"a Universal Representation Of Rules"

[Rade]

If we make no assumptions at all, then we know absolutely nothing about the circumstances being explained

Why do you say this ? Suppose a circumstance, I fall down the steps because I step on a marble. Why do I need to make any "assumptions" about the circumstance ? Seems like a clear fact that (1) I fall down the stairs, (2) the marble was the cause. It is an event anyone could communicate clearly without use of any assumption. The problem you are having explaining yourself here is that you need to first define what you mean by "assumption" and what you mean by "knowledge of circumstance". After we have those two new definitions in terms of "Doctordick Speak", then we can decide if what you claim about the relationship of assumption to knowledge to explanation is based on correct reasoning.

 

DD said..."The first [assumption] is that an explanation to be represented exists"

This is nonsense. Many circumstances have absolutely no explanation at all, they just happen. Being in love is a circumstance, and it is a circumstance that can exist with absolutely no "explanation" that exists. Not all circumstances can be represented mathematically by an explanation. Given this fact, there is no reason for anyone to continue reading any further, for, as you say, your presentation is based on two assumptions, and the first is false.

[Bombadil]

The first point of the above is, if I know an explanation and the language required to communicate it I certainly know the circumstances that explanation explains and the ontological elements required by that explanation. Thus laying out those circumstances I wish to represent is quite a simple job. I can define every one of the ontological elements required by that explanation in that known language and attach a different numerical label to each every definition. It follows that the circumstance of interest can be represented by a collection of those numerical labels. (They are neither continuous nor infinite.)

 

[math](i_1,i_2,i_3,\cdots,i_n)[/math]

 

So I can obviously represent those circumstances; however, that representation utterly fails to satisfy the requirements I have set for my representation. My goal is to design a representation which makes no assumptions whatsoever. The representation just given is capable of representing only a specific explanation: i.e., it makes the assumption that the given explanation is the only one I want to represent. That is a constraint on the representation which fails to be entirely general.

 

It fails to be general only from the prospective that it requires that the elements are the elements in the explanation that assigned the labels, in fact every explanation has such a representation. The real problem seems to be that this representation cannot be communicated because it is unique to the explanation and so can only represent what the explanation is representing with it.

 

The “x” index used to represent the underlying noumena being represented by the ontological elements labeled by the “i” index provides a mechanism for bridging that constraint. In order for you to understand how the “x” index manages that feat. Let us consider two different explanations for exactly the same circumstances. Either explanation can be represented by the notation given,

 

[math](i_1,i_2,i_3,\cdots,i_n);[/math]

 

however, a number of difficulties arise. One, the second explanation may require different ontological elements and a different language to express those ontological elements. Thus the “i” indexes used in the two explanations may refer to totally different ontological definitions. But both explanations are explaining the same known circumstances thus every “i” index from one explanation can be tied to an “i” index from the other explanation (they are referring to exactly the same collection of noumena). Which noumena is being referred to is identified by the numerical label “x”.

 

That is the x labels are a set of labels we agree to use to refer to the ontological element. And we can communicate the x label and by doing so we can then use the i label used in our explanation. The problem that I have here is where do the x labels come from then, are we just assuming that such a representation exists.

 

This leads to the possibility that an i label may well refer to multiple x labels, this is not really a problem as the i labels are supplied purely by the explanation and so considering two elements to be the same is really a question of how things are being explained. We simply allow the possibility that two different x labels will refer to the same i label we can think of them as the same element. The possibility that more then one x label will be referred to by a single i label is also possible and we can simply allow this possibility as well.

 

The possibility that two i labels will refer to a single x label also exists, this is somewhat more problematic as, if we try to use the x label which i label is being referenced is not known from the given information, this can be solved by adding the tau axis and giving each element a unique location in this new x,tau coordinate system so that the i labels will refer to elements with different tau labels.

 

In essence, once we introduce a hypothetical element (which is required by the explanation under discussion) that hypothetical element must be viewed as being as real as any of the other elements the explanation explains. Your complaint that rotation can remove meaning of the “x” axis from the representation is no more meaningful than to note that the opening orientation omitted meaning from from the “tau” axis. It is the “x,tau” plane which is required by the explanation. The fact that “tau” is not part of the “known information” is no different from the fact that the Tortoise moves: it is an essential part of the explanation and must be handled as if it is real in analyzing the consequences of that explanation.

 

Then is it also impossible to say that there doesn't exist explanations in which the x axis has no meaning and the tau axis is the axis of interest?

 

The point is that the tau axis is necessary for distinguishing elements apart and so is a necessary consequence of how you are defining an explanation. It supplies a necessary constraint on the elements of interest. Without it there is no way to distinguish between different elements being at the same x location.

 

My complaint is not will a rotation remove meaning from the x axis but rather I want to know if this symmetry or more precisely will rotations in the x,tau plane give meaning to the tau axis in such a way that we can no longer ignore the location on the tau axis when considering the probability of an element being at some location or set of locations in the x,tau plane.

 

The infinite limit in the x case is not so trivial. Extending F to the limit of infinite data would cause the x variables to be continuous and that continuity brings a bit of a problem into procedure of adding hypothetical elements. The single most significant step in generating that table of F was adding hypothetical elements such that all circumstances represented in the table were different. When the number of elements in that table are extended to infinity, we run directly into Zeno's paradox. We cannot list an infinite number of cases thus, in the limit, we cannot know that every x argument in every listed circumstance is different from every other x argument in that circumstance. The argument for hypothetical elements being able to differentiate between circumstances fails.

 

The issue here is and what Zeno's paradox is all about is not that the x axis is continuous but rather that if we assume that it is it implies that there is in fact an infinite amount of information that must be kept tracked of and by definition we can't list an infinite amount of information. Weather or not the variable is continuous is not an issue, as to include every possibility we must allow for it to be continuous.

 

In fact there is no way to prove if it is continuous or not as to do so would require an infinite amount of information and there is no way that we can make such a list.

 

The point is that whatever is necessary in our explanation can't be proven to be right or wrong as it is an integral part of our explanation. So if our x axis is allowed to be continuous we can't prove otherwise much as the tau axis is a required part of our explanation we can't prove that is doesn't exist because it is needed for the definitions that are being used.

 

In this case this has the effect of meaning that to allow a continuous x axis and to insure that all elements will be included in the information we must make a further assumption about them. The assumption that we are going to use is to make any elements that may be listed an infinite number of times asymmetric with respect to exchange.

[Doctordick]

Hi Bombadadil, Sorry I have been so slow to respond.

 

It fails to be general only from the prospective that it requires that the elements are the elements in the explanation that assigned the labels, in fact every explanation has such a representation.

Yes, that is true. Your further comment seems to express an understanding of the difficulty; however, it doesn't really seem clear to me.

 

The real problem seems to be that this representation cannot be communicated because it is unique to the explanation and so can only represent what the explanation is representing with it.

I would have said that my original purpose was to establish a universal representation of any circumstances without making any assumptions. The representation, as it is defined at this point, can only represent a specific known explanation.

 

The real problem revolves around the fact that we are trying to represent something we do not know. That is a fundamental difficulty almost impossible to get around. How can we possibly represent something without knowing anything about what it is we are representing; we simply have no way of generating the lists I refer to.

 

...if I know an explanation and the language required to communicate it I certainly know the circumstances that explanation explains and the ontological elements required by that explanation.

That is essentially an “if-then” statement that is quite clearly not applicable without assuming we know what we are representing. I said I wanted to assume nothing, thus that is not a usable prescription.

 

The solution is to comprehend that I never said I was going to “explain all explanations”. What I said was that I was looking for constraints implied by the definition of an explanation, quite a different thing. Since my interest is discovering implied constraints, I am free to look at abstract representations of known explanations; we have lots and lots of such things to examine. That process is not limited by the assumption of no information and, as I point out, it implies some rather interesting relationships.

 

Thus it is that I examine two explanations of exactly the same circumstances. Now I don't actually lay out what these explanations explain but as you have stated above, since I have assumed that I do know them, I can perform the label listing operation defined by the notation and examine the possible variations which have to be handled.

 

That is the x labels are a set of labels we agree to use to refer to the ontological element. And we can communicate the x label and by doing so we can then use the i label used in our explanation. The problem that I have here is where do the x labels come from then, are we just assuming that such a representation exists.

The x labels are the unknown and undefined ontological elements (that is why I use “x” to refer to them) and I have no knowledge of what they actually refer to. However, I do have the two explanations which explain the two identical circumstances. When I am comparing the two explanations (and using the representation to represent one), I can (for the sake of comparison) temporally assign labels from one explanation to the x while I look at the representation of the other via the “i” labels. The explanation actually has nothing to do with the labels I choose and everything to do with handling the various complications which arise in such a representation.

 

This leads to the possibility that an i label may well refer to multiple x labels, this is not really a problem as the i labels are supplied purely by the explanation and so considering two elements to be the same is really a question of how things are being explained. We simply allow the possibility that two different x labels will refer to the same i label we can think of them as the same element. The possibility that more then one x label will be referred to by a single i label is also possible and we can simply allow this possibility as well.

This appears to be an accurate accounting of the various possible complications inherent in the circumstance being represented.

 

The possibility that two i labels will refer to a single x label also exists, this is somewhat more problematic as, if we try to use the x label which i label is being referenced is not known from the given information, this can be solved by adding the tau axis and giving each element a unique location in this new x,tau coordinate system so that the i labels will refer to elements with different tau labels.

Essentially what I was saying though the idea that these labels can be seen as positions in an x,tau coordinate system can not be taken to imply the existence of that space. The “existence” of such a space only arises if one allows the number of labels to become infinite: i.e., the possible circumstances to be examined constitute a continuous range.

 

Then is it also impossible to say that there doesn't exist explanations in which the x axis has no meaning and the tau axis is the axis of interest?

This question is somewhat incoherent. First, neither the x axis nor the tau axis is ever without meaning; they are both nothing more or less then a way of displaying that finite number of indices used to represent the finite number of elements used in the two explanation. The “x” numerical label, by definition labels a known ontological element (known within the explanation being represented) whereas the “tau” label is hypothetical. The fact that it is hypothetical has nothing to do with the character of the representation and everything to do with the calculation of your expectations.

 

When you go to calculate your expectations you must handle the fact that, “you don't know and have no way of knowing the correct label to apply to the circumstance whose expectation you are calculating.” This is not a problem of the representation, it is a problem associated with calculating those expectations. Exactly the same problem may arise in calculating your expectations regarding the numerical labels associated with the “x”; there may exist one or more such labels where you do not know exactly what label should be applied to that specific element.

