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"a Universal Representation Of Rules"


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#1 Doctordick

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Posted 19 December 2010 - 01:09 PM

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]

 

Understanding this post depends upon the reader possessing a competent understanding the the two previous posts, “Laying out the representation to be solved” and “Conservation Of Inherent Ignorance!

I will summarize those two posts here; however, that summary will omit some subtle issues covered in the original posts. The post “Laying out the representation to be solved” essentially lays out a specific defined notation which is capable of representing any conceivable circumstances.

Step I of my presentation is the simple fact that any circumstance can be represented via a notation consisting of a collection of numerical indices expressed by [math](x_1,x_2,\cdots,x_n,\cdots,t)[/math]. The central question here is, does there exist any communicable circumstance conceivable which cannot be represented by a computer file? (As an aside, if it is not communicable, there can be no reason to discuss it.) Any computer file can certainly be written as a collection of such packets of numerical references.

Step II of that post is little more than pointing out that one's expectations (engendered by understanding an explanation) can be seen as a collection of probabilities of truth specified for each and every conceivable circumstances: i.e., if you understand an explanation, your expectations of truth for any specified circumstance can be represented by a number bounded by zero and one (the definition of a probability). Thus it is that one's understanding of any explanation can be represented by the mathematical function: i.e., it is no more than the conversion of one set of numbers into another (circumstances into probability).

 

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

 

Where the form, [math]\vec{\Psi}^\dagger \cdot \vec{\Psi}[/math], is little more than a way of handling the requirement that [math]0\leq P\leq 1 [/math] by definition. Thus it follows that every explanation conceivable can be mapped into a function of the form, [math]\vec{\Psi}[/math].

The post “Conservation Of Inherent Ignorance” essentially takes note of the fact that arguments, [math](x_1,x_2,\cdots,x_n,\cdots,t)[/math], of that function are no more than numerical labels for the significant elements underlying that explanation and are thus absolutely and totally arbitrary: i.e., the definitions lie in the explanation, not in the actual numerical labels used to represent them. This leads to the validity of what is called “shift symmetry” in the representation of the arguments of [math]\vec{\Psi}[/math]. That fact yields a constraint on acceptable form of the the functional representation: i.e., it requires that [math]\vec{\Psi}[/math] must obey the following equations:

 

[math]
\sum^n_{i=1} \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

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

The final required constraint: representing "Rules" mathematically

There exists one additional constraint upon the function [math]\vec{\Psi}[/math] which should be seen as very important. That constraint has to do with the fact that every explanation (save one) expresses “rules” which must be obeyed. If we are going to bring that set of rules into a mathematical form such that no assumptions are made as to what the rules are we have a rather difficult task to accomplish. I will now present an approach which will accomplish that result.

 

Step I: An examination of the “what is” is “what is” explanation.

 

The “what is” is “what is” explanation is essentially an explanation which expresses no rules: i.e., in essence the implied expectations are “what happens” is “what happens” and no understanding is possible. The idea is that the information upon which it is based “what is” (what I have defined to be “the past”) has utterly no bearing upon what should be expected. Nevertheless, the case is interesting as it still expresses knowledge of the past, “what is”. Thus it is perhaps the simplest situation to analyze regarding the problem of expressing that past in a clear mathematical form. We may not have any solid expectations but we still apparently have, in our minds eye, a defined past: i.e, the fundamental elements referred to by the “i” index are defined as are the circumstances defined by the “t” index.

That past can be seen as a collection of circumstances, [math](x_1,x_2,\cdots,x_n)[/math] indexed by “t”. Since it is the past (what is presumed to be known: i.e., “what is”), the probability of each of those circumstances is clearly “one” and all other possibilities have a probability of “zero” (they are not part of “what is”). That is the extent of our knowledge of the situation.

Since the number of elements with probability “one” is finite, we can certainly list them in a file along with their specified probability. This is quite analogous to, what in ancient days (prior to computers) was referred to as a tabular representation of a function (note that, in this mental picture, both “x” and “i” indices have been assigned). Clearly, the function represented by that table is identical to the function supposedly representing the explanation no matter what that explanation might be. Since this representation has nothing to do with the explanation itself (other than the fact that what is known is defined and represented) some interesting questions arise directly from the representation itself.

My first question here concerns the issue of recovering the “t” index. If the “t” index were to be omitted, could we establish such an index from the table of circumstance? It should be recognized here that the actual value of that index is immaterial. Regarding the “what is” is “what is” explanation, the past is what the past is and the order you put the circumstances in has utterly no bearing on the issue. Thus the only issue of importance here is that every supposed “circumstance” must have a different attached index.

If every circumstance in our table is different from every other, there is no problem; each must have a different “t” index. A problem occurs when we have two identical circumstances. That situation is a little more complex than it appears to be on first examination. In constructing the table, actual values are assigned to the x and i numerical labels (what they are supposed to be referring to is, at the moment, immaterial) by whoever it is that is representing their explanation; however, the symmetries discussed in the “Conservation of Ignorance” post must still be applicable as the assignment of those labels is arbitrary.

Since the “t” index separates every circumstance into an explicitly different case, the shift symmetry can be used to set one “x” index to be the same in every explicit circumstance and scale symmetry can be used to set a second to be the same. In essence, it is not the actual values of the assigned x indices but rather the internal patterns which are significant. Thus, as stated above, a problem arises when two circumstances are represented by identical patterns.

That situation can be removed via the introduction of “hypothetical elements”: i.e., elements not actually part of the information standing behind the explanation but rather, elements presumed to exist by the explanation. (Note that their existence is implied by the existence of identical circumstances themselves; otherwise the identical circumstance would create no problems.) It should be clear that it is always possible to add hypothetical elements sufficient to make every explicit circumstance in the table different.

A rather interesting characteristic of the table as constructed reveals itself. From the original table together with the added “hypothetical elements”, a new table can be constructed where the “t” index is omitted and is instead represented by the function, [math]t(x_1,x_2,\cdots,x_{n+k})[/math], which is the value of the “t” index associated with represented circumstance without that "t" index. Thus we can construct a new table where the value of “t” index can be see as embedded in the underlying circumstances themselves. Since the index “t” is now (via the addition of hypothetical elements) embedded in the new table, this new table of circumstance (sans the “t” index) is, in a sense, equivalent to the original table. The “t” index has been replaced by those hypothetical elements required to make every circumstance unique.

Exactly this same procedure can be used to produce a table expressing a function which yields the value of some removed “x” index. For example, if we remove the [math]x_1[/math] index from all circumstances in the new table and set the function represented by that table to be exactly [math]x_1[/math], we then have a table representing the function

 

[math]
x_1=g(x_2,x_3,\cdots,x_{n+q}).
[/math]

 

As in the first case, we must insure that every pattern [math](x_2,x_3,\cdots,x_{n+q})[/math] is unique. That result can be accomplished by adding “hypothetical elements” to the collections representing the circumstances of interest. The real thing of significance here is that we know the function [math]x_1=g(x_2,x_3,\cdots,x_{n+q})[/math] exists. If that function exists, there exists another function of great interest. Define [math]F(x_1,x_2,x_3,\cdots,x_{n+q})[/math] to be

 

[math]
F(x_1,x_2,x_3,\cdots,x_{n+q})=x_1-g(x_2,x_3,\cdots,x_{n+q}).
[/math]

 

That function clearly vanishes for every valid entry to the associated table of circumstances (which, by the way, include all the required hypothetical elements)

Note that the table representing that function still has exactly the same number of entries as did the original table which represented the information upon which our explanation is based so it is still a finite table; however, since the collection of all possible circumstances (the collection for which our explanation was to yield our expectations) is infinite, the function representing our explanation is still essentially wide open: i.e., in order to obtain expectations for circumstances not represented in the table, we must perform some kind of interpolation based upon the constructed table.

