What can we know of reality?

Qfwfq · 10-18-2007

Quote:

Originally Posted by Doctordick

You've done it nine times in your latest post.

That's a form of laziness known as the copy-paste effect!

Quote:

Originally Posted by Doctordick

I think you are concerned with the issue of solutions and not the issue of constraints.

Actually I see both as ways of discussing more or less the same thing, the factor (K(...)) which multiplies $i\psi(...)$ is no more nor less than the rate of change in phase ( $\frac{d\varphi}{da}$ in the case of the shift symmetry).

Quote:

Originally Posted by Doctordick

We are dealing with a linear differential equation here.

The one in P is, this does not imply the one in $\psi$ must consequently be.

Quote:

Originally Posted by Doctordick

Essentially, all we are talking about here is a shift in phase between two different solutions (a shift in phase is a rotation in the complex plane).

Yes, but one nitpickin' detail: constant K is the specific case of global phase shift, the shift symmetry (or any one that's on P) doesn't imply this, it may be local.

Quote:

Originally Posted by Doctordick

What do you say to laying the issue aside until after we have discussed the general solutions of the constraint expressed by my fundamental equations:

No problem discussing these, as being a specific case and of course one of interest. Our disagreement is all a matter of necessary and sufficient condition.

${\rm A}\Rightarrow {\rm B}$ (A implies B)

A is a sufficient condition for B, B is a necessary condition for A.

${\rm Dic}0\Rightarrow{\rm Dic}K\Rightarrow{\rm Q}\Rightarrow{\rm P}$

Your tighter constraint implies your less stringent one, both imply mine, all three imply the symmetry in terms of P. Of course, if asserts A and B are:

$\psi\in A$ and $\psi\in B$

the implication $\psi\in A\Rightarrow\psi\in B$ is equivalent to:

$B\subset A$ (which shows how constraints are related to sets of solutions).

Quote:

Originally Posted by Doctordick

All I think Anssi needs to comprehend the situation is to understand how the introduction of the anti-commuting factors alpha and beta allow recovery of the four essential constraints. Essentially that solutions to the above equations satisfy the four constraints deduced and that all solutions which satisfy those four constraints are also solutions to the fundamental equation.

Oui, bon, allez. I'd like to see these steps better detailed, rather than work it out myself, I think it would be a quicker way to see if your argument is satisfactory, in the case you consider, and then perhaps understand the more general case.

I hope no wrong slashes have slipped into my equations.

HydrogenBond · 10-18-2007

The original question was, what can we know of reality? One thing we do know, science has not reached the end of the line or else there would be no room for speculation and all scientists would be historians. That being said, what we percieve to be reality may not be reality at all. Or only part of what we percieve to be scientific reality is reality and will remain at steady state. But anoither part is out of touch with reality and will continue to evolve. But we can't tell one from the other, yet.

One of the litmus tests of scientific validity is math. But the math is only as reliable as the assumptions it is based on. One can make math do almost anything one wants, even if it is out of touch with reality. Let me give an example. Say I assumed that gravity was due to the repulsion of matter by space. This is not true, but it used for a demonstration. Someone skillful in math could use this assumption and sort of do a reciprical of the existing equations, to come up with a math model . The result will be able to make predictions and could be used to put a man on the moon. But it is not in touch with reality, yet supported with math. If we extrapolate from there and this advance math made a good prediction, is this deeper relationship reality?

What this means, if reality is important, math is not the final judge. Math can used to support both reality and conceptual illusions. What is more fundamental is the conceptual analysis, before the math. Good conceptual analysis has to be consistent with reality observation. At the same time, reality observation, needs to be in context of the largest system.

For example, say we had a photo of a pond in the back of someone's yard. We can see the pond, house and other identifying objects. If we zoom in to only the pond, this same observation is no longer certain, because one does no longer has the context of the big picture. The singular reality observation, can now be a lake, pond or ocean. It can be in the north, south, east or west, any time of the year, etc. This is the reality problem faced by conceptualizing from the point of view of specialization. It can see something closely, but not always in the context of the big picture. Yet the close view is so detailed, one assumes what we see is reality. Math is a willing accomplise in all this, with math able to support anything.

For example, say we determine this is a lake near the Adirondac Mts, based on water color and transparency data. Based on that, one now has certain assumptions of what they expect to happen If this doesn't happen as often as we expect, then we need to add statisitical/chaotic theory. If the big picture was also seen, then this chaos may have an explanation. What I am saying is random and chaos is a good litmus test for reality, the more it exsits, the farther from reality our assumptions are getting.

Let me give an historical example. Before Newton was hit on the head with the apple, that got his mental gears of gravity in motion, the affect of falling objects was percieved to be far more random. Sometimes, big things fell faster, and sometimes small things. The theory of chaos and random could have come in handy at that time to explain this. Once the rational relationships of gravity appeared, than all that chaos went away. Before Newton, nobody had big enough perpective to see a trend. But each theory could have had legs with the proper use of chaos.

To add to the confusion, the human mind is both rational and irrational. If both are working at the same time, the result be can rational explanations of the irrational. Or irrational explanations for the rational.

Doctordick · 10-25-2007

Qfwfq, I am sorry for being so slow to respond. Your last post dismayed me quite a little. One of your comments made it quite clear that you do not understand the central problem under discussion. Anssi understands the problem but his mathematics is currently insufficient to understand my constraint mechanisms precisely. The line in your post which made your lack of understanding so clear was ,

Quote:

Originally Posted by Qfwfq

Yes, but one nitpickin' detail: constant K is the specific case of global phase shift, the shift symmetry (or any one that's on P) doesn't imply this, it may be local.

You simply don't seem to understand the fundamental question that I am attacking. Please don't feel bad; it seems that no one comprehends what I am talking about (except of course Anssi as he seems to have come to the same question I am concerned with completely on his own). Personally, I simply do not know how to make the issue clear to someone who has never thought of it. The circumstance reminds me of something I read many years ago. It was a paper written by some ancient Greek (I don't read Greek so what I read was an English translation of the original). Though it wasn't exactly mentioned, the paper concerned the definition of speed. The writer pointed out that, if one person was faster than another, he would cover the same distance in less time or, on the other hand, he would be able to cover a greater distance in the same time. The paper continued to make a number of different comparisons in an attempt to clarify what the writer was talking about. What became quite clear was that the idea of “speed” was not a concept the writer held as obvious. Today, everyone (except maybe some very primitive peoples out of touch with modern gadgets) understand exactly what one means by speed and it is difficult for us to comprehend that confusion could ever have existed.

