What can we know of reality?

AnssiH · 11-21-2007

Sorry it has taken a while. I've had little bit time to look at this every now and then and it's taken a while to gather some sort of overall picture from multiple posts, regarding the "fundamental equation" and those anticommuting entities. (Funny sidenote, the finnish terminology for algebra - which I'm more familiar with - is sometimes quite different from english, and I didn't even realize what "multiply through by $\alpha_{qx}$ " meant until I saw it done in post #83, simply because "multiply through" doesn't translate directly to finnish terms :P )

Anyhow, I've made enough progress to start asking some questions, trying to make sure I understand things correctly.

---QUOTE post#89---
There is a different alpha for every index $x_i$ or $\tau_i$ and there is a $\beta_{ij}$ for every pair of points in that space used to represent our ontological elements.
------

I suppose when you say "every pair of points", that doesn't refer to an "x,tau pair", but instead every possible "ontological element pair" you could compose from a specific present, i.e. 4 elements, each expressed as an (x,tau) point, could be used to compose 6 different pairs, i.e. they yield 6 different $\beta_{ij}$ elements?

Also, tell me if I understood the following correctly;
$\alpha_{qx}\beta_{ij} = -\beta_{ij}\alpha_{qx}$

The above is so because $\alpha_{qx}$ can never be the same element as $\beta_{ij}$ , so invariably $[\alpha_{qx} , \beta_{ij}] = 0$

And thus:
$\alpha_{qx}\beta_{ij} + \beta_{ij}\alpha_{qx} = 0$
can always rearrange to
$\alpha_{qx}\beta_{ij} = -\beta_{ij}\alpha_{qx}$

Was that correct?

---QUOTE post#89---
There is no “ $\delta_i$ " in the fundamental equation. I suspect you are referring to the “Nabla” (looks like an upside down capital delta).
Remember, $\vec{\nabla_i}$ was define to be a vector differential
$\frac{\partial}{\partial x_i} \hat{x} +\frac{\partial}{\partial \tau_i} \hat{\tau}$ and $\vec{\alpha_i}$ was defined to be $\alpha_{ix}\hat{x} + \alpha_{i\tau} \hat{\tau}$ thus that first sum becomes:
-----

Yes I was referring to "nabla"

I am forced to obsess on the details since I need to learn math as I go on, and I am puzzled by couple of things with the above.

x hat and tau hat appear in the definitions of "alpha" and "nabla", but never in the equations where you use those definitions. So I figure that's because they essentially mean "multiply by one", and thus can be simply removed. In fact I don't understand why they appear in the definitions in the first place; is it just a convention to say these elements are conceived as vectors?

Related to this, for "nabla", in the definition itself there is no function which is to be differentiated, right? I mean, x hat is not something you can differentiate.

And finally, following the algebraic rule:
(a + b) * (c + d) = ac + ad + bc + bd

I would have thought the first sum is:

$\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i} = \alpha_{1x}\frac{\partial}{\partial x_1}+\alpha_{1x}\frac{\partial}{\partial \tau_1}+\alpha_{1\tau}\frac{\partial}{\partial x_1}+\alpha_{1\tau}\frac{\partial}{\partial \tau_1}+\cdots+\alpha_{nx}\frac{\partial}{\partial x_n}+\alpha_{nx}\frac{\partial}{\partial \tau_n}+\alpha_{n\tau}\frac{\partial}{\partial x_n}\alpha_{n\tau}\frac{\partial}{\partial \tau_n}$

What am I missing?

Running out of time to write this post now, but let it be said I'm starting to have some vague idea about the sign change and the term that arises when "i=q", but I need to do some more "obsessing".

Thank you for your patience.

Oh, noticed your post while posting this, and just a quick comment:

Quote:

Originally Posted by Doctordick

What you are presenting is a specific epistemological construct. You are clearly trying to comprehend how that specific construct is to be seen as a ”what is”, is “what is” table. Understanding how a specific epistemological construct is to be represented in a ”what is”, is “what is” table is exactly the same problem as trying to understand a language or reality itself (that understanding requires a specific epistemological construct which itself needs to be understood). I am simply not concerned with how that all is to be achieved. The only issue of interest to me is that it can be done!

Actually I understand what you mean by "it can be done" and why that is the relevant point (as long as it can be done, one way or another, the constraints ought to hold).

Quote:

To hold that the epistemological construct cannot depend on patterns in the collection of symbols is simply not correct.

Yeah, isn't it instead the opposite that is correct; The construct must depend on the patterns exclusively because the labels were arbitrary to begin with?

So, what I sad was, it seems the confusion lies in thinking of the probability function as something that relies on some specific labels (the way I described), when instead its behaviour cannot be a function of specific labels but of the patterns found from the table. Isn't that correct?

About "space", even though in PF I was asking about whether you've conceived some specific (albeit "arbitrary") way to map some specific epistemological view (or in this case some definition of space), you notice it didn't stop me from plowing onwards with the actual topic that I never got an answer

While writing the previous post I was kind of assuming it might be little bit too complicated issue to discuss on the side.

