose any method of explaining anything. That is the work of the person providing the explanation. All I am going to do is deduce the logical consequences which are required by my definition. The issue under discussion at this moment is no more than your willingness to accept my definition. From that perspective, what you have said above can only be taken as an example of something you don't believe is consistent with that definition. So let's look at it and try to resolve what you are trying to say in terms my defined components. :)...

 

...If I understand what you are trying to say here, the answer would be yes. The analytical proposition is that "an explanation" is a method of obtaining expectations from known information. No more, no less! What I am concerned with is exactly what can be deduced from that proposition. I have experienced the construction of that deduction and would like to communicate the construct to you. ;)

 

Have fun -- Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

 

__Yes, I was joking about never. I *am* trying to understand this & see if I have a better handle now.

___I either implied a deuction wasn't consistent with the definition by error or other ignorance. I accept the definition; deduce away. Thanks for your patience with my slow understanding.;)

[Doctordick]

I accept the definition; deduce away. Thanks for your patience with my slow understanding.B)

Well, thank you very much. I might be wrong but this seems to be the first time anyone seemed to grasp what I was trying to show. B) Just to make sure I have all my ducks in a row, let me first summarize some aspects of my method. I am going to define what I mean by the terms I use: i.e., set forth what are commonly called analytical propositions (things true by definition). I will use common words for many of the things I define because I happen to know where the thing is heading and the words I use will turn out (in the end) to be very analogous to the common meanings accepted by everyone. However, you should always bear in mind the fact that what I am presenting is actually completely abstract and would be just as true were I to use nonsense words to identify these mental constructs. (I use common words because the human brain seems to handle large volumes of information more easily when the reference tokens are familiar and the path we are headed down is not short.)

 

What I am trying to say here is that anytime I make a statement (something which I present as a deduction) where there is any doubt at all that the statement follows from what has already been presented, you should carefully examine the step in terms of the definitions actually given and not depend on your intuitive understanding of the terms (the fact that the human brain is able to handle large volumes of familiar information can be, and often is, used to slip sloppy logic past an unwary listener and we need to be very careful to avoid such).

 

I will also deviate occasionally from the pure presentation of an abstract step with a side discussion of why I am moved to make the step. Please note that, when I do that, it is not to defend the validity of the step but is rather to create sign posts in the quagmire we are stepping into so that we can quickly trace and retrace our steps to assure their deductive validity. Having said all that, I will reassert the definitions I have already presented in this thread; please either acknowledge your understanding or point out the difficulties you see.

 

1. I have defined an explanation to be a method of obtaining expectations from known information. This definition is to be taken as an analytical proposition from whence my deductions are to emanate. I do hold that it is a valid expression of the common meaning of the concept of "an explanation". (Turtle has already said he accepts this.)

 

2. I have defined the abstract sets A to be an arbitrary set, B to be a finite collection of elements of A, and C to be a finite collection of sets B. Those definitions are totally abstract analytical propositions. These sets were created in order to represent what I feel are important aspects of all explanations. A symbolizes what is to be explained. C symbolizes the information upon which the explanation is based (what we know) and B symbolizes a possible change in C.

 

3. The set C' is a fictitious set which will be more carefully defined later. At this moment, it is introduced to accommodate the fact that there might be (and probably is) a significant difference between "what we know" and "what we think we know".

 

What is important here is that our explanations, though they must be based on C will, in general, include elements which are presumed to exist and are not necessarily actual elements of C. Please note that the introduction of C' does not require the existence of fictitious information as C' could certainly be NULL; however, it is quite presumptuous to assume nothing in your explanation is fictitious. Having introduced C' I must also introduce B' to accommodate change in C'.

 

An important issue here is that, if the existence of any part of C' is inconsistent with your expectations (what your explanation leads you to expect), then your explanation can be rejected out of hand. The point being that there can exist no information within the C+C' set which can invalidate an acceptable explanation implies that there can exist no mechanism which can differentiate between C and C'. On the other hand, though there can exist no mechanism to differentiate between the two, they must nonetheless obey very different logical constraints: C is fixed and one has no power to alter it in any way while C', being fictitious, can be whatever happens to be convenient to the explanation. Because these two sets play such an important role, I am going to name them so I can refer to them easily. The elements of B (the constituent elements of C) I will call "knowable data" (in the sense that it must be part of any reliable theory) while B' (the constituent elements of C') I will call "unknowable data" (in the sense that it cannot be proved to be necessary: it is always possible that there may exist a reliable theory which does not include this data). See message #53 on page #6 of this thread for my first attempt to introduce my idea of "knowable" and "unknowable" information.