What this means is that [math]\vec{\Psi}(x_1,x_2,x_3,\cdots,x_n,t)[/math] is still a totally open function, except for the fact that the probability can not be inconsistent with any case represented by the table upon which our explanation is based (otherwise the explanation would be flawed) and also yield exactly the same expectations as the represented explanation for every circumstance not known (including consistency with each “t” index given the absence of all circumstances greater than or equal to that “t” index). On the other hand, we now know that there must exist a function [math]F(x_1,x_2,x_3,\cdots,x_{n+q})[/math] which vanishes for every valid circumstance. Again, all we have is a finite table of that function and the actual function itself must be obtained via interpolation.

The “what is” is “what is” explanation clearly fulfills all the specified requirements: i.e., it is a valid flaw free explanation of anything. The function “F” must vanish for all valid circumstances and, since the explanation presumes absolutely anything is deemed possible, “F” must also vanish for all other circumstances, not just the known circumstances. Since there are an infinite number of possibilities and all are equally possible, the probability of any given circumstance must be zero. And, since any circumstance is possible, the result of any experiment (observation of a new “t” circumstance greater than the previously known “t”) is totally consistent with the predicted expectations no matter what happens.

Clearly [math]F\equiv 0[/math] represents the function F required by the “what is” is “what is” explanation. The only real problem with that explanation is that it is not a particularly valuable explanation of anything.

 

Step II: Extension to more valuable expectations.

 

It should be clear here that what we actually desire is a function “F” which vanishes for every possible valid circumstance and is non-zero for every invalid circumstance. Any reasoning person should comprehend that there exists no way to guarantee that any function can be known to satisfy such a proposition as doing so would require one to be “all-knowing” and that would require an infinite amount of information.

Any attempt to discover the correct algorithm, that would be [math]F(x_1,x_2,x_3,\cdots,x_n)[/math] which vanishes only for all possible valid circumstances, is doomed to failure. For example, consider the fact that a mathematical fit can be made to any finite collection of known data plus any random additional data. It follows directly from that possibility that there must exist an infinite number of functions that fit the known data exactly. This means that we can expect no more than undefendable approximations to truth so long as the data available is finite.

Meanwhile there is a side issue which needs to be brought up somewhere and this is perhaps the best place. When people start reading about my notation, [math](x_1,x_2,x_3,\cdots,x_n)[/math], they invariably presume that this can be seen as a set of points on the x axis. That presumption is inherently false as each of the indices [math]x_i[/math] is actually a numerical label and not a measurement of any kind. On the other hand, given a specific assigned set of such numerical labels, it is mentally convenient to think of it as a set of points on the x axis. When it comes to actual facts, such a mental mapping can not exist.

 

Doctordick, on 05 Aug 2010 - 2:24 PM, said:
I know that almost everyone who reads this is going to jump to the conclusion that the collection of arguments represented by the specific circumstances [math](x_1,x_2,\cdots ,x_n)[/math] can be seen as a set of points on the x axis. That assumption is patently false as I will show in a following post which I will title "A Universal Representation of Rules".

 

The problem is that in any case where [math]x_i=x_j[/math], mapping the information onto the x axis loses information as the existence of multiple elements vanishes from the data. On the x axis, [math]x_i[/math] and [math]x_j[/math] will map to the same point: i.e., a collection of points on the x axis can not represent such a circumstance. I bring this difficulty up here because we have, above, just discussed a means of overcoming this problem.

A visual picture of the data would be nice, if we could create such a thing without making any presumptions concerning the explanation. In essence, we need a way of visually displaying multiple points with identical x values. There is a very simple way to display such a thing. It can be done by adding “hypothetical data”, or, in this case a hypothetical [math]\tau[/math] axis perpendicular to the x axis: i.e., allowing every [math]x_i[/math] point to be represented by the point [math](x_i,\tau_i)[/math] in an x, tau space.

It should be realized that, having added hypothetical variables (both here and in the discussion above) a very serious question arises. We are as free to assign numerical [math]\tau_i[/math] labels as we were to assign the numerical [math]x_i[/math] labels. The problem arises when we consider the mathematical means to be used to calculate the probability of specific circumstances implied by our explanation. The hypothetical elements discussed above may or may not exist and the mechanism to handle them is quite straight forward. If they actually exist, values will eventually appear. Until that time, in calculating probabilities of specific circumstances, we must integrate [math]\vec{\Psi}^\dagger\cdot\vec{\Psi}[/math] over all possibilities regarding these hypothetical elements. In contrast to that, the underlying [math]\tau_i[/math] data have been completely fabricated and it should be clear that no actual value can ever be known. It should be seen that this clearly requires that the probability calculations must always be integrated over all possibilities regarding these variables. Other than that requirement, the representation is really no different from the earlier “hypothetical elements”.

Returning to the discussion prior to the addition of the tau axis, it is interesting that we can assert the following:

 

QuoteThe algorithm we are searching for may vary from time to time but it must depend on the data received to date and the method of determining it must be independent of time. I hold that the above is the only valid statement of any problem confronting scientific analysis.


Any attempt to bestow structure on any solution of any problem beyond that contained in the above statement is to presume facts neither evident or defendable!

 

The “rules” standing behind our explanation can be mathematically expressed by

 

[math]
F(\vec{x}_1,\vec{x}_2,\vec{x}_3,\cdots,\vec{x}_n)=0
[/math]

 

where [math]\vec{x}_i \equiv x_i\hat{x}+\tau_i\hat{\tau}[/math] in the hypothetical x, tau space. All we really know is that such a function must exist. So long as our table of data is finite, there will exist an infinite number of functions which will fit the bill exactly.

However, there does exist a subtle possibility here. In finding a function which fits the finite table we have constructed, there does exist the possibility that a proposed function is indeed the correct function. We cannot prove that function is correct but, since it fits all the known data, neither can we prove it is wrong. What is interesting about this possibility is that we can examine some of the consequences to be expected and the difficulties to be handled as the above table representation expands towards infinity.

There is one very important issue which arises in such an examination. That is the fact that of the increasing number of circumstances in the table. Certainly, so long as the number of elements defining a circumstance and the number of circumstances themselves are finite the procedure defined above can be accomplished; however, no matter how many we have we must always admit of the possibility of one more circumstance. That is the very definition of infinite. If we do indeed have the correct function, the relationships used cannot be destroyed by the continuity implied by that infinite result. This places some subtle constraints on F.

In generating our “what is” is “what is” table for the function F, we added hypothetical elements. We added the hypothetical tau axis in order to allow representation of identical positions on the x axis. One of the consequences of that step was that, in calculating the probabilities of our expectations, we had to integrate over all [math]\tau_i[/math] values. That essentially says that the tau axis plays no role in the development of F: i.e., it is not an aspect of adding hypothetical elements necessary to make every circumstance in the table of known circumstances different. So there is no issue regarding the extension of the continuity of the tau variables.

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.

Once again, the problem has a simple solution: all we need do is require the function [math]\vec{\Psi}[/math] be asymmetric with respect to exchange of any pair of elements. Mathematically, that means that for any i,j pair,

 

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

 

Note that, in the above, only the arguments [math]x_i[/math] and [math]x_j[/math] are shown; all the rest are presumed the same as before and therefore not necessarily shown.

Notice that [math]\vec{\Psi}=0[/math] whenever [math]x_i=x_j[/math] as zero is the only number equal to its negative. This type of asymmetry is exactly what stands behind what is called.

 

What it guarantees is that no two elements in this x, tau space can be in the same place for a specific t index (remember the x indices are mere labels and when they are the same what they represent must be identical). Another way to express the same thing is to assert that all hypothetical elements used to generate F must obey Fermi-Dirac statistics. This will eliminate the problem with continuity of x and the existence of F.

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.