Anyway, I didn't mean to confuse matters

I'm guessing a lot of people are thinking right now "but how could any of this be of any use if we don't know how to map any worldview onto it", then let me say it is sufficient if you pick up that it would be possible to map any sort of worldview this way (just might be complicated!), and as long as that is just possible, you can investigate whether the constraints ought to hold, and consequently what (epistemological) significance that would have.

...and now I really need to run.

-Anssi

Qfwfq · 11-21-2007

Quote:

Originally Posted by Doctordick

Off we go again. You have brought up “Fock space” at least three times already.

I knew that would irritate you Dick, sorry, but I just couldn't help it!

Quote:

Originally Posted by Doctordick

Can't you comprehend that the idea of “Fock space” is an epistemological construct?

Of course it is!

I did not say that "what is, s what is" equals Fock space, I simply saw the nexus, of which you seem to be aware.

Quote:

Originally Posted by Doctordick

This implies we need to understand physics (an epistemological construct).

No it doesn't. That isn't the meaning of Slowikowski's words. By "origin of the Fock space concept" he simply meant from where folks got the bright idea. Notice one of the last sentences in that page: "Last but not least there exist certain interesting models of Fock spaces within social science."

The helpful insight was just a little step toward getting a grip on the mechanism of how you represent an ontology table and deduce those constraints and the fundamental equation.

Quote:

Originally Posted by Doctordick

Start defining them and you are off subject!

I was only trying to get a grip, I would have understood just as well if Anssi had talked about a pear and a koala instead of proton and electron.

What happened to that avatar you were so pleased with in the past? Remember? The old professor in front of the blackboard? You could alter it to have an old-style teacher's cane in one hand, so as not to allow pupils to exchange remarks!

Buffy · 11-21-2007

Quote:

Originally Posted by AnssiH

Quote:

Originally Posted by Doctordick

To hold that the epistemological construct cannot depend on patterns in the collection of symbols is simply not correct.

Yeah, isn't it instead the opposite that is correct; The construct must depend on the patterns exclusively because the labels were arbitrary to begin with?

The conflict here is between the notion that "the labeling is completely arbitrary" and "the resulting patterns *must* obey shift symmetry."

If we can stop for a moment from associating with any meaning--even a vague "this vector represents an epistemological construct"--except for pure math, then there is no requirement that a "completely arbitrary mapping" *must* "obey shift symmetry." The vector can be of any number of dimensions and has no restrictions on its structure or internal relationships. Referring back to Dick's example, it could obey anti-symmetric or non-associative properties.

If it truly is "completely arbitrary" there can be no mathematical certainties about it without creating *assumptions*.

Now if this shift symmetry is supposed to be *imposed* by some sort of restrictions (i.e. "assumptions") that go along with the statement "this vector represents an epistemological construct" then we need to understand how that vector's structure and labeling are restricted by that statement.

Its exactly that point that is causing all the confusion here and why it is so hard to grant Dick his fundamental equation: Unless we understand this most fundamental mapping--from the "arbitrary vector representation" onto "an arbitrary *MODEL* of an epistemological construct (note that this is a meta-definition and not a demand that it be mapped onto the "real world" or any specific example)--there is no way to resolve the central question of whether or not the equation can be considered "valid" and therefore support its admittedly possibly profound conclusions.

This last point is important Anissi, because when you say:

Quote:

Originally Posted by AnssiH

I'm guessing a lot of people are thinking right now "but how could any of this be of any use if we don't know how to map any worldview onto it", then let me say it is sufficient if you pick up that it would be possible to map any sort of worldview this way (just might be complicated!), and as long as that is just possible, you can investigate whether the constraints ought to hold, and consequently what (epistemological) significance that would have.

I think I can at least speak for both Q and myself in saying that any activity into even *partially* or *conceptually* "mapping onto the real world" is useful as a mechanism for *understanding* the model, but its *not* the *point*! In fact the really interesting implications have to do with *models* *because* of the fact that Dick correctly points out, actually enumerating and mapping all those elements is computationally impossible! What it might tell us is something about models in general that goes directly to the title of this thread: what we "know" has to do with the *models* we create to *explain* reality, and if there are limits or hidden structures to these models, we can dramatically alter how we "judge" (assign a probability to (!)) them!

Fundamentally I think that what Dick is trying to do here is *fascinating* but I admit I don't get his fundamental equation, and that's why we continue to get stuck here: it just seems to me to be a violation of itself by claiming no restrictions, but not being valid without them (even from a purely mathematical viewpoint).

So, Dick, I have to admit that your last response to me still did not get across the reasons why you think that the fundamental equation *must* obey shift symmetry, precisely because it can be arbitrary and obey any number of perfectly valid, "pure" mathematical rules. I can see how a specific set of labeling rules or structure--again completely unmapped and non-meaningful, but not "arbitrary" or "random" in the strict sense of those terms--could provably require the equation to obey shift-symmetry. But as far as I can see, you can't have *both* of these concepts together using *only* mathematics: you either need to define the vector structure or the equation is valid only under certain assumptions. And either way, you need to justify the structure or the assumptions, or there are no conclusions to be drawn.