The issue here is that the indices $(x_1.\tau_1,x_2,\tau_2,\cdots,x_n,\tau_n,$ and t) are nothing more than arbitrary numerical labels used to refer to ontological elements. They constitute an undefined language containing the information to be explained. In any logical deduction based upon those ontological elements, nothing can be introduced which violates the arbitrariness of those assignments (what I believe Anssi refers to as semantics) as, if you make an assumption which violates that arbitrariness, it amounts to asserting you know something about what the assignments mean. I have essentially defined only one basic concept; I have defined what I mean by time. Now if you were to define what you meant by a term, as I did with my definition of “time”, that would be another story: i.e., you could use the concept “local” but only after you had defined what you meant by the term. The issue is that you cannot talk about ontological elements being “local” without providing me with a method of determining which ones qualify as local, the point being that, your definition must be applicable even under the arbitrary reassignment of all of those labels. You should understand here, that I can talk about position on the x axis because that is nothing more than a representation of arbitrary label: that is to say, the labels can be arbitrarily shuffled throughout the collection of ontological elements without violating that definition.

This brings up another issue which I am not sure you understand. I often comment that you are concerning yourself with the solutions and not the constraints on the solutions. The solution here is to find the function $\vec{\psi}$ which yields the probability distribution for those ontological elements identical to the probability distribution yielded by the explanation that $\vec{\psi}$ is to represent. That is why I use the vector notation I use. I refer to $\vec{\psi}$ as a function because, given any set of arguments, that index set $(x_1.\tau_1,x_2,\tau_2,\cdots,x_n,\tau_n,t)$ , it must yield another set (in a totally different abstract space by the way) consisting of $(\psi_0,\psi_1,\psi_2,\psi_3,\cdots,\psi_q)$ where the number q is an open issue. This representation can represent an explicit mathematical function, an arbitrary computer program or even a simple look up table. The central issue being that no relationship capable of generating any specific collection of expectations is omitted. This representation is capable of yielding any result so your solution (that would be your epistemological solution), no matter what it is, is representable by the expression $\vec{\psi}$ . The issue of finding such a solution is an epistemological problem and is of no interest here. What I am concerned with is the fact that our ignorance places some subtle constraints on a flaw-free epistemological solution. I believe I have discovered a way of expressing some very specific constraints in a mathematically exact form.

I showed, it detail, that this required (under the fact that arbitrary reassignment of numerical labels can have no consequences on the solution) that shift symmetry demands that the following expressions are true.

$\frac{P(x_1+a,x_2+a,\cdots,x_n+a,t)-P(x_1,x_2,\cdots,x_n,t)}{a}=0$ ,

$\frac{P(\tau_1+b,\tau_2+a,\cdots,\tau_n+a,t)-P(\tau_1,\tau_2,\cdots,\tau_n,t)}{a}=0$

and

$\frac{P(t+a)-P(t)}{a}=0.$

The form of those expressions being exactly the fundamental nugget of the definition of a derivative leads directly (via some common mathematics) to the requirements that, if the range of possibilities available to those labels is extended to the entire continuum, the following relationships must be valid.

$\sum_{i=1}^n\frac{\partial}{\partial x_i}P(x_1,x_2,\cdots,x_n,t)=0$ , $\sum_{i=1}^n\frac{\partial}{\partial \tau_i}P(\tau_1,\tau_2,\cdots,\tau_n,t)=0$ and $\frac{\partial}{\partial t}P(t)=0:$

i.e., no valid itemized data underlying any epistemological solution can invalidate that expression.

If P, the method of obtaining expectation probabilities, is represented by $\vec{\psi}^\dagger \cdot \vec{\psi}$ (a representation used because it can represent absolutely any mechanism for obtaining P) then the above can be reduced to a constraint on that $\vec{\psi}$ .

$\sum_{i=1}^n\frac{\partial}{\partial x_i}\vec{\psi}(x_1,x_2,\cdots,x_n,t)=0$ , $\sum_{i=1}^n\frac{\partial}{\partial \tau_i}\vec{\psi}(\tau_1,\tau_2,\cdots,\tau_n,t)=0$ and $\frac{\partial}{\partial t}\vec{\psi}(t)=0:$

together with an interesting collection of associated relationships expressing essentially the same constraint.

$\sum_{i=1}^n\frac{\partial}{\partial x_i}\vec{\phi}(x_1,x_2,\cdots,x_n,t)=iK_x\vec{\phi}$ , $\sum_{i=1}^n\frac{\partial}{\partial \tau_i}\vec{\phi}(\tau_1,\tau_2,\cdots,\tau_n,t)=iK_\tau \vec{\phi}$ and $\frac{\partial}{\partial t}\vec{\phi}(t)=im \vec{\phi}$

via the simple mechanism of defining $\vec{\phi}=e^{iK_x * ( x_1+x_2+\cdots+x_n)}e^{iK_\tau * (\tau_1+\tau_2+\cdots+\tau_n)}e^{imt}\vec{\psi}$ . As both of us realize, the introduction of the term $e^{iK}$ has utterly no impact upon the resultant P. Your position was that K could be any function of those indices without effecting the resultant P and I balked. At the time I was very disturbed by the nature of $\vec{\phi}$ so defined. To me it seemed quite obvious that the result violated shift symmetry. After a little examination of the details of possible $\vec{\psi}$ I concluded you were right (the nature of $\vec{\phi}$ was such that any relationship desired could be obtained). What I failed to pick up on was that you had deflected my interest from the fundamental issue. It is not just P which must satisfy shift symmetry but rather the whole issue of any aspect of any epistemological solution. No valid procedure exists which can add information to that which is to be explained and failure to accommodate shift symmetry on the reference labels to those ontological elements is sufficient cause to reject any epistemological solution.

The final constraint (see my exposition on the function I call F) which I introduced involved using Dirac delta functions (which together with a proper collection of invalid ontological elements could eliminate all circumstances not in the base data): i.e., such an F is capable of constraining the valid ontological elements to whatever they were (that ”what is”,is “what is” table to be explained) by simple inclusion of “invalid” ontological elements to eliminate the unwanted possibility.

$\sum_{i \neq j}\delta(x_i -x_j)\delta(\tau_i -\tau_j)\vec{\psi}=0$

I then asserted that, under my definitions of alpha and beta together with the vector representation of the partial derivatives (the meaning of the symbol $\vec{\nabla}$ ),

$\left\{\sum_i \vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\psi} = K\frac{\partial}{\partial t}\vec{\psi} = iKm\vec{\psi}.$

generates exactly the same constraints expressed above. You appear to want that assertion justified in detail.

Quote:

Originally Posted by Qfwfq

Oui, bon, allez. I'd like to see these steps better detailed, rather than work it out myself, I think it would be a quicker way to see if your argument is satisfactory, in the case you consider, and then perhaps understand the more general case.

Suppose we have found a solution to the above equation which yields exactly the probability distribution required to match a specific flaw-free epistemological solution proposed as an explanation of reality (our “valid” ontological elements on which that explanation is to be based). Call that solution $\vec{\Psi}$ . The first point is that the above expression is actually two equations (note the two equal signs). The final relationship is exactly relationship expressed by the shift symmetry constraint given above as $\vec{\phi}$ and its dependence on t. The $\vec{\psi}$ version (the one where the differential vanishes) is easily retrieved by multiplying $\vec{\Psi}$ by the factor $e^{-iKmt}$ which we know does not impact the probabilities yielded as P. So the shift constraint on the index t is exactly the constraint imposed by that expression.