 

4. Any explanation is based on two logically different collections of information (knowable and unknowable) which obey very different constraints: "knowable" information is fixed while "unknowable" information is actually part of the explanation (and thus a free variable) not actually part of what is being explained.

 

What is important here is that the abstract deductions concerning the character of "unknowable" data are not the same as the abstract deductions concerning the character of "knowable" data and that this fact has nothing to do with any ability to identify which is which. I only mention this because a lot of philosophers can't seem to handle this fact. They seem to hold that things which follow different rules must, by virtue of following those different rules, be physically identifiable as different. Yet, not one of them has ever defended that proposition to me satisfactorily.

 

It should be clear that we need a way of referring to those elements of A which are in our "knowable" data (likewise for those hypothetical elements of our "unknowable" data). In any case, with regard to any actual extant explanation, the common mechanism is to use a name with a defined meaning to identify or refer to those elements; however, the process of developing and defining those names is central to expressing one's understanding the data: i.e., it is part and parcel to to a specific explanation itself and is obtained via the process of induction, exactly the issue we wish to avoid. We are only interested in the absolute truths which may be deduced from the definition of "an explanation", and not concerned with any particular explanation. In order to bypass that conundrum, I will simply label them with numbers. In order to map our logical deductions into a particular explanation, we then need only to assert that a given number will map into the defined name the explainer has assigned to the element referred to by that number. Thus it is that the label "i" (a number arbitrarily assigned) will refer to a particular element of A (or actually an element of B, a known member of A since those are the only elements of A one actually knows).

 

What is very important here is that the actual number "i" assigned to that element of B is of utterly no significance; not so long as we are indifferent to any actual explanation. Remember, we are only concerned here with the absolute truths which can be directly deduced from the definition of "an explanation". This insignificance of assignment of numbers has some very important consequences. Essentially these numbers can be seen as a coded stand in for the eventual defined names to be assigned to these elements once a theoretically possible explanation is conceived of.

 

Meanwhile, we cannot proceed farther without coming to grips with the difficulty of setting forth a completely general representation of the explanation itself: i.e., that method of obtaining expectations from known information. Clearly, in this deduction, the known information (consisting of the collection of "knowable" and "unknowable" data) will be represented by the elements of C+C', that would be the entire collection of known sets B+B'. But each such set is a collection of elements which have been labeled with a number therefore, each set B+B' is represented by a set of numbers.

 

Now the explanation is to yield our expectations; essentially that means that the explanation is to yield our expectations that a specific B+B' is indeed a collection of elements of A. Or somewhat more exact terms, as B+B' represents change in our knowledge, we are speaking of a the B+B' we would expect following a given state of knowledge (a specific collection of sets B+B'). This means that any specific explanation should assert some order to the changes. Since the B were already defined to be finite, we know that they may be ordered (that is a basic theorem of sets) so let us simply attach the index "t" to specify the order the explanation is going to presume.

 

Clearly I am using the letter "t" to specify the order because time is what we usually call the sequenced change in our knowledge; but, for the moment, "t" is no more than an abstract index and is to be arbitrarily assigned for the simple purpose of ordering our changes in knowledge and nothing else. It follows then that the method of obtaining our expectations can be seen as nothing more or less than a mathematical function of many variables. If unity represents we expect it, and zero represents we don't then our expectations can always be represented by P(B(t)). That is, our explanation will tell us the probability that we think a particular specific set B will belong in that ordered list at index "t". Note that I am not asserting that this will be what is normally called a common function, it may very well be no more than a table of your expectations (the explanation itself must tell you how to achieve that); all I am saying is that it can be represented in the notation of a mathematical function.

 

I will stop here for the moment to give you a chance to comprehend what I have said. Please read it carefully and make sure I have not slipped any subtle presumption into the stream of analysis. It would be nice if it were all totally clear to you but I sincerely doubt that will happen. I will presume that, if you have no questions, you haven't read it carefully. Don't just trust me; drag me over the coals.

 

Have fun -- Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

[Turtle]

___So an abstract algebra on sets of sets of sets by whose manipulations we may determine an ordered method for reliably determing the veracity of elements of C, as well as increase the cardinality of C.

___I see no internal inconsistencies in the setup or terms you have described. In my own words now without looking, an explanation is a method of obtaining our expectations? Further you intend to codify that method?