As a side note (at this point), since it was the asymmetry under exchange which generated the required vanishing of identical positions in x, tau space, the absence of this asymmetry (or exchange symmetry) must be the characteristic of those additional elements which serve only to yield different probabilities. In essence, an infinite number of exchange symmetric elements may be added to the mix in order to adjust the calculated probabilities to the probabilities implied by the explanation. As opposed to the earlier elements which caused F to fit the underlying data, these additional elements must obey
.

 

Step III: Some subtle additional constraints on the form of [math]F(\vec{x}_1,\vec{x}_2,\vec{x}_3,\cdots,\vec{x}_n)[/math].

 

As we still have an infinite number of possibilities which fully fulfill the requirements of a flaw-free explanation, it is valuable to examine possibilities which which can be eliminated through the symmetry requirements discussed in the “Conservation of Ignorance” post. First, the same shift symmetry which exists in x must also exist in the hypothetical tau axis. That fact leads to the constraint on [math]\vec{\Psi}[/math] that

 

[math]
\sum^n_{i=1} \frac{\partial}{\partial \tau_i}\vec{\Psi}(\tau_1,\tau_2,\cdots,\tau_n,t)=im\vec{\Psi}(\tau_1,\tau_2,\cdots,\tau_n,t)
[/math].

 

where the arguments [math]x_i[/math] still exist but have not been explicitly written down. By defining [math]\vec{\nabla}_i=\hat{x}\frac{\partial}{\partial x_i}+\hat{\tau}\frac{\partial}{\partial \tau_i}[/math] and [math]\vec{k}=k\hat{x}+m\hat{\tau}[/math] the required conservation constraint implied by x and tau shift symmetry can be written in a two dimensional form

 

[math]
\sum^n_{i=1} \vec{\nabla}_i\vec{\Psi}(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t)=i\vec{k}\vec{\Psi}(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t)
[/math].

 

There is also another very subtle consequence of shift symmetry which concerns the form of the arguments of F. The existence of shift symmetry in both the x and tau dimensions (since we are now viewing the circumstances as a collection of points in the x, tau space) means that the origin must be a free parameter: i.e., changing the presumed origin in that space yields no consequences in the evaluation of F. This means that the information contained in the set of arguments [math](\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n)[/math] is identical to the information contained in the set of arguments consisting of the entire collection of differences between [math]\vec{x}_i[/math] and [math]\vec{x}_j[/math].

If we have all [math]\vec{x}_i[/math] arguments for a particular circumstance, the construction of all [math]\vec{x}_i -\vec{x}_j[/math] for that same circumstance is a trivial problem. Likewise, if we have all [math]\vec{x}_i -\vec{x}_j[/math] arguments for a particular circumstance, the construction of all [math]\vec{x}_i[/math] is rather easily achieved so long as position of the origin is a free parameter as. It may not be as trivial a problem as the reverse but anyone with a decent understanding of algebra should find the process quite straight forward.

That fact implies that we can rewrite our table of known F=0 points (known as a function of [math]\vec{x}_i[/math] arguments) as a new table of known F=0 points as a function of [math]\vec{x}_i -\vec{x}_j[/math] arguments. Adding scale symmetry to the representation (remember, these numbers are nothing more than numerical labels) there is another very important consequence of these symmetries.

We now have F being expressed in the x, tau space in terms of the differences [math]\vec{x}_i -\vec{x}_j[/math]. Let me again bring up the possibility of guessing the correct function F. If we have indeed guessed the correct function then the predicted expectations for unknown circumstances will be correct all the way out to that infinite collection of circumstances. This fact can be seen in a slightly different perspective: only the correct function will continue to be correct through out the entire process. This implies another required constraint.

The correct function must vanish for every specified point (i.e., the points allowed by the rule being represented by F) in that two dimensional space. The integration over all tau dependence has to do with the calculation of expectations, and not with the rule F is to represent. Thus ignoring how that representation was achieved, seen merely as a function defined over that x tau space, rotation in the plane of that space cannot change the function (all we really have is a set of points which are being used to define that function).

But rotation will convert tau displacement into x displacement. Since tau displacement is an entirely hypothetical component, F simply can not depend upon the actual tau displacement and by the same token neither can F depend upon actual x displacement. Since we have converted F into a function of distances between points, this essentially says that F can not depend upon the actual magnitude of these separations. This should be quite reasonable as, since we are talking about mere numerical labels, multiplication of all labels by some fixed constant cannot change what is being represented.

Either F simply vanishes and we have no rules (and the “what is” is “what is” explanation is the only valid explanation) or rules actually exist. If rules do indeed exist, F can not vanish for all circumstances: i.e., there must exist some circumstances which are impossible and [math]F\neq 0[/math] must be true for those circumstances. The only integrable function which does not depend upon the magnitude of its argument and still has a non zero value for some argument is the Dirac delta function [math]\delta(x)[/math], commonly defined as follows:

 

[math]
\int_a^b\delta(x-c)dx=1
[/math]

 

only if the range of integration includes c and is zero if the range of integration does not. The value of the Dirac delta function is clearly zero everywhere except when the argument is zero; in which case it must be infinite. It is usually defined as the limit of an integrable function who's graph has a fixed area (unity) as the width of the non zero region goes to zero.

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.

The Final Conclusion

 

At this point I have uncovered three specific mathematical constraints implied by the symmetries embedded in the representation of an explanation requiring expectations given by [math]P(x_1,x_2,\cdots,x_n,t)[/math] where

 

[math]
\int P(x_1,x_2,\cdots,x_n,t)dV_x= \int \vec{\Psi}^\dagger(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t) \cdot\vec{\Psi}(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t)dV_{x\tau}
[/math]

 

where [math]dV_x[/math] and [math]dV_{x\tau}[/math] represent the abstract differential volume to be integrated over (both hypothetical elements and possible ranges of presumed valid elements). It is [math]\vec{\Psi}[/math] which represents the explanation.

From the analysis I have presented, the three required constraints are as follows.

 

[math]
\sum^n_{i=1}\vec{\nabla}_i\vec{\Psi}(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t) = i\vec{k}\vec{\Psi}(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t)
[/math]

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

 

and the constraint required by there being rules behind circumstances which are possible: i.e., the requirement that there exist a function F which will discriminate between what circumstances can and cannot occur.

 

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

 

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]

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.)

 

QuoteIt is a trivial matter to convert a solution of [math]\sum_i\frac{\partial}{\partial x_i}\Psi_0 =0[/math] into a solution of [math]\sum_i\frac{\partial}{\partial x_i}\Psi_1 =iK_x\Psi_1[/math]. Simple substitution will confirm that if [math]\Psi_0[/math] is a solution to the first equation,[math]\Psi_1=e^{\sum_j \frac{iK_x x_j}{n}}\Psi_0\;\;\left(where \;\;i=\sqrt{-1}\right)[/math] is a solution to the second. Exactly the same relationship goes for the equation on tau.

 

The equation of interest is the following:

 

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

 

Note that [math]\delta(\vec{x}_i -\vec{x}_j)\equiv \delta(x_i -x_j)\delta(\tau_i -\tau_j)[/math].

It is almost trivial to prove that the above equation satisfies the constraints expressed above. First, the right hand relationship divided by K is exactly the constraint

 

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

 

on each component of [math]\vec{\Psi}[/math] in the abstract vector space of interest. I will explicitly show the algebra necessary to the remainder of the proof.