To weave a wall to hem us in,

Buffy

Doctordick · 11-22-2007

Quote:

Originally Posted by AnssiH

Sorry it has taken a while.

No problem at all. I am fully aware that your first priority is earning a living and I wouldn't want you to think there is anything more than enlightenment to be found here. But enlightenment can be fun. All in all, I think you are doing an excellent job of picking up the mathematics. And I am very well aware of the nuances of translation from one language to another. I think most people (and that very much includes me) in the United States are severely short changed by the lack of interest in learning multiple languages.

My mothers family emigrated from Germany in the 1800's and a lot of people in my home town were German (our next door neighbor didn't speak English). Most of my mothers family spoke both English and German but my mother absolutely would not allow German to be spoken in our house (her attitude was “we are Americans and we speak English”). I can understand her attitude and in many respects I think she did the right thing for the time but I also feel it would have been nice to have learned two languages back when it was easy.

Quote:

Originally Posted by AnssiH

I suppose when you say "every pair of points", that doesn't refer to an "x,tau pair", but instead every possible "ontological element pair" you could compose from a specific present, i.e. 4 elements, each expressed as an (x,tau) point, could be used to compose 6 different pairs, i.e. they yield 6 different $beta_{ij}$ elements?

Absolutely correct. The term “point” here refers to a point in a specific $(x,\tau)_t$ plane. Remember, a tau index was attached to every x index for the sole purpose of assuring that multiple occurrences of a specific numerical reference x (and that would be in a specific epistemological construct to be displayed in the ”what is”, is “what is” table) could be represented as a point on the x axis without loosing quantity information.

Your comment concerning why the alpha beta commutation yields $\alpha_{qx}\beta_{ij} = -\beta_{ij}\alpha_{qx}$ is absolutely correct.

Quote:

Originally Posted by AnssiH

Yes I was referring to "nabla"

I am forced to obsess on the details since I need to learn math as I go on, and I am puzzled by couple of things with the above.

x hat and tau hat appear in the definitions of "alpha" and "nabla", but never in the equations where you use those definitions.

Here you are missing a very important point. The notation $\hat{x}$ denotes a unit vector pointing in the x direction and $\hat{\tau}$ denotes a unit vector pointing in the tau direction. Another common method of representing a vector in mathematics is to simply bold the symbol.

Quote:

Originally Posted by AnssiH

So I figure that's because they essentially mean "multiply by one", and thus can be simply removed. In fact I don't understand why they appear in the definitions in the first place; is it just a convention to say these elements are conceived as vectors?

Their use allows us to write “vector” equations. The symbol $\vec{\nabla}$ (which you will often also see represented as a bold $\nabla$ ) is the vector representation of a partial derivative. Being a vector in the $(x,\tau)_t$ plane, the representation has an x component (which is the partial with respect to x) and a tau component (which is the partial with respect to tau).

Quote:

Originally Posted by AnssiH

Related to this, for "nabla", in the definition itself there is no function which is to be differentiated, right? I mean, x hat is not something you can differentiate.

That is correct, all x hat does is specify the direction of the component.

Quote:

Originally Posted by AnssiH

And finally, following the algebraic rule:
(a + b) * (c + d) = ac + ad + bc + bd

Your interpretation here is erroneous. In vector relations, there is a thing called the “dot” product (or often, the “inner” product). The “dot” product between two vectors is defined in terms of the components: $\hat{x}\cdot\hat{x}=\hat{\tau}\cdot\hat{\tau}=1$ and $\hat{x}\cdot\hat{\tau}=\hat{\tau}\cdot\hat{x}=0$ , Suppose we are given two vectors $\vec{A}$ and $\vec{B}$ where, expressed as components in an (x,y) space, these vectors are $\vec{A}=A_x \hat{x}+A_y \hat{y}$ and $\vec{B}=B_x \hat{x}+B_y \hat{y}$ . This example is analogous to what you are trying to talk about when you wrote (a + b) * (c + d) = ac + ad + bc + bd (a would be $A_x$ , b would be $A_y$ , c would be $B_x$ and d would be $B_y$ ); and, if this is to be a vector “dot” product, your use of the star should be replaced with a "dot". The star stands for regular multiplication and is not equivalent to a vector “dot” product. Thus the proper expansion of the expression you quote should be,

$(a\hat{x}+b\hat{y})\cdot(c\hat{x}+d\hat{y})=ac(\hat{x}\cdot\hat{x})+ad(\hat{x}\cdot\hat{y})+bc(\hat{y}\cdot\hat{x})+bd(\hat{y}\cdot\hat{y})=ac+bd$

The reason this inner or dot product is defined the way it is, is that it turns out that the result is independent of the orientation of the coordinate system. Look at two coordinate systems centered at exactly the same point but one rotated by some angle $\theta$ with respect to the other. Call the first an (x,y) representation and the second a (p,q) representation. The components of some arbitrary vector $\vec{A}$ in the (x,y) representation are easily obtained via $A_x= \hat{x}\cdot \vec{A}$ and $A_y= \hat{y}\cdot \vec{A}$ . Likewise, the components in the (p,q) representation are $A_p= \hat{p}\cdot \vec{A}$ and $A_q= \hat{q}\cdot \vec{A}$ . The same thing can be done for an arbitrary vector $\vec{B}$ . Thus, we can express the dot product in either representation.