___B)

[Turtle]

I'm trying to draw a Venn diagram of this. First, do you think it is suitable, & second is all of A in C? Thanks.B)

[Doctordick]

I'm trying to draw a Venn diagram of this. First, do you think it is suitable, & second is all of A in C? Thanks.B)

Venn diagrams are pretty well always useful to display intersection of sets but I think that, here, you have made some subtle but significant errors. A, B and C have been defined in a very specific way and Venn diagrams do not really lend themselves to displaying information about the intersections of these sets. First, Venn diagrams are only valid when the elements of the various sets are the same. In this case they are not necessarily the same. C is defined to be a finite collection of sets B thus the elements of C are not the same as the elements of A unless all of the sets B each consist of but a single element of A, a rather special case. Also, as B is defined to be a collection of elements taken from A it is possible for B to contain multiple identical elements while, at the same time, the entire set of elements in A may consist of unique elements only, a subtle but interesting circumstance we might discuss after I have finished my deductions.

 

Also, if all of A is in C then absolutely everything which can be known about A (the thing we are trying to explain) is known and the idea of "an explanation" sort of vanishes. What purpose would such an explanation serve? (There is also another problem with this particular concept which we can only discuss at the end of the deduction.)

 

Now, to go back to your previous post:

___So an abstract algebra on sets of sets of sets by whose manipulations we may determine an ordered method for reliably determing the veracity of elements of C, as well as increase the cardinality of C.

No, not a method for determining that veracity but rather a logical way of representing the method no matter what that method might be. A subtle but important difference. Actually, I won't bother to include representing unreliable explanations (that would be explanations which are internally inconsistent) but that is a little further down the road.

___I see no internal inconsistencies in the setup or terms you have described. In my own words now without looking, an explanation is a method of obtaining our expectations? Further you intend to codify that method?

Yes and but not exactly "codify" but rather represent it so that I can talk about the details without constraining the range of possibilities in any way. Remember, I want to examine what can be deduced from the definition of an explanation and none of this has any impact on understanding the universe other than to point out some god awful assumptions within the foundations of science.

 

On the other hand, I suppose you could call it "codify" if you are thinking in terms of a computer programmer deciding on the way he wants to represent various aspects of information on a machine which can only specify yes or no answers (bit values). The problem is not trivial and the solutions are not unique. For example, think of the various file protocols for text, pictures, sounds etc. Specific problems can be found in most all of them: i.e., aspects things which can not be represented in that particular protocol (I believe they all fail to represent smells). What kind of picture file would you use if you were interested in extremely large variations in resolution across a scene (say at the atomic level in one area together with national resolution elsewhere)?

 

Science has always approached the problem of understanding phenomena by trying to focus in on what they consider to be the important or relevant aspects. In some great sweeping way, that is what I have done by focusing in on "an explanation". But, in another way, I think that is the single broadest, all encompassing problem which can be conceived of and I absolutely do not wish to limit my examination of it in any way. The single most important aspect of my attack is that no possibilities be inadvertently discarded by the representation. This is totally in opposition to the standard scientific attack: their approach is to establish reasonable constraints upon what will be examined while mine is to avoid putting any constraints on the possibilities at all.

 

As an example of my open ended attack, let me bring up that mathematical function which is to represent the method of obtaining our expectations. It follows from the definition of the problem that we are only interested in algorithms which yield, as a result of their application to a given set of numbers, a real number bounded by zero and one. That is, the result must be interpretable as an estimate of the probability we expect that particular B(t). This is a particular subset (not the entire set) of "all possible mathematical functions" thus we need to be very careful to assure that we do not omit any possibilities.

 

Thus it is that I propose the following mathematical operation be used to simplify the circumstances. Instead of looking at the function P(B(t)) directly, I will instead let P(B(t)) be the result of of the mathematical operation commonly referred to as a "normalized" inner product of some mathematical function with its own complex conjugate. Here I have defined a mathematical function as any operation which carries one set of numbers (called the argument) into another set of numbers (called the function). The inner product would be the sum of the positive definite squares of that generated set of numbers (the complex conjugate is defined so as to yield a positive definite result). And normalization is division by the total integral of that inner product over the entire range of its arguments (or sum, if the number of possibilities is a discrete set). Essentially finding a factor which will make the sum or integral over all possibilities exactly equal to one. The result will always be a real number bounded by zero and one but the important issue is actually somewhat simpler than that: what is important is that the result should always be interpretable as a probability.

 

If you are interested in rigor (and you should be), there are three basic issues which should be clarified here; all related to the specific nature of the integral referred to above.

 

The first is division by zero: since every element of the inner product is positive definite, a zero sum can only occur if every element vanishes and, in that case, the "normalization" step need not be implemented as its only purpose was to constrain the result to be not greater than one. (Note that, in this case, all probabilities vanish identically so it isn't really a problem.)