 

In order to make that issue a little clearer to you, suppose you have a simple circumstance where you want to use your explanation (which is correctly represented in all its glory) to determine the expectation for some specific outcome. So “Rade” is always good for a simple minded example:

 

Suppose a circumstance, I fall down the steps because I step on a marble. Why do I need to make any "assumptions" about the circumstance ? Seems like a clear fact that (1) I fall down the stairs, (2) the marble was the cause.

That is a rather simple minded representation of the circumstances. A correct representation of the circumstance would require including billions upon billions of other elements. The steps stand on something which are part of another collection of things which is in turn part of many other describable entities. A correct representation would require a specific description of what “I” refers to, what fall down means what steps are and what a marble is together with more elements than I could enumerate in a lifetime.

 

Essentially, Rade has assumed that none of these aspects of the circumstance are significant. That is fine, if he wants to know the expectation under the assumption that these aspects are insignificant, the correct approach would be to estimate the probability each and every one of these additional elements (all the elements he has omitted) would have on his expectations.

 

Now the means of doing that is quite straight forward. To do it correctly, one would have to take every possible variation on each element (together with the probability of that specific variation) and the impact of that variation on the elements of the circumstances he knows. Summing the outcome over every possibility consistent with his explanation would give the correct result. Now, if one “assumes” that the impact of every variation possible is the same and sums over all possibilities, the correct answer for all omitted elements is “one”. The subtly here is that one times the expectation you would have obtained by ignoring these element is exactly the expectation you would have obtained by ignoring these elements and that is exactly what Rade has done. The problem was that, when he made that post, it was totally beyond his comprehension that he was doing that. As far as I am aware, he absolutely never takes the trouble to think anything out, that is why he is back on my ignore list.

 

So the central theme here is that, from a representation perspective, hypothetical is no more than saying, “I have no idea as to what the correct label is” whenever I go to estimate the implied expectations”. As far as knowing whether of not the specified element really exists, there is no information on that issue at all; if there were, that information could be used to prove it did or didn't exist. If you can prove it doesn't exist, the explanation is invalid!

 

By the way, in modern physics, the presumption is always made that, if you can't prove it doesn't exist, it exists. That is exactly the policy taken by religious faiths and is in fact the real conflict between science and religion. How can the scientist presume something exists when he can not prove the existence is necessary (how come he gets an out and the religionist doesn't?) Both parties point to their experiences in predicting outcomes to justify their beliefs. Since the Middle ages, most religious arguments have been shown to be tautological constructs whereas scientific arguments have not. But this is all beside the present point.

 

My complaint is not will a rotation remove meaning from the x axis but rather I want to know if this symmetry or more precisely will rotations in the x,tau plane give meaning to the tau axis in such a way that we can no longer ignore the location on the tau axis when considering the probability of an element being at some location or set of locations in the x,tau plane.

Rotations in a representation don't add or remove meaning to things. That meaning resides in the explanation being represented. All I am concerned with is being able to represent all possible circumstances and examining variations which do not change the meanings implied in the explanation.

 

Weather or not the variable is continuous is not an issue, as to include every possibility we must allow for it to be continuous.

Why don't we say, “if we want our explanation to be valid all the way to an infinite amount of information, we can't assume it isn't continuous”. Proving it one way or the other is obviously impossible; which is exactly your comment.

 

In fact there is no way to prove if it is continuous or not as to do so would require an infinite amount of information and there is no way that we can make such a list.

 

...

 

In this case this has the effect of meaning that to allow a continuous x axis and to insure that all elements will be included in the information we must make a further assumption about them. The assumption that we are going to use is to make any elements that may be listed an infinite number of times asymmetric with respect to exchange.

Well, I really don't like you referring to that as an assumption. It really isn't. Requiring the function [math]\vec{\Psi}[/math] to be asymmetric with respect to exchange with regard to some of those arguments will indeed solve the problem we have run into; however, it isn't an assumption as any function can be written as a sum of symmetric and asymmetric components. I tried to google the relationships and found nothing directly presenting the issue. It should be kind of obvious to you that it has to be true. If you still have difficulties, let me know and I will try to explain the issue to you.

 

Again I am sorry I took so long -- Dick

[Bombadil]

The real problem revolves around the fact that we are trying to represent something we do not know. That is a fundamental difficulty almost impossible to get around. How can we possibly represent something without knowing anything about what it is we are representing; we simply have no way of generating the lists I refer to.

 

On the other hand without a representation something we can examine how can we have any idea of what it is that we are trying to represent.

 

Which is exactly the problem that you get around when you say.

 

The solution is to comprehend that I never said I was going to “explain all explanations”. What I said was that I was looking for constraints implied by the definition of an explanation, quite a different thing. Since my interest is discovering implied constraints, I am free to look at abstract representations of known explanations; we have lots and lots of such things to examine. That process is not limited by the assumption of no information and, as I point out, it implies some rather interesting relationships.

 

That is we are not really interested in actual explanations or representations but rather it is the constraints that are implied by the definition of an explanation that we are interested in. We still have to deal with the question of if we are dealing with a general definition or not. But the only real way for this to be a problem is if someone can come up with a definition that your definition doesn't have an equivalent explanation of and that can't include your definition, of course we would have to agree that it is in fact an explanation and not something else as well.

 

Essentially what I was saying though the idea that these labels can be seen as positions in an x,tau coordinate system can not be taken to imply the existence of that space. The “existence” of such a space only arises if one allows the number of labels to become infinite: i.e., the possible circumstances to be examined constitute a continuous range.

 

I get the impression that maybe we are talking about different things here. I am suggesting that the (x,tau) labels are a space in the sense that we are free to define a set of operations that we can use on them, not in the sense that they form a continues set on their own rather they only form a set of unknown ontological elements that we use the (x,tau) variables to represent.

 

The problem here is that if we want to be able to manipulate the labels (say by defining a rotation in a (x,tau) plane), I can see no way of doing this without defining some kind of relationships between x and tau that will form a plane or even just a set of operations that can be performed on x or tau. The point is, won't these relationships form some type of group. All that I am saying is don't we have to define a mathematical structure to the representation of (x,tau). Otherwise ideas like when you suggest taking a derivative seem doomed to fail because there is no where to define the limit in x or tau.

 

Well, I really don't like you referring to that as an assumption. It really isn't. Requiring the function [math]\vec{\Psi}[/math] to be asymmetric with respect to exchange with regard to some of those arguments will indeed solve the problem we have run into; however, it isn't an assumption as any function can be written as a sum of symmetric and asymmetric components. I tried to google the relationships and found nothing directly presenting the issue. It should be kind of obvious to you that it has to be true. If you still have difficulties, let me know and I will try to explain the issue to you.

 

Perhaps assumption was a bad choice of words, I am seeing it as a simple and elegant solution to the problem, actually I think that it wouldn't take much effort to prove that such a statement is true, all that would be needed is to say that since two of these elements will never occupy the same location the probability of them occupying the same location is zero so that

 

[math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)=-\vec{\Psi}(\vec{x}_j,\vec{x}_i)=0 [/math]

 

if [math] \vec{x}_i = \vec{x}_j[/math] and since we are only interested in

 

[math] P(x_1, x_2, x_3,\cdots, x_n, \cdots, t) \equiv \vec{\Psi}^\dagger(x_1, x_2, x_3,\cdots, x_n, \cdots, t)\cdot\vec{\Psi}(x_1, x_2, x_3,\cdots, x_n, \cdots, t) [/math]

 

the sign of [math] \vec{\Psi}(\vec{x}_i,\vec{x}_j) [/math] has no effect so that we may define them to be antisymmetric. The only problem that I have is isn't this going to replace one symmetry with another one. Of course since we are only interested in the constraints placed on the explanation it seems like this really cannot limit [math] P(x_1, x_2, x_3,\cdots, x_n, \cdots, t) [/math] in any way.

 

 

Again I am sorry I took so long -- Dick

 

I didn't think that you took that long, if any one took a wile to respond its me.

[AnssiH]

Just couple of comments which may be helpful.

 

The real problem revolves around the fact that we are trying to represent something we do not know. That is a fundamental difficulty almost impossible to get around. How can we possibly represent something without knowing anything about what it is we are representing; we simply have no way of generating the lists I refer to.

 

On the other hand without a representation something we can examine how can we have any idea of what it is that we are trying to represent.

 

Think about this; it is possible to generate expectations about some information whose real meaning remains unkown (via inductive reasoning), but in doing so, the information, or those expectations, need to be expressed in some terminology. The form (or "language") of that terminology is completely immaterial, as long as the expressed expectations themselves are valid.

 

Note that we cannot actually check the validity of the terminology of our explanation, we can only check the validity of the expectations it provides. Albeit, it is a common error to think that the validity of those expectations also proves the validity of the rules; that is just a case of ignoring the infinite number of other ways to express the same expectations; it is a case of confusing our own representation of reality with the actual reality (or to confuse "the map with the territory").

 

So here's the important bit;

 

Perhaps one good way to look at this is to first think about the fact that it is possible to generate expectations about some data without ever getting to actually know its meaning, by interpreting it in means of defined entities whose existence is hypothetical.

 

...

 

That is we are not really interested in actual explanations or representations but rather it is the constraints that are implied by the definition of an explanation that we are interested in. We still have to deal with the question of if we are dealing with a general definition or not. But the only real way for this to be a problem is if someone can come up with a definition that your definition doesn't have an equivalent explanation of and that can't include your definition, of course we would have to agree that it is in fact an explanation and not something else as well.

 

His definition of explanation is not "something that explains the meaning of undefined information". It is "a procedure which will provide rational expectations for hypothetical circumstances."

 

Note again, to provide expectations does not entail to provide the meaning of, it just entails that some hypothetical meaning is provided to represent the information, i.e. the existence of each defined entity is hypothetical.

 

That is the central issue behind all the philosophical talk about ontology and epistemology, and that is exactly what we mean when we supposedly "explain" something. If we already knew what it was, we would not have anything to explain. If we don't know what it is at all, it is essentially undefined, and our explanation of it can never be checked in terms of "absolute meaning". Our explanation is very much a procedure for providing expectations, with absolute freedom to choose whatever terminology we want to, as long as it provides valid expectations and is self-coherent.

 

You can probably understand this commentary about the issue;

http://scienceforums.com/topic/22957-the-most-critical-question/page__view__findpost__p__308604

 

You also asked about continuity and DD made comments about Zeno's paradox. To continue with what I said above, note that the actual data behind our explanations consists of a finite amount of circumstances or finite amount of information, and it is up to the choice of terminology whether some information is taken to mean there exists something continuous there; it is up to the explanation to interpret something in terms of "Achilles passing the Tortoise". At no point there exists data that contains explicit information of anything passing anything, or anything moving anywhere, or anything existing at all. It is the terminology which constitutes definitions (or rather assumptions) of continuous things, but they always refer to some set of finite information.