Not only can we express the dot product in either representation but we can also express the unit vectors, represented by the “hat” notation, in either representation: $\hat{p}=\cos{\theta}\hat{x}-\sin{\theta}\hat{y}$ and $\hat{q}=\sin{\theta}\hat{x}+\cos{\theta}\hat{y}$ (assuming positive $\theta$ is a clockwise rotation). Now, if we write down the vector dot product in the (p,q) representation,

$\vec{A}\cdot\vec{B}=A_pB_p+A_qB_q=\left\{\hat{p}\cdot\vec{A}\right\}\left\{\hat{p}\cdot\vec{B}\right\}+\left\{\hat{q}\cdot\vec{A}\right\}\left\{\hat{q}\cdot\vec{B}\right\}=$

$\left\{(\cos{\theta}\hat{x}-\sin{\theta}\hat{y})\cdot\vec{A}\right\}\left\{(\cos{\theta}\hat{x}-\sin{\theta}\hat{y})\cdot\vec{B}\right\}+\left\{(\sin{\theta}\hat{x}+\cos{\theta}\hat{y})\cdot\vec{A}\right\}\left\{(\sin{\theta}\hat{x}+\cos{\theta}\hat{y})\cdot\vec{B}\right\}$ .

We can then rearrange the terms and get

$\vec{A}\cdot\vec{B}=(\cos^2{\theta}+\sin^2{\theta})\left\{\hat{x}\cdot\vec{A}\right\}\left\{\hat{x}\cdot\vec{B}\right\}+(\sin^2{\theta}+\cos^2{\theta})\left\{\hat{y}\cdot\vec{A}\right\}\left\{\hat{y}\cdot\vec{B}\right\}$

$=\left\{\hat{x}\cdot\vec{A}\right\}\left\{\hat{x}\cdot\vec{B}\right\}+\left\{\hat{y}\cdot\vec{A}\right\}\left\{\hat{y}\cdot\vec{B}\right\}=A_xB_x+A_yB_y$

which is exactly $\vec{A}\cdot\vec{B}$ expressed in the (x,y) representation. The only reason I went through all of that was to convince you that the dot or inner product is indeed something defined by the vectors themselves and not by the particular coordinate system chosen to represent those vectors. In my presentation so far, that issue is not important; however, down the road we will be confronted with a situation where the fact becomes a very important aspect of the analysis and you should understand that the product is not dependent upon the actual axes used to represent the vector.

I might also comment that the dot product is a scaler (not a vector) and that it is exactly this same kind of product I used to turn an arbitrary mathematical function (expressed as $\vec{\psi}$ , a vector in an abstract multidimensional space) into a probability (a number and not a vector). Oh, another name often used for the inner product or dot product is “a scalar product”. Mathematics is as full of synonyms as is any language :P .

Quote:

Originally Posted by AnssiH

So, what I sad was, it seems the confusion lies in thinking of the probability function as something that relies on some specific labels (the way I described), when instead its behavior cannot be a function of specific labels but of the patterns found from the table. Isn't that correct?

Absolutely!

Quote:

Originally Posted by AnssiH

About "space", even though in PF I was asking about whether you've conceived some specific (albeit "arbitrary") way to map some specific epistemological view (or in this case some definition of space), you notice it didn't stop me from plowing onwards with the actual topic that I never got an answer

While writing the previous post I was kind of assuming it might be little bit too complicated issue to discuss on the side.

I would rather put it off the definition for some subtle reasons but the issue is not really any different from my definition of time. Time was defined in terms of changes in “what was known” but was eventually laid out as a parameter on a t axis in my Euclidean $(x,\tau,t)$ representation of the ontological elements. My definition of space will arise in a very similar manner from those x and tau indices.

Quote:

Originally Posted by AnssiH

Anyway, I didn't mean to confuse matters

I'm guessing a lot of people are thinking right now "but how could any of this be of any use if we don't know how to map any worldview onto it", then let me say it is sufficient if you pick up that it would be possible to map any sort of worldview this way (just might be complicated!), and as long as that is just possible, you can investigate whether the constraints ought to hold, and consequently what (epistemological) significance that would have.

You are absolutely correct. Based on any conventional view of science, there is absolutely nothing to be expected from this analysis. As you say, “how could any of this be of any use if we don't know how to map any world-view onto it", it absolutely can be of no use. That is exactly why the solutions to my equation are so fascinating; they go totally against any reasonable expectations.

Quote:

Originally Posted by Qfwfq

The helpful insight was just a little step toward getting a grip on the mechanism of how you represent an ontology table and deduce those constraints and the fundamental equation.