So let us step on to the three other constraints which are expressed in the first equation. If $\vec{\Psi}$ is a solution to that equation then $\vec{\Psi}$ is also a solution to

$\left\{\sum_i \alpha_{qx}\vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}\alpha_{qx}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\Psi} = iKm\alpha_{qx}\vec{\Psi}$

as all I have done is multiply through by $\alpha_{qx}$ . Now $\alpha_{qx}$ has been defined by two expressions; first by its commutation properties, $[\alpha_{ix} , \alpha_{jx}] \equiv \alpha_{ix} \alpha_{jx} + \alpha_{jx}\alpha_{ix} = \delta_{ij}$ which demands that $\alpha_{qx} \alpha_{ix}= -\alpha_{ix}\alpha_{qx} + \delta_{qi}$ . Likewise, $\alpha_{qx} \alpha_{i\tau}= -\alpha_{i\tau}\alpha_{qx}$ and $\alpha_{qx} \beta_{ij}= -\beta_{ij}\alpha_{qx}$ (no delta element appears). As neither alpha or beta are functions of x, $\tau$ or t, we may be assured that $\vec{\Psi}$ will also be a solution to

$\left\{\sum_i -\vec{\alpha}_i \cdot \vec{\nabla}_i \alpha_{qx}+ \sum_{i \neq j}-\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \alpha_{qx}\right\}\vec{\Psi} +\frac{\partial}{\partial x_q}\vec{\Psi}= iKm\alpha_{qx}\vec{\Psi}$

as commutation merely changes the sign and adds one additional term (that partial with respect to $x_q$ sans alpha or beta) when i happens to be q. Finally, factoring out the term $\alpha_{qx}$ we can sum the above equation over q and we still have an equation which is satisfied by $\vec{\Psi}$ ; however, when we perform that sum we get the following result

$\left\{\sum_i -\vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}-\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\sum_q\alpha_{qx}\vec{\Psi} +\sum_q\frac{\partial}{\partial x_q}\vec{\Psi}= iKm\sum_q\alpha_{qx}\vec{\Psi}.$

At this point, the second portion of the definitions of alpha and beta come in to play.

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

Thus it is that only one term in the above expression is non zero:

$\sum_q\frac{\partial}{\partial x_q}\vec{\Psi}=0$

and it follows that our solution $\vec{\Psi}$ obeys the shift symmetry constraint on the $x_i$ which was originally deduced as necessary. Exactly the same algebraic procedure, working with $\alpha_{q\tau}$ , will yield the fact that $\vec{\Psi}$ obeys the shift symmetry constraint on the $\tau_i$ arguments. Finally, if one multiplies through by $\beta_{qp}$ and commutes it to the right, the act will yield nothing more than a negative sign except for two very specific cases (where q=i and p=j or q=j and p=i). Each of those cases will pull out a single term from the sum with the Dirac delta functions: that is, $\delta(x_q -x_p)\delta(\tau_q -\tau_p)$ and $\delta(x_p -x_q)\delta(\tau_p -\tau_q)$ (again, sans any beta). When the resulting equation is summed over $q \neq p$ all terms vanish except the ones just discussed and one is left with

$\sum_{q\neq p}\delta( x_q -x_p)\delta(\tau_q -\tau_p)\vec{\Psi} +\sum_{q\neq p}\delta( x_p -x_q)\delta(\tau_p -\tau_q)\vec{\Psi}=0$

but the two sums given are exactly the same so the result is exactly twice the single sum and one can divide by two and obtain exactly the original constraint that the Dirac delta function was created to impose.

It follows immediately that any solution to my fundamental equation which yields exactly the same probability distribution as a specific flaw-free epistemological solution will exactly fulfill the shift symmetry constraints required by our ignorance of reality. And finally, since the equation is a first order linear differential equation, any sum of solutions is a solution.

$\left\{\sum_i \vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\psi} = K\frac{\partial}{\partial t}\vec{\psi}.$

Except for the Dirac delta functions, the above equation is, for all practical purposes a wave equation (if you have to have a mental picture, the delta function things can be seen as essentially having the same impact on the equation as infinitesimal massless dust motes): i.e., it has an infinite number of solutions which essentially can be seen as propagating waves constrained only by the boundary conditions which have not been specified. It follows that any particular desired solution can be constructed via a sum of those “propagating waves”. There exists no probability distribution which cannot be seen as a sum of solutions to that equation.

At this point, the only constraints placed on one's epistemological solution are the constraints imposed by shift symmetry. What I have presented is no more than a different way of viewing the problem confronting us. In fact, the reaction "so what" would be very appropriate as there is nothing here to impose any constraint on one's epistemological solution except that of shift symmetry in the representation and the assertion that a sufficient number of invalid ontological elements can make the rule “no two identical elements exist” sufficient to define the “valid” ontological elements which are to be explained. Both of which are pretty obvious and certainly not to be taken seriously as neither have any really serious consequences. Or maybe I should say, at least no apparent serious consequences; we certainly cannot know if we do not examine the consequences.

Now, if you let me know what part of what I have said you have difficulty with, I will do my best to clarify the issues.

To Anssi, you might be interested in the fact that last weeks issue of “Science News” had an article called “Shifty Talk” concerning the process of word evolution. I loved one line,”our results indicate that languages can evolve in such an orderly fashion that simple mathematical descriptions capture their behavior.” One might say, they are simply working on another “explanation” of phenomena previously considered unfathomable.

Have fun – Dick

AnssiH · 10-26-2007

Hello. Sorry for the delays, finally had time to read the new replies and have time to start figuring out the last steps to the "fundamental equation which $\vec{\psi}$ must obey"

Quote:

Originally Posted by Doctordick

You seem to me to understand post #51 pretty well; the big issue now is can you follow the second half of post #72. What follows is the central point of what I am talking about.

$\alpha_{qx}$ , commute it through the various alpha and beta elements in the equation and then sum the result over q.” I am not sure of your familiarity with commutation so I thought I might point out the following.

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

can be rearranged to show that $\alpha_{ix}\alpha_{jx} = \delta_{ij} -\alpha_{jx}\alpha_{ix}$ which implies

$\alpha_{qx}\alpha_{ix} = \delta_{iq} -\alpha_{ix}\alpha_{qx}$ and $\alpha_{qx}\beta_{ij} = -\beta_{ij}\alpha_{qx}$

(look at the defined commutation of alpha with beta).
---END OF QUOTE

Well I know what commutativity means as in being able to change the order of some elements without changing the end result. But I could not figure out what is going on above. I've been trying to figure out what all the math notation means by reading post #42, the end of #77, and now the further clarification at the end of #83. But once again my limited familiarity with math leaves me with far too many shaky assumptions

I don't even know what would be meaningful questions, so I'll just try to probe everything without even bothering to provide my own quesses;

What does "alpha element" or "beta element" refer to?
What does it mean that there is "ix" or "jx" suffix to such an element?
I notice the symbol $\delta$ , does it refer to dirac delta function here also? What does the ij refer to there, and how does it turn into "iq" later (i.e. what does iq mean)?