 

Note that various assumptions about continuity have led to various inconsistencies within our explanations of reality. A main example perhaps the ultraviolet catastrophy, which led to new definitions; quantification of certain electromagnetic properties, and eventually to quantum mechanical definitions.

 

-Anssi

[Doctordick]

Hi Bombadil, reading your post over, I get the impression that you pretty well understand the abstract situation I am trying to work with; however, I think a few points could be clarified.

 

 

AM said:

 I get the impression that maybe we are talking about different things here. I am suggesting that the (x,tau) labels are a space in the sense that we are free to define a set of operations that we can use on them, not in the sense that they form a continues set on their own rather they only form a set of unknown ontological elements that we use the (x,tau) variables to represent.

 

Yes, we are both definitely talking about two different things here and that can be a confusing conversation. In the original discussion of numerical reference indices, there are two very distinct concepts being brought up: first, the idea of using numbers as reference labels attached to the finite number of concepts being referred to and second, the fundamental nature of the source of those labels (the set of real numbers). Saying that these labels are selected from the infinite continuum of real numbers is not at all the same as saying the things being referred to are infinite and the two concepts must be kept separate.

Likewise, when I bring in the tau index (for the purpose of marking different occurrences of the same x label) that presumes another set of real numbers totally different from the first set of real numbers. Since all tau labels are paired with a specific x label, the actual label for a given element now consists of two numbers. Whereas the original label was to be chosen from the range of real numbers, the new label must be chosen from the range of two paired numbers. Mathematicians often refer to that first range as a line and the range defined by the collection of all possible pairs of numbers is generally called a Euclidean plane. Any discussion of a Euclidean plane brings in the concept of rotations (essentially what tau points are associated with what x points): i.e., once again the range from which the labels are chosen is infinite; however, this in no way implies the number of labels required is infinite.

That difference must not be allowed to confuse the significant issues. Just as “shift symmetry” does not require the x indices to be infinite, the rotational symmetry of the x, tau plane implies nothing about how these numbers are to be chosen or that the range and configuration of these numbers have anything to do with what is being represented.

Again, you are confusing two very different things here. The existence of the function [math]\vec{\Psi}[/math] is guaranteed by virtue of the fact that the explanation exists: i.e., there exists a method of obtaining expectations for specific defined circumstances. Putting that together with the general representation of those circumstances shows that shift symmetry (in those numerical labels) requires:

 

Otherwise ideas like when you suggest taking a derivative seem doomed to fail because there is no where to define the limit in x or tau.

 

 

[math]
P(x_1,x_2,\cdots,x_n,t)=P(x_1+a,x_2+a,\cdots,x_n+a,t)
[/math]

 

 

for all possible choices of “a”. That very fact guarantees that the function P yields the same result for all a: i.e.,

 

 

[math]
P(x_1+a+\Delta a,x_2+a+\Delta a,\cdots,x_n+a+\Delta a,t)-P(x_1+a,x_2+a,\cdots,x_n+a,t)=0
[/math]

 

 

Which tells us something very important about the nature of the function P. If “a” is a shift parameter (identically applied in all arguments), the derivative of P with respect to “a” must vanish. Division by [math]\Delta a[/math] does not alter the truth of the above assertion but the limit of the result as [math]\Delta a[/math] goes to zero turns out to be a very important mathematical operation. This tells us nothing about the possible continuity of the arguments of P; it is no more than a required characteristic of the function P. The further work I developed guarantees that the function [math]\vec{\Psi}[/math] must be bound by the requirement that

 

[math]
\sum_{i=1}^n\frac{\partial}{\partial x_i}\vec{\Psi}(x_1,x_2,\cdots,x_n,t)=ik\vec{\Psi}(x_1,x_2,\cdots,x_n,t)
[/math]

 

And, once again, this has nothing at all to do with the continuity of the arguments but relates only to the nature of the function itself.

 

You sort of have the whole thing backwards. There is a simple procedure for creating a function which is asymmetric under exchange from a function which is not. First you must understand that any function of many variables may be seen as a function of two variables (just see all the other variables as constants; essentially exactly what is done when one speaks about defining partial differentiation).

If you start with any function of two variables, f(x,y), interchange the two variables and subtract that from the original function, divide the result by two and one has a new function of the form

 

 

Perhaps assumption was a bad choice of words, I am seeing it as a simple and elegant solution to the problem, actually I think that it wouldn't take much effort to prove that such a statement is true, all that would be needed is to say that since two of these elements will never occupy the same location...

 

[math]

F(x,y)=\frac{f(x,y)-f(y,x)}{2}
[/math]

 

Notice that F(x,y)=-F(y,x): i.e., it is asymmetric with respect to exchange of those two arguments. Given this new function, do the same with any other two arguments getting exactly the same result with regard to the new set. Just keep doing that over and over until you have exchanged every possible pair. At that point, you have generated a function which is asymmetric with respect to exchange of any pair of arguments.

Now interchange the two variables in the original function and add them together. Dividing by two one obtains a function of the form

 

[math]
G(x,y)=\frac{f(x,y)+f(y,x)}{2}
[/math]

 

Notice that G(x,y)=G(y,x): i.e., it is symmetric with respect to exchange of those two arguments. If this procedure is continued over and over until you have exchanged every possible pair, the resultant function will be symmetric with respect to exchange of any pair of arguments. But, at each step, G(x,y)+F(x,y) = f(x,y): i.e., when you finish the procedure the resultant sum will still be the original function. It follows that the original function can be seen as a sum of two parts, one symmetric with respect to exchange and the other antisymmetric with respect to exchange.

When I said you had the whole thing backwards I was referring to your assertion of

 

[math]
\vec{\Psi}(\vec{x}_i,\vec{x}_j)=-\vec{\Psi}(\vec{x}_j,\vec{x}_i)=0
[/math]

 

as a starting point. That is the conclusion. Clearly, if the function is asymmetric with respect to exchange,

 

[math]
\vec{\Psi}(\vec{x}_i,\vec{x}_j)=-\vec{\Psi}(\vec{x}_j,\vec{x}_i)
[/math]

 

the value of the function must vanish if the two arguments are the same. If [math]\vec{x}_i = \vec{x}_j[/math] there is utterly no difference between the two functions and the only number equal to its own negative is zero: i.e., the probability of [math]\vec{x}_i = \vec{x}_j[/math] must vanish identically.

 

 

 

Of course since we are only interested in the constraints placed on the explanation it seems like this really cannot limit [math] P(x_1, x_2, x_3,\cdots, x_n, \cdots, t) [/math] in any way.

 

 

Oh yes it can. We cannot actually constrain [math]\vec{\Psi}[/math] in any way (except for validity: i.e., it is required to yield the known information). Getting the expectations of a specific explanation consistent with all possible circumstances means that we must allow a symmetric [math]\vec{\Psi}[/math] into the picture (remember, the function was defined to be taken from the set of "all possible functions"; however, the elements referenced by the symmetric portion of [math]\vec{\Psi}[/math] can not be technically known: i.e., they must essentially be hypothetical. This means that any actual evaluation of the related expectations needs to take into account all possibilities. From a quantum-mechanical perspective it reflects the fact that Bosons (the standard name given to things with symmetric wave functions) can not be localized as individual entities. Something physicists don't really like to talk about much; that is why they always speak of them in terms of “Fields”.

But we will get into that kind of stuff down the road.

Have fun -- Dick

 

Of course since we are only interested in the constraints placed on the explanation it seems like this really cannot limit [math] P(x_1, x_2, x_3,\cdots, x_n, \cdots, t) [/math] in any way.Quote

Edited May 27, 2014 by Doctordick

[Bombadil]

Likewise, when I bring in the tau index (for the purpose of marking different occurrences of the same x label) that presumes another set of real numbers totally different from the first set of real numbers. Since all tau labels are paired with a specific x label, the actual label for a given element now consists of two numbers. Whereas the original label was to be chosen from the range of real numbers, the new label must be chosen from the range of two paired numbers. Mathematicians often refer to that first range as a line and the range defined by the collection of all possible pairs of numbers is generally called a Euclidean plane. Any discussion of a Euclidean plane brings in the concept of rotations (essentially what tau points are associated with what x points): i.e., once again the range from which the labels are chosen is infinite; however, this in no way implies the number of labels required is infinite.

 

While I don't have a problem believing it can we be sure that anything can be represented by use of real numbers. I suspect that it can be done as I know that the complex numbers can be shown to be equivalent to a real plane in two dimensions. But can we be sure that there is no representation that can't be obtained from some real space.

 

Actually even if such a thing exists it seems to be an insignificant issue as we would only need to define some kind of operation that we could use as addition and we could keep going.

 

Notice that G(x,y)=G(y,x): i.e., it is symmetric with respect to exchange of those two arguments. If this procedure is continued over and over until you have exchanged every possible pair, the resultant function will be symmetric with respect to exchange of any pair of arguments. But, at each step, G(x,y)+F(x,y) = f(x,y): i.e., when you finish the procedure the resultant sum will still be the original function. It follows that the original function can be seen as a sum of two parts, one symmetric with respect to exchange and the other antisymmetric with respect to exchange.

 

But what about the norm of F(x,y) that is what we are interested in isn't it? We are only interested in [math] \vec{\Psi} [/math] because of how we have defined its norm so, by making it antisymmetric or symmetric to exchange of elements have we limited how it's norm can behave in any way other then at the location of other elements?

 

The problem that I am having is that I don't see the necessity of either symmetry except in neighborhoods of the zeros of [math] \vec{\Psi} [/math] unless there is also the issue of distinguishing between elements. That is, that since we can't distinguish between different elements we have to have

 

[math] P(x_i,x_p,\cdots,t)=P(x_p,x_i,\cdots,t) [/math]

 

which actually makes a lot of sense as otherwise P would no longer be a function of patterns of elements, and the question of what element is where would have meaning, which implies we had to distinguish the order of the elements from the start, which would make us ask the question of how we distinguished the elements in the first place, without knowing the resulting explanation.

 

Actually now that I think about it I can't think of any other possibility other then that P is symmetric with exchange of its elements, although I don't remember reading this anywhere, although it seems that it must be true.