I suppose I do not have a clear idea of what you do and don't understand. I have gone back and analyzed the thrust of your comments on most all of my posts and have come to the conclusion that the difficulty I have with your responses is that they invariably seem to confuse the difference between theory and fact. You don't seem to comprehend that what I am putting forth is not a theory and you keep putting forth theories as possible interpretations of what I am saying; for example, look at the title of Slowikowski's outline, “Fock space theory and its applications in analysis, algebra and physics”. If you want me to, I can give you many many examples where you bring in theories and ask me what my work has to say about them. The correct answer is “nothing”; theories are called theories because they can not be proved. They are based upon inductive conclusions: “by guess and by golly” propositions found to be, apparently, consistent with what we think we know. That simply is not a valid approach to the issues I am trying to discuss.

Quote:

Originally Posted by Qfwfq

What happened to that avatar you were so pleased with in the past? Remember? The old professor in front of the blackboard? You could alter it to have an old-style teacher's cane in one hand, so as not to allow pupils to exchange remarks!

I wouldn't say I was “pleased” with that avatar. I just stuck it in after someone posted it as a supposed picture of me. I just figured the best way to get rid of the “picture” was to use it as an avatar for a while. Actually I found the thing kind of dumb.

Quote:

Originally Posted by Buffy

The conflict here is between the notion that "the labeling is completely arbitrary" and "the resulting patterns *must* obey shift symmetry."

Not the resulting patterns; but rather, the epistemological constructs created from this information.

Quote:

Originally Posted by Buffy

If we can stop for a moment from associating with any meaning--even a vague "this vector represents an epistemological construct"--except for pure math, then there is no requirement that a "completely arbitrary mapping" *must* "obey shift symmetry."

Of course not. That is a misrepresentation of the issue under discussion. The issue is that the epistemological construct can not be a function of the specific numerical mappings chosen to refer to the ontological elements on which that epistemological construct is built.

Quote:

Originally Posted by Buffy

If it truly is "completely arbitrary" there can be no mathematical certainties about it without creating *assumptions*.

Again, I have no argument with this statement at all.

Quote:

Originally Posted by Buffy

Now if this shift symmetry is supposed to be *imposed* by some sort of restrictions (i.e. "assumptions") that go along with the statement "this vector represents an epistemological construct" then we need to understand how that vector's structure and labeling are restricted by that statement.

It's not imposed; it is just there and it cannot be ignored. We have two very simple issues here. A, in our left hand is the information which constitutes “what is to be explained” (what we have to work with is finite and incomplete; always changing: see my definition of time). B, in our right hand is our explanation which yields our expectations concerning new information. Somehow, by a means beyond the interest of this presentation, we start with A and achieve B.

The symmetry arises from the simple fact that the process can not depend upon the specific numerical labels we use to refer to the elements of information which go to make up A. That's it! There is no more to the circumstance than that!

If the elements of information are referred to with numerical labels and B can be achieved, then the expectations can be expressed as a function of those numerical labels (it can be represented as a mathematical function). Since those expectation can not be a function of how the numerical labels are assigned, but rather, as Anssi has commented, must be a function of the actual information which goes to make up A. It follows that, if we are going to express our expectations though the use of those numerical labels, the form of the function used must display shift symmetry. There is no way out of the conclusion; it follows from the simple fact that we are using arbitrary numerical labels. At the same time, there exists no set of elemental information which cannot be referred to with an arbitrary set of numerical labels so the conclusion must be universal.

Quote:

Originally Posted by Buffy

Unless we understand this most fundamental mapping--from the "arbitrary vector representation" onto "an arbitrary *MODEL* of an epistemological construct (note that this is a meta-definition and not a demand that it be mapped onto the "real world" or any specific example)--there is no way to resolve the central question of whether or not the equation can be considered "valid" and therefore support its admittedly possibly profound conclusions.

Now here I disagree with you completely. The relationship still exists even if we cannot actually generate a specific procedure for getting from A to B. And, since the problem of generating such a procedure, even for apparently simple circumstances, is so complex as to be next to impossible, it is a total waste of time to worry about the issue.

When I took my Ph.D. candidacy exam, there was one rather simple problem on exam: “Prove that a radially oscillating electric charge will not radiate!” The answer is quite simple. In a vacuum, the electric and magnetic vectors of EM radiation are perpendicular to the propagation direction. Center your coordinate system on the center of the oscillating charge then calculate the electric vector at some large radius at a specific time “t” related to the supposed oscillation. Then remove the radial component as it is not part of the “radiation” (it's a static component). The component perpendicular to r yields the electric field of the radiation itself. Having calculated that component, look at a mirror image of the problem and redo exactly the same calculation. The answer in the second case will be a vector pointing in exactly the opposite direction. But there is no difference whatsoever between the two problem (the mirror image is identical to the original) so the only valid answer is zero. The point being that knowing how to do the calculation is not needed at all. It is simply a consequence of the fact that the electric vector is perpendicular to the propagation direction: i.e., the information necessary to establish a radiation solution simply isn't there. Information not in the specified problem cannot be in the solution.