 

The second is division by infinity: this leads to an expectation of zero for all possibilities. There is nothing fundamentally unreasonable in a zero result; however, if the number of theoretical possibilities is infinite this result is not actually what one is interested in. More to the point (in such a case) would be comparing one collection of possibilities to another. Again, in this case, normalization is not needed as the only requirement is that the result (in this case a ratio of two mathematical expressions) can be interpreted as a probability ratio. Again, the only important issue is that the result must be interpretable as a probability; all possible mathematical functions remain eligible as possibilities.

 

The third problem has to do with the existence of the integral. If P(B(t)) exists, then there exists at least one normalized function who's square will yield that function: that would be the square root of P(B(t)). If that function does not exist, there exists no method of generating one's expectations and the explanation is moot as it does not satisfy our definition of an explanation. That is, the function under examination is not correct as it will not fix the probability of all internally consistent possibilities at "one". If all functions which exist fail this test, no reliable explanation exists.

 

The central issue of this meandering is to make sure that no possibilities are ignored. We can now make a fifth analytical proposition:

 

5. The expectations from an explanation are defined to be the normalized inner product of a mathematical function with its complex conjugate.

 

Now we can merely assert that every explanation of anything is equivalent to some mathematical function. The problem of finding an explanation is totally equivalent to finding a mathematical function (from the set of all possible mathematical functions) which will produce the reliable expectations which are desired.

 

Just as an aside, I have clearly laid out the form of my representation of the explanation because I know what that particular form leads; however, I also assert that under this representation there exist no conceivable relationship which is excluded. Just as a realistic example of the astonishing range which can be represented by such a notation consider the mathematical function where the argument is the machine language code for a particular question and functional result is the machine language code for everything which has ever been written about that question. One could call it "the Google Function".

 

The concepts I have defined now provide a set of basic axioms from which logical deductions can be made. Again I ask for either agreement or complaint before I go off into that swamp of deduction as there are only two reasons anyone could disagree with my deduced results: either they can show a specific error in the logic itself (which should be relatively straight forward challenge) or they are, for some reason, refusing to accept one or more of those axioms.

 

Have fun -- Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

[Turtle]

Also, if all of A is in C then absolutely everything which can be known about A (the thing we are trying to explain) is known and the idea of "an explanation" sort of vanishes. What purpose would such an explanation serve? (There is also another problem with this particular concept which we can only discuss at the end of the deduction.)

 

___Here is what I make of things now. I disagree that "an explanation" disapears just because all of A is in C, that is to say 'what we are trying to explain' is "already known". I get the general sense that who is explaining (or *what* is explaining in the context of your functional recursive algorithm, is only as good as who, or what, is understanding or *receiving* the explanation. In other words, the expectational value in an explanation is its unfolding path in time & it only grows larger in regards to how many receivers accord it attention & as everything from our frame of reference is moving ahead in time( the ordination of your sets), we may expect to always increase our expections.

___Thanks as always for the patients Doc:)

[Doctordick]

___Here is what I make of things now. I disagree that "an explanation" disapears just because all of A is in C, that is to say 'what we are trying to explain' is "already known".

I know exactly where you are coming from but nevertheless assert you are wrong anyway. Unless you are able to prove that you are all knowing you must always allow for the fact that there might be elements of A not in C so the question is moot at best and certainly no concern of ours; I only inserted my comment because it happens to be true, not because I needed it in the deduction. For the time being, don't worry about it; as I said there are other problems with the concept which will only become clear once the consequences of the deduction are understood :xx:

 

Meanwhile, we need to talk about a very important relationship between symmetries and conserved quantities first laid out in detail via a theorem proved by Emmy Noether sometime around 1915. The essence of the proof can be found on John Baez's web site. This is a fundamental theorem accepted by everyone. The problem is that very few students think about the underpinnings of the circumstance but rather just learn to use it. :D

 

You will hear many professors (and publications for the general public) simply state that "symmetry arguments are the most powerful arguments which can be made" without explaining what makes them so powerful. They usually give fairly simple examples and walk the student through, displaying the result as a self evident conclusion. These examples almost always begin with the phrase, "assume we have [such and such] symmetry". Notice the opening to John Baez's proof starts exactly the same way. I will give him credit as he at least he tells you what he means by a symmetry. Symmetry is another of these things that is "understood" on an intuitive level without much thought. :naughty:

 

What I would like to point out is that any symmetry is essentially an expression of a specific ignorance: i.e., something very specific is not known. For example, mirror symmetry means that there is no way to tell the difference between a given view of a problem and its mirror image: in effect you are in a state of enforced ignorance as to which view is being presented. Shift symmetry, the symmetry which Noether's theorem can be applied to yield conservation of momentum, arises if shifting the origin of your coordinate system has no impact on the nature of the problem: i.e., the information as to where the origin must be is unavailable to you. In a careful examination, every conceivable symmetry can be seen as a statement of some specific instance of ignorance.