 

Oh yes it can. We cannot actually constrain [math]\vec{\Psi}[/math] in any way (except for validity: i.e., it is required to yield the known information). Getting the expectations of a specific explanation consistent with all possible circumstances means that we must allow a symmetric [math]\vec{\Psi}[/math] into the picture (remember, the function was defined to be taken from the set of "all possible functions"; however, the elements referenced by the symmetric portion of [math]\vec{\Psi}[/math] can not be technically known: i.e., they must essentially be hypothetical. This means that any actual evaluation of the related expectations needs to take into account all possibilities. From a quantum-mechanical perspective it reflects the fact that Bosons (the standard name given to things with symmetric wave functions) can not be localized as individual entities. Something physicists don't really like to talk about much; that is why they always speak of them in terms of “Fields”.

 

Do you mean that on a mathematical level we can't distinguish between two bosons?

 

I have to wonder at this point will this symmetry manifest as a force that will keep all antisymmetric elements away from each other and cause symmetric elements to clumped together?

 

Also might it not be interesting to derive from the fundamental equation the equations for just symmetric and just antisymmetric elements or are these going to have the same form as the fundamental equation?

 

There is a subtle thing going on here. The existence of F is a consequence of our ability to add hypothetical elements which will make every entry to the “what is” is “what is” table unique. The possibility of also adding hypothetical elements which lend nothing to that end also exists. The subtle consequence is that these elements may have nothing to do with establishing the existence of F but none the less influence the form of F. We once again come to the conclusion that there are most probably an infinite number of functions F which fit the given information exactly but yield different probabilities for the new (or unknown) data: i.e., there exist many different explanations even in the continuous infinite limit.

 

So after we have added all of the antisymmetric elements and we know that all values of t have a unique set of elements associated with them, can we then add any number of symmetric elements without worrying if each set of elements can be distinguished from each other?

 

Since [math]\delta(x)[/math] only has value for x=0, a power series expansion of F around a distribution satisfying F=0 implies that F may be written

[math]

F=\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0.

[/math]

 

Thus it is that we come to the conclusion that any appropriate collection of rules can be expressed in terms of those hypothetical elements which can exist and that interactions at a distance in our hypothesized space can not exist. As an aside, it is interesting to note that Newton, in his introduction to his theory of gravity, made the comment that it was obvious that interactions at a distance were impossible. I have always wondered exactly what he had in mind when he said that. I take it to mean that, although field theories make some excellent predictions, they cannot be valid in the final analysis and are only an approximation to the correct result.

 

But this is only true for the possible [math]\vec{x}_i[/math]. That is this equation will define the set of possible [math]\vec{x}_i[/math] values. On its own though isn't this a rather weak function in the sense that it is only a discreet function and combined with the idea of having only a finite number of elements will create situations in which only a small number of possibilities have been removed.

 

Isn't the possibility of an element being in any location where F =0 still the same? Or is this all going to create a situation of constructive and destructive interference between waves that will in fact have a noticeable influence on the likelihood of where elements will be found?

[Doctordick]

Hi Bombadil,

I was slow to answer your latest post because I really don't know how to answer you. Most of your questions lead me to think that you have made some of the same errors in your thinking that seem to confuse most everyone who has posted replies to my threads. I made a few comments to you concerning aspects of modern physics which I thought were consistent with conclusions I reached but did not ever intend for you to take these as any support to my deductions. When I made those comments, I was under the impression you understood the nature of my proof. Now, I am not so sure. If your confusion arose because of those comments, I apologize.

There are two very different issues which I have presented in my threads. I tried to keep those issues in different threads but I did occasionally make connecting comments. The issue of my fundamental equation and its proof can be found in the following threads: “Laying out the representation to be solved.”,
 

Remember, I am talking about “any possible explanation”, not about any specific explanation and the notation must be able to handle any possible set of circumstances.

 

A central issue concerning the purpose of that thread apparently missed by most everyone but Anssi.

Conservation Of Inherent Ignorance!,
 

 

[math]
\sum^n_{i=1} \frac{\partial}{\partial x_i}\Psi(x_1,x_2,\cdots,x_n,t)=ik\Psi(x_1,x_2,\cdots,x_n,t)
[/math].

...And...

[math]
\frac{\partial}{\partial t}\Psi(x_1,x_2,\cdots,x_n,t)=iq\Psi(x_1,x_2,\cdots,x_n,t)
[/math].

 

 

The central issue of that thread was the universal power of “symmetry” arguments and how the specified necessary symmetries could be embedded in the definition of [math]\vec{\Psi}[/math]. Anyone familiar with advanced mathematics should be aware of the fact that differential equations define the form of mathematical functions. The validity of the above equations is required by the described symmetries.

and, finally, "a Universal Representation Of Rules”
 

This is the final crux of my proof of the fact that any flaw free explanation can be represented by a mathematical function which is required to satisfy my “fundamental equation”:

[math]
\left\{\sum_i \vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}\beta_{ij}\delta(\vec{x}_i -\vec{x}_j) \right\}\vec{\Psi} = K\frac{\partial}{\partial t}\vec{\Psi}.
[/math]

 

 

The comment, “any flaw free explanation can be represented by a mathematical function which is required to satisfy” that equation, is apparently overlooked by most people. They seem to think the proof is asserting something beyond that issue.

You have taken my comments about fermions and bosons to have something to do with the proof. They do not! They were in there for one reason only: to inform the reader that the mathematical consequences of exchange symmetry are a seriously important issue and not trivial.

In this proof, exchange asymmetry is used for one reason only: as a mechanism to guarantee no identical index reference will exist in any circumstance specifying known information even when the volume of known information is allowed to extend to infinity. That issue has to do with our ability to construct that table of F as a function of those circumstances from the the table of known values of [math]\vec{\Psi}[/math].

Although we cannot ever consider and infinite amount of “known information” we must always allow for more than we are currently explaining. That relationship is itself the very definition of infinity; thus whatever intellectual mechanisms we use in the proof must not fail as the volume of information goes to infinity.
 

Those three posts constitute the proof of my equation

 

Nothing outside those OP's is required to prove my equation is required by the definition of “an explanation”. That being said, I will make a few comments on your post.
 

... can we be sure that there is no representation that can't be obtained from some real space.

 

You are presuming the existence of “some real space”. The only presumption we are working with in my proof is, “communication” (a language) can be represented with a finite number of index-able definitions. If some explanation can't be communicated, I have no interest in it and I do not understand why you would concern yourself with such a problem.
 

But what about the norm of F(x,y) that is what we are interested in isn't it?

 

The procedure described has absolutely no impact on normalization. Normalization is established after the algebraic form of the function is established: i.e., we have a supposed solution. Remember the only issue bearing on normalization that we don't want the integral over the arguments to ever exceed “one” as otherwise it cannot be interpreted as a probability. The normalization is a procedure applied to the entire function not a characteristic at a point. The word “norm” is modern physics jargon referring to the resulting dot product (which is to yield the probability density of a specific circumstance). This is an exact example of your pulling in issues of modern physics which are of no interest to the proof.
 

The problem that I am having is that I don't see the necessity of either symmetry except in neighborhoods of the zeros of [math] \vec{\Psi} [/math] unless there is also the issue of distinguishing between elements. That is, that since we can't distinguish between different elements we have to have

[math] P(x_i,x_p,\cdots,t)=P(x_p,x_i,\cdots,t) [/math]

which actually makes a lot of sense as otherwise P would no longer be a function of patterns of elements, and the question of what element is where would have meaning, which implies we had to distinguish the order of the elements from the start, which would make us ask the question of how we distinguished the elements in the first place, without knowing the resulting explanation.

 

That whole paragraph is completely confusing. I don't think you understood what I was doing. There is no necessity of either symmetry so long as the number of presumed known elements is finite. I have introduced hypothetical elements in order to assure that I can (by that introduction) create a table of additional interesting (and important) functions. The creation of hypothetical elements are used to assure that all circumstances are unique thus allowing me to use that table of “known circumstances” to generate tables of these other functions which agree with the known information. The existence of the function [math]t(x_1,x_2,\cdots,x_n)[/math] consistent with the known information is thus assured.

This whole operation is concerned with the constraints implied by the definition of “an explanation”. I laid out my representation of circumstances and defined the function [math]\vec{\Psi}[/math] in such a way that “if I knew a specific explanation of a collection of specific circumstances” I could obtain the same expectations from that function as I did from the explanation. Where does this phrase “without knowing the resulting explanation” come from? I get the impression that you missed the whole point of the exercise.
 

Actually now that I think about it I can't think of any other possibility other then that P is symmetric with exchange of its elements, although I don't remember reading this anywhere, although it seems that it must be true.

 

I have no idea what you are talking about here. I think you will have to go back and look at what I am doing and why I am doing it. I don't think you understand the issues.
 

Do you mean that on a mathematical level we can't distinguish between two bosons?

 

The issues you concern yourself are outside the interest of the proof. It was probably a terrible thing that I even brought up the issue of “non-localization” of bosons as I think it just confused you. I apologize.
 

I have to wonder at this point will this symmetry manifest as a force that will keep all antisymmetric elements away from each other and cause symmetric elements to clumped together?

 

Understand the proof! Issues outside the proof are of no interest. You should comprehend that the idea “force” plays no role in the proof.
 

Also might it not be interesting to derive from the fundamental equation the equations for just symmetric and just antisymmetric elements or are these going to have the same form as the fundamental equation?

 

Again this sentence makes no sense to me. You appear to be confusing solutions of the equation with the equation. At this point, we have no concern with the solutions at all. All possible solutions consistent with the definition of an explanation must be included. We are concerned only with constraints implied by the definition of “an explanation”.
 

So after we have added all of the antisymmetric elements and we know that all values of t have a unique set of elements associated with them, can we then add any number of symmetric elements without worrying if each set of elements can be distinguished from each other?

 

Again you appear to be talking about “solutions”. Don't worry about “exchange symmetric” functions. The constraint is only that [math]\vec{\Psi}[/math] must satisfy my differential equation. This symmetric/asymmetric issue concerns only the construction of that table of the function whose roots must be the known information.
 

But this is only true for the possible [math]\vec{x}_i[/math].

 

That is all we are concerned with.
 

... and combined with the idea of having only a finite number of elements ..

 

What, did you miss the entire discussion of handling things as the volume of known information goes to infinity? If all we are looking for are the constraints implied by the definition of “an explanation”, all possibilities must be handled and, in the limit of infinite known information, the simple procedure given for creating a table of hypothetical rules constraint fails. I think you are looking at aspects of the problem way ahead of the original problem being solved without understanding how and why the approach I present will work.
 

...will create situations in which only a small number of possibilities have been removed.