First (from the left) multiply the equation of interest by [math]\alpha_{kx}[/math]. In the original equation, whatever k is chosen, that explicit term appears only once: i.e., the term where i=k. By definition, that operator anti-commutes with every alpha and beta operator in the entire equation except for [math]\alpha_{ix}[/math]. For that specific term (when i=k) [math]\alpha_{kx}\alpha_{ix}=1-\alpha_{ix}\alpha_{kx}[/math]: thus, what happens is that every term of the left hand side of that equation simply changes sign and one additional term is generated (the specific term where i=k is duplicated without an alpha operator). The result of the multiplication is (after [math]\alpha_{kx}[/math] is commuted to the far right so as to operate directly on [math]\vec{\Psi}[/math])

 

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

 

If one then sums that resulting equation over k, every term will vanish (because of the fact that the sum over the alpha operators taken over all elements vanishes) except for that single term, [math]\frac{\partial}{\partial x_k}\vec{\Psi}[/math] which lacks any alpha or beta operator. The final result, as a consequence of that sum over k, becomes,

 

[math]
\sum_k \frac{\partial}{\partial x_k}\vec{\Psi}=0.
[/math]

 

Exactly the same thing happens when we multiply the original equation by [math]\alpha_{k\tau}[/math] and again sum over k. These two operations taken together yield exactly the constraint

 

[math]
\sum^n_{i=1}\vec{\nabla}_i\Psi(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t) = i\vec{k}\Psi(\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n,t)
[/math]

 

when [math]\vec{k}=0[/math]: i.e., when the sum over the momentum in the x, tau space vanishes.

Left multiplication of the original equation with the [math]\beta_{kl}[/math] operator followed by a sum over i and j (where [math]i\neq j[/math] ) results in exactly the final constraint.

That is, we may state unequivocally that it is absolutely necessary that any algorithm which is capable of yielding the correct probability for observing any given pattern of data in any conceivable problem to be explained must obey the relation deduced above, which constitutes my fundamental equation:

 

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

 

This constraint follows from the definition of "an explanation" and nothing else. If anyone finds fault with that deduction, please let me know.

Have fun – Dick

PS This is not actually the end of the road here. There are a number of additional conclusions which can be proved which are quite interesting. One is a rather explicit reason for viewing the universe as a three dimensional spacial entity.


Edited by Doctordick, 02 June 2014 - 03:13 AM.


#2 HydrogenBond

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Posted 20 December 2010 - 09:34 AM

Here is a problem.

Say you took a situation like Archimedes sitting in his tub and his Eureka moment for understanding displacement in water. The king thought he was being cheated due to silver being mixed with gold, and he needed proof. Archimedes thought about the weight difference, and found the needed experiment in the tub; Eureka!. He saw how he displaced water when sitting in the tub and then inferred that since silver is less dense, he could compare displacement of equal weights of a known gold object to a lighter silver/gold fake. The fake should be fluffier and displace more water.

Once you know his solution it seems obvious. Although anyone might have experienced the tub displacement, nobody considered what it meant until it was pointed out by Archimedes. Necessity was the mother of this invention, since it created the need to know, narrowing down a random search. If he simply went into his lab and tried random things, he may or may not have reached this simple solution. Or if another person had been given the task, they may have overlooked the obvious. The equation can have more that one solution for any problem.

Maybe once knowledge is defined, making use of that is more predictable.

#3 Rade

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Posted 24 December 2010 - 11:27 AM

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]

any conceivable problem to be explained must obey the relation deduced above, which constitutes my fundamental equation

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


-------
Comment: It looks like you left out the Delta (i) in your second equation ?

Also, for consistency I would suggest you include the final [math]K\frac{\partial}{\partial t}\vec{\Psi}=iKq\vec{\Psi}[/math] in your first presentation (the upper one) of your fundamental equation. Something called a fundamental equation should not change form from one place to another in the presentation. Decide what the equation is, then stick to it throughout--advice offered to help with clarity of presentation.

-------
I think it would be great if you can next provide a new thread of the important applications to science of this unique presentation.

#4 Rade

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Posted 25 December 2010 - 09:34 AM

Doctordick. I have a comment about the "what is is what is" (WI=WI) explanation. You comment that this type of explanation has no rules. So, "what happens is what happens" and no understanding about what did happen and why it happened or what will happen in the future is possible. Yet, the (WI=WI) explanation does express knowledge of the past (there is a set of events that did happen). Thus the (WI=WI) explanation represents knowledge (the past of what did happen) without understanding (why what happened did happen).

I would like to present an example of a (WI=WI) explanation and welcome your comment.

It is a basic question in philosophy to explain "what exists". There are many such explanations of the question. One such explanation (the one I use), appears to represent a (WI=WI) explanation. So, if someone asks the question, "please explain to me what exists", the (WI=WI) explanation I use to answer the question would be "existence exists". In our minds eye we have a finite number of possible things or events that exist from our past experience of living (we call these our past). A (WI=WI) explanation of these past things or events is that we have no idea why they happened, or how, and we have no expectations for the future. What we "know" is that they did happen and now are part of our past experience. What exists is what exists (i.e.; what is is what is), and although we cannot have any "understanding" of ["what" exists] (in specific), we can have "knowledge" of [what "exists"] (existence itself) via our accumulated past experiences. Thus we can conclude that the statement "existence exists" is a (WI=WI) explanation to the question: (what exist ?).

Edited by Rade, 27 December 2010 - 08:20 AM.


#5 Rade

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Posted 27 December 2010 - 08:18 AM

Necessity was the mother of this invention, since it created the need to know, narrowing down a random search.

Yes, off topic for this thread, but, is it not interesting how natural selection (i.e., the non-random reproduction of genotypes)acts as a non-knowing mother (mother earth) of invention, the necessity being for a species to survive over time. For Archimedes the need to know (to add to his past experiences) was a need to solve a problem (how to make the King happy), and rational thinking (a non random process) was the process used. For species in nature, the need to know (to add to past experiences)is a need to solve a problem (how to survive), and natural selection (a non random process) is the process used. Of interest is that both explanations must obey the fundamental equation of Doctordick, well, at least I think he would agree.

#6 Qfwfq

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Posted 28 December 2010 - 07:09 AM

Of interest is that both explanations must obey the fundamental equation of Doctordick, well, at least I think he would agree.

He has never actually proven this. In the OP, at best he shows that a solution [imath]\vec{\Psi}[/imath] to his FE complies with his initial constraints, but what he claims is the other way around:

That is, we may state unequivocally that it is absolutely necessary that any algorithm which is capable of yielding the correct probability for observing any given pattern of data in any conceivable problem to be explained must obey the relation deduced above, which constitutes my fundamental equation:


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


This constraint follows from the definition of "an explanation" and nothing else. If anyone finds fault with that deduction, please let me know.

So, how can this claim be actually proven? The argument provided doesn't even attempt to support its necessity, only it being sufficient. Has nobody noticed?

#7 Doctordick

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Posted 29 December 2010 - 01:21 PM

He has never actually proven this. In the OP, at best he shows that a solution [imath]\vec{\Psi}[/imath] to his FE complies with his initial constraints, but what he claims is the other way around:

What I have claimed is that, under my definition of an explanation, the representation I propose for what is being explained makes no presumptions as to what the explanation might be and furthermore must be bound by the constraints shown. In addition, there must exist a collection of mathematical functions which yield exactly the same expectations as any proposed explanation of what is being explained. Thus I assert that any solution to my equation which yields the same expectations as a given specific explanation can be used to represent that explanation. What else is there to the issue?

Now Rade wants to disagree with my definition. I am fine with that but all it really means is that we are not communicating. I have only a vague idea of what qualifies as an explanation in his mind. I certainly can not say that his explanations must satisfy my equation so long as what he is talking about differs from what I am talking about.

So, how can this claim be actually proven? The argument provided doesn't even attempt to support its necessity, only it being sufficient. Has nobody noticed?

Qfwfq, you have things exactly “bass ackwards”. If one is willing to work with my definition of an explanation, then my equation is a direct mathematical deduction and is thus very definitely a necessary constraint. On the other hand I have nothing to support it being sufficient. There may very well be additional constraints which should be applied; however, as far as I have been able to determine, it seems to be sufficient. Of course, even forty years ago I was not cognizant of a lot of physics and I am certainly not qualified to say what relationships have been proved since then.