Actually I am so royally lost at this point already that I can't make any sense of the rest of the post yet either. I am probably missing knowledge about some standard mathematical definitions, just I have no idea what those are and how to find material about them :P

Quote:

The circumstance reminds me of something I read many years ago. It was a paper written by some ancient Greek (I don't read Greek so what I read was an English translation of the original). Though it wasn't exactly mentioned, the paper concerned the definition of speed. The writer pointed out that, if one person was faster than another, he would cover the same distance in less time or, on the other hand, he would be able to cover a greater distance in the same time. The paper continued to make a number of different comparisons in an attempt to clarify what the writer was talking about. What became quite clear was that the idea of “speed” was not a concept the writer held as obvious. Today, everyone (except maybe some very primitive peoples out of touch with modern gadgets) understand exactly what one means by speed and it is difficult for us to comprehend that confusion could ever have existed.

The issue here is that the indices

and t) are nothing more than arbitrary numerical labels used to refer to ontological elements. They constitute an undefined language containing the information to be explained. In any logical deduction based upon those ontological elements, nothing can be introduced which violates the arbitrariness of those assignments (what I believe Anssi refers to as semantics)

In a sense, yes you could say that. Arbitrariness of assignments and the consequent arbitrariness in the defined behaviours and properties of things.

I would word it this way; What I sometimes refer to as "semantical worldview" is a worldview where such concepts as "speed" can be sensical only by the way they relate to other concepts such as "distance" or "time", where those concepts are sensical only how they relate to yet another concepts such as "location" or "change", and in the end the set of concepts only validate each others but not the ontological nature of reality. I.e. where concepts are understood through other concepts.

Referring to the gravity example that HydrogenBond mentioned; if such a worldview where "gravity was due to the repulsion of matter by space" was able to provide us with all the predictions as, say GR, then it would be just as true or untrue as GR. We should just say the concepts/elements these views consists of are a handy way to map the behaviour of reality around us.

That reminds me of a disturbing comment I heard from some physicist in some documentary regarding the string theory. The comment was something to the effect of "it appears that the string theory can never be proven or disproven by observation, so is it physics, or just philosophy?" The first obvious point is of course some idea being physics or philosophy applies to any view which claims a specific ontology. (Why did he suppose that some pre-existing view was the correct one and string theory was just a philosophical bastardization of that correct view is beyond me)

The second and more relevant point is that any single idea of a "vibrating string" you can ever fathom in your mind is completely and utterly based on the semantical concepts you have built about reality, i.e. those things like "location" or "speed" or "acceleration" that you understand by the way you have defined them in terms of other semantical (= independently undefendable) concepts.

I have nothing against models like string theory, but when it goes so far as to people actually start claiming that there must be ontologically real strings that vibrate in 11 dimensions (also taken as "ontologically real" dimensions, whatever "dimension" means! Get it?) is exactly as naive as saying we are conscious because there is a conscious homunculus in our mind. The predictive side of it is pure science, but the ontological mental image of it is pure religion.

Anyhow, Qfwfq referred to "local shift symmetry" and Doctordick jumped at it, so I thought I'd try to clarify things on my own part as well and say that the shift symmetry is not referring to shift inside some "semantically defined thing" (like "space"), but it just refers to shift among the labels used to refer to "ontological elements" (arbitrary features in a raw data whose meaning is unknown). A shift symmetry of labels inside one's worldview, so to speak.

It could be you had realized this and had something specific in mind when you referred to locality though, but I can't be sure. I'm afraid I need to figure the details of the math better myself before I can really say more :P Hopefully you can stick around as your comments have been helpful. Oh, and thank you about the explanations at the post #79 btw, I still need to go through them with thought though.

-Anssi

Qfwfq · 10-26-2007

Quote:

Originally Posted by Doctordick

You simply don't seem to understand the fundamental question that I am attacking.

Actually I think I understand well enough, aside from details, but what I need isn't a lecture in modern mathematics. I fully expected you would be able to catch on to my use of terms, global and local, just as you are using the notion of symmetry. I simply meant phase dependent coordinate values versus a same one for all of them. After all, it is the terminology of gauge symmetry, which is somewhat akin to the phase arbitrarity due to going from P to $\psi$ .

BTW I don't think that Greek guy was confused by the notion of velocity, more that he was working to give the intuitive notion a precise definition for philosophical purposes. Your assumption that he was confused is somewhat like the many people who, reading your arguments, suppose you must have no connection with reality.

The rest of your arguments appear to mean that you are changing your assumption of shift symmetry from:

$\sum_{i=1}^n\frac{\partial}{\partial x_i}P(x_1,x_2,\cdots,x_n,t)=0$ , $\sum_{i=1}^n\frac{\partial}{\partial \tau_i}P(\tau_1,\tau_2,\cdots,\tau_n,t)=0$ and $\frac{\partial}{\partial t}P(t)=0:$

to the tighter:

$\sum_{i=1}^n\frac{\partial}{\partial x_i}\vec{\psi}(x_1,x_2,\cdots,x_n,t)=0$ , $\sum_{i=1}^n\frac{\partial}{\partial \tau_i}\vec{\psi}(\tau_1,\tau_2,\cdots,\tau_n,t)=0$ and $\frac{\partial}{\partial t}\vec{\psi}(t)=0:$

and:

$\sum_{i=1}^n\frac{\partial}{\partial x_i}\vec{\phi}(x_1,x_2,\cdots,x_n,t)=iK_x\vec{\phi}$ , $\sum_{i=1}^n\frac{\partial}{\partial \tau_i}\vec{\phi}(\tau_1,\tau_2,\cdots,\tau_n,t)=iK_\tau \vec{\phi}$ and $\frac{\partial}{\partial t}\vec{\phi}(t)=im \vec{\phi}$

I'll examine your last post when I can, I don't think I'l be able this weekend so it depends on how hectic thing will be net week.

Doctordick · 10-26-2007

Quote:

Originally Posted by AnssiH

Well I know what commutativity means as in being able to change the order of some elements without changing the end result. But I could not figure out what is going on above. I've been trying to figure out what all the math notation means by reading post #42, the end of #77, and now the further clarification at the end of #83. But once again my limited familiarity with math leaves me with far too many shaky assumptions

I don't even know what would be meaningful questions, so I'll just try to probe everything without even bothering to provide my own quesses;

Don't worry, you have made your difficulties clear. Your only problem is the dearth of mathematical knowledge; a problem easily remedied. Remember when I said that mathematics was the invention and study of self consistent systems? Well this is just another one of those systems which have made their way into mathematics.