 

I again get the feeling you are concerned with solutions not with constraints.
 

Isn't the possibility of an element being in any location where F =0 still the same? Or is this all going to create a situation of constructive and destructive interference between waves that will in fact have a noticeable influence on the likelihood of where elements will be found?

 

Again with solutions!! Forget about solutions to my equation; that is not what I am talking about. I am talking about structuring the equation such that it exactly reflects the correct constraints implied by the definition of an explanation.

I think you need to go back to the beginning and make sure you understand what I do and why I do it at every step in the derivation. Sorry to be so hard on you but I think the problem is that you think the purpose of my work is to show you how to find solutions and it isn't. My equation is a tautological construct having nothing to do with reality.

Many of my other posts are concerned with the astounding fact that, as far as I can tell, modern physics is no more than a restatement of my equation. That has everything to do with the solutions but is of utterly no consequence without a proof that the equation is valid.

But developing and understanding my equation has nothing to do with the solutions. Don't start worrying about solutions until you understand why my equation has to be valid. I made the mistake of thinking you pretty well understood the whys and wherefores of that deduction and got into other issues I should never have brought up.

Go read my posts in the “Exactly What Is Tractatus?” thread. That thread originally had no connections with my proof but a lot of the misunderstood issues in my proof came up in some of the replies. If any part of any of my posts in that thread don't make perfect sense to you let me know on this thread and I will attempt to clarify whatever issues bothered you.

I hope I haven't upset you -- Dick

 

Edited May 27, 2014 by Doctordick

[AnssiH]

Hi Bombadil. Admittedly, some things in the OP can be hard to interpret unambiguously. I thought I would also try and offer some clarification.

 

The issue behind the OP is that, any explanation that is capable of generating useful expectations, also defines some rules as to how the defined elements are expected to behave. In expressing the universal constraints, we obviously don't want to make assumptions regarding what those rules would be. But we do want to express one important universal constraint; a constraint that can't be violated by any explanation that expresses some rules, regardless of what those rules themselves are.

 

As you already know, that universal constraint is [imath]\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0[/imath]. It is important to understand that it does not in itself tell us anything about what the rules of an explanation are, it is just something that a self-coherent explanation can't violate. Just like the symmetry arguments, it will only play a role in the final analysis, when we explore logically possible relationships between defined entities (i.e. what sorts of logical consequences one definition places on other definitions, as long as those definitions are required to be logically consistent with each others).

 

It is also important to understand the arguments behind that constraint, so I will explain them in my own words; sometimes a different perspective on the same thing is clarifying. I am writing this under the assumption that you have read the OP, so now you can compare your understanding to my comments, and hopefully clear out some ambiguities.

 

The "what is, is what is" analysis.

 

This is an explanation that would always interpret any newly accumulated piece of information as evident of an entirely new entity, as oppose to interpreting information in terms of the same entities persisting between circumstances.

 

You can think of "what is, is what is" as each circumstance being seen as some set of x values (the "i" index of each value is somewhat meaningless since each is unique through the entire accumulated information anyway), and the entire "past" is just collection of such circumstances. It is somewhat like a straightforward list of raw information.

 

Now the question arises, using this explanation, what would it mean to have expectations? Expectations beyond the accumulated "past" are meaningless, but what about a function that generates correct expectations about the known "past"?

 

What that means is a function that would tell you the probability that some specific x value exists in some specific circumstance, given the rest of the x values of that circumstance. Think of it this way; if you omit one x value from a circumstance, then the entire rest of that circumstance is the context for that x value; if you have a valid explanation, then the context of an element is what tells you what to expect the missing element to be. (or in this case, what the missing x-value is)

 

Since we are only talking about the known past, the function needs to yield either a probability of "1" or "0" for any query; either our query agrees with the known information, or it doesn't.

 

Thus a valid explanation of this terminology means you'd have a function that can recover any missing x value from the entire known past, given its context.

 

Now the subtleties. Imagine 2 circumstances that are entirely identical, except for one x-value. If you omit that one x-value, no mathematical function can tell from the rest of the x-values which one is missing. I.e. you would have two different x-values whose context is the same. It would appear as if that same context is to be taken as evidence of two different x-values. But we are talking about known past; it would be invalid for the rules to yield a 50% chance of either value.

 

Note that if you have accumulated two different circumstances that suggest as if the same context can surround two different elements, that in itself can just be seen as implying the existence of some unaccounted elements (i.e. "what caused the difference"). Otherwise your rules would be time-dependent.

 

There may or may not actually exist such an unaccounted cause, or perhaps reality just is random. But that is entirely irrelevant. What matters is that even if information has been accumulated in such manner as to create this problem, it is always possible to assign "hypothetical elements" to each circumstance so to ultimately make any index recoverable (so to be able to build a valid explanation in this terminology).

 

At the risk of making unwanted implications, let it be said that these types of "implied elements" are very typical in our everyday explanations too; if something behaves in unexpected manner, your original expectations were a function of how you interpreted the context (the situation), and the unexpected behaviour can be seen as implying that something unaccounted was affecting the situation. At the same time, we can't actually see what is the underlying noumenaic information underneath our interpretation of reality, and thus it is not really a meaningful question to ask which noumenaic elements are "implied" and which are "known".

 

But back to the analysis. During the "Step I" in the OP, we have only proven that it is possible to represent the rules of any set of known past information (i.e. to recover the missing values from their context), as long as it is a finite set that we want to explain.

 

Universal constraint on rules

 

Moving away from the "what is, is what is" explanation onto more general arguments, all useful explanations define elements that exist persistently over multiple circumstances. They may easily define rules and elements in a manner where those elements undergo continuous motion or have other continuous parameters. Or in terms of DD's notation, some explanations may contain rules, which treat the x-variables as if they represent continuous paths of defined elements, moving through multiple circumstances.

 

Suppose you have two otherwise identical explanations, except where one explanation sees its persistent elements as always "skipping" from one value to the next, the other one sees the values as transitioning in continuous manner.

 

The one with "skipping" elements would not have any problems in recovering any missing value given its context, but the one containing rules with continuous transitions, could contain representable (interpolated) circumstances where two different elements would cross each others. It would be possible to represent a situation where two values have come together, and in terms of recovering elements from their context, information would be lost. I.e, it would just mean that the rule definitions would be in some way logically incoherent; how could two different values also be the same value?

 

In other words, the statement;

 

[math]

F=\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0.

[/math]

 

can be seen as stating, that if an explanation violates this equation, its rules contain a logical self-conflict. It doesn't say anything about reality, it just says something about what is logically possible in terms of rule definitions in our "explanations".

 

One last subtlety to this story is the points about fermions and bosons, but as DD said, maybe you shouldn't worry about those things yet. Those are very specific definitions anyway, and at this point it just has got some curiosity value that exchange symmetric elements in quantum theory affect expectations via all their possible paths; they don't have defineable classical paths associated to themselves. I'm very shaky on that subject myself, but nevermind that, just concentrate on understanding the rest of the arguments to the fundamental equation, things will get more interesting in the various derivations of modern physics.

 

-Anssi

[Bombadil]

I was slow to answer your latest post because I really don't know how to answer you. Most of your questions lead me to think that you have made some of the same errors in your thinking that seem to confuse most everyone who has posted replies to my threads. I made a few comments to you concerning aspects of modern physics which I thought were consistent with conclusions I reached but did not ever intend for you to take these as any support to my deductions. When I made those comments, I was under the impression you understood the nature of my proof. Now, I am not so sure. If your confusion arose because of those comments, I apologize.

 

I suspect this may be a case of a little knowledge being a dangerous thing. The truth is that whatever I might know about Mathematics I know quite little about modern physics and it has been quite some time since I had much interest in the philosophy that physicists try to use to explain it to others that are not physicists.

 

Actually many of the things that physics seems to be based on seem more like magic then deductions, in the simple sense that the only defense seems to be that, it looks like what physics says it should look like, so physics must be right.

 

As a result when you start talking about things from physics and based on the nature of your deduction I can only conclude that you are trying to bring attention to something that your derivation is implying and that you are trying to compare it to something in physics. This is perhaps seldom your intent and I suspect that my trying to understand what you are pointing out has side tracked us more then once. For the time being it would perhaps be best if such things are brought up after we have finished this deduction and perhaps in a thread where such discussion belongs as I really don't know enough about physics for such comments to have any meaning to me except in the context of your derivation.

 

As for some of my comments I may have gotten carried away but I am starting to wonder what some of the things that you are pointing out mean to the behavior of an arbitrary explanation and not just that we need to, and how we can include such things in your derivation and there is no thread yet where such questions have come up as part of the purpose of the thread they have merely came up as a topic of no interest or not came up at all.

 

An explanation is a procedure which will provide rational expectations for hypothetical circumstances.

 

Remember, I am talking about “any possible explanation”, not about any specific explanation and the notation must be able to handle any possible set of circumstances.

 

That is, we are interested in the constraints that must be placed on any procedure for obtaining expectations for any possible circumstance without placing any constraints on what circumstances are being examined or making any assumptions about what is being explained.

 

The last part I think is confused by most people and where many people try to find a counter-diction. That is, it seems to me that everyone that tries to make a counter example or suggest that there is something that your deduction can't be equivalent to, are assuming what is being explained.

 

While you have not assumed what is being represented and you have allowed the possibility that what is being represented may be different then what people would consider to be observed, and all that you require is that any future information is consistent with the explanation or the explanation is flawed. Further more the idea of observing has no place in your deduction as there is no need to define anything and your deduction is in many ways based on the idea of what can be known without defining anything.

 

Clearly before such a thing can be done we must find a notation that is capable of representing any possible circumstance which is what much of “laying out the representation to be solved” is about.

 

Your second thread “Conservation Of Inherent Ignorance!” then shows how such a representation of elements implies things that appear to have no influence on the representation since there is no way to find them to include them in the representation. But in fact they define a set of symmetry's that any explanation must obey if it is not going to make assumptions. That is, appear to know things that it can't possibly know about the representation.

 

Then "a Universal Representation Of Rules" gives a method by which any possible rule can be represented and a means of including all of the deductions in a single equation.

 

As you already know, that universal constraint is [imath]\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0[/imath]. It is important to understand that it does not in itself tell us anything about what the rules of an explanation are, it is just something that a self-coherent explanation can't violate. Just like the symmetry arguments, it will only play a role in the final analysis, when we explore logically possible relationships between defined entities (i.e. what sorts of logical consequences one definition places on other definitions, as long as those definitions are required to be logically consistent with each others).