 

The fundamental issue behind the power of symmetry arguments is the fact that information which is not available can not be produced by any algebraic procedure. It is a characteristic of mathematics that everything is deduced from a set of axioms; a proof amounts to a specific procedure which demonstrates that some piece of information is contained in a particular set of axioms. That being the case, how were we able to solve the problem above for specific expressions of q when changing q has no impact on the problem? The answer lies in Noether's theorem. There must be another relationship which relates the range of possibilities for q (the transformations Baez refers to) to the various specific solutions. In shift symmetry, this required relationship is conservation of momentum; in rotational symmetry, the required relationship is angular momentum.

 

The above can be seen as a means of obtaining information from ignorance and I have had professors actually make that statement. This is why symmetry arguments are often referred to as the most powerful argument which can be made. But let's think about that for a moment. Noether's theorem is a mathematical result and, as such, cannot produce anything which is not contained in the axioms. Ignorance cannot be the true source of our result; it must be arising from some other source. In many respects, Noether's theorem may be seen as a subtle result of conservation of ignorance. Symmetries arise in a problem because the structure of the representation of the problem contains an element which cannot be known: that is, the information which the structure explicitly represents exceeds the information available.

 

Take for example, the consequences of not knowing the "center of our universe" (the origin of the coordinate system used to represent positions in our problem solving). If we don't know where the origin is, we don't know the particular value of any position. It follows that there is a different solution for every possible position of that origin. The other side of the coin is, if we were actually able to find a solution some positional problem (say x as a function of t) we can clearly take that particular solution and deduce exactly where the origin was.

 

At that point, we have information which was not available in the original problem given to us and something is logically inconsistent. Conservation of momentum is a mathematical relationship on that solution which makes all the various solutions (the collection of solutions, each of which would independently allow deduction of a different origin) equivalent to one another. It is the constraint on the solution which provides the required reduction in information. What I am giving you here is no more than a different perspective on Noether's theorem. :D

 

My purpose in stating things in such a strange way is to bring out the obvious inconsistencies implied by presuming we know things we cannot possibly know (setting up a coordinate axis when we don't know where the origin is). Remember, my sole purpose here is to establish the parameters on my thoughts which will assure me that I am not inadvertently presuming to have information I do not have. Noether's theorem is an excellent example of how easy such a thing can happen and I don't think the common presentation brings that single most important issue to the forefront.

 

The axiom behind Noether's theorem is essentially, we are ignorant of something. When we set up our coordinate system, that ignorance is not explicitly displayed and blind usage of the coordinate system ignores that embedded ignorance. It follows that there must exist a constraint on the solution which will yield up that same ignorance in our final results. It is that underlying relationship which requires Noether's theorem to be valid. This is the relationship I am trying to bring to your attention, not the solution of a particular problem.

 

I will stop here in order to give you an opportunity to discuss whatever confusion emanates from that indirect and rather unconventional presentation of the essence of Noether's theorem. :xx:

 

___Thanks as always for the patients Doc:)

Actually I should thank you for your patients as my thoughts are far more trying to follow than are yours.

 

Have fun – Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

[Turtle]

I know exactly where you are coming from but nevertheless assert you are wrong anyway. Unless you are able to prove that you are all knowing ...

 

I am not all knowing, I am all learning. I have learned what you propose, because you explanined it to me, I compared it with what I learned before, & I learned what is the join. I then communicated - explained what I learned 'till you now understand it. You know exactly where I'm coming from you wrote as quoted above.

That you refuse to believe it even that you know it, is the problem you have to resolve. The problem I had to resolve & we all have to resolve.

Keep digging Doc.

Best Regards, Turtle

[Doctordick]

I am not all knowing, I am all learning. I have learned what you propose, because you explanined it to me, I compared it with what I learned before, & I learned what is the join. I then communicated - explained what I learned 'till you now understand it. You know exactly where I'm coming from you wrote as quoted above.

