Opinions on the fundamental nature of reality.

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#1 Doctordick

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Posted 20 March 2006 - 08:37 PM

Since the validity of my "Fundamental Equation" doesn't seem to interest the people who could varify it, I thought I might comment on the implications. If the equation is false, then of course none of the implications are meaningful; however, suppose it is true just for the fun of it. The first thing is that, as Qfwfq has commented on many occasions, being an abstract logical construct, it has absolutely nothing to do with reality. On the other hand, it has everything to do with how reality could be interpreted. It is nothing more or less than a possible way to interpret information such that the interpretation is always guaranteed to be perfectly consistent with the information. And that is not an easy thing to do when the volume of the information to be dealt with becomes excessively large.

The equation does nothing more or less than provide constraints on predicting expectation consistent with what is known. You have all taken these tests where you have been given a list of numbers and asked what the next number is. The object is to find a pattern and deduce the next number. The goal of understanding anything is finding patterns which you can use to predict what will come next. Anyone familiar with mathematics knows full well that there exists an infinite number of ways to mathematically fit any finite set of known numbers. The process of mathematically fitting a set of given numbers is most easily seen in a graphics presentation. In such a case one is essentially performing a process commonly called “interpolation”. To see the range of possibilities I would suggest the http://en.wikipedia..../Interpolation'>wikipedia entry on the subject.

There are probably an infinite number of ways to fit a collection of data all of which become unstable when the data doesn't actually follow method. That is why experimental data fits are usually done by what is called "least square" methods.

In many respects, my equation is actually little more than another way of fitting a set of points. Dealing as it does with probability of a number and not with the actual number, it has a number of freedoms not found in methods which requiring an exact fit. One of the effects is that the expectations predicted by the equation are "uncertain". That hits on two fronts; first the predictions are not uniquely fixed and second it doesn't assume the data on which the predictions are made are exactly known. Those are certainly valid aspects of reality.

Now let's look at the simplicity of the equation. As Qfwfq has commented, it bears a striking resemblance to Quantum Mechanics equations (one of the single most dependable theories by the way). I think that those who know a little about Quantum will support the fact that the equation is exactly what one would expect of a universe which consisted of nothing except massless infinitesimal spinners. That's a pretty simple universe! Well, simple except for the far reaching consequences of the extreme numbers involved and the great number of possibilities in the Feynman sums of virtual exchange phenomena. There's a whole lot of room there. What is actually astounding is the fact that I can show that most all of modern physics comes directly out of that equation; all the way from Classical Mechanics to General Relativity. It unites modern physics in much the same way Maxwell's equations united E & M a hundred and forty years ago.

Actually it is roughly analogous to Newton's simplification of astronomical orbits. It just happens to be my great misfortune to be born during the existence of a powerful well indoctrinated academy wholly convinced the need for their extremely complex collection of theories is absolute and incontrovertible. They react to me about how I think the great followers of Ptolomy with their celestial spheres mechanically driven by complex cycles and epicycles would have reacted to the idea that the moon was just plain falling? They would find the idea just too simple minded to consider. How about a little help shaking their cages?

Have fun -- Dick

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

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Posted 21 March 2006 - 07:32 AM

Well! itried to go through your paper on the net, and I must confess it was not easy, although I have significant training in maths (I had taken a PG course in topology while I was a chem. graduate student). But, what so ever I could comprehend is as follows: ( I hope you will indulge with me, go through it and let me know if that is what you mean)

"You trying to find some logical framework for the scientific explanations offered in response to various questions that arise during scientific discourse. According to you, whenever a question is raised, a part of the knowledge framework is already present. The question arises because the curious is not able to connect the various knowledge elements present in the questions and is either unaware of several other knowledge elements that are relevant. Therefore a scientific explanation should clearly point to these elements or to some new elements that are required.":confused:

This is my comprehension, it may be your idea or not I am not very sure. I often indulge with teenagers, trying to satiate their natural curiosities about science. The statergy I normally adopt, is to cunter question the curious about the various elements in his question. In the process I try to help him/her to find remember connections between them. Often I have to tell them about some concepts and phenomena they may be unaware of.

I am often successful.

Perhaps my practice is the reason, I comprehended your theory in the way I did.