Under the normal understanding of mathematics, ab (meaning the multiplication of b by a) is one of the central issues of, addition and multiplication. Under the original definition of multiplication, ab is identical to ba (the difference between the two notations is called “commutation”). Well, it turns out that you can define a situation where ab = -ba (it's called “anti-commutation”). In the simple case of anti-commutation, it is quite clear that ab+ba must be exactly zero. But that brings up the issue of what happens when a=b; in that case, one has the result aa+aa=2aa=zero (a rather simple proof that all these “anti-commutating elements are zero). The somewhat subtle way out of the problem (a method of inventing a new mathematical system) is not to define ab+ba to equal zero but rather to define it to be zero only when a and b are different. Following this tack, the common definition for the result when they are the same is that aa+aa=2aa=1. This definition leads to a whole new collection of internally consistent mathematical relationships. One result of rather significant importance is the whole field of “spin” (obtained in analogy to angular momentum) which I am sure you are familiar. Notice that the common definition which I have just given you yields $a^2=\frac{1}{2}$ . I presume that strikes a bell; I am sure you are familiar with the “spin 1/2” entities in modern physics.

Quote:

Originally Posted by AnssiH

What does "alpha element" or "beta element" refer to?

These are no more than mathematical concepts similar to the square root of -1 called i. Their magnitude is another variable as anything can be multiplied by an ordinary number. That is, all of the additional properties are encompassed in the definition that ab+ba=zero and aa+aa=1. That symbol $\delta_{ij}$ is nothing more than a symbol for the situation just defined: $\delta_{ij}$ is defined to be one if i=j and zero otherwise. We are talking about nothing more than another "internally consistent mathematical structure".

Quote:

Originally Posted by AnssiH

What does it mean that there is "ix" or "jx" suffix to such an element?

Those are “subscripts” not suffixes. The subscript “ix” means that the alpha is associated with the x component of the ith element and nothing more. Look at it this way, $\alpha_{ix}$ means that the “anti-commuting element being referred to has to do with the x component of the ith element; it is no more than a reference notation. $\alpha_{ix}$ means we are referring to the ith (that is a very specific case) alpha associated with the x axis. Clearly, you should understand that the qth alpha refers to a different alpha; however, we are suming over i. It should be obvious to you that, in that sum (which is over all i) there will be one case wher i=q. In that case (when i=q) the reference $\alpha_{ix}$ and the reference $\alpha_{qx}$ refer to exactly the same element: we are then confronted with the fact that $\alpha _{ix}\alpha_{qx}$ is not $-\alpha _{qx}\alpha_{ix}$ but rather is $-\alpha _{qx}\alpha_{ix}+1$ . Think about it!

Quote:

Originally Posted by AnssiH

I notice the symbol $delta$ , does it refer to dirac delta function here also? What does the ij refer to there, and how does it turn into "iq" later (i.e. what does iq mean)?

As I have said, that symbol $\delta_{ij}$ is nothing more than a symbol for the situation just defined: $\delta_{ij}$ is defined to be one if i=j and zero otherwise. The “ij” refers to a particular pair of references $(x_i$ and $x_j)$ , which are numerical labels for specific ontological elements.

Quote:

Originally Posted by AnssiH

Actually I am so royally lost at this point already that I can't make any sense of the rest of the post yet either. I am probably missing knowledge about some standard mathematical definitions, just I have no idea what those are and how to find material about them :P

We are using these “subscripts” to refer to a specific set of numerical labels (which have been arbitrarily assigned). We are only trying to generate rational constraints on these arbitrary assignments: essentially, the assignments cannot contain information not available in the patterns of those “valid” ontological elements (a pretty loose statement so long as no definition of those ontological elements exists).

Quote:

Originally Posted by AnssiH

I would word it this way; What I sometimes refer to as "semantical world view" is a world view where such concepts as "speed" can be sensible only by the way they relate to other concepts such as "distance" or "time", where those concepts are sensible only how they relate to yet another concepts such as "location" or "change", and in the end the set of concepts only validate each others but not the ontological nature of reality. I.e. where concepts are understood through other concepts.

It is the sensibility of the entire construct which is of interest here.

Quote:

Originally Posted by AnssiH

I Referring to the gravity example that HydrogenBond mentioned; if such a world view where "gravity was due to the repulsion of matter by space" was able to provide us with all the predictions as, say GR, then it would be just as true or untrue as GR. We should just say the concepts/elements these views consists of are a handy way to map the behavior of reality around us.

Once again, you are talking about solutions (epistemological constructs) not the constraints on the underlying ontology.

Quote:

Originally Posted by AnssiH

Why did he suppose that some pre-existing view was the correct one and string theory was just a philosophical bastardization of that correct view is beyond me,

The answer is clearly that he is approaching the problem with a preconceived answer.

Quote:

Originally Posted by AnssiH

I have nothing against models like string theory, but when it goes so far as to people actually start claiming that there must be ontologically real strings that vibrate in 11 dimensions (also taken as "ontologically real" dimensions, whatever "dimension" means! Get it?) is exactly as naive as saying we are conscious because there is a conscious homunculus in our mind. The predictive side of it is pure science, but the ontological mental image of it is pure religion.

All I can say is that you are absolutely correct.

Quote:

Originally Posted by AnssiH

Anyhow, Qfwfq referred to "local shift symmetry" and Doctordick jumped at it, so I thought I'd try to clarify things on my own part as well and say that the shift symmetry is not referring to shift inside some "semantically defined thing" (like "space"), but it just refers to shift among the labels used to refer to "ontological elements" (arbitrary features in a raw data whose meaning is unknown). A shift symmetry of labels inside one's world view, so to speak.

An exact statement of the difficulty.

Quote:

Originally Posted by Qfwfq

Actually I think I understand well enough, aside from details, but what I need isn't a lecture in modern mathematics. I fully expected you would be able to catch on to my use of terms, global and local, just as you are using the notion of symmetry. I simply meant phase dependent coordinate values versus a same one for all of them. After all, it is the terminology of gauge symmetry, which is somewhat akin to the phase arbitrarity due to going from P to $psi$ .

Qfwfq, I know you are a bright fellow but you are failing to attack the central problem here. Suppose you were given a seriously macroscopic series of symbols (which were totally undefined) and you wanted to come up with an explanation of that series of symbols; what would your attack on the problem be? Until you can take that problem seriously, you miss the entire basis of my approach.

Quote:

Originally Posted by Qfwfq

BTW I don't think that Greek guy was confused by the notion of velocity, more that he was working to give the intuitive notion a precise definition for philosophical purposes. Your assumption that he was confused is somewhat like the many people who, reading your arguments, suppose you must have no connection with reality.

I did not say the writer was confused, what I said was that the idea of “speed” was not a concept the writer held as obvious, quite a different thing.