I think you misunderstood my comment a bit. When I said I knew where you were coming from, I was referring to the specific disagreement you brought up. And, even there, my comment was really only a colloquial response indicating that, prior to my understanding of the deduction which is to follow, I would have made very much the same response. Even with that assertion I could very well be in error as to your motivation. When it comes to the rest of your comments, I can only interpret them as best I can with the limited information available to me. Until I reach the stage where your responses no longer surprise me, I have no way to conclude that your understanding of what I said corresponds (or can be mapped into) my understanding of that same information. The issue is that we are both working with finite partial information and depending very strongly on a great many inductive conclusions; whatever we interpret the other as saying, there is still a strong possibility of misinterpretation. :(

 

I originally started this thread in order to hammer out the necessity of separating inductive and deductive conclusions. As I have tried to say, we cannot deduce anything without accepting a number of inductive conclusions. As Popper and others have commented, the reliability of inductive conclusions vary and it behooves us to be aware of that reliability. Mathematics, as I have said before, is essentially the invention and study of internally consistent systems. The intention of mathematicians is to make mathematics as reliable as is possible (setting that goal above all others). As a group, they seldom have any interest in the applicability of their work, but rather the issue that the deductions be error free (that they are indeed internally consistent systems). However, their axioms and procedures were arrived at via induction from "realistic" observations and, as any competent philosopher will comment, can be held as suspect as any other inductive conclusions (but, nevertheless, among the most reliable available).

 

So my position is a tad askew of the norm. I say, ok, let us accept some inductive conclusions to build a starting point, but let us keep that structure as small and as reliable as possible. If you can come up with a start point smaller and more reliable than mine which will produce more usable constructs, then, by all means do so. I choose mathematics as reliable and small because that is essentially the goal of the historic mathematical community. I will simply depend on their results under the realization that they are much more competent than I am and have had much more time to clean up their act. But mainly I accept mathematics because of the great agreement about exactly what the symbols and operations mean (that is to say, the probability that your mathematical results of following my instructions will, if not exactly the same as mine, still map into mine in a very reliable manner). That is, we should be able to communicate!

 

Logical deduction is also a widely examined and accepted construct, held by many to be a branch of mathematics, so I also accept logic as a basic tenet of my arguments. In addition to mathematics and logic, I add my definition of an explanation (which has two components: the information and the method), my definition of those components (A,B,C,B' and C'), the characteristics of those components (called knowable and unknowable or real and fictitious, which ever you prefer) and a definition of the method of obtaining an expectation (the normalized inner product described earlier). All these have been achieved and defined via inductive construction.

 

To this is added my element labeling convention: the elements of B will be labeled with numbers I will call x_i and the elements of C which I will label with t, yielding a notation for a specific expectation P(B(x₁,x₂,...,x_n, t ). With these tools and these tools alone, I will move off into that swamp of confusion I spoke of before. Nothing discussed beyond this point has anything at all to do with reality so you should not try and relate it to anything else you know. We will be out of sight of reality for a good while and the only course is to examine every step of the logic very carefully.

 

The first step is to look at the way those numerical labels have been generated. It should be recognized that utterly no method has been mentioned. They are no more than simple labels. If an explanation of A (in the conventional sense) exists, then the explanation is based upon defined names for the elements of A. Among all possible ways of numerically labeling those elements, there must exist a set of numerical labels which will map 1::1 with those defined names. In fact, this must be true for all possible explanations of A. Actually, we can go a bit farther than that: there must exist an infinite number of ways of numerically labeling those elements such that the labels will map 1::1 with those defined names. If one set exists, merely pick an arbitrary number and add that number to every specific label in that set. The new set will map into those defined names quite as well as the original set. Clearly there are an infinite number of ways of selecting that arbitrary number to be added.

 

[Well, I was surprised that this forum does not implement html subscripts so I am not sure what the preferred method of showing such things is. I will therefore just use the html directives _-- and let the reader rewrite the expression if he has sufficient interest to examine what is said. More to the point, I do not understand why a "science" forum fails to implement LaTex or some other mechanism for displaying mathematical expressions as they are somewhat central to science of all kinds.] B)

 

The above situation is exactly what is called a shift symmetry. So my first deduction is that a shift symmetry exists within the numerical labeling called for in the explanation. This shift symmetry has some important consequences. Go back to that explanation which exists and look at the expectations it produces: the function P(B(x₁,x₂,...,x_n, t) . Adding some number "a" to every numerical label expressed there does nothing except shift the mapping. This implies that it must be possible to map exactly that same function into one where the numerical labels on the elements have been relabeled so as to refer to exactly the same B with the expression: P(B(x₁+a,x₂+a,...,x_n+a, t). The point being that the expectations are a function of the actual changes in C denoted by the specific B's and that fact does not require a unique collection of numerical labels.

 

This fact tells us something very important about the internal structure of that mathematical function which serves the purpose of yielding the expectations of that explanation. P(B(x₁,x₂,...,x_n, t) must be identical to P(B(x₁+a,x₂+a,...,x_n+a, t) or, more to the point, their difference is zero. If you divide that difference by a, you obtain a very common expression. In fact, that expression in the limit as a goes to zero is exactly the definition of the differential with respect to a of the mathematical function which yields the appropriate expectations. The conclusion is that a differential with regard to the shift parameter must be zero.