 

It is not the issue of setting up a representation of rules that I am having a problem with as it seems clear at this point that if we can include an infinite number of elements then we can represent any rule that we want. Of course this means that we must find a way that we can add an infinite number of points and still insure that no two points will occupy the same location. Making the points that we add antisymmetric when exchanged will definitely do this. But is it necessary? Just setting

 

[math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)=0 [/math]

 

if [math]\vec{x}_i=\vec{x}_j[/math] seems sufficient and so far there is proof that adding to this isn't just going to add an unneeded constraint.

 

For instance what if [math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)[/math] satisfied all of the requirements including [math] \vec{\Psi}(\vec{x}_i,\vec{x}_i)=0 [/math] except it has no exchange symmetries for at least one set of elements? Can such a function exist and if so can we be sure that another function that has the desired exchange symmetries will supply the same expectations?

 

There seems to be anther possible view that I tried to bring up in my last post but it probably got confused in all of the confusion that post brought up.

 

That point is, couldn't we have started the discussion of exchange symmetry by pointing out that we are making no assumptions concerning what each [math]x_i[/math] represents in [math] (x_1, x_2 ,x_3, \cdots ,x_n,t) [/math] and so the original order of listing is irrelevant and in fact we are only using this order for the subscripts as a convenience. They are really the i labels that you defined in the first place so they may even repeat.

 

This seems to mean that if we truly exchanged two different elements in the representation then the new representation is equivalent to the old representation and our expectations must be the same. This would mean that the original function [math] P(x_1, x_2, x_3,\cdots, x_n, \cdots, t)[/math] must be symmetric with exchange of its elements.

 

This would imply that every element must have one of the symmetries that you are requiring. This seems to make sense to me but if it's not true, why not? Wouldn't any other possibility mean that we have defined the elements in [math] (x_1, x_2 ,x_3, \cdots ,x_n,t) [/math] to have some meaning and so treated them differently based only on the order in which we included them in the representation.

 

What, did you miss the entire discussion of handling things as the volume of known information goes to infinity? If all we are looking for are the constraints implied by the definition of “an explanation”, all possibilities must be handled and, in the limit of infinite known information, the simple procedure given for creating a table of hypothetical rules constraint fails. I think you are looking at aspects of the problem way ahead of the original problem being solved without understanding how and why the approach I present will work.

 

You have pointed out a means by which we can insure that no two elements occupy the same location as the number of elements go's to infinity but what about the question of uniqueness of each set of elements referenced by t. Don't we still need a means by which we can insure that no to instances are the same.

 

If we truly allow the possibility that an infinite number of elements are to be included don't you have to define a means by which we can insure that no two instances are the same as we can no longer assume that we can add elements in such a way as to avoid the possibility that two instances are the same. We cant even insure that we can add more element or are we going to allow the possibility that two instances are the same.

[Doctordick]

I know quite little about modern physics and it has been quite some time since I had much interest in the philosophy that physicists try to use to explain it to others that are not physicists.

 

To my knowledge, physicists make very little effort to defend their ideas from a philosophic perspective.

 

Physicists have a very low opinion of philosophy in general and your perspective of the issue is quite accurate:

 

Actually many of the things that physics seems to be based on seem more like magic then deductions, in the simple sense that the only defense seems to be that, it looks like what physics says it should look like, so physics must be right.

 

You have hit the nail right on the head.

 

As a result when you start talking about things from physics and based on the nature of your deduction I can only conclude that you are trying to bring attention to something that your derivation is implying and that you are trying to compare it to something in physics.

 

The only significant issue I keep trying to bring up is exactly that spurious defense of physics put forth by physicists. My proof is purely a mathematical deduction from my definition of an explanation. The fact that, in a perfect language, where circumstances can be represented by a collection of “simples” and understanding can be expressed by probable expectations, any explanation of anything can be expressed as a mathematical function.

 

I want to make it as clear as possible that I have Absolutely No Intentions of finding any explanations of anything. I am concerned only with constraints implicitly imposed upon an explanation by the definition of the concept “an explanation” and nothing else.

 

I hope this clarifies exactly what the expression [math]\vec{\Psi}(x_1,x_2,\cdots,x_n,t)[/math] stands for.

This is perhaps seldom your intent and I suspect that my trying to understand what you are pointing out has side tracked us more then once. For the time being it would perhaps be best if such things are brought up after we have finished this deduction and perhaps in a thread where such discussion belongs as I really don't know enough about physics for such comments to have any meaning to me except in the context of your derivation.

 

You are correct. We should not mix the two.

 

As for some of my comments I may have gotten carried away but I am starting to wonder what some of the things that you are pointing out mean to the behavior of an arbitrary explanation and not just that we need to, and how we can include such things in your derivation and there is no thread yet where such questions have come up as part of the purpose of the thread they have merely came up as a topic of no interest or not came up at all.

 

The only application to “the behavior of an arbitrary explanation” is the fact that any explanation of anything which is inconsistent with the underlying facts of physics and mathematics is an inconsistent explanation. Or, of more interest, it can be seen as a proof that a full comprehensive explanation of all aspects and expectation of “physical reality” (being essentially no more than a solution of my equation) will also explain anything else of interest. Or, as a physicist would put it, physics underlies chemistry, chemistry underlies biology, biology underlies anthropology, etc., etc., etc.,.

 

That is, we are interested in the constraints that must be placed on any procedure for obtaining expectations for any possible circumstance without placing any constraints on what circumstances are being examined or making any assumptions about what is being explained.

 

No, we are not concerned with constraints “placed on any procedure for obtaining expectations” in any way at all. All I am saying is that, if the “simples” underlying the language being used to to express an explanation can be used as mere labels (that is they represent no underlying complex of information) to represent circumstances then the explanation can be seen as represented by a mathematical function which obeys my equation.

 

The last part I think is confused by most people...

 

Everything I say is confused by most people.

 

While you have not assumed what is being represented and you have allowed the possibility that what is being represented may be different then what people would consider to be observed, and all that you require is that any future information is consistent with the explanation or the explanation is flawed. Further more the idea of observing has no place in your deduction as there is no need to define anything and your deduction is in many ways based on the idea of what can be known without defining anything.

 

Yes, as I have commented many times, what I have presented is a tautology and has no internal content. It rests on one very simple assertion:

 

It follows that every explanation of any collection of circumstances can be represented by the notation.

 

[math]

P(x_1,x_2,\cdots, x_n, t)

[/math]

 

Clearly before such a thing can be done we must find a notation that is capable of representing any possible circumstance which is what much of “laying out the representation to be solved” is about.

 

Absolutely correct!

 

Your second thread “Conservation Of Inherent Ignorance!” then shows how such a representation of elements implies things that appear to have no influence on the representation since there is no way to find them to include them in the representation. But in fact they define a set of symmetry's that any explanation must obey if it is not going to make assumptions. That is, appear to know things that it can't possibly know about the representation.

 

Then "a Universal Representation Of Rules" gives a method by which any possible rule can be represented and a means of including all of the deductions in a single equation.

 

You are correct!

 

Making the points that we add antisymmetric when exchanged will definitely do this. But is it necessary? Just setting

 

[math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)=0 [/math]

 

if [math]\vec{x}_i=\vec{x}_j[/math] seems sufficient and so far there is proof that adding to this isn't just going to add an unneeded constraint.

 

Here you are missing a very important point. You are introducing a constraint on [math]\vec{\Psi}[/math] not necessarily required by “all” explanations. That is exactly the issue I brought up in my original construction of the representation.

 

Upon discovering a specific mathematical constraint is necessary, the problem arises of guaranteeing that it is impossible that the found constraint is a consequence of that mathematical function being a probability. That constraint is not a constraint on an explanation but is rather, only a constraint on the representation I have chosen.

 

If we can open up the range of possible functions to All mathematical functions, that difficulty vanishes. We need to have a way of satisfying that constraint due to the definition of probability without constraining the representative function in any way.

For instance what if [math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)[/math] satisfied all of the requirements including [math] \vec{\Psi}(\vec{x}_i,\vec{x}_i)=0 [/math] except it has no exchange symmetries for at least one set of elements? Can such a function exist and if so can we be sure that another function that has the desired exchange symmetries will supply the same expectations?

 

Ah, but can you prove that? I think I could give you a counter-example. In fact, I think you are skirting the counter example in your reasoning in the last paragraph.

 

If we truly allow the possibility that an infinite number of elements are to be included don't you have to define a means by which we can insure that no two instances are the same as we can no longer assume that we can add elements in such a way as to avoid the possibility that two instances are the same. We cant even insure that we can add more element or are we going to allow the possibility that two instances are the same.

 

You are once again confusing two very different issues (issues I think I have brought up to you in the discussion of symmetries; remember the rotations in the x, tau space?). Adding hypothetical elements in order to insure that our table entries of [math]\vec{\Psi}[/math] are unique is quite different from specifying the actual entries in that table which represent our supposed known information. One thing hypothetical elements possess is that we actually do not know the truth of their existence (they are not really in the table supporting our explanation). It is the presumption of truth in their existence which requires they be different (presumed actual different entries in that table).

 

Thus it is that we cannot defend the existence of these hypothetical elements except via the theory which requires them. When the door to hypothetical elements is opened, it is also opened to elements which need not be in the table of known information and are not required to make those entries unique. One could ask, for what purpose would we want to add such elements? The answer is quite simple; such additions can make a significant difference in calculating the probability of our expectations: i.e., make changes in the structure of [math]\vec{\Psi}[/math]. As I said earlier, we don't want to put any constraints on our explanation not required by the definition of an explanation.

 

Boundary conditions are a separate matter as they are required for the function to fit the known information; but that is another issue of no concern to us unless we have reason to solve that differential equation which defines [math]\vec{\Psi}[/math] (one could say, “which defines our explanation”).

 

A lot of people, unfamiliar with solving differential equations, don't realize that differential equations are really definitions of functions: how those functions change when the arguments change (which is what we are calling our circumstances).

 

Have fun -- Dick

Edited May 27, 2014 by Doctordick

[AnssiH]

As you already know, that universal constraint is [imath]\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0[/imath]. It is important to understand that it does not in itself tell us anything about what the rules of an explanation are, it is just something that a self-coherent explanation can't violate. Just like the symmetry arguments, it will only play a role in the final analysis, when we explore logically possible relationships between defined entities (i.e. what sorts of logical consequences one definition places on other definitions, as long as those definitions are required to be logically consistent with each others).