Quote:

Originally Posted by Qfwfq

The rest of your arguments appear to mean that you are changing your assumption of shift symmetry from: [probability to general epistemological constructs]

And I would say yes to that conclusion. Please, if you would, analyze the problem of explaining an undefined series of symbols, references, data inputs, whatever and coming up with a procedure for predicting the next sequence. It is a simple problem and it deserves careful analysis.

Have fun -- Dick

AnssiH · 10-28-2007

Quote:

Originally Posted by Doctordick

Under the normal understanding of mathematics, ab (meaning the multiplication of b by a) is one of the central issues of, addition and multiplication. Under the original definition of multiplication, ab is identical to ba (the difference between the two notations is called “commutation”). Well, it turns out that you can define a situation where ab = -ba (it's called “anti-commutation”). In the simple case of anti-commutation, it is quite clear that ab+ba must be exactly zero. But that brings up the issue of what happens when a=b; in that case, one has the result aa+aa=2aa=zero (a rather simple proof that all these “anti-commutating elements are zero). The somewhat subtle way out of the problem (a method of inventing a new mathematical system) is not to define ab+ba to equal zero but rather to define it to be zero only when a and b are different. Following this tack, the common definition for the result when they are the same is that aa+aa=2aa=1. This definition leads to a whole new collection of internally consistent mathematical relationships.

Okay, thanks, that was helpful.

Quote:

One result of rather significant importance is the whole field of “spin” (obtained in analogy to angular momentum) which I am sure you are familiar. Notice that the common definition which I have just given you yields $a^2=frac{1}{2}$ . I presume that strikes a bell; I am sure you are familiar with the “spin 1/2” entities in modern physics.

I know what particle spin refers to and how such a concept was conceived and what it's for, but I am not familiar with the mathematical side of it so I must say $a^2=\frac{1}{2}$ does not strike a bell

Quote:

These are no more than mathematical concepts similar to the square root of -1 called i. Their magnitude is another variable as anything can be multiplied by an ordinary number. That is, all of the additional properties are encompassed in the definition that ab+ba=zero and aa+aa=1.

Hmm... Are you saying that $\alpha$ and $\beta$ are defined by ab+ba=0 and aa+aa=1 ? Are the A's and B's referring to "alpha" and "beta" there? Or are you saying rather that $\alpha$ and $\beta$ don't have any standard definitions but in this case they are completely defined by the relationships you gave in post #42?

I cannot explain how tricky it is for me to try and figure these things out, especially as the information is spread across about 4 different posts by now :P

Quote:

That symbol $delta_{ij}$ is nothing more than a symbol for the situation just defined: $delta_{ij}$ is defined to be one if i=j and zero otherwise. We are talking about nothing more than another "internally consistent mathematical structure".

That part I understood.

Quote:

Those are “subscripts” not suffixes. The subscript “ix” means that the alpha is associated with the x component of the ith element and nothing more. Look at it this way, $alpha_{ix}$ means that the “anti-commuting element being referred to has to do with the x component of the ith element; it is no more than a reference notation.

I understood that part too. Except for whether $\alpha_{ix}$ gets some real value depending on what that X is. This question popped into my head when I was looking at the definition from post #42: $\vec{\alpha}_i = \alpha_{ix}\hat{x} + \alpha_{i\tau} \hat{\tau}$

I am moving on very thin ice here :P I am 100% certain that I have understood something completely topsy turvy. A standard textbook explanation of the common usage of "alpha" and "beta" would be nice to have :P (any links, anyone?)

Quote:

$alpha_{ix}$ means we are referring to the ith (that is a very specific case) alpha associated with the x axis. Clearly, you should understand that the qth alpha refers to a different alpha; however, we are suming over i. It should be obvious to you that, in that sum (which is over all i) there will be one case wher i=q. In that case (when i=q) the reference $alpha_{ix}$ and the reference $alpha_{qx}$ refer to exactly the same element: we are then confronted with the fact that $alpha _{ix}alpha_{qx}$ is not $-alpha _{qx}alpha_{ix}$ but rather is $-alpha _{qx}alpha_{ix}+1$ . Think about it!

The above I understand but it doesn't get me far while trying to understand post #42... Actually before that let me get back to:

---QUOTE---
$[\alpha_{ix} , \alpha_{jx}] \equiv \alpha_{ix} \alpha_{jx} + \alpha_{jx}\alpha_{ix} = \delta_{ij}$
can be rearranged to show that $\alpha_{ix}\alpha_{jx} = \delta_{ij} -\alpha_{jx}\alpha_{ix}$ which implies

$\alpha_{qx}\alpha_{ix} = \delta_{iq} -\alpha_{ix}\alpha_{qx}$ and $\alpha_{qx}\beta_{ij} = -\beta_{ij}\alpha_{qx}$
(look at the defined commutation of alpha with beta).
Thus all that happens as $\alpha_{qx}$ is commutated through an alpha or a beta is a sign change except when q=i. In that case, the $\delta_{iq}$ picks up one additional term with no alpha or beta.
---END OF QUOTE

I now understand that up to the point the first $\beta$ appears. I don't know where it comes from or why is it there all of a sudden. Since you advice to "look at the defined commutation of alpha with beta", I take it that's probably $\alpha_{qx}\beta_{ij} = -\beta_{ij}\alpha_{qx}$ , but it doesn't cause any switches to be thrown in my head

Also I don't know what it means to "commutate $\alpha_{qx}$ through an alpha or a beta"...

After spending this day trying to figure these things out (and that is why this is such a messy post), I must say I feel like I'm struggling more in this step than I have in any earlier steps. Right now I don't know where to start unraveling all these things. Other people feel free to help also!

Btw, the first sum over i in that fundamental equation, that refers to the delta sub i also, and not just the alpha?

-Anssi

Qfwfq · 10-29-2007

Quote:

Originally Posted by Doctordick

I did not say the writer was confused, what I said was that the idea of “speed” was not a concept the writer held as obvious, quite a different thing.

Alright, let's put it that way, just that you had said:

Quote:

Originally Posted by Doctordick

Today, everyone (except maybe some very primitive peoples out of touch with modern gadgets) understand exactly what one means by speed and it is difficult for us to comprehend that confusion could ever have existed.

In short, I assume the ancient Greek was defining the notion without the use of predefined concepts of direct proportionality and of derivative. Today's students, when beginning kinematics, already know at least the first of those two and usually also the second one. They are however philosophical definitions rather than something obvious from familiarity with modern gadgets.

Quote:

Originally Posted by Doctordick

Please, if you would, analyze the problem of explaining an undefined series of symbols, references, data inputs, whatever and coming up with a procedure for predicting the next sequence. It is a simple problem and it deserves careful analysis.