In the case we are talking about, the shift symmetry is a simple consequence of the fact that no information exists which can be used to specify what numerical labels are to be used. That information arises only when we have a specific epistemological solution and is, in fact, nothing but pure inductive hypothesis. The same phenomena exists even if the ontological elements and the epistemological solutions are expressed in ordinary common language; however, the use of common language precludes examination of the consequences since (at least presently) there exists no mechanisms in ordinary language syntax which can express such coherent specific label alterations. The situation is simply beyond any comprehension expressible in ordinary language. It should be clear that failure to provide a specific mapping is certainly not an issue here. The significant point is that the information necessary to establish a unique labeling system simply isn't there. If that is true, then the expression

$\vec{\psi}(x_1+a,\tau_1,x_2+a,\tau_2,\cdots, x_n+a,\tau_n,t)=\vec{\psi}(x_1,\tau_1,x_2,\tau_2,\cdots, x_n,\tau_n,t)$

must be true. That simple change in the labeling system cannot invalidate B, our explanation which yields our expectations concerning new information.

Most people have no comprehension of such a difficulty because they tend to believe the meanings of words are contained in the words themselves and simply can not comprehend the possibility of any difficulty with their understanding of that meaning or that an alternate interpretation can exist.

Quote:

Originally Posted by Buffy

So, Dick, I have to admit that your last response to me still did not get across the reasons why you think that the fundamental equation *must* obey shift symmetry, precisely because it can be arbitrary and obey any number of perfectly valid, "pure" mathematical rules. I can see how a specific set of labeling rules or structure--again completely unmapped and non-meaningful, but not "arbitrary" or "random" in the strict sense of those terms--could provably require the equation to obey shift-symmetry.

It is the epistemological solution ( $\vec{\psi}$ ) which, when expressed as a function of numerical labels, must obey shift-symmetry. The equation is nothing more than an enforcer of that constraint.

The equation also enforces elimination of unobserved information via specific relationships with “invalid” ontological elements. I notice you don't mention that issue. Am I making an error when I presume you understand it?

I also suspect you are having the same difficulty Qfwfq is struggling with: you are confusing theory with deduction. What I am presenting is a deduction from definition, not a theory. If you miss that point, you miss the essence of the entire presentation.

Have fun -- Dick

Qfwfq · 11-23-2007

Quote:

Originally Posted by Doctordick

I suppose I do not have a clear idea of what you do and don't understand. I have gone back and analyzed the thrust of your comments on most all of my posts and have come to the conclusion that the difficulty I have with your responses is that they invariably seem to confuse the difference between theory and fact. You don't seem to comprehend that what I am putting forth is not a theory and you keep putting forth theories as possible interpretations of what I am saying; for example, look at the title of Slowikowski's outline, “Fock space theory and its applications in analysis, algebra and physics”. If you want me to, I can give you many many examples where you bring in theories and ask me what my work has to say about them. The correct answer is “nothing”; theories are called theories because they can not be proved. They are based upon inductive conclusions: “by guess and by golly” propositions found to be, apparently, consistent with what we think we know. That simply is not a valid approach to the issues I am trying to discuss.

There has come to be a bit of a variety in the usage of the word theory and also there's no point in taking Slowikowski's words as the Bible. I mentioned Fock space considering it a definition and nothing more. If Slowikowski's course is about "Fock space theory" then he's doing something a tad more than what I was talking about, still it is a mathematical theory, the meaning of which has nothing whatsoever to do with the kind of inductive reasoning you are talking about. Yet again I see what our problems in communicating are due to. In mathematics, a theory is a conceptual construct starting with definitions, and perhaps axioms, which really just set requisite properties for calling something a certain name; it then procedes deductively from these and is not to be confused with a conjecture (even less one about reality) and it makes no sense to say it can't be proven.

The whole thing you are discussing is a logical-mathematical framework, one which requires modern mathematical notions, you should try to keep your language in line with that of modern mathematics. Therein lies my need for examples, specific case or whatever may help me to see what you do and don't mean by things you say.

Quote:

Originally Posted by Doctordick

I wouldn't say I was “pleased” with that avatar.

It would however suit you all the better with a teacher's cane in the hand, when you decide what we may and may not say.

Doctordick · 11-23-2007

Quote:

Originally Posted by Qfwfq

There has come to be a bit of a variety in the usage of the word theory and also there's no point in taking Slowikowski's words as the Bible.

I didn't mean to do that. All I was trying to do was to clarify our communication problems. You have to see that I am having as big a problem understanding your difficulties with what I say as you and Buffy are having trying to understand me. To me, the underlying basis of my argument is so obvious as to be unmistakable and yet, to date, except for Anssi, no one seems able to pick up on the issue. Perhaps I am wrong and you do understand what I am talking about but I certainly don't get that impression when I read your responses.

Quote:

Originally Posted by Qfwfq

Yet again I see what our problems in communicating are due to. In mathematics, a theory is a conceptual construct starting with definitions, and perhaps axioms, which really just set requisite properties for calling something a certain name; it then procedes deductively from these and is not to be confused with a conjecture (even less one about reality) and it makes no sense to say it can't be proven.