 

However, there is another way to represent such a shift. It can be seen as the consequence of a simple change in arguments of the form z_i=x_i+a . That case can be seen as a particular incident of a general change of variables and one can write the differential with respect to a as identical to a sum over i of the partials with respect to z_i times the partial of z_i with respect to a. But the partial of z_i with respect to a in this case is identically one. Since this change from x_i to z_i amounts to no more than a simple change in name of the variables, it must be true that the sum over i of the partials with respect to x_i of the mathematical function which yields our expectations must vanish.

 

This is exactly where I apparently lose every supposedly competent scientist; they invariably stop listening at this very juncture. I suspect your reaction will be the same as theirs. Their immediate reaction seems to be, "that can't possibly be true!" Anyone can quite easily come up with a thousand reasons why such a proposition can not be valid and they utterly refuse to even consider the matter further. Any rational person's first reaction is to set up some simple explanation where they can tabulate their expectations and then demonstrate that the tabulated function fails to obey the rule just supposedly deduced. And there are thousands of such examples which can be generated.

 

That is why I made so much noise about moving off into that swamp of confusion out of touch with reality. The error made in all those examples which are presented is the same in every case. When those simple examples are set up, the circumstances conceived of to set up that "simple explanation" will invariably include information accepted as properly interpreted which was achieved by induction.

 

Take the trouble to set up a simple "realistic" example (absolutely anything will do; the simpler the better). When you do that, you will very probably give specific names to the involved elements of interest to your explanation. Now, what most people invariably forget is that C constitutes all (and that is absolutely all) of the information necessary to generate that "simple" explanation. In your "simple" example you must take the trouble to make sure that no required information is omitted from C. That information includes sufficient information to explain the meaning of every name you used (including every reference required to provide those explanations).

 

It is the fact that every concept used in that "simple" example must be defined by the way in which C is constructed from the associated B's (no definitions of anything may be presumed known). Once you are seriously into constructing a the set C representing your "simple" example you will inevitably discover your example is not "simple" at all. Of greatest significance is that the process of trying to construct C will lead you down that same trail of infinite regression Kriminal99 has been complaining about. Either that, or you will have to resort to induction on C in order to establish the meanings of the names given to the elements in your example. Either way, that "simple" example will not prove my contention invalid. :confused:

 

In other words, don't deny this initial deduction because your intuition tells you it is wrong; point out any specific errors you think exist in my logic. Perhaps you can bring up a problem I hadn't foreseen. B)

 

Sorry about the delay in my response but it is the Christmas season and retired life can get full quickly. :)

 

Have fun – Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

[Turtle]

I think you misunderstood my comment a bit. When I said I knew where you were coming from, I was referring to the specific disagreement you brought up. And, even there, my comment was really only a colloquial response indicating ....that, prior to my understanding of the deduction which is to follow, I would have made very much the same response. Even with that assertion I could very well be in error as to your motivation. When it comes to the rest of your comments, I can only interpret them as best I can with the limited information available to me. Until I reach the stage where your responses no longer surprise me, I have no way to conclude that your understanding of what I said corresponds (or can be mapped into) my understanding of that same information. The issue is that we are both working with finite partial information and depending very strongly on a great many inductive conclusions; whatever we interpret the other as saying, there is still a strong possibility of misinterpretation. :)

 

I originally started this thread in order to hammer ...the necessity of separating inductive and deductive conclusions.

Sorry about the delay in my response but it is the Christmas season and retired life can get full quickly. :confused:

 

Have fun – Dick

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

 

Happy Holidays to You & Yours Doc! I am making my own fun as it were too.

So, I have quoted some sections of your last post & also placed red highlights on those items I think germain. What has led me to my current understanding - including significantly this discussion;thank you Doc - is that since we have to take time to resolve disagreements into agreements, we have first come to a single agreement; that is, we must agree on the "definition" of "one". I have below the "definition" of "one" from my Websters Illustrated Encyclopedic Dictionary Edition 1991 [under Fair Use I have posted & paraphrased].

Happy Holidays Again, Turtle

 

one___adjective. 1. Designating a single entity, unit, object, or being. 2. Characterized by unity; of a single kind or nature; undivided. 3. Designating a person or thing that is contrasted with another or others. 4. Designating a specified but indefinate thing or time. 5. a. Designating a certain person,

especially a person not previously known or mentioned. b. Designating an indefinate time or occassion. 6. A or an. Used informally as a substitute for the indefinate article for emphasis. 7. Single in kind; alike or the same. 8. Being unique of a specified or implied kind.