I'm trying to follow your analysis. Don't think I don't get what you're doing, it's only the how that I've been unable to follow in detail. Now I'm not reasoning confined in a groove. Here's a cute illustrative example which you might find amusing especially if you can guess the game:

A spy is sent to gather intelligence for the planners of an attack against a castle, they need to know how to fake a patrol returning to the castle and pass ID. The spy manages to hear 3 of the exchanges between the guard corps and the patrol commander. After the halt and the patrols ID being called out, the guard says a number and the patrol commander replies with another number; the first time guard calls out 6 and the reply is 3, the second time 18 is answered with 9, the third time 12 is answered with 6. At that point the spy supposes he has got the game and the planners are also convinced and sort out all other details. When the carry it out, on a very dark night, the guard seems unalarmed until his call of 10 is answered with 5, at which the whole guard corps immediately leaps into action and surrounds them. What should the patrol have replied to 10?

Doctordick · 10-30-2007

Quote:

Originally Posted by AnssiH

I know what particle spin refers to and how such a concept was conceived and what it's for, but I am not familiar with the mathematical side of it so I must say aa = 1/2 does not strike a bell

Sorry about that. I am again being led off subject by questions which really amount to outside baggage. These things are usually introduced in a very different manner; in a manner implied by the physicist's concept of reality and the mathematics he has discovered applicable. That is really counter to the problem I am looking at as the problem of interest must be looked at from a position of complete ignorance. All we are really concerned with here is the fact that such a counter intuitive thing (anti-commuting entities) can be the basis of a internally consistent system. The only reason I use it is because it allows me to express four different constraints in a form which appears to be a single differential equation. I personally regard it as a convenient mathematical trick; no more than another mathematical operation which can be defined and which provides a valuable service (such as adding or multiplying or taking derivatives are valuable mathematical procedures, this is no more than another).

Quote:

Originally Posted by AnssiH

Hmm... Are you saying that $alpha$ and $beta$ are defined by ab+ba=0 and aa+aa=1 ?

No, I am not. The issue here is anti-commutation itself and the fact that we can define such things where the elements being defined need not be zero.

Quote:

Originally Posted by AnssiH

Or are you saying rather that alpha and beta don't have any standard definitions but in this case they are completely defined by the relationships you gave in post #42?

That is correct. The definitions given in post #42 completely define them.

Quote:

Originally Posted by AnssiH

I understood that part too. Except for whether $alpha_{ix}$ gets some real value depending on what that X is. This question popped into my head when I was looking at the definition from post #42: $vec{alpha}_i = alpha_{ix}hat{x} + alpha_{itau} hat{tau}$

No, they are merely “things” that anti-commute with one another. 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 $(x,\tau)$ space used to represent our ontological elements. Think of them as subtle complications inserted into that differential equation; complications which have the power to force the solutions to that equation to obey the original constraints and serve no other purpose. As I said, a mere mathematical trick which allows me to write the constraints in one apparently simple differential equation.

Quote:

Originally Posted by AnssiH

I am moving on very thin ice here :P I am 100% certain that I have understood something completely topsy turvy. A standard textbook explanation of the common usage of "alpha" and "beta" would be nice to have :P (any links, anyone?)

You will find no textbook explanation of “the common usage of 'alpha' and 'beta'" as there is none (common usage that is). If you insist, you might take a look at “Pauli's spin matrices”. I doubt you will find that presentation any more meaningful than mine. His definitions constitute a matrix representation of anti-commuting entities. If you go any deeper, you will see the physics reasoning behind using them. I really don't care for that presentation because it requires one to believe the physics presentation is a valid representation of reality; a highly presumptive place to start. I think my presentation is much more to the point of the problem we are trying to solve. As I have said many times, I define mathematics as the invention and study of internally consistent systems and these alphas and betas are no more than defined mathematical entities which obey some strange rules (the exact rules are given in post #42 ). What it is or why it is doesn't really bear on the issue here. All that is important is that it serves the purpose for which it was introduced: it allows me to write my constraints in what appears to be one simple equation.

Quote:

Originally Posted by AnssiH

Also I don't know what it means to "commutate $alpha_{qx}$ through an alpha or a beta"...

Order of the terms in that differential equation is an important issue. Note that $\frac{\partial}{\partial x_i}\vec{\psi}$ is a meaningful expression (it means “take the derivative of $\vec{\psi}$ ”) but that $\vec{\psi}\frac{\partial}{\partial x_i}$ is not; what is the function which is to be differentiated? The difference between those two expressions is the order of the terms themselves. The symbol $\sum_i \alpha_{ix}\vec{\psi}$ has been defined to be zero so, if we can achieve that symbol set via defined algebraic operations on my fundamental equation, then we can replace it by zero. Now when we multiplied by $\alpha_{qx}$ , it was clearly on the wrong side of $\alpha_{ix}$ ; in order to get in directly on $\vec{\psi}$ (in order to be able to replace it with zero, after we sum over q), it had to be commuted with $\alpha_{ix}$ and $\frac{\partial}{\partial x_i}$ . We had to “commute it through those terms” so that, when we summed over q, it would be in the explicit form, $\alpha_{qx}\vec{\psi}$ (that is, nothing is between the alpha sub q and the $\vec{\psi}$ and we can directly use the fact that the sum is defined to be zero.

Quote:

Originally Posted by AnssiH

Btw, the first sum over i in that fundamental equation, that refers to the delta sub i also, and not just the alpha?

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:

$\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i} = \alpha_{1x}\frac{\partial}{\partial x_1}+\alpha_{1\tau}\frac{\partial}{\partial \tau_1}+\alpha_{2x}\frac{\partial}{\partial x_2}+\alpha_{2\tau}\frac{\partial}{\partial \tau_2}+\cdots+\alpha_{nx}\frac{\partial}{\partial x_n}+\alpha_{n\tau}\frac{\partial}{\partial \tau_n}$ .

If you multiply that expression by $\alpha_{qx}$ you will have:

$\alpha_{qx}\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i} = \alpha_{qx}\alpha_{1x}\frac{\partial}{\partial x_1}+\alpha_{qx}\alpha_{1\tau}\frac{\partial}{\partial \tau_1}+\alpha_{qx}\alpha_{2x}\frac{\partial}{\partial x_2}+\alpha_{qx}\alpha_{2\tau}\frac{\partial}{\partial \tau_2}+\cdots+\alpha_{qx}\alpha_{nx}\frac{\partial}{\partial x_n}+\alpha_{qx}\alpha_{n\tau}\frac{\partial}{\partial \tau_n}$ .