Now that, on the other hand, seems to be a clear statement of the circumstance I am trying to communicate which I would ordinarily take as an indication that you do understand what I am doing.

Quote:

Originally Posted by Qfwfq

The whole thing you are discussing is a logical-mathematical framework, one which requires modern mathematical notions, you should try to keep your language in line with that of modern mathematics.

Well I might be willing to do such a thing except for two rather overwhelming factors: first, I have been outside the academic field for close to forty years and am not at all familiar with current jargon and second, I am an old man and it's not easy to teach an old man new tricks (I would no longer refer to my mind as particularly sharp). It has to be simple or I am suspicious of it and heaping lots and lots of meaning in a symbol seems to me to be an intellectually dangerous thing to do. The complexity of the concepts behind the symbol requires complete understanding and that is a difficult thing to be sure of. You can easily end up making assumptions you don't even realize you have made. Even simple issues often tread into subtle assumptions if you are not careful.

Quote:

Originally Posted by Qfwfq

Therein lies my need for examples, specific case or whatever may help me to see what you do and don't mean by things you say.

I am at a loss to provide the examples you desire because I have no idea as to what the stumbling block is. On a positive note, I will say that the conversation has ground my thoughts down to the fundamental essence of the thing and that certainly makes this dialog worthwhile to me.

It seems to me that one must comprehend, if the absolute entirety of the things you have to work with can be referred to via a finite set of numerical labels and your expectations can expressed in a numerical manner (the probability of new things) then your method of getting from the given information to the solution qualifies as a defined mathematical function. The fact that the numerical labels are utterly immaterial implies all kinds of symmetries within that process, only one of which have I so far begun to touch upon here.

I have merely named the mathematical function $\vec{\psi}$ and asserted that the fact that the solution can not depend upon the specific assignment of those labels implies:

$\vec{\psi}(x_1+a,\tau_1,x_2+a,\tau_2,\cdots, x_n+a,\tau_n,t)=\vec{\psi}(x_1,\tau_1,x_2,\tau_2,\cdots, x_n,\tau_n,t),$

which is the very definition of shift symmetry; and Buffy baulks! I have no understanding of the difficulty there. I wish someone could make it clear to me why this isn't a factual deduction

.

Quote:

Originally Posted by Qfwfq

It would however suit you all the better with a teacher's cane in the hand, when you decide what we may and may not say.

I have no intention of putting any such constraint on anyone here. I am sorry if I have given that impression.

Have fun -- Dick

Rade · 11-24-2007

Quote:

Originally Posted by Doctordick

We have two very simple issues here. A, in our left hand is the information which constitutes “what is to be explained” (what we have to work with is finite and incomplete; always changing: see my definition of time). B, in our right hand is our explanation which yields our expectations concerning new information. Somehow, by a means beyond the interest of this presentation, we start with A and achieve B.

I have a question. Suppose I have in my left hand (at A) all the parts of a 10 piece puzzle, and in my right hand (at B) an explanation on paper of how to put the parts together. Each of the 10 pieces has a label (1...10). The explanation at B uses these labels directly, nothing arbitrary, but clearly the explanation is very complex because it must allow for all combinations of placement of pieces. But, it matters not which piece (1...10) is placed first, then second, etc--we reach the expectation (complete puzzle) from any sequence.

Now, clearly, what I have at A is finite and complete, never changing. And what I have at B is complete explanation to meet expectation. So my question--it is then correct to conclude that your approach is not needed to understand this example, that is, to start with A and achieve B, since your approach requires that what is at A is "always incomplete, always changing" (as you say) ?

Doctordick · 11-24-2007

Rade, I am sorry but you totally misinterpret my statement, “in our left hand is the information which constitutes 'what is to be explained' (what we have to work with is finite and incomplete; always changing: see my definition of time)”. What you propose as a possibility, in your left hand, is not at all “the information which constitutes the totality of 'what is to be explained'”; it is rather, a mere reference to a that information posed in the form which already presumes a very complex solution (your acquired world-view). The actual information which stands behind your supposed solution is probably so voluminous that, if you lived to be a thousand years old, you would not have sufficient time to write down a complete list of references to that specific information. Think about giving the problem to a new born infant keeping track of every piece of information impacting his (or her) knowledge until he (or she) solves the problem. Keeping track of every bit of specific information he (or she) experiences in the first day is probably beyond comprehension and I doubt the solution would arrive in less than two or three years.

If the only information ever acquired were the mere ten symbols you present, I doubt a solution resembling the one you have in mind would occur in the history of the universe. I don't think you even begin to comprehend the nature of the problem I am talking about here.

Quote:

Originally Posted by Rade

So my question--it is then correct to conclude that your approach is not needed to understand this example, that is, to start with A and achieve B, since your approach requires that what is at A is "always incomplete, always changing" (as you say) ?

My approach is not needed because your example is simply not an example of what I am talking about and has absolutely nothing to do with the issue of “incompleteness” or “always changing”. My comment as to the information "always incomplete, always changing" has to do with the validity of the solution, any solution, and is there to encompass the fact that we are not all knowing (the future, what we do not know, may well invalidate the most perfect flaw-free explanation of what we think we know).