So, what I have proved is: if one accepts my definition of an explanation (and the meaning of my notation), my equation represents a required constraint on any explanation. Regarding sufficiency, I have nothing to offer beyond showing that special relativity, general relativity, Shrödinger’s equation, classical mechanics, Dirac’s equation, Maxwell’s equations and some important nuclear equations can be deduced by making some rather standard approximations. It should be realized that approximations have to be made as my equation is what is usually referred to as a many body equation and thus general solutions are not (for the most part) feasible.

My only real contention is that there is enough evidence here to conclude that the issue of sufficiency should be looked into. Enough said?

Have fun -- Dick

#8 Rade

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Posted 29 December 2010 - 04:33 PM

Now Rade wants to disagree with my definition. I am fine with that but all it really means is that we are not communicating. I have only a vague idea of what qualifies as an explanation in his mind. I certainly can not say that his explanations must satisfy my equation so long as what he is talking about differs from what I am talking about.

Hello Doctordick. Not sure why you said I want to disagree with your definition of explanation ? I am more than willing to accept your definition to see where it leads.

In my mind, what qualifies as an explanation would be a procedure of communication that uses an interconnected combination of facts, laws, inferences, tested hypotheses, and theories to provide clear understanding of some unknown or not understood circumstance.

This definition of explanation differs from yours because it does not specify that the goal of the procedure of communication is to "yield rational expectations". My definition includes your "procedure" and "circumstance".

So, while your goal of explanation is to yield rational expectations for some hypothetical circumstance, my goal is to yield clear understanding of something unknown or not understood for some circumstance.

I try to understand your definition of explanation because you have presented it mathematically.

I see no reason my definition also could not be presented mathematically. So, your approach is to start with the assumption that any circumstance that needs an explanation can be represented as a set {X1,X2,X3...Xn} where (Xi) is a numerical label in reference to a specific fundamental element of the circumstance.

In my definition, what would be needed is a minimum set of interconnected information {F, L, I, H, T} required to yield clear understanding of some circumstance, where F=fact, L=law, I=inference, H=hypothesis, T=theory.

Here is an example of how my definition of explanation could be applied in conjunction with yours. Suppose you are asked to 'explain' why an object released exactly 4.9 meters above the Earth will collide with the Earth in exactly 1 second. If you already know the answer why, no explanation is needed. Using my definition of explanation there would be a minimum set of facts and recognized laws of nature that would be required to clearly explain the circumstance. Using your definition of explanation, whatever explanation the use of my definition provides, it must obey your Fundamental Equation. If not, could you please explain why not.

#9 Qfwfq

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Posted 30 December 2010 - 06:38 AM

What I have claimed is that, under my definition of an explanation, the representation I propose for what is being explained makes no presumptions as to what the explanation might be and furthermore must be bound by the constraints shown. In addition, there must exist a collection of mathematical functions which yield exactly the same expectations as any proposed explanation of what is being explained. Thus I assert that any solution to my equation which yields the same expectations as a given specific explanation can be used to represent that explanation. What else is there to the issue?

Dick, what I posted above concerns what you put into the OP of this thread and nothing else; it refers to your premises and your conclusions. There is absolutely no point twisting it around. I wasn't challenging your claim on grounds of information theory or entropy or any other matrhematical of physical consideration, as I've done in other threads.

Qfwfq, you have things exactly “bass ackwards”.

No I don't. You have always claimed that your premises imply the equation, and you say:

If one is willing to work with my definition of an explanation, then my equation is a direct mathematical deduction and is thus very definitely a necessary constraint.

You have not in any manner shown this.

On the other hand I have nothing to support it being sufficient.

Yet, this is the only contention that your actual argument makes, in the part headed as The Final Conclusion, starting from: "It is almost trivial to prove that the above equation satisfies the constraints expressed above."

I don't see any fault in your argument in support of this statement I quote, and your presentation has definitely improved even though it still displays want of clarity here and there, but you seem to make some ad hoc choices and certainly you adopt a mighty sleight of hand when you turn the argument for the above statement into the claim of necessity. You confuse the roles of necessary and sufficient. Could you at least be clear about what you claim to be proving?

Regarding sufficiency, I have nothing to offer beyond showing that special relativity, general relativity, Shrödinger’s equation, classical mechanics, Dirac’s equation, Maxwell’s equations and some important nuclear equations can be deduced by making some rather standard approximations.

Here it almost seems you admit not having proven necessity (although calling it sufficiency). My opinion is that these things you are offering as support are due to the fact that you choose the Lie algebra, even though it doesn't follow of necessity from your premises. I also hold reserves about the manner in which you introduce the requisites of exchange symmetry and antisymmetry.

I'll try to be clear:

In The Final Conclusion, after summing up your premises, introducing a Lie algebra you write:

[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].

Let's call all up to here a premise [imath]\cal{P}[/imath] and call your FE (which you write at that point) [imath]\cal{F}[/imath]. Your claim has alway been like:

[math]\cal{P}\Rightarrow\cal{F}[/math]

but your actual argument after that point only supports:

[math]\cal{F}\Rightarrow\cal{P}[/math].

#10 Doctordick

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Posted 30 December 2010 - 03:03 PM

Qfwfq, you use the term “ad hoc” quite liberally and I get the impression you don’t clearly understand the meaning of the term or else perhaps presume the average reader will not understand your intentions. I have asserted many times that I define mathematics to be the invention and study of internally consistent systems. As such, the addition and use of any defined mathematical relationship can not be taken to be “ad hoc”; no more than the use of addition or multiplication can be characterized as “ad hoc”. In my opinion you are merely grabbing at straws which happen to be obscure to the average person reading this forum in order to defend your refusal to look at the actual details of my proof.
 

Dick, what I posted above concerns what you put into the OP of this thread and nothing else; it refers to your premises and your conclusions.

The proof is a three part construct. To omit parts one and two is to neglect the definitions of the terms and expressions being used. I am of the opinion that you desire to ignore the definition of those terms and expressions for the simple purpose that it allows you to insert (surreptitiously) your own definitions in order to confuse the important issues.

Section one has to do with the meaning of my indices i, x, and t and the meaning of [math]\vec{\Psi}[/math] such that the expression [math]\vec{\Psi}(x_1,x_2,\cdots,x_n,t)[/math] can represent any possible explanation. Section two uses the fact that the indices are defined to be mere arbitrary numerical references implies the existence of certain symmetries and then uses those symmetries to conclude the asserted differential constraints must be true. Dropping either from the arguments allows you to misdirect the course of the proof towards the manufactured flaw of “additional premises”..
 

I don't see any fault in your argument in support of this statement I quote, and your presentation has definitely improved even though it still displays want of clarity here and there, but you seem to make some ad hoc choices and certainly you adopt a mighty sleight of hand when you turn the argument for the above statement into the claim of necessity. You confuse the roles of necessary and sufficient.

I think you are once again attempting to take advantage of the lack of education of the people on this forum in order to cast doubt on my assertions. The three (or four if one includes tau as a separate index) differential constraints I deduce are certainly “required” by my definitions laid out earlier. It follows that you must be referring to my supposed “ad hoc” use of the anti-commuting operators defined by


[math]
\left\{\sum_i \vec{\alpha}_i \right\}\vec{\Psi}= \left\{\sum_{i\neq j}\beta_{ij}\right\}\vec{\Psi}= 0
\quad\quad where \quad\quad\vec{\alpha}_i = \hat{x}\alpha_{ix}+\hat{\tau}\alpha_{i\tau}[/math].

 

Which is no more “ad hoc” than introducing the defined indices “x” or “[math]\tau[/math]” used to represent other “required” aspects of my equation: these are no more than defined mathematical operators..
 

Let's call all up to here a premise [math]\cal{P}[/math]

To call the proof up to that point a “premise” is to totally misrepresent the situation. You are once again dropping out the definitions and conclusions put forth in sections one and two of the proof. The differential constraints expressed are required and the anti-commuting operators are defined; neither can be characterized as premises.

The only complaint which could be raised at this point would be, do all solutions which satisfy the given implied constraints (prior to my use of anti-commuting operators to create a single equation) also satisfy my final equation. Or, in your notation, (calling the separate constraints “C”), does my proof that [math]\cal{F}\Rightarrow\cal{C}[/math] also imply that [math]\cal{C}\Rightarrow\cal{F}[/math].

Any decent mathematician would immediately see that the expression [math]\cal{C}\Rightarrow\cal{F}[/math] is a trivial conclusion. A Rather trivial Fourier transforms exist which will provide [math]\Psi_0[/math] given [math]\Psi_1[/math] from the original set of constraints and that [math]\Psi_0[/math] is seen to be a solution to my equation by direct substitution (the equation simply reduces to 0+0=0). As I presume you are a half way decent in your understanding of Fourier transforms, I can only conclude that your complaint is no more than an attempt to discourage others from considering examining proof.
 

Here it almost seems you admit not having proven necessity (although calling it sufficiency). My opinion is that these things you are offering as support are due to the fact that you choose the Lie algebra, even though it doesn't follow of necessity from your premises.

Addition does not follow of necessity from my premises! Addition follows from the definitions of mathematics and there is no more need for me to prove the consequences of using anti-commuting operations than there is for me to prove the consequences of addition. It is a useful operation and nothing more.
 

I also hold reserves about the manner in which you introduce the requisites of exchange symmetry and antisymmetry.

Those operators are requisites of my fundamental equation in exactly the same way addition of terms is a requisite of that equation. Now you wouldn’t suggest “reserves about the manner” in which I introduced the requisites of addition in my equation. You would be laughed off the forum if you did because most everyone here understands the issues addition brings up. They don’t all entirely understand anti-commutation and that leaves you an opening to confuse them.

I also “USE” the nature of exchange anti-symmetry to preserve a very essential aspect of “adding” hypothetical elements such that the validity of the procedure does not vanish when an infinite number of such elements are added. If this step is omitted the power of “adding” hypothetical elements (essential to my definition of the “rules”) vanishes with the continuity which appears when the number of possibilities goes to infinity: i.e., there would then, in such a case, exist explanations which could not be represented by my notation. They are thus a required aspect of my definition of an explanation.

In the final analysis, my definition of an explanation plus the definitions of various constructs I use in my proof end up satisfying the two fundamental requirements underlying the answer to the question, “what follows from the definition of an explanation and nothing else?” Any solution to my fundamental equation satisfies my definition of an explanation and, by construction, no possible explanation has been omitted from representation (that goes to the "and nothing else" phrase).

I would also like to comment that, in my attack, I have defined every aspect of my representation for very clear reasons. I have inserted no “ad hoc” elements; by ad hoc I mean elements not required by the nature of the information to be represented. I put this in contrast to the ad hoc insertion of “space” and “time” in the standard representation of problems to be explained. My development of the concepts of space and time are direct consequences of developing an unconstrained representation of the underlying information to be explained.

Have fun -- Dick

 


Edited by Doctordick, 26 May 2014 - 04:28 AM.


#11 Qfwfq

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Posted 31 December 2010 - 07:03 AM

Dick, if any person reading our exchanges is ignorant, it's their tough bananas and I am not here to take some advantage of it. What a lame excuse! It is rather sly to make out that anyone who agrees with me must be ignorant and anyone else ought to agree with you, as if there were no chance of the other way around.

I have asserted many times that I define mathematics to be the invention and study of internally consistent systems.

Great, because that's how most any modern mathematician sees it! But... errr:

As such, the addition and use of any defined mathematical relationship can not be taken to be “ad hoc”; no more than the use of addition or multiplication can be characterized as “ad hoc”.

This isn't enough. If you want to support the claim that B follows from A and nothing else, you must provide an argument reliant only on the claimed premises without slipping in any other ones that don't follow of necessity from A. That's how all mathematicians see it.

To omit parts one and two is to neglect the definitions of the terms and expressions being used. I am of the opinion that you desire to ignore the definition of those terms and expressions for the simple purpose that it allows you to insert (surreptitiously) your own definitions in order to confuse the important issues.

I wasn't omitting them, I simply stated that I was NOT raking in considerations from other topics when making that point. Do you understand the English Language?

Dropping either from the arguments allows you to misdirect the course of the proof towards the manufactured flaw of “additional premises”.

If you cannot provide a good argument by which your previous premises lead to the Lie algebra that you introduced, you have no point in making this accusation.

To call the proof up to that point a “premise” is to totally misrepresent the situation.

No Dick, you're missing my point. I might re-state it like: If we grant you all up to there and call it [math]\cal{P}[/math] (or whichever letter you prefer), then in the last bit you only argue for one thing and not for the other.

Any decent mathematician would immediately see that the expression [math]\cal{C}\Rightarrow\cal{F}[/math] is a trivial conclusion. A Rather trivial Fourier transforms exist which will provide [math]\Psi_0[/math] given [math]\Psi_1[/math] from the original set of constraints and that [math]\Psi_0[/math] is seen to be a solution to my equation by direct substitution (the equation simply reduces to 0+0=0). As I presume you are a half way decent in your understanding of Fourier transforms, I can only conclude that your complaint is no more than an attempt to discourage others from considering examining proof.

I must suppose you are by far overestimating me here, in presuming me to be "a half way decent" in it, because I really don't get your whole point here. Since you're the one who doesn't want ignorant, education deprived folks like myself to be confused, why the heck did you not spell this out in the first place, in your presentation?

Addition does not follow of necessity from my premises! Addition follows from the definitions of mathematics and there is no more need for me to prove the consequences of using anti-commuting operations than there is for me to prove the consequences of addition. It is a useful operation and nothing more.

Addition is one of the many things required by your premises, hence tacitly incorporated into them. The anti-commuting operators are not. I could submit a purported very simple proof of, say, the Riemann hypothesis or the Poincarè conjecture, Beal conjecture or Fermat's last, to be published, that works by introducing a relation or property that doesn't appear to have counterexamples but without proving it, and reply to the referee's objections just as you argue here. Would you call them pigheaded for not accepting it? After all, I would simply be introducing a useful thing. Sure enough, up until someone finds a counterexample, I could even say it works!

Those operators are requisites of my fundamental equation in exactly the same way addition of terms is a requisite of that equation.

I have no objection against you saying they are a requisite of your equation --in whatever manner-- but this does not support your equation unless I can get a clearer view of how these constraints arise from the initial premises, rather than being choices further to them. This dumb *** would need to sort out all those tricky details, leading up to it, but why the heck should I if you can't even show me how the final part is actually what you claim it to be?

In any case Dick, you ended your OP by asking people to let you know if they find fault with your deduction. You therefore should not react so badly if somebody does this, whether they are pointing to an error, a missing argument or a lack of clarity that prevents understanding.

#12 Doctordick

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Posted 31 December 2010 - 12:55 PM

As I have said many times, I take mathematics to be a given collection of internally consistent structures. I leave arguments as to the validity of those structures to the experts. I merely make use of them!

This isn't enough. If you want to support the claim that B follows from A and nothing else, you must provide an argument reliant only on the claimed premises without slipping in any other ones that don't follow of necessity from A. That's how all mathematicians see it.

Then please explain to me how anti-commutating operations differ from other mathematical operations other than by the definitions themselves.

My opinion is that these things you are offering as support are due to the fact that you choose the Lie algebra, even though it doesn't follow of necessity from your premises.

It is you who keep bringing up Lie algebra, not me. All I do is define a set of specific anti-commuting operators and then multiply the left hand sides of my various constraint equations by those defined operators. Exactly what is your complaint? Are you asserting that my premise (that an explanation is a mechanism for producing expectations) makes the existence of anti-commuting operators impossible? You bring up the issue of counter examples. Exactly what is the counter example of an anti-commuting operation? Or are you suggesting that defining that set of anti-commuting operators is not a possibility?

And exactly what do you have in mind with your comment, “support the claim that B follows from A and nothing else”? Are you trying to say that multiplication by those defined operators does not yield what I write down? Those operators are defined mathematical operations and B multiplied by those operators yields A. No other information is required to come to that conclusion. I suspect it is the wealth of undefendable conclusions you have built in your mind which are bothering you; not my logic: i.e., it is a direct consequence of your failure to follow my logic.

I wasn't omitting them, I simply stated that I was NOT raking in considerations from other topics when making that point. Do you understand the English Language?

You have made it quite clear in various earlier posts that you had no intention of making any examination of my logic in those first two sections. My complaint is that you have no idea as to what the terms in my equation mean and thus your arguments tend towards being totally facetious.

I must suppose you are by far overestimating me here, in presuming me to be "a half way decent" in it, because I really don't get your whole point here. Since you're the one who doesn't want ignorant, education deprived folks like myself to be confused, why the heck did you not spell this out in the first place, in your presentation?

In my opinion, I did;

(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.)

:i.e., all we are talking about here are different frames of reference.

Addition is one of the many things required by your premises, hence tacitly incorporated into them. The anti-commuting operators are not.

And what about integration and/or differentiation? Get off your high horse; any defined mathematical operation could be used. I could divide my equation by cosh(k) and get a new equation. If the original equation was valid, the second would be valid. (B would follow from A). I don’t do it because the result wouldn’t be useful. If I had used such a relationship, would you claim my premises didn’t require it? Put a little thought into your comments Qfwfq.

I could submit a purported very simple proof of, say, the Riemann hypothesis or the Poincarè conjecture, Beal conjecture or Fermat's last, to be published, that works by introducing a relation or property that doesn't appear to have counterexamples but without proving it, and reply to the referee's objections just as you argue here. Would you call them pigheaded for not accepting it? After all, I would simply be introducing a useful thing. Sure enough, up until someone finds a counterexample, I could even say it works!

There is a big difference between a conclusion and a definition. Are you attempting to say that I have to prove that anti-commuting operations can exist? I will simply say that the mathematicians seem to have decided such a thing can exist and who am I to argue with them.

In any case Dick, you ended your OP by asking people to let you know if they find fault with your deduction. You therefore should not react so badly if somebody does this, whether they are pointing to an error, a missing argument or a lack of clarity that prevents understanding.

Well I guess I expected a little more careful thought from you and I am sorry if you find me reacting badly.

Have fun -- Dick

#13 Rade

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Posted 31 December 2010 - 06:35 PM

Hello Doctordick. A simple question. Thinking about your definition of 'explanation'. How does this differ from your definition of 'prediction" ? Would not a 'prediction' be a procedure for yielding rational expectations for some hypothetical circumstance ? It seems to me that your fundamental equation is an equation of prediction, and not explanation, for there can be many types of explanation that your equation does not apply to, yet it must apply to all types of prediction ? What am I not understanding ? Also, have a happy new year :D

#14 Doctordick

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Posted 31 December 2010 - 09:06 PM

How does this differ from your definition of 'prediction" ? Would not a 'prediction' be a procedure for yielding rational expectations for some hypothetical circumstance ?

I don’t think I ever defined “prediction”, but if you were going to make a prediction would you not propose something consistent with your expectations?

It seems to me that your fundamental equation is an equation of prediction, and not explanation, for there can be many types of explanation that your equation does not apply to, yet it must apply to all types of prediction ?

Again you simply seem to refuse to use my definition of “an explanation”. Your definition of an explanation eludes me. What you seem to have in mind is a vague and complex idea far beyond logical analysis.

You ask what you are not understanding. I am of the opinion that you completely missed the thrust of my post “Defining the nature of a rational discussion!

In that post, I essentially define “logical “ as a process where the axioms are specifically and accurately laid out and the conclusions are deduced from those axioms. As it has been my experience that most people have a great tendency to identify “rational” with that process, I am concerned that the combined concepts make “rational thought” impossible: i.e., the combination makes the creation of axioms “irrational”. If the axioms are irrational, how can such logical thought be classified as rational.

It is for that reason I tried to introduce what I call “squirrel thought”:

If one holds that only logical thoughts are rational, then scientific progress becomes impossible since any deductions must be based on things presumed to be valid without reason (those axioms one starts with) and that is certainly irrational. However, I hold that there is a second kind of rational thought which needs to be clearly understood. Call it intuition, Zen or whatever you prefer; I will give it my own name as, though it is what is commonly referred to as intuition or Zen, I don't want to include some of the common connotations of those terms.

I will use the adjective "squirrel" (my own creation) to classify thought which is not "logical". (I do this because I think the word has some valuable applicable connotations.) If one has ever watched squirrels in the tree tops, they will see those squirrels making life and death decisions without pause; and usually the correct decisions. Have you ever seen a squirrel run full tilt down a thin branch (the branch bending under his weight) jumping out into empty air to catch a thin branch on another tree ten or twelve feet away? Very rarely do they miss their mark (actually I have never seen an error, but my wife says she has). How do they do this? Most people would agree that they manage this feat by intuition but few would call it Zen.

With regard to that post, no one has seemed to grasp what I was trying to say. In particular, you seem to make a lot of your decisions as to what is reasonable through your emotional reaction to the expressed issues. As I say, that is certainly a “rational” approach but it clearly does not qualify as “logical”: i.e., you have not laid out your axioms in an accurate and specific manner.

Since I seem to have your attention, there is another issue you should think out carefully. It might help you comprehend what I am talking about. By what means do you come to the conclusion that you understand something? Does that conclusion not arise when begin to be unsurprised by events associated with whatever it is you feel you understand? That is, isn’t surprise a pretty fair sign of the fact that something was misunderstood? If that is the case, then understanding is clearly related to “expectations”: i.e., understanding an explanation is clearly related to the nature of your associated expectations.

And thus we arrive back at my definition of an explanation.

Happy New Year -- Dick

#15 Rade

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Posted 02 January 2011 - 09:51 PM

I don’t think I ever defined “prediction”, but if you were going to make a prediction would you not propose something consistent with your expectations? Again you simply seem to refuse to use my definition of “an explanation”.

Hi Doctordick. Your response does not make sense to me. I did agree to use your definition of explanation, but, the reason for my question, is that I can find no logical difference between your definition of the word 'explanation', from what you imply would be your definition for the word 'prediction'. So, as you said, one would expect that a valid definition for the word 'prediction' would be = " a procedure for yielding rational expectations for a hypothetical circumstance ". OK, this fine. But, as you can clearly see, this is exactly the same definition you use for explanation--I am saying no more or no less than this factual observation.

So, are you saying there is absolutely no difference between these two concepts ?:

1. Explanation (as a concept)
2. Prediction (as a concept)

If your answer is yes, fine, but some explanation why would be helpful, because the dictionary does not define these two concepts as being the same.

Your definition of an explanation eludes me. What you seem to have in mind is a vague and complex idea far beyond logical analysis.

Well, I am sorry you find it complex, and vague it may be, but it is completely logical.

In that post, I essentially define “logical “ as a process where the axioms are specifically and accurately laid out and the conclusions are deduced from those axioms.

I would not use the term axiom, I would use the term 'premise'. The purpose of logic is to distinguish correct reasoning from incorrect reasoning. Reasoning is a mental activity called inferring. To infer is to draw conclusions from premises. In place of the word premise you can use data, information, facts. However, a premise has a different meaning than axiom.

In particular, you seem to make a lot of your decisions as to what is reasonable through your emotional reaction to the expressed issues. As I say, that is certainly a “rational” approach but it clearly does not qualify as “logical”: i.e., you have not laid out your axioms in an accurate and specific manner.

Well, I am sorry that is how you feel, and it is of concern because I view the issue as being of extreme importance, because many false arguments result because they are based on false premises.

Since I seem to have your attention, there is another issue you should think out carefully. It might help you comprehend what I am talking about. By what means do you come to the conclusion that you understand something? Does that conclusion not arise when begin to be unsurprised by events associated with whatever it is you feel you understand? That is, isn’t surprise a pretty fair sign of the fact that something was misunderstood? If that is the case, then understanding is clearly related to “expectations”: i.e., understanding an explanation is clearly related to the nature of your associated expectations.

I view understanding as a type of integration of concepts, there being some minimum set of concepts (entities, events) needed to say you understand. I do not see that all surprise results from misunderstanding, it also can result from lack of focus. So, suppose you are cutting the onion while talking to your wife about some family event, and you cut off the tip of your finger. I think it logical to conclude you would be surprised by the event, but the surprise would not mean that something was not understood, I mean, clearly you understand that the purpose of cutting the onion is that the finger is not removed in the process. Now, your expectation of the onion cutting circumstance is that the finger not be removed in the process, the explanation you would give to anyone that asked about your finger was that you did not focus on the task, not that you lacked understanding. So, I appreciate your example, but I think your attempt to link surprise, explanation, understanding is off the mark.

Edited by Rade, 03 January 2011 - 12:30 AM.


#16 Rade

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Posted 03 January 2011 - 01:45 AM

:
If one holds that only logical thoughts are rational, then scientific progress becomes impossible since any deductions must be based on things presumed to be valid without reason (those axioms one starts with) and that is certainly irrational. However, I hold that there is a second kind of rational thought which needs to be clearly understood. Call it intuition,...

I do not agree that an axiom is presumed to be valid "without reason". Look at your Webster New World Dictionary...so....axiom is defined as "in logic and mathematics, a statement that needs no proof because its truth is obvious, self evident proposition". Therefore, to begin a deduction with an axiom is not in any use of the word, irrational.

Second, your so-called squirrel thinking is not a type of "rational thinking". It is, as you say, a type of "intuition thinking" that is part of the subconscious. Again, Webster defines intuition as "the immediate knowing of something without conscious use of reason". So, when the squirrel see the branch it (1) knows the branch exists (2) knows how much energy to use to reach the branch, and it knows all the aspects of these without any rational (conscious) thinking (use of reason). Doctordick, it is not that people do not understand what you are saying on this topic, it is more the case they do not agree with your definition of words as presented in dictionary.

I will use the adjective "squirrel" (my own creation) to classify thought which is not "logical".

But, this is not the case. The use of intuition by both squirrels (to jump branch to branch) and humans (to create works of art, music, etc.) is logical, even if it is not an immediate act that is rational or derived from conscious thinking. Your philosophy on this issue makes no sense to me. Edit: I know you read Kant. Kant does not view squirrel thinking using intuition as not logical. In CPR-65, here is what he had to say about squirrel thinking ..."All thought must directly or indirectly relate ultimately to intuitions, and therefore, with us, to sensibility, because in no other way can an object be given to us"

...jumping out into empty air to catch a thin branch on another tree ten or twelve feet away? Very rarely do they miss their mark (actually I have never seen an error, but my wife says she has). How do they do this?

Many, many aspects of animal behavior (as well as human) have a genetic basis, and much behavior is outside rational thinking. But I think you already know this, so I am confused by the question.

#17 Qfwfq

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Posted 03 January 2011 - 07:15 AM

Dick, I will get off my high horse after you've got off yours.

As I have said many times, I take mathematics to be a given collection of internally consistent structures. I leave arguments as to the validity of those structures to the experts. I merely make use of them!

That's exactly what I've been noticing. Trouble is you manifest having no idea of how mathematics can and can't be used for drawing conclusions about reality.

Then please explain to me how anti-commutating operations differ from other mathematical operations other than by the definitions themselves.

First, you read the text of mine that you quoted there. Read it properly and understand it. It referred to the sentence (bold mine) at the end of your OP:

This constraint follows from the definition of "an explanation" and nothing else.


You claim "and nothing else" but you throw the Lie algebra in, without proving it being consequential to the initial premise.

It is you who keep bringing up Lie algebra, not me.

Really!? Dick, read your very own OP of this very thread. I don't have too much time to waste, not on nonsense at least.

Are you asserting that my premise (that an explanation is a mechanism for producing expectations) makes the existence of anti-commuting operators impossible? You bring up the issue of counter examples. Exactly what is the counter example of an anti-commuting operation? Or are you suggesting that defining that set of anti-commuting operators is not a possibility?

Strawman fallacy. You should be quite aware I meant nothing of this sort, by simply reading my point properly. The only thing I maybe didn't spell out was about counterexamples but you should be a big enough boy to see the point (at least going by what you claim about yourself, but you don't always show). The hypothetical counterexamples in my analogy stand for cases which aren't solutions of your equation, but comply with the initial premises and not with the choice of Lie algebra.

You have made it quite clear in various earlier posts that you had no intention of making any examination of my logic in those first two sections.

I did not make this clear at all, it is your accusation against me. Au contraire, I more than once said what I thought about that part of your presentation and that I was waiting for this part, with some vague memories of past versions of it not being clear enough for critical analysis. Both you and Anssi kept ignoring this.

My complaint is that you have no idea as to what the terms in my equation mean and thus your arguments tend towards being totally facetious.

Goodness, it has taken years for your incoherent ramblings to assume any semblance of meaning. Currently, it seems to be your own version of the first part (representation of a state) of the von Neumann quantum formalism and a specific symmetry. Your version however calls in cause considerations of well-known topics of aesthetics and metaphysics but the views that you and Anssi expose, with the claim of being logical and mathematical, are so ill defined that they might as well be based on Zen Buddhism instead of logic and math, and the contentions run into serious difficulty with considerations about known mathematical topics.

In my opinion, I did;

(Note that this is actually no constraint on the problem as, once we have a solution \vec{\Psi} expressed in that space, a simple Fourier transform can be used to produce the solution in any other frame of reference.)


:i.e., all we are talking about here are different frames of reference.

If you really want people to take your presentation seriously, you are being quite sloppy. If you want to claim the "and nothing else" then prove that you do not lose generality by making that choice of Lie algebra (as well as improving your clarity a bit, up to that point). Show me how the choice of Lie algebra is no more than a choice of reference frame. Make sure you don't confuse the matter of "which reference frame" with that of "which transformation represents a given change of reference frame".

And what about integration and/or differentiation? Get off your high horse; any defined mathematical operation could be used. I could divide my equation by cosh(k) and get a new equation. If the original equation was valid, the second would be valid. (B would follow from A).

You do not prove such a general equivalence and I remain convinced that the choice in question is what leads to the possibility of working your FE to imply the Dirac equation. If you had proper understanding, you would be aware that it is exactly the Dirac algebra that serves the purpose of obtaining a Schrödinger equation (i. e. choosing a Hamiltonian) suited for a particle in Minkowskian spacetime, in the half-spin case. A non equivalent --yet self-consistent-- choice of Lie algebra would not give the description of what is physically observed (save for the non relativistic approximation of course).

According to your logic, Dirac would have claimed instead that the Minkowskian nature of spacetime follows (with other things) from the very basics of the von Neumann quantum formalism and nothing else! According to your terminology, folks would be calling the representation of state in this formalism "an explanation". :doh:

There is a big difference between a conclusion and a definition. Are you attempting to say that I have to prove that anti-commuting operations can exist? I will simply say that the mathematicians seem to have decided such a thing can exist and who am I to argue with them.

Dick, again, I don't say you must prove they exist, mathematically. The thing is that you don't state them in the initial premises which you claim your FE to follow from "and nothing else". If you can't prove they follow of necessity, you ought to explicitly include them into your premises.