Using the definition of its commutation with $\alpha_{ix}$ , you may commute it with the explicit alphas there and obtain:

$\alpha_{qx}\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i} = -\alpha_{1x}\alpha_{qx}\frac{\partial}{\partial x_1}-\alpha_{1\tau}\alpha_{qx}\frac{\partial}{\partial \tau_1}-\alpha_{2x}\alpha_{qx}\frac{\partial}{\partial x_2}-\alpha_{2\tau}\alpha_{qx}\frac{\partial}{\partial \tau_2}+\cdots-\alpha_{nx}\alpha_{qx}\frac{\partial}{\partial x_n}-\alpha_{n\tau}\alpha_{qx}\frac{\partial}{\partial \tau_n}+\frac{\partial}{\partial x_q}$

Since it commutes with the partial dirivative, the final result can be written:

$\alpha_{qx}\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i} = \left\{\alpha_{1x}\frac{\partial}{\partial x_1}+\alpha_{1\tau}\frac{\partial}{\partial \tau_1}+\alpha_{2x}\frac{\partial}{\partial x_2}+\alpha_{2\tau}\frac{\partial}{\partial \tau_2}+\cdots+\alpha_{nx}\frac{\partial}{\partial x_n}+\alpha_{n\tau}\frac{\partial}{\partial \tau_n}\right\}(-\alpha_{qx}) +\frac{\partial}{\partial x_q}$ ,

Which can be written:

$\alpha_{qx}\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i} = \left\{\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i} \right\}(-\alpha_{qx}) +\frac{\partial}{\partial x_q}$ ,

That last single term, all by itself arises when i=q and that event only occurs once;

You might be confused by exactly what these i,j,q,p,l,k subscripts are all about. They are no more than letters standing for the appropriate x and tau reference labels to be in the sum. The term being summed is $\vec{\alpha_i}\cdot \vec{\nabla_i}$ and each “i” yields a different term in that sum but every term is operating on the same $\vec{\psi}$ .

Quote:

Originally Posted by Qfwfq

Here's a cute illustrative example which you might find amusing especially if you can guess the game:

I don't understand why you are putting that down. Those kinds of questions on intelligence tests always bugged the hell out of me. Not that I couldn't pick up on the series they wanted but rather that it was awfully presumptive of them to think that their answer was “correct”. Anybody with any sense at all knows that there are an infinite number of series with exactly the same first n terms and a totally different n plus first term. Based on the information available to him, five is a fine expectation. Any answer could be wrong; as I have said many times, “the future is what we do not know”!

What is significant here is that I have no idea as to why you brought up that little story. It certainly is not an example of the problem I am discussing. I suspect very strongly that it is no more than evidence that you have no idea of what I am doing. It reminds me very much of a post Rade made in Feburary of this year on “physicsforums.com”. You can see my response to that post here. If you can understand that response, you might be a lot closer to understanding what I am doing than I think you are. It might be worth reading. Rade's base of information to be explained was at least presented as undefined. Your example is chock full of assumptions (presumed valid ontological concepts) which you take no trouble to explicitly list (because the size of such a list would probably be beyond accomplishment).

Now let's make the problem more analogous to what I am talking about. Let the spy obtain a billion exchanges between the guard corps (sounds, motions, light signals, ... “smells”, etc) but he is dealing with total aliens and has utterly no idea of what meanings the aliens attach to these things. How would you suggest he decide what the billionth and first exchange should be? What you don't seem to comprehend is that the answer to that question is the result of an epistemological solution to the series, a subject of no interest to me. All I am concerned with is the constraints I can place on the “interpretation” problem; which happen to be exactly the constraints I have specified. Think of it this way, if I give the problem to a million people and after a thousand years they individually (without communicating with one another) come up with a solution which exactly matches the known exchanges which they all believe is the simplest solution possible. Now amongst all those solutions, let us say that there are a good number of which are actually the same solution: i.e., the only difference is the particular symbols they used to refer to the specific exchanges. Think about it, do you seriously think that there exist no constraints on the “simplest interpretation” of those symbols? That is, the interpretation to be placed on the symbols these people use to represent those specific exchanges?

If you can not follow what I just said, you do not understand the problem I am talking about.

Have fun -- Dick

Qfwfq · 10-31-2007

Quote:

Originally Posted by Doctordick

Those kinds of questions on intelligence tests always bugged the hell out of me. Not that I couldn't pick up on the series they wanted but rather that it was awfully presumptive of them to think that their answer was “correct”. Anybody with any sense at all knows that there are an infinite number of series with exactly the same first n terms and a totally different n plus first term. Based on the information available to him, five is a fine expectation. Any answer could be wrong; as I have said many times, “the future is what we do not know”!

But it's not that kind of question on intelligence tests at all. That is an assumption of yours which is not in line with how I posed the matter and shows how you so often tend to capsize people's meaning when you interpret what they post.

Quote:

Originally Posted by Doctordick

What is significant here is that I have no idea as to why you brought up that little story.

To show, as I said, that I know what you mean about assumptions and reasoning in the groove.

Quote:

Originally Posted by Doctordick

Your example is chock full of assumptions (presumed valid ontological concepts) which you take no trouble to explicitly list (because the size of such a list would probably be beyond accomplishment).

That depends on what you mean by "assumptions"; in a story (or fictitious example) the truth is what is posited as such. The guard corps and the patrols are ontological elements and thus valid, not presumed, and the convention between them is thus correct. The spy must guess it and the plan's success relies upon his reckoning, which is either right or wrong. The only assumption in my story is made by the spy, which turns out to be wrong.

Many a person hearing the riddle, after the exchanges between the real patrols and before the ending, will tend to assume the criterion being "divide by two" but this turns out to be wrong, a hasty leap to a seemingly obvious conclusion. One assumption remains typical, and much more radicated, and I would expect you to know which I'm alluding to. Can't you see my example is actually an illustration many of the things you always say? It shows exactly what you so sternly replied to it!

By the way the criterion, which the riddle obviously leaves unspecified, isn't easy to guess due to the radicated assumption I alluded to and yet is very simple. It's quite amusing to find it out and I would expect you to find it highly entertaining if you knew it. The answer to 10 should have been 3, the answer to 16 would be 7 and the answer to 14 would be 8. Want more clues?

Quote:

Originally Posted by Doctordick

Let the spy obtain a billion exchanges between the guard corps (sounds, motions, light signals, ... “smells”, etc) but he is dealing with total aliens and has utterly no idea of what meanings the aliens attach to these things.

Now that comes into my castle example more than you believe!

Quote:

Originally Posted by Doctordick

Think about it, do you seriously think that there exist no constraints on the “simplest interpretation” of those symbols? That is, the interpretation to be placed on the symbols these people use to represent those specific exchanges?

I don't think that there exist no constraints on the “simplest interpretation” of those symbols, at all. The first problem is in defining a "simplicity function" over the whole space of interpretations and minimizing it, not an easy task. We usually go by an intuitive attribution of "simplicity" to such things.

Quote:

Originally Posted by Doctordick

If you can not follow what I just said, you do not understand the problem I am talking about.

I follow what you said but still struggle to follow the details of putting it into practice, including justification of your fundamental equation. I would be somewhat interested in knowing how it could attack the riddle, given an available source of kosher answers to chosen numbers. Give me a number, reasonable for the guards and patrol to work out, and I'll give you the correct reply for it.

What can we know of reality?

Hypography?

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