Have fun -- Dick

Doctordick · 11-25-2007

Hi Buffy,

I was thinking about the problem you seem to be having with my assertion of symmetries in $\vec{\psi}$ . Perhaps there is another way I can present the issue that might be clearer to you. Again, with this “left hand”, “right hand” picture of the circumstance. In your left hand you have the collection of ontological elements, the collection of “noumenons" or perhaps a collection of experiences if you wish. What name you give to this collection is fundamentally immaterial. It is nothing except a reference label for what we are talking about. It is elemental in the sense that it is irreducible; it is the basis upon which your understanding of the circumstance rests; the raw foundation of information which is to be explained by your epistemological solution. As such, in the absence of that explanation, it is totally undefined data.

Quote:

Originally Posted by Anssi

The way I see it, a "noumenon" is referring to the reality behind a "phenomenon" we are subjectively aware of (I.e. it is contrasted by "phenomenon"). The reason Kant was using that concept was to refer to the idea that being subjectively aware of some phenomenon is a case of having mentally categorized reality, and the actual ontological reality behind that mental idea is not captured by that categorizing (which results into what we call "phenomena" and "things" so to speak")

There are two very fundamental aspects of this data which cannot be denied. First, no matter what “noumenons” stand behind our understanding, and how well our understanding explains and defines those “noumenons”, the possibility exists that new information will arise to invalidate that understanding. And, second, the number of these “noumenons” on which our understanding is built is finite as we cannot intellectually consider each and every specific case of an infinite set (that is essentially the definition of infinity: if the number is infinite, no matter how many we consider we haven't finished).

So, since the number standing behind our epistemological solution is finite, we can attach numerical reference labels to them. It has to be understood that definitions for these labeled entities are not at all inherent in the “noumenons” themselves but flow directly from the epistemological construct which was created to explain them. All I am trying to do here is to clarify exactly what is in your left hand per this presentation.

Now, in your right hand is the result of your epistemological solution, the result of your explanation of those “noumenons”. The result of your explanation is that you have expectations concerning what additional information will arise. So long as your expectations are consistent with your experiences (the new “noumenons” as defined by your epistemological construct) you will consider your explanation correct or “flaw-free”. So these expectations can be cast as a probability that a specific collection of “noumenons” will or will not become part of that finite set you actually know (the set upon which your understanding is based). That probability can be seen as a normalized magnitude of a vector in an abstract space, $\vec{\psi}$ , which is capable of representing absolutely any transformation from one set of numbers to another set: i.e., the representation omits no procedure whatsoever.

So, if you can explain the “noumenons” in your left hand, you can produce a table of expectations in your right hand. Since we have used numerical labels for the “noumenons” in your left hand, the procedure for producing that table in your right hand can be seen as some specific mathematical function defined by $\vec{\psi}$ as a function of the specific collection of “noumenons” about which the probability is desired. Essentially, based on the collection of “noumenons” in our left hand, we can produced a table of expectation given by

$\vec{\psi}(x_1,\tau_1,x_2,\tau_2,\cdots, x_n,\tau_n,t)$

in our right hand (note that this is the very definition of $\vec{\psi}$ ). You should understand that I have put this in this “left hand”, “right hand” representation because I want to remove any reference at all concerning how this is to be accomplished. All I am concerned with here is the fact that given one, you can achieve the other. At that point I make the assertion that

$\vec{\psi}(x_1+a,\tau_1,x_2+a,\tau_2,\cdots, x_n+a,\tau_n,t)=\vec{\psi}(x_1,\tau_1,x_2,\tau_2,\cdots, x_n,\tau_n,t),$

which is, of course, a statement of shift symmetry itself. You baulk; so let us look at the possibilities. Essentially, your claim is an assertion that the equal sign does not belong there. If I presume your assertion is factual, it implies that, if I know the procedure for getting from the information in my left hand (the numerically labeled noumenons) to the expectations in my right hand (the mathematical function $\vec{\psi}$ of those noumenon labels) and I perform the relabeling (adding a to each and every noumenon label in both hands) then the procedure no longer yields the same probabilistic table.

If simple relabeling of the noumenon argument labels destroys the solution procedure, then that solution procedure must depend on how these noumenons are labeled and that means the labeling certainly isn't arbitrary.

Essentially, what you are saying is that the problem of explaining the noumenons in your left hand depends on how you label the things you are explaining. If that is the case, where are you to get the information as to the proper labeling procedure? Supposedly, your solution is based upon those undefined noumenons in your left hand and nothing else.

Think about it -- Dick

Majik · 11-30-2007

You seem to be trying to make a distinction between ontology in the left hand and epistimology in the right hand. But what if our epistimology were derived solely from inescapable logic? Would we really then need to even look in our left hand? Would we really need to do experiements? If physics were derived from logic alone, wouldn't we "know" without even looking what IS real?

What can we know of reality?

Hypography?

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