 

It is not the issue of setting up a representation of rules that I am having a problem with as it seems clear at this point that if we can include an infinite number of elements then we can represent any rule that we want. Of course this means that we must find a way that we can add an infinite number of points and still insure that no two points will occupy the same location. Making the points that we add antisymmetric when exchanged will definitely do this. But is it necessary? Just setting

 

[math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)=0 [/math]

 

if [math]\vec{x}_i=\vec{x}_j[/math] seems sufficient and so far there is proof that adding to this isn't just going to add an unneeded constraint.

 

I'm not sure I'm following what you are saying, but isn't your statement;

 

[math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)=0 [/math] if [math]\vec{x}_i=\vec{x}_j[/math]

 

exactly the same as implementing the constraint [imath]\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0[/imath]?

 

If so then you are just commenting on the subtleties in the arguments that got us there? If so, I personally would say your understanding seems sufficient to continue to the next step, and return here later if the subtleties still worry you.

 

-Anssi

[Bombadil]

Not meaning to take so long to reply but I am quite busy right now and I'm not sure how much longer it's going to last.

 

Upon discovering a specific mathematical constraint is necessary, the problem arises of guaranteeing that it is impossible that the found constraint is a consequence of that mathematical function being a probability. That constraint is not a constraint on an explanation but is rather, only a constraint on the representation I have chosen.

 

So the issue here is not that we want all explanations to satisfy the constraint of being symmetric or antisymmetric but rather there exists a means by which any function that cannot be used as an explanation due to its value at the same location as some other element has a simple means by which we can transform it into a function that is antisymmetric or symmetric and so can be seen as being an explanation.

 

If we took this a step further and said that if there are an infinite number of elements we can't prove that the function can be used as an explanation then it makes sense to apply this to any elements that must satisfy the desired rules. In this case I can understand making some elements antisymmetric but I can't understand making some elements symmetric since there seems to be nothing gained over just leaving them as they were.

 

But this just leads me back to asking. if we can know there is no other way of converting a function that can't be seen as an explanation into one that can, that is not equivalent to the one you are using. Or is this really beside the point as. if we can find such a transformation then we will use it as well and if not then we will have to use the transformation that you are suggesting in order to remain consistent with the definition of an explanation that we are using?

 

You are once again confusing two very different issues (issues I think I have brought up to you in the discussion of symmetries; remember the rotations in the x, tau space?). Adding hypothetical elements in order to insure that our table entries of [math]\vec{\Psi}[/math] are unique is quite different from specifying the actual entries in that table which represent our supposed known information. One thing hypothetical elements possess is that we actually do not know the truth of their existence (they are not really in the table supporting our explanation). It is the presumption of truth in their existence which requires they be different (presumed actual different entries in that table).

 

So we are in fact talking about two different types of elements. Elements that we are basing our explanation on. These elements we presume to be real elements such that any explanation that we are interested in must explain there behavior, and presumed elements that are not needed except that they are used to modify our actual expectations. The second type of elements are what we may look at as being needed for a theory but we have no reason to assume that they exist beyond that. It is only the first kind of elements (the ones that we are assuming to be real) that must be unique for all cases of interest and since there can be only a finite number of such elements we can add a finite number of elements to this set to insure that it is.

 

I'm not sure I'm following what you are saying, but isn't your statement;

 

[math] \vec{\Psi}(\vec{x}_i,\vec{x}_j)=0 [/math] if [math]\vec{x}_i=\vec{x}_j[/math]

 

exactly the same as implementing the constraint [imath]\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0[/imath]?

 

If so then you are just commenting on the subtleties in the arguments that got us there? If so, I personally would say your understanding seems sufficient to continue to the next step, and return here later if the subtleties still worry you.

 

Well I would think it would be the same as

[imath] \vec{\Psi} \sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0 [/imath]

but only because I am allowing [math]\vec{x}_i=\vec{x}_j[/math]. And yes I am getting the impression that there are just a few subtleties here that I am starting to suspect are only giving a problem because I don't know the basis for or the defense for using only them.

 

These three mathematical constraints can be cast into a single mathematical constraining relationship via a rather simple mathematical trick. If one defines the following mathematical operators (both the definition of “[a,b]” and the specific alpha and beta operators):

[math]

[\alpha_{ix},\alpha_{jx}]\equiv \alpha_{ix}\alpha_{jx}+\alpha_{jx}\alpha_{ix}=\delta_{ij}

[/math]

 

[math]

[\alpha_{i\tau},\alpha_{j\tau}]=\delta_{ij}

[/math]

 

[math]

[\beta_{ij},\beta_{kl}]=\delta_{ik}\delta_{jl}

[/math]

 

[math]

[\alpha_{ix},\beta_{kl}]=[\alpha_{i\tau},\beta_{kl}]=0

[/math]

 

where [math]\delta_{ij}[/math] equals one if [math]i=j[/math] and zero if [math]i\neq j[/math]. This requires these mathematical operators to anti-commute with one another and requires their squares to be one half. These mathematical constructs are closely related to what is called Lie algebra (pronounced, “lee” after Sophus Lie). At the moment, we are only concerned with the anti-commutation property as it allows us to mathematically wrap all four of the above constraints into a single equation for [math]\vec{\Psi}[/math]

 

I'll presume that such an operator is consistent and so there is no mathematical issues with using it. Actually this seem quite reasonable to me. Although I have to wonder about calling it a mathematical trick as due to your ease of using it I suspect that it is used elsewhere and you seem to be hinting that it is although for what I really don't know.

 

Although I would like to know what I should be thinking of these as, I've always thought that they must be matrix's but I don't know if this is a good way to think of them as using them seems to imply that [math] \vec{\Psi} [/math] is in the same space as the mathematical operators that you are defining.

 

All we need do is require the constraint on both alpha and beta operators that their sums over all elements of every circumstance be zero; explicitly,

[math]

\left\{\sum_i \vec{\alpha}_i \right\}\vec{\Psi}= \left\{\sum_{i\neq j}\beta_{ij}\right\}\vec{\Psi}= 0

[/math]

 

where [math]\vec{\alpha}_i = \hat{x}\alpha_{ix}+\hat{\tau}\alpha_{i\tau}[/math]. (Note that this vector construct lies in the x, tau space, not in the abstract space of [math]\vec{\Psi}[/math].) If we then make the simple constraint that we are working with [math]\vec{\Psi}[/math] expressed in the specific x, tau space where the sum of the “momentum” of all the elements in every circumstance is zero. (Note that this is actually no constraint on the problem as, once we have a solution [math]\vec{\Psi}[/math] expressed in that space, a simple Fourier transform can be used to produce the solution in any other frame of reference.)

 

This though I have to wonder about, how can we know that it is not going to effect the form of [imath] \vec{\Psi} [/imath] isn't the way that you are using them in the fundamental equation going to define the mathematical space that [math] \vec{\Psi} [/math] is defined in. Furthermore. how do we know that [imath]\vec{\alpha}_i \vec{\Psi}=0[/imath] is not true for some i Wouldn't this mess things up as the sum no longer has to be over all of the elements? What if this happened for one of the beta operators?

[Doctordick]

Not meaning to take so long to reply but I am quite busy right now and I'm not sure how much longer it's going to last.

 

Don't worry about it. We all have more important things to do than this stuff.

 

So the issue here is not that we want all explanations to satisfy the constraint of being symmetric or antisymmetric but rather there exists a means by which any function that cannot be used as an explanation due to its value...

 

You could pretty well stop right there. I am interested in finding the constraints embedded in “the definition of an explanation” and I am really not interested in any other issues. The issue paramount in the quote you brought up is the fact that I don't want to make any other constraints of any kind. If every possible function can be interpreted as an explanation (that is it gives expectations for every conceivable circumstances) then there are no constraints arising from the function notation. The “symmetric/antisymmetric“ issue is no more than a division of all possible functions into two different types. I use that fact to conclude that all “non-hypothetical elements” must be represented by antisymmetric [math]\vec{\Psi}[/math]. Note that does not mean that all elements represented by an antisymmetric function are not hypothetical (the whole kit and caboodle could by hypothetical; that philosophical position is called Solipsism and it is a well know fact that you cannot prove Solipsism is false). But beyond that, if all functions qualify as explanations, the symmetric functions can not be ignored. If you ignore them you are inserting constraints which are not due to the definition of an explanation.

 

If we took this a step further and said that if there are an infinite number of elements we can't prove that the function can be used as an explanation then

 

Any function can be used as an explanation. The only issue of interest is that the boundary conditions are satisfied (the information we think we know to be true). All a differential equation tells us is how the function changes as one moves away from the boundary conditions. So it is the fact that the function fits the boundary conditions which determines if the explanation it corresponds to is valid or not. Explanations can certainly be wrong, there is nothing in the definition which says they have to be correct! But we really have little interest in explanations which we know are wrong! That essentially has nothing to do with what I am talking about.

 

it makes sense to apply this to any elements that must satisfy the desired rules. In this case I can understand making some elements antisymmetric but I can't understand making some elements symmetric since there seems to be nothing gained over just leaving them as they were.

 

It has to do with the fact that [imath] \vec{\Psi} \sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0 [/imath] allows the invention of hypothetical elements to enforce any possible set of rules and nothing else.

 

If we can know there is no other way of converting a function that can't be seen as an explanation into one that can, that is not equivalent to the one you are using.

 

I have already come up with two!

 

 

[math]

P=\vec{\Psi}\cdot\vec{\Psi}[/math] (which doesn't work very well)

 

and

 

[math]P=\vec{\Psi}^\dagger\cdot\vec{\Psi}[/math] (which works quite well by generating internal correlations in that vector representation).

 

You come up with another and it may very well have significant implications.

 

So we are in fact talking about two different types of elements. Elements that we are basing our explanation on. These elements we presume to be real elements such that any explanation that we are interested in must explain there behavior, and presumed elements that are not needed except that they are used to modify our actual expectations. The second type of elements are what we may look at as being needed for a theory but we have no reason to assume that they exist beyond that. It is only the first kind of elements (the ones that we are assuming to be real) that must be unique for all cases of interest and since there can be only a finite number of such elements we can add a finite number of elements to this set to insure that it is.

 

You are talking about the issue of finding explanations. I am not concerned with that as it has nothing to do with finding the constraints embedded in “the definition of an explanation”.

 

Well I would think it would be the same as

[imath] \vec{\Psi} \sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0 [/imath]

but only because I am allowing [math]\vec{x}_i=\vec{x}_j[/math].

 

The Dirac delta function, [imath] \delta(\vec{x}_i-\vec{x}_j) = 0 [/imath] unless the argument is zero; so that expression is always zero, even when [math]\vec{x}_i\neq\vec{x}_j[/math].

 

And yes I am getting the impression that there are just a few subtleties here that I am starting to suspect are only giving a problem because I don't know the basis for or the defense for using only them.

 

Because you are jumping ahead of the problem. The problem here is to express the constraints embedded in “the definition of an explanation”. You are worrying about what these constraints imply. That is an issue to be discussed down the road; after you understand the proof that my fundamental equation is valid.

 

I'll presume that such an operator is consistent and so there is no mathematical issues with using it. Actually this seem quite reasonable to me. Although I have to wonder about calling it a mathematical trick as due to your ease of using it I suspect that it is used elsewhere and you seem to be hinting that it is although for what I really don't know.

 

It is just a trick because it actually does nothing except make the equations look simple. It amounts to a notational issue.

 

Although I would like to know what I should be thinking of these as, I've always thought that they must be matrix's but I don't know if this is a good way to think of them as using them seems to imply that [math] \vec{\Psi} [/math] is in the same space as the mathematical operators that you are defining.

 

There are at least three different “mathematical” spaces here and you best keep them in mind. There is the (x,tau) space (later to become an (x,y,z,tau) space); the abstract vector space represented by the notation [math]\vec{\Psi}[/math] and the two dimensional space used to think of matrix's. That third one is only necessary if you want to create those mathematical operators as objects in a “real” representation in which case, every element of of [math]\vec{\Psi}[/math] must also be represented in that space (that is [math]\vec{\Psi}[/math] exists in the mathematical representation use to implement that anti-comutation). The whole thinq here is to make the relationships look simple. If we actually go to solve the thing, we are actually working with a whole slew of equations. That is why I call it a trick!

 

This though I have to wonder about, how can we know that it is not going to effect the form of [imath] \vec{\Psi} [/imath] isn't the way that you are using them in the fundamental equation going to define the mathematical space that [math] \vec{\Psi} [/math] is defined in. Furthermore. how do we know that [imath]\vec{\alpha}_i \vec{\Psi}=0[/imath] is not true for some i Wouldn't this mess things up as the sum no longer has to be over all of the elements? What if this happened for one of the beta operators?

 

Look at the definition of those anti-commuting operators. That is all you need to know about them. You seem to confuse implementing a definition with using the entities being defined. Those are different issues. Just accept the fact that the definitions I gave can be implemented (that is a math problem and those mathematical elements have been around since quantum mechanics was invented).

 

Have fun -- Dick

Edited May 27, 2014 by Doctordick

[Bombadil]

You could pretty well stop right there. I am interested in finding the constraints embedded in “the definition of an explanation” and I am really not interested in any other issues. The issue paramount in the quote you brought up is the fact that I don't want to make any other constraints of any kind. If every possible function can be interpreted as an explanation (that is it gives expectations for every conceivable circumstances) then there are no constraints arising from the function notation. The “symmetric/antisymmetric“ issue is no more than a division of all possible functions into two different types. I use that fact to conclude that all “non-hypothetical elements” must be represented by antisymmetric [math]\vec{\Psi}[/math]. Note that does not mean that all elements represented by an antisymmetric function are not hypothetical (the whole kit and caboodle could by hypothetical; that philosophical position is called Solipsism and it is a well know fact that you cannot prove Solipsism is false). But beyond that, if all functions qualify as explanations, the symmetric functions can not be ignored. If you ignore them you are inserting constraints which are not due to the definition of an explanation.

 

But why would we think that all of the elements that we are interested in can be classified into “symmetric/antisymmetric“ elements. How do we know that we not inserting constraints by not including other possibilities at least for hypothetical elements.

 

The only defense for this that I can think of is that all that we are really interested in is that we can represent any possible rule. That is, that we can add elements so that

[math] F=\sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0. [/math]

is true and determines the possible locations that an element can be no matter what they may be.

 

At this point perhaps we should make sure that we agree at least as to what a rule is doing, from how you have used it in the opening post I would say that we are determining a rule by saying that “ a rule is determined by a function F that is zero for any possible location for a element” this seems to agree with what we are doing and it would make since then that all that we would need to be able to do is add antisymmetric elements to satisfy any possible rule.

 

In a sense a rule is just telling us that a particular set of elements is not possible. But how do we define a rule.

 

It has to do with the fact that [imath] \vec{\Psi} \sum_{i\neq j} \delta(\vec{x}_i-\vec{x}_j) = 0 [/imath] allows the invention of hypothetical elements to enforce any possible set of rules and nothing else.

 

But what about those functions where those elements were neither symmetric or antisymmetric, are we just going to ignore the possibility that such functions might exist. Or do we have some defense for not using them? Is it really just a question of “if we can enforce any possible rule and what elements we have to add makes no difference to us?”

 

Look at the definition of those anti-commuting operators. That is all you need to know about them. You seem to confuse implementing a definition with using the entities being defined. Those are different issues. Just accept the fact that the definitions I gave can be implemented (that is a math problem and those mathematical elements have been around since quantum mechanics was invented).

 

For the time being I'm going to leave this subject as I suspect that a satisfactory understanding of it would take us to far off topic for the time being and accept that it is nothing more then a defining of a new space that serves no more purpose and has no effect other then to give us a single equation to work with.

[Rade]

Quote of DoctorDick: But we really have little interest in explanations which we know are wrong! That essentially has nothing to do with what I am talking about.

 

Suppose you present this explanation for a circumstance: "the neutron is charge neutral because it has an electron and proton +(anti)neutrino that cancel charge". This would have been the explanation about the neutron a physics teacher would have presented to a class in the late 1930's. But this is not the modern day 3-quark inside neutron explanation of the circumstance. So, do we then conclude that we "know" the explanation about the neutron from the mid 1930's is "wrong", that we "know" the quark theory of the Standard Model is "NOT WRONG" ? Of course not. No physicist would make such a claim to have complete and certain knowledge about the structure of the neutron. Modern day observation on electric dipole moment in the neutron raise questions about the Standard Model explanation. Also, this is a good example why there is GREAT INTEREST in science in past explanations of circumstances that differ from present day explanations, for the simple fact that a scientist NEVER CLAIMS TO KNOW WITH CERTAINTY that a past explanation is wrong and what they "know" at present is correct.

 

If you disagree, please explain how you "know is wrong" with certainty ANY explanation.

[Doctordick]

Hi Bombadil,

I am sorry I failed to catch your post. I seldom read this forum any more as there seems to be very little competent posting going on.
 

But why would we think that all of the elements that we are interested in can be classified into “symmetric/antisymmetric“ elements. How do we know that we not inserting constraints by not including other possibilities at least for hypothetical elements.

 

We are looking at solutions to a first order linear differential equation. One thing one can be quite confident of is the fact that any sum of solutions is also a solution. It follows that any solution can be broken into a symmetric solution plus an antisymmetric solution and absolutely any function is still allowed. We have inserted no constraints.
 

In a sense a rule is just telling us that a particular set of elements is not possible. But how do we define a rule.

 

No, a rule which disallowed any set of elements would be an undefendable constraint. The rule we need must allow all “known” information and, at the same time, make no constraints upon the truth. I am looking at constraints “required by the definition of an explanation”. Most all actual explanations make use of hypothetical constraints above and beyond what is logically required.

The existence of those hypothetical constraints constitute the very essence of any theory. That is why experimentalists perform experiments: to see if the constraints implied by the theory are indeed valid. My rule (that there must exist a function [math]F(x_1,x_2, \cdots, x_n)[/math] which vanishes for all “known” circumstances) implements the exact constraint that “the known data are known” and nothing more.

Of course, every actual explanation (in order to be valid: i.e., not disprovable by experiment) must also be consistent with a very similar constraint: the presumed known data must be consistent with what is known. You need to comprehend that this constraint is quite different from what my constraint expresses.

I introduced the tau index for a very important purpose. In order to assure that all [math]x_i[/math] could be explicitly separated from one another without requiring [math]x_i \neq x_j [/math]. If we cannot do that, we cannot represent those elements as points on a line without losing information.

You should comprehend that representation of known information as points on a line (or in a space) is a rather convenient (and universal) mental representation of reality. That does not make such a representation "real"!

So the tau axis is entirely “hypothetical” (not a real representation of any “known information”) and as such allows us the step of adding hypothetical information all the way out to infinity.

The original rule was, just add a hypothetical element in order to make sure no two circumstances are identical but, mathematically speaking, that rule cannot be extended to an infinite amount of known information. That is where anti-symmetry was introduced and, if you review the discussion carefully you will see that it applies only to “known information”. It does not apply to “presumed known information”. Hypothetical elements can be represented with symmetric functions.

The inverse idea should be seriously examined: being represented by an antisymmetric function in no way guarantees an element is not hypothetical.

What is interesting about that fact is that, since it does not apply to hypothetical knowledge, the resulting representation covers all explanations including solipsism. That result is a rather surprising indicator that that the constraints I develop are a consequence of “the definition of an explanation” and nothing more. It is a well known fact that solipsism (that everything is a figment of your imagination) can not be disproved.
 

But what about those functions where those elements were neither symmetric or antisymmetric, are we just going to ignore the possibility that such functions might exist.

 

As I have proved elsewhere, any function can be represented as a sum of symmetric and antisymmetric function so nothing has been ignored.
 

For the time being I'm going to leave this subject as I suspect that a satisfactory understanding of it would take us to far off topic for the time being and accept that it is nothing more then a defining of a new space that serves no more purpose and has no effect other then to give us a single equation to work with.

 

I am of the opinion that you have fallen into exactly the same mental trap that has apparently confused everyone except Anssi. You are presuming that what I am presenting is a theory of some sort. It isn't! It is exactly what I have been representing it to be from the first day: a representation of the constraints implied by the definition of an explanation and nothing more.

That is the issue I attacked and solved back in the 1960's. It wasn't until around 1980 that I began to find ways of developing solutions to my “fundamental equation” and discovered the rather surprising result that almost everything of modern physics turned out to be a direct consequence of the definition of an explanation and nothing more: i.e., that modern physics itself was a tautology!

The modern conflict between relativity and quantum mechanics does not exist in my representation and I am of the opinion that the actual reason for that conflict is that Einstein's theory of General relativity is an invalid explanation. That is exactly why I am considered a crackpot as only a crackpot would consider thinking a genius like Einstein could be wrong.

I would like to point out that Qfwfq (my most serious adversary) has admittedly never examined my deduction of the "fundamental equation". His only concern has been to give examples which, in his mind, contradict my conclusions. What he fails to comprehend is that an example which contradicts my equation contradicts modern physics. I know that is a subtle point but one would expect anyone claiming expertise in modern physics would understand the issue.

Have fun -- Dick

 

Edited May 27, 2014 by Doctordick