#3 Doctordick

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Posted 21 March 2006 - 03:42 PM

( I hope you will indulge with me, go through it and let me know if that is what you mean)

I will try!

The first step is to define an explanation. Unless you define what you mean by "an explanation" how can you expect others to know what you mean by the expression?

You don't seem to buy my definition, "an explanation" is "a method of obtaining expectations" from "given known information".

The following is an example of "an explanation" I could have given to my wife Sunday morning specifying what my categories C and D refer to. The explanation begins with given information which can be seen as fitting different categories brought up in my paper: category C, actually true; category D, a fabrication which can not be disproved without further information or perhaps a mix of the two. For an example of C,"I came home drunk Saturday night." An example of D, "two guys held me down and poured whiskey down my throat". Any method of obtaining expectations has two parts, the information on which it is based (maybe C, maybe D or maybe a mix) and the rule (already understood by my wife) as to what expectations go with that given information: i.e., drunks are not expected to conform to decent social behavior.

Maybe that hypothetical example is not a good explanation but it is an explanation and if my wife had understood it when I came home her expectations would have been exactly what happened. Essentially what I am saying is that every explanation is “a method of obtaining expectations”: i.e., a story leading to specific results! It's my story and I'm sticking to it!

I have done the best I can to make my definition clear. Would you do me the favor of telling me what you mean by "an explanation"?

Have fun -- Dick

#4 hallenrm

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Posted 22 March 2006 - 01:15 AM

Thanks DoctorDic!

I see no difference between your POV and mine.Let me now put forward my story.

A child asks me, Why is there a rainbow in the sky? Now, if he is familiar with the various phenomena around light, I can try to give the standard explanation about the formation of light, but in case she does not, she will not buy my explanation, till I am able to demonstrate it (additional information (maybe C, maybe D)). If she has an open mind she will be perhaps convinced, otherwise she may prefer the conventional explanation that rainbows are the acts of gods.

#5 Doctordick

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Posted 22 March 2006 - 04:49 PM

I see no difference between your POV and mine.

Does that mean that you accept my definition of "an explanation"? That it is a method of obtaining one's expectations from known information?

Let me now put forward my story.

I don't understand why you gave your story. Are they supposed to be examples of "explanations" which don't fit my definition or "explanations" which do fit my defintion.

From my perspective, both are explanations. I don't know that I would think of them as being of the same quality; but that is a totally different issue. You seem to regard the "standard explanation" as better but I won't hold you to that opinion. "Rainbows are the acts of gods", is none the less an explanation. Of course, in both cases, you omit to bring forth a lot of information bearing on the question. If one accepts the explanation, "Rainbows are the acts of gods", that acceptance usually includes a lot more information than you give here. As I said the explanation is a method of obtaining your expectations. In this case one's expectations are identical to what they expect the gods to do and, if I am to understand their explanation, it requires I understand what they think the gods want to do. They could think the gods are capricious or maybe the gods like to paint the sky after a rain. Whatever their thoughts are, the explanation is a method of generating their expectations.

There is nothing absolute, nothing the ultimate truth, nothing eternal, everything is transitory. There is more then enough scope for you and me to explore, to find the next level of truth, it only depends on our will to do so.

Ah but what is really in your past is eternal and unchanging; all you can do is add to it or forget parts of it; you cannot actually change anything that is already there.

Have fun -- Dick

"The simplest and most necessary truths are the very last to be believed."
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#6 Erasmus00

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Posted 22 March 2006 - 07:11 PM

I don't have a ton of time right now to go write a very detailed response, but I did read through your paper.

It seems to me like you recover quantum mechanics only because you WANT to recover quantum mechanics. If you had chosen to represent your explanation in a different way, it seems you could have easily recovered classical mechanics.

Once you have decided that any explanation can be represented by a wave function analog (your psi function), you recover "conservation of momentum" by exploiting the same translation symmetry expected of a free particle in quantum mechanics.

Now I ask, what if our set of information A cannot be mapped continously, but must be mapped discretely to the number line? How then can we use a wave function psi to represent our "explanation?"

Sorry if this isn't making much sense, I'll go into more detail sometime next week when I have a bit of time. Forgive the haste.
-Will

#7 Doctordick

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Posted 22 March 2006 - 09:18 PM

I don't have a ton of time right now to go write a very detailed response, but I did read through your paper.

Thank you and a detailed response is not necessary. Either you understood what I was saying or you misinterpreted something. If you understood it all no response is really necessary and if you misinterpreted something, you might as well stop with the first difficulty.

It seems to me like you recover quantum mechanics only because you WANT to recover quantum mechanics.

Let us say that I understand the fundamental reasons quantum mechanics works. And you are right, the vanishing of that derivative is closely related to conservation of momentum. In quantum mechanics the relationship arises because of translation symmetry which is exactly the consequences of the fact that no particular point has preference over any other. In my presentation, where I convert element references to numerical notation, exactly the same symmetry exists as no particular number has preference over any other.

In any language, no particular symbol has any inherent preference over any other but the fact is only usefully handled in a numeric representation. So that fact, thought of as "conservation of momentum" is embedded in any explanation of anything. It is a fundamental truth.

If you had chosen to represent your explanation in a different way, it seems you could have easily recovered classical mechanics.

My only real answer to that is, show me. I personally am convinced it is not true (except as an approximation to truth).

Once you have decided that any explanation can be represented by a wave function analog (your psi function)

I don't think the word "decided" is accurate there. I might say, once I defined expressing expectations by probability ... . It seems to me that once one decides that any expectation can be expressed via probabilities, the representation of probabilities by a vector inner product is pretty general. I don't think you can come up with a probability function which cannot be so represented.

Now I ask, what if our set of information A cannot be mapped continuously, but must be mapped discretely to the number line? How then can we use a wave function psi to represent our "explanation?"

Now you ask a very serious question; one which requires you to really understand the fundamental underlying nature of an explanation. The number of elements in A may be finite but you can not prove that. The fact that, no matter how much you know of A, you must allow for the possibility of an element of A you have never encountered, requires you to consider A to be an infinite set (that is the very definition of infinity). It follows that a valid explanation must yield a result for an infinite number of possible B(tk)'s. Remember, the actual information you have to work with is C, a finite collection of B(tk)'s.

The equation does nothing more or less than provide constraints on predicting expectation consistent with what is known. I am sure you have taken one of these tests where you have been given a list of numbers and asked what the next number is. The object is to find a pattern and deduce the next number. The goal of understanding anything is finding patterns which you can use to predict what will come next. Any computer scientist knows any information can be reduced to a collection of numbers. And anyone familiar with mathematics knows full well that there exists an infinite number of ways to mathematically fit any finite set of known numbers. So given this tremendous quantity of pure numbers, just how should one go about deciding what number should be next? Clearly for any symbol, language or number, the absolutely best one can really do is to make an estimated of the probability for a given symbol based on the collection of symbols you have already seen. What we are looking for is a rational way of predicting expectations consistent with what is known, and our method must be reasonable with unbelievably large amounts of information. Think about the common approaches to such a problem.

One common procedure used to fit a set of points to a power series. Now a power series can be made to fit any finite volume data but the problem is that it is quite often a poor predictor for new data (a poor model of the situation). The fault with a power series fit is that the predictions are highly unstable and only give correct answers when the data actually are explicitly represented by a power series. There are probably an infinite number of other ways to fit a collection of data; all of which become unstable when the data doesn't actually follow that method. That is why experimental data fits are usually done by what is called "least square" methods.

My equation is actually little more than another way of fitting a set of points. Dealing as it does with probability of a number and not with the actual number, it has a few freedoms not found in methods which require an exact fit. One of the effects is that the expectations predicted by the equation are "uncertain". That hits on two fronts; first the predictions are not uniquely fixed and second it doesn't assume the data on which the predictions are made are exactly known. Those are certainly valid aspects of reality.

And finally, if you are asking me if an explanation which presumes A consists of a finite set of discrete elements (thus a discrete set of number references) can be represented by my equation, the answer is yes. Psi becomes what is often called a salt and pepper function (it's not continuous) and the number of elements in D become infinite. And, lastly, the vacuum of the real space representation (that is the absence of real elements) becomes essentially infinitely hard: i.e., it is totally filled with virtual element pairs of all possible energy and the exchange forces become so high that nothing can move (the whole collection becomes one infinitely hard object). A somewhat unrealistic circumstance.

Sorry if this isn't making much sense, I'll go into more detail sometime next week when I have a bit of time. Forgive the haste.

I like haste. What I hate is no response at all. A simple complaint about a single factor is a wonderful response. And your complaints made a lot of sense.

Have fun -- Dick

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

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Posted 23 March 2006 - 01:48 AM

.........................
..........
Ah but what is really in your past is eternal and unchanging; all you can do is add to it or forget parts of it; you cannot actually change anything that is already there.

"The simplest and most necessary truths are the very last to be believed."
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Ah!!! but my total past is not eternal, it is changing every moment. Nor am I eternal, so how can be my present past?

#9 Qfwfq

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Posted 23 March 2006 - 03:54 AM

My equation is actually little more than another way of fitting a set of points.

Perhaps we are gradually getting somewhere, but I still don't see why you keep comparing your method to ones which "require an exact fit".

The least squares method is the case of a more general method, when the distribution for each measurement value is known to be gaussian. More in general one can maximize the product of probability densities, since the measurement results are independent events. How does your psi compare with this? Is it even more general? In what way?

I was also wondering if your talk about symmetry and conservation intends to be an improvement on, or a replacement of, Nöther's theorem, or what. I don't see a need for this. Or were you just saying that Emmy's all-fundamental theorem holds in your formalism?

#10 fivish

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Posted 24 March 2006 - 03:53 AM

If you write in such a way that nobody can understand you, whats the point?
Writing hundreds or thousands of words that have no clear meaning wastes everybodys time. I have read many documents and books on all sorts of scientific theories, but they are all flawed by being overly long. No theory should need more than a pamphlet let alone a book.

Reality is very simple.....

Stuff exists.
'I' experience the stuff thorugh my senses.
'I' interact with the stuff with my physical being.

We will never know what 'stuff' is or why its here.
We will never know why 'I' am me and not another 'I'.

Reality is just stuff. Real stuff.

Now go get a life.....................

#11 infamous

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Posted 24 March 2006 - 09:17 AM

Reality is just stuff. Real stuff.

This is a science forum fivish, so please support your opinions with some evidence.

1. Define stuff
2. Define real

Now go get a life.....................

I suggest that you read our FAQ and rules page fivish, suggesting that someone should "go get a life" sounds a bit insulting to me and considering your post count, I don't think you have earned the right to correct our membership as yet.................Infy

#12 Doctordick

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Posted 25 March 2006 - 07:37 PM

Perhaps we are gradually getting somewhere

As far as I can see we are gradually getting nowhere. I have a suspicion you are grossly over estimating the significance of my comment. "My equation is actually little more than another way of fitting a set of points". That comment goes to nothing more than a general description of the fundamental content of the equation; analogous to bringing up a sheep to a shepherd in an attempt to communicate the idea of "one". I only put it there to dissuade people from the unwarranted illusion that I was putting forth some method of obtaining conclusions; those too dense to comprehend that a model of "an explanation" is a model of the abstract concept "an explanation" created to logically display its structure and subtle nuances, and is not a model of "what the explanation explains". I can only guess that your real purpose in this endeavorer is to set things up for some gross misinterpretation of what I am talking about so you can create some contrived straw man to tear to pieces.

My equation constitutes nothing more than a convenient representation of an absolute constraint upon any internally consistent explanation of anything (not the only one by the way; there are others). I have no idea where your argument with its validity lies. If you won't accept my definition of "an explanation" just let me know. I am not insisting you accept my definition but I would certainly like to know where you think the flaw is and what you think you mean when you use the term "an explanation".

And, furthermore, I cannot comprehend your position. Is it your position that there are no constraints on a valid explanation? If there are no constraints, how does one tell an invalid explanation from a valid one. I really feel as if I am talking to MagnetMan in another guise here. Read my response to him as I get the feeling it applies to you just as well.

, but I still don't see why you keep comparing your method to ones which "require an exact fit".

I had no idea that I had made a penchant for comparing my work to things which "require an exact fit".

The least squares method is the case of a more general method, when the distribution for each measurement value is known to be gaussian. More in general one can maximize the product of probability densities, since the measurement results are independent events. How does your psi compare with this? Is it even more general? In what way?

I have no idea how what you are talking about has any bearing at all on anything I have said other than material for your straw man.

I was also wondering if your talk about symmetry and conservation intends to be an improvement on, or a replacement of, Nöther's theorem, or what. I don't see a need for this. Or were you just saying that Emmy's all-fundamental theorem holds in your formalism?

If you don't see a need for "symmetry arguments" you clearly have no comprehension of what I am doing (which actually seems to be very much a fact). I was aware of "symmetry arguments" long before I ever heard of Emmy Nöther and none of the symmetry arguments I ever heard as a student invoked her name or her perspective. I have no idea which came first, "symmetry arguments" or Nöther's theorem but I really doubt the theorem arose from an intellectual vacuum.

Sometimes, when I read comments of the so called experts on this forum, I am very much reminded of something said by Kenneth McLeish concerning medieval scholarship. "... scholars became archivists rather than innovators, museum curators rather than original thinkers. ... European universities became filled with magnificently reconstructed texts which everyone revered but no one bothered to relate to living beings who had created and enjoyed them in the first place, and with experts who knew the size and shape of each brick in every ancient wall, but could neither see the walls themselves or understand their purpose." I think he is saying "they couldn't see the woods for the trees".

Nevertheless, for those who are ignorant of "symmetry arguments" and/or Nöther's theorem, the relationship between symmetries and conserved quantities was laid out in detail through 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 fundamental physics accepted by everyone. The problem is that very few students think about the underpinnings of the circumstance but rather just learn to use it.

It is not necessary to know Nöther's theorem in order to appreciate the power of symmetry arguments. Many professors 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: Next, suppose the Lagrangian L has a symmetry, meaning that it doesn't change when you apply some one-parameter family of transformations sending q to some new position q(s). At least he tells you what he means by a symmetry. Symmetry is another of these things that is commonly "understood" on an intuitive level without much thought. :redface:

What I would like to point out is that any symmetry is essentially an expression of a specific ignorance. 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 yields conservation of momentum via Noether's theorem, 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 not available in the statement of a problem cannot exist in the solution of the problem: no algebraic procedure can produce a result unless that result is implicit in the given starting point. 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 is it possible to solve a problem for specific expressions of q when changing q has no impact on the problem? The solution is actually quite simple: there must be another relationship implicit in the statement of the problem which relates the range of possibilities for q (the transformations Baez refers to) to the various possible specific solutions. In shift symmetry, this required relationship is conservation of momentum; in rotational symmetry, the required relationship is angular momentum.

I have actually heard professors say that symmetry arguments are a means of obtaining information without invoking any assumptions and this is what makes them the most powerful arguments which can be made. But let's think about that for a moment. Noether's theorem is a mathematical result and, as such, cannot possibly produce anything which is not implicitly contained in the axioms. Ignorance cannot be the true source of our result; it must be arising from some other source. Its real source is the fact that we are using a mental image of the problem which requires more information than is available and, in order to represent the problem, one must assume some value for the unavailable information.

Since any specific solution must embody a specific value for that information (position of the origin for example in shift symmetry), that solution is only valid if changing all q produces no changes in any relationships. Look at the definition of a differential with respect to q of any function related to that solved problem. The differential is defined to be the limit as a goes to zero of the function evaluated at q+a minus the function evaluated at q. That means that our solution is only valid if the differential with respect to q vanishes. It is nothing more or less than an unstated axiom inherent in our presumed representation.

Any competent computer programmer is well aware of the fact that any language may be expressed in terms of a numerical code and that shift symmetry exists within that code: i.e., adding a specific number to every existing representation of every symbol of that language. It follows that one cannot create a language without the axiom that a differential related to that shift must vanish. It is just hard to express that idea within the confines of the presumed solution (the language itself) because people tend to think that the meanings of words are inherent to the words themselves and that there is no such thing as shift symmetry in a human language. The fact that this presumption is false is evident in the existence of pig Latin which is no more than a case of every word having a specific shift in representation. The translation into pig Latin makes utterly no change in the meaning being expressed and the difference in meaning (a function of what is said) vanishes perfectly.

Now, as to how what I have said relates to Nöther's theorem, I will only comment that I do not hold myself to be an authority on the theorem. I will bow to Baez's authority and according to him, "if someone claims Nöther's theorem says "every symmetry gives a conserved quantity", they are telling a half-truth. The theorem only applies to certain classes of theories."

Have fun -- Dick

"The simplest and most necessary truths are the very last to be believed."
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#13 Erasmus00

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Posted 25 March 2006 - 09:51 PM

Alright,before I make any more comments, I thought I should be sure I understand the claim. There seem to be several parts:

1. Any set of information can be mapped to some part of a hilbert space (or at least an inner product space?) You call this your set A.

2. What we desire from what you call an explanation, is that (with an explanation) given ANY points contained in A, we can build the whole of A using our explanation. You call these given points C.

3. Next, you attempt to establish constraints on your explanation. You introduce a set D, which contains information presumed true. You assert (though I don't know if I follow the logic) that D can be mapped to that part of our hilbert space that isn't occupied by A. These constraints lead to:

4. Now, I'm unsure if your equation is supposed to be the explanation or not. Given a collection of points C, can we use the equation to drive us around Hilbert space untill we have built up all of A?

Also, I'd be interested in seeing your "explanation" model used on a particularly simple set A(say an explanation of the integers, or a dihedral group).
-Will

#14 HydrogenBond

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Posted 26 March 2006 - 01:44 PM

hallenrm; A child asks me, Why is there a rainbow in the sky? Now, if he is familiar with the various phenomena around light, I can try to give the standard explanation about the formation of light, but in case she does not, she will not buy my explanation, till I am able to demonstrate it (additional information (maybe C, maybe D)). If she has an open mind she will be perhaps convinced, otherwise she may prefer the conventional explanation that rainbows are the acts of gods

This sort of hits at the heart of reality. Reality come down to what can be percieved and understood. It also demonstrates that the best theories can be useful at all levels of perception and understanding. If one was to explain to a child or to the world's smartest person that the planets go around the sun in orbits. This is reality because we all see and understand the same thing.

As theory or explanations become more schewed to either direction, i.e., toward the child or toword the expert, those in the middle often have a problem; maybe it is not reality. The leoprocaun and rainboy is good for children while other space/time dimensions is good for the physics buff, but these perceptions of reality lose everyone in the middle because they defy common sense because they have no everyday proof.

Don't get me wrong science needs to inquire and speculate to improve our understanding. But it also has an obligation to stay in the mainstream reality. If I look at the universe around me I see hydrogen and electrons, stars and galaxies, not quarks, strings, dark energy. Call me old fashion but what I see is also what the majority of humans see. Even the child or expert can be made to see this. This is reality.

Mathemathics is something that can get to the point where only a handful can truly see. Almost everyone else has to rely on second hand opinion to be made to see, without being able to rely on one's own sensory systems or common sense. This may not be reality but only the perception of a small group. I personally believe if the guy in the streets can not reach the same gist indendantly using some basic arguments and some data, it is probally not entirely reality.

When I was younger I had a bit of a problem buying into special reativity. What changed me was actual data of the half-life of particles getting longer at near light speeds in a particle accelerator. The proof was in the pudding instead of some mathematical abstraction. It became more than a abstraction to me because common sense said it refected reality.

#15 Doctordick

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Posted 26 March 2006 - 08:09 PM

Will,

I am sorry but I don't think we are communicating. It seems to me that you are jumping to unwarranted conclusions without knowing exactly what is meant by what I say or without understanding what I am doing. In addition, you seem to have missed the point that I am modeling the abstract concept "an explanation" created to logically display its structure and subtle nuances, and am not a modeling "what the explanation explains", a totally different thing.

1. Any set of information can be mapped to some part of a Hilbert space (or at least an inner product space?) You call this your set A.

You seem to be jumping the gun here. The set A is not being mapped into any space at all; what is being mapped into a space are a collection of references to the elements of A which go to make up C (which is what is known of A). Secondly, referring to the space as a Hilbert space is rather extreme considering that the space being used is no more than a simple three dimensional Euclidean space. The only purpose the sets A, B and C serve is to give me a way of referring to the fact that explanations are based on incomplete information and that support for an explanation arrives through successful prediction. A very simple concept but difficult to display in an analytically general manner.

2. What we desire from what you call an explanation, is that (with an explanation) given ANY points contained in A, we can build the whole of A using our explanation.

You seem to have the cart before the horse here. Nowhere do I say anything about what we want from the explanation. What I said is that I defined an explanation to be a method of obtaining our expectations of a specific B(t). Of course, a "good" explanation would be one which was consistent with C (what is known): that would be the fact that the existence of a particular B(t) in C would be consistent with what our expectations would have been if C consisted of all B prior to t. In plane English, that would be our expectations (given our explanation) are consistent with our experiences. Again a rather simple concept which looks complex when expressed analytically.

You call these given points C.

Backwards again. C is what is known. The points in (x,tau,t) space are no more than references to the elements taken from A which comprise the description of what is known.

3. Next, you attempt to establish constraints on your explanation.

I am afraid I don't know where you are in paper and have no idea what you are referring to here. You seem to be looking at some distorted overview and missing the details. I would prefer for you to look at this thing one line at a time and make sure you understand each line as it comes up.

You introduce a set D, which contains information presumed true.

It doesn't "contain" the information presumed true; it is the information presumed to be derived from A which is not actually derived from A. Of course there exists no way of determining what is C and what is D but a general model of the concept of an explanation certainly must allow for the fact that some of the things we think are necessary parts of A are not really necessary. The only way we can know they are not is if their existence eventually turns out to be inconsistent with with the explanations; however, until that moment occurs, C and D must be perfectly indistinguishable.

You assert (though I don't know if I follow the logic) that D can be mapped to that part of our hilbert space that isn't occupied by A.

The logic is quite simple (and does not require the complex concept of Hilbert space). The number of elements of A making up any element of C is finite and the number of elements of C is finite so the total number of points plotted in that (x,tau,t) space is finite. That leaves an infinite number of points in the (x,tau,t) unused and available to represent any references to elements of D.

Again, I do not know what constraints you are referring to.

4. Now, I'm unsure if your equation is supposed to be the explanation or not.

The explanation is Psi, a function, the definition of which constitutes the procedure for obtaining expectations. The equation expresses the constraints on the explanation. What you must remember is that all information upon which the explanation is based is included in C and D. Most common explanations hold a lot of information as already having been explained.

Given a collection of points C, can we use the equation to drive us around Hilbert space until we have built up all of A?

The only thing I can get from this question is that you seem to think that I am modeling "what the explanation explains".

Also, I'd be interested in seeing your "explanation" model used on a particularly simple set A(say an explanation of the integers, or a dihedral group).

Again, you seem to be confusing modeling the concept "an explanation" with modeling a particular explanation. What ever explanation of "the integers" you have in mind, it can be cast into my model; however, to do so would do nothing to clear up your difficulties as I don't think you understand what I am doing.

And a "simple" representation of your explanations in my model certainly would not clear up the necessity of my equation at all. The necessity of that equation arises directly from the fact that C must include absolutely all information upon which the explanation is based. If the explanation is to be in English, C must include sufficient information to deduce the English language and there is no way to think of that problem as "simple".

I think what you need to do is start with the paper and discuss each sentence (one sentence at a time) to make sure you understand exactly what I am saying. Once you understand exactly what I am doing, I think you will find the logic compelling and straight forward.

Looking forward to your response -- Dick

#16 Erasmus00

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Posted 26 March 2006 - 09:34 PM

I have cleared up some of my confusion, especially regarding the meaning of your set D. Also, at least part of the last bit was a simple misunderstanding, I was using A, B and C to be, interchangabely, our initial sets of information AND the portion of hilbert space in the image of the mapping of each set to the space. Sorry for the confusion.

And a Hilbert space is simply a space defined with a positive deffinate inner product. The 3D space we are using must have such a function inner product for psi to make any sense.

And a "simple" representation of your explanations in my model certainly would not clear up the necessity of my equation at all. The necessity of that equation arises directly from the fact that C must include absolutely all information upon which the explanation is based. If the explanation is to be in English, C must include sufficient information to deduce the English language and there is no way to think of that problem as "simple".

What I was requesting was that you apply your model of an explanation to something ALREADY mathematical, not because I believe your equation unneccesary but because I'd like to see it applied to something.

For instance, the integers have a natural mapping to the number line, so why not give a concrete example by using your equation to work through an explanation of the integers. (not in English, but in mathematics).
-Will

#17 hallenrm

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Posted 27 March 2006 - 01:53 AM

Now. I think, is the time to draw the attention of the viewers of this thread to my earlier thread "the language of elites