___noun. 1. a. The first cardinal number; the first positive whole number after zero. b. A symbol representing this such as 1, I, or i. 2. A size or thing designated as one. 3. A single person or thing; a unit. 4. The first in a series. 5. A one dollar bill or coin. b. One hour after midnight or noon. 7. British Informal. A humorous or jocular person. 8. -a rightone. British Informal. A fool or nuisance.

___Pronoun. 1. A certain person or thing; someone or something. 2. Any person or thing; anyone or anything. 3. a. Any person representing the same, usually priviledged social class as the speaker. b. The speaker. 4. A single person or thing among persons or things already known or mentioned. -at one. In accord or unity. -in one. At the first in a single attempt. -one and

all. Everyone. -one another. Each other. Used to describe a reciprocal relation or action. -one by one. Individually and in succession.

[Doctordick]

So, I have quoted some sections of your last post & also placed red highlights on those items I think germain.

Well, I don't find them germain at all! I definitely get the impression you have no interest in the deductions I am talking about. B)

What has led me to my current understanding - including significantly this discussion;thank you Doc - is that since we have to take time to resolve disagreements into agreements, we have first come to a single agreement; that is, we must agree on the "definition" of "one".

I thought that I had made myself clear but I apparently failed. I pointed out that, in the absence of inductive conclusions, one has absolutely nothing to work with. In view of that I accepted without question certain inductive results which I listed in details; mathematics being one of the first.

But mainly I accept mathematics because of the great agreement about exactly what the symbols and operations mean (that is to say, the probability that your mathematical results of following my instructions will, if not exactly the same as mine, still map into mine in a very reliable manner). That is, we should be able to communicate!

Apparently you are not willing to accept these inductive constructs: per your opening discussion of the definition of one. I can only take your response to be a simple avoidance of the issue under discussion. Apparently we are not able to communicate. B)

 

Perhaps you do not comprehend the fact that dictionary definition of anything is worthless unless the meaning of every word in the definition is understood: i.e., any language is an inductive construct. The meaning of the word "one" is immaterial to my discussion. Your ability to perform the mathematical operations surrounding the symbol "1" is the material issue.

In other words, don't deny this initial deduction because your intuition tells you it is wrong;

Well, apparently you didn't, you just rejected it out of hand for no reason at all. Sorry to hear that! :confused:

 

I had expected more from you – Dick B)

 

"The simplest and most necessary truths are the very last to be believed."

by Anonymous

[Turtle]

Perhaps you do not comprehend the fact that dictionary definition of anything is worthless unless the meaning of every word in the definition is understood: i.e., any language is an inductive construct. "The simplest and most necessary truths are the very last to be believed."

by Anonymous

 

I fully comprehend the self-referencing construction of the/A dictionary. Start anywhere, in any dictionary, & look up every word the first definiton uses. Mark each word you read the definition of. Continue until all words have a mark. Nothing is left out. Completely self-similar; self-referencing; a container of all its parts. One. One labrynth. Go to your dictionary & find the difference between a labrynth & a maze. A maze has dead ends, wheras a labrynth does not. Perfect.

Kome on dok; think outside the tetrahedron. Are we debating, or are we knot?

Have fun yourself,

Turtle

[questor]

i have tried to follow and become interested in this conversation and i'm having trouble. once Doc has explained his thesis to Turtle, what will Turtle do with the information ? if we must consult the dictionary every time we have a conversation, there will have to be hand trucks for all the Webster's

Unabridged. in addition, everyone will have to follow Doc's demands for the proper phrasing and nuances of the word game. is there a great truth here that is unfolding ? or is this just semantic word games ?

[Turtle]

i have tried to follow and become interested in this conversation and i'm having trouble. once Doc has explained his thesis to Turtle, what will Turtle do with the information ? if we must consult the dictionary every time we have a conversation, there will have to be hand trucks for all the Webster's

Unabridged. in addition, everyone will have to follow Doc's demands for the proper phrasing and nuances of the word game. is there a great truth here that is unfolding ? or is this just semantic word games ?

 

Right on time Questor. You have struck the nail squarely on the head. It is not a word "game" in the sense that it is a great truth. I followed all doc's argument right to the root, the kernel, the "nut" of it. Which is, each person has to answer the question "Do I exist" for themselves." He tried to disqualify that root, which is why when I asked if all of A is in C on a Venn diagram. In order to make his deduction work, you have to deny you exist. I didn't, I don't, I wont.

By all means use whatever version of a dictionary you have, what ever language, edition, etc.. Take your time using the technique I described. Read the definition of just one word a day, & mark it so you know you were there. Then "spread out" as Mo used to say in the Three Stooges Films.

Make haste slowly. Tease out the nuts; then roast 'em & crack 'em & eat 'em.:hihi: