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# What can we know of reality?

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Would it be correct to say that F is a function which "tests" every element of a "present" with some function f? Or is it more proper to just say it is "any function which returns 0 when its input with a full present"?

Neither really. All I am saying here is that, if a mechanism exists to recover any specific (x,tau) point in the ”what is”, is “what is” table (which includes all invalid ontological elements necessary to the explanation expressed by the table), then that mechanism itself can be used to define a function who's roots are exactly the entries in the table. It is essentially a proof that there exists a "rule" function F who's roots define the table.

Okay, and I wish to solve one more ambiguity that caused some headache.

Even though you express the proof as:

$F((x,\tau)_1,(x,\tau)_2, \cdots , (x,\tau)_n) = \vec{(x,\tau)}_n - \vec{f}((x,\tau)_1,(x,\tau)_2, \cdots , (x,\tau)_{n-1}) \equiv 0$

...it does NOT mean that a specific function needs to employ that mechanism (difference between missing element and "f"):

$\vec{(x,\tau)}_n - \vec{f}((x,\tau)_1,(x,\tau)_2, \cdots , (x,\tau)_{n-1}) = 0$

(Or rather a mechanism where you'd take the sum over every possible "f"-recovery and expect it to be 0... I could't figure out how to express that in math :)

But instead, the F just needs to be:

$F((x,\tau)_1,(x,\tau)_2, \cdots , (x,\tau)_n) = 0$

I suppose this is so, because the specific F...

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

...does not seem to employ such "difference between f and missing element" mechanism in any sense.

So now I think I am starting to understand what the point was with that "dirac delta function F". I suppose most specific F=0 functions would not include all the points as ontological elements.

.

$\sum_{i \neq j} \delta(x_i - x_j) \delta(\tau_i - \tau_j) \vec{\psi}(x_1,\tau_1,x_2,\tau_2, \cdots , x_n, \tau_n, t) =0$

is an equation which must be obeyed by psi under the rule that no two ontological element references can appear twice: i.e., psi must be zero if the sum over Dirac delta functions is not zero. I don't think it can be put any clearer than that.

And $\psi$ (psi) was defined as:

A function which yields a non-zero probability only for the correct points to valid ontological elements. So it would be zero by definition if the set was incorrect. And consequently that equation you gave looks valid to me.

-Anssi

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Hi Doc! And Happy Birthday as well. :cheer: Very nice to see you have an astute new friend interested in your philosophy discussions. :)

Welcome to Hypography AnssiH. I have enjoyed reading your post. :cup:

I do have a question in all this for both of you now, and that is what part -if any- does the physiology of the brain play in what we can know? :hihi: :cup:

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Hi Doc! And Happy Birthday as well. :cheer: Very nice to see you have an astute new friend interested in your philosophy discussions. Thanks for your thoughts. I have always been quite surprised by your lack of interest considering your great interest in the patterns collections of numbers can create.
I do have a question in all this for both of you now, and that is what part -if any- does the physiology of the brain play in what we can know? :hihi: :cup:
I am afraid you miss the entire thrust of my argument; but don't feel bad as it seems to be outside everyone's comprehension. Only Paul Martin (you can find him on physicsforums also) and Anssi Hyytiäinen seem to have grasped an idea of what I am talking about. Anssi struck the nail on the head with this comment:
Simply put, the discussion is about:

Logical constraints that any model/theory/world-view must conform to if they are internally coherent

(i.e. when they do not contain logical self-contradictions)

What I am presenting is a very straight forward but complex deduction with far reaching consequences. My problem is that I cannot get anyone to stick to the issues of significance for anything but a short time. When you are trying to explain something from dead scratch (and any “model/theory/world-view” is an explanation of some sort) you must begin with “nothing” defined. When you say, “what part -if any- does the physiology of the brain play”, you are presuming the idea (in essence a model of something), “the physiology of the brain” is a real defined component of a flaw free explanation. Maybe it is, maybe it isn't, but certainly the assumption that it must be valid component of your world-view has no place in the type of logical deduction I am laying out.

What difference does the physiology of your brain make? It still has to solve the same problem: i.e., come up with an explanation of its existence sans self contradiction. (Note that any self contradictory explanation fails as an explanation.)

Anssi, I will get back to you later today. I have some things I have to get done and I want to put some thought into my response. There are a number of subtle issues which must be clearly understood before my fundamental equation can actually be seen as truly universal. Once we get past that, I will lay out the first solution I discovered.

Have fun -- Dick

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Welcome to Hypography Anssi! :cheer:

Unfortunately I can't afford much time to follow your great input, whether here or elsewhere, I certainly agree that it is a valid philosophical discussion and I hope you will be helpful in analysing the basis of Dick's fundamental equation. Meantime, I must also fully agree with:

These logical constraints are NOT leading us towards any specific ontology. If anything, they indicate that many sorts of ontologies can always be built without sacrificing the internal coherence of our worldviews (i.e. they would be "semantically different" but "logically the same")
...and points after it, it's a good statement of the modern view of math and formal systems and the fact that they are valid regardless of reality, which may be described in different equivalent ways.

What we call reality is, of course, what we perceive through our senses and hence take for granted is reality but actually is only the realm of phenomena. We get to the point where we ca only distinguish between descriptions that do and those that don't match up with phenomena.

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...it's a good statement of the modern view of math and formal systems and the fact that they are valid regardless of reality, which may be described in different equivalent ways.

What we call reality is, of course, what we perceive through our senses and hence take for granted is reality but actually is only the realm of phenomena. We get to the point where we ca only distinguish between descriptions that do and those that don't match up with phenomena.

...which leads to the conclusion that you can have consistent formal systems, and you can have "reality" but never in the twain shall they meet: any mapping between "perceived reality" and a "formal system" requires assumptions and therefore lies outside the realm defined by our eminent and irascible doctor.

This of course would lead to an answer to this thread's title of "nothing," thus begging the question--asked many times previously in this thread--what are the "profound implications" if there are an infinite number of possible models whose relationship to "reality" is entirely subjective?

I can see a justification for sociopathic behavior coming out of this, but little else. So I will take my feeble little mind and shut up for now. To quote Mr. Turtle, "carry on."

Models my reality but no one else's,

Buffy

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Thanks for your thoughts [Turtle]. I have always been quite surprised by your lack of interest considering your great interest in the patterns collections of numbers can create.

My pleasure Doc. Please don't mistake my lack of replies for a lack of interest. As you have alluded to my math threads, your interest is implied and I note with some irony how frequently you reply to them. :cup:

I am afraid you miss the entire thrust of my argument; but don't feel bad as it seems to be outside everyone's comprehension. Only Paul Martin (you can find him on physicsforums also) and Anssi Hyytiäinen seem to have grasped an idea of what I am talking about. Anssi struck the nail on the head with this comment:

What I am presenting is a very straight forward but complex deduction with far reaching consequences. My problem is that I cannot get anyone to stick to the issues of significance for anything but a short time. When you are trying to explain something from dead scratch (and any “model/theory/world-view” is an explanation of some sort) you must begin with “nothing” defined. When you say, “what part -if any- does the physiology of the brain play”, you are presuming the idea (in essence a model of something), “the physiology of the brain” is a real defined component of a flaw free explanation. Maybe it is, maybe it isn't, but certainly the assumption that it must be valid component of your world-view has no place in the type of logical deduction I am laying out.

I'm looking forward to reading your ongoing discussion with Anssi in order to improve my comprehension. My mention of physiology did not imply 'flaw-free' anything, but perhaps if I explain what prompted it. We often hear that the brain has some number of neurons with some number of connections, and that the connections change as we learn over time. Inasmuch as your equation(s) has a time component and constrains associations/interconnections, I wondered if the same limits of the brain further constrain your limits. In other words, could we know more of what your equation constrains if we had larger brains and/or more time? Or is it that "we" have nothing to do with it, and the question is really "what can be known about reality?"

Feel free to reply or not; I'll be over here in the corner reading. :turtle: :esmoking:

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Actually Buffy I wouldn't put "perceived reality" outside of what Kant calls "a priori" knowledge. The problems are really in reaching ontology. Normal people just take basic things for granted and survive, if their perception is "normal" i. e. that which allows and aids survival in the environment. However...

...but little else.
there are nonetheless folks that think about these things (and don't rule out the odd physicist too).
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any mapping between "perceived reality" and a "formal system" requires assumptions and therefore lies outside the realm defined by our eminent and irascible doctor.
The realm I have defined consists of “the constraints on an internally self consistent” explanation and that issue cannot be settled without thought.
This of course would lead to an answer to this thread's title of "nothing," thus begging the question--asked many times previously in this thread--what are the "profound implications" if there are an infinite number of possible models whose relationship to "reality" is entirely subjective?
Once again I am confronted with someone who “knows the correct answer” without examining the problem at all. If you see a flaw in my logic, please point it out; otherwise, as most everyone feels compelled to inform me that nothing can be gained by thinking the thing out, it really serves no purpose to raise that battle flag again.
As you have alluded to my math threads, your interest is implied and I note with some irony how frequently you reply to them. ;)
We both have a very strong interest in the patterns numbers can display; however, our interests are far different. From looking at your work, it seems to me that your interest is in the variety of patterns which are possible whereas my interest would be, “what do all these patterns have in common”. That is why I posted that thread ”A simple geometric proof with profound consequences”. What that proof shows is that every possible pattern in one, two or three dimensions can be seen as a projection of a rotated minimal unitary n dimensional equilateral polyhedron onto that one, two or three dimensional space. They are all patterns of exactly the same thing. From your approach, examining individual patterns, no matter how many patterns you examine, there exists another you have not yet examined. The difficulty with that attack is that the number of possibilities is infinite and there is no end to the procedure. If your purpose is merely to keep your mind busy looking at the beauty of the possible patterns, it's perhaps a nice pass time; however, if your interest is to understand those patterns, as mine is, your approach is little more than a delightful distraction. All I am saying is that our interests are evidently quite different and everyone needs to pursue their own interests.
We often hear that the brain has some number of neurons with some number of connections, and that the connections change as we learn over time.
That statement amounts to an explanation and may or may not be a flaw-free explanation. It makes the presumption that the common world-view is correct. My approach is to make no such presumptions. And finally, Qfwfq, I am certainly an “odd” physicist!

Anssi, please stop me if I comment on anything which is not absolutely clear to you.

The function [imath]\vec{\psi}[/imath] is open to be absolutely any function. The only constraint on [imath]\vec{\psi}[/imath] is that its normalized scaler product, [imath]\vec{\psi}^\dagger \cdot \vec{\psi}[/imath] must be the probability your flaw-free explanation gives for a specific set of reference indices [imath](x_i,\tau_i)[/imath] for a given t index. If your explanation yields such expectation (probability estimates) then a method of achieving them exists. That proves the function [imath]\vec{\psi}[/imath] exists. The probability so defined, cannot be a function of the particular symbols (read numeric labels) but rather must be a function of the entire set taken as a whole. This implies the existence of what is normally called a shift symmetry and such a shift symmetry requires the following behavior of the function [imath]\vec{\psi}[/imath].

$\sum_i \frac{\partial}{\partial x_i}\vec{\psi}(x_1,\tau_1,x_2,\tau_2, \cdots , x_n, \tau_n,t) = iK_x\vec{\psi},$

$\sum_i \frac{\partial}{\partial \tau_i}\vec{\psi}(x_1,\tau_1,x_2,\tau_2, \cdots , x_n, \tau_n,t) = iK_\tau\vec{\psi},$

and

$\frac{\partial}{\partial t}\vec{\psi}(x_1,\tau_1,x_2,\tau_2, \cdots , x_n, \tau_n,t) = iK_t \vec{\psi}.$

Finally, through extended additions of “invalid” ontological elements, I have proved that there always exists a collection of such “invalid” ontological elements such that the entire ”what is”, is “what is” table is specified by the “rule” F=0 where

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

Please note that the adjective “invalid” does not mean the references to those ontological elements do not obey the rule but rather, they are not required by reality but merely by the explanation itself, a subtly different issue. It follows that only the [imath]\vec{\psi}[/imath] which indeed yields the correct probabilities for your flaw-free explanation will satisfy the equation

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

These are four constraints which must be satisfied by the [imath]\vec{\psi}[/imath] flaw-free explanation which yields expectations consistent with the ”what is”, is “what is” table which defines the results of our flaw-free explanation. At this point, there are only a couple of steps to obtaining the fundamental equation which [imath]\vec{\psi}[/imath] must obey: i.e.,

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

First, we must define each of the various mathematical expressions in that equation. The alpha and beta expressions stand for anticommuting elements obeying the following relationships:

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

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

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

[imath][\alpha_{ix}, \beta_{kl}]=[\alpha_{i\tau}, \beta_{kl}] = 0 \text{ where } \delta_{ij} =

\left\{\begin{array}{ c c }

0, & \text{ if } i \neq j \\

1, & \text{ if } i=j

\end{array} \right.

[/imath]

The following two expressions are defined as [imath]\vec{\alpha}_i = \alpha_{ix}\hat{x} + \alpha_{i\tau} \hat{\tau}[/imath] and [imath]\vec{\nabla}_i =

\frac{\partial}{\partial x_i} \hat{x} +\frac{\partial}{\partial \tau_i} \hat{\tau}[/imath]. A little algebra will show that any solution of that “fundamental equation” will satisfy the four constraints required by a flaw-free explanation under the simple additional constraint that:

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

All one need do is multiply the fundamental equation through by the term [imath]\alpha_{qx}[/imath], commute it through the various alpha and beta elements in the equation and then sum the result over q. The commutation properties on the original multiplication will yield only one term without an [imath]\alpha_{qx}[/imath]; that will be the partial with respect to x sub i when i is equal to q. When the result is summed over q all the terms with an [imath]\alpha_{qx}[/imath] will sum to exactly zero, leaving the result,

$\sum_q \frac{\partial}{\partial x_q}\vec{\psi} = 0.$

Now, this isn't exactly [imath] \sum_q \frac{\partial}{\partial x_q}\vec{\psi} = K_x \vec{\psi}[/imath]; however, it is very easy to show that, if we have a solution (call it [imath]\vec{\psi}_0[/imath]) which yields the case [imath]K_x =0[/imath] the solution which yields a specific nonzero K_x (call it [imath]\vec{\psi}_1[/imath]) is a simple transformation of the solution [imath]\vec{\psi}_0[/imath]:

$\vec{\psi_1}= e^{\sum_{j=0} ^n iK_x \frac{x_j}{n}} \vec{\psi}_0.$

Anyone familiar with standard QM will recognize this transformation as exactly the transformation necessary to correct from the center of momentum solution of a many body problem to one where the center of momentum is not at rest in the coordinate system. This means that the fundamental equation is only valid in in one specific Euclidean coordinate system: that would be the coordinate system where [imath] \sum_i \frac{\partial}{\partial x_i}\vec{\psi} = 0.[/imath]. This can be seen as quite analogous to Newton's “inertial” coordinate system in that his equations simplify to F=ma in that coordinate system. Likewise, my fundamental equation is much simpler in the particular coordinate system where [imath] \sum_i \frac{\partial}{\partial x_i}\vec{\psi} = 0.[/imath]. Thus this is a constraint on the coordinate system to be used, not actually a problem with achieving the represented deduced constraints.

Exactly the same analysis (using [imath]\alpha_{q\tau}[/imath] or [imath]\beta_{ij}[/imath] respectly) will yield the remaining constraints.

Unless I have made a typo in the LaTex I have written, this pretty well sets up the fundamental equation; however, there are a couple of subtle issues which need to be addressed before we proceed to the problem of solving that equation.

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I do have a question in all this for both of you now, and that is what part -if any- does the physiology of the brain play in what we can know? :esmoking: The problem is that investigating the physiology of the brain, and drawing an interpretation of your investigation, depends on your worldview. IOW being able to probe the structure of the brain depends on having a good model for a wide variety of things/phenomena.

As an arbitrary example, think about how do we know what is the "speech area" of the brain? What experiments we had to conduct, and how did we understand the meaning of those experiments? If you backtrack your "knowledge tree", you pretty soon need to ask yourself, how does the brain interpret the meaning of ANY sensory data. How does it know that certain pattern is the sound "meaow", or the "sight of an apple".

Let me stress at this point that the idea of this discussion is not so much to talk about specific models (i.e. models that attempt to explain the functions of the brain), but to investigate what is the logical procedure that allows something (like "the brain") to take a stream of completely unknown data (like "the sensory data"), and deduce some meaning from that data. (Some meaning that hopefully has got something to do with the actual reality So, what I refer to as "the worldview" is not just the knowledge you are conscious of, but all the "information" that you are not consciously aware of, but that allows the brain to infer meaning from the patterns. (As an abritrary example, you are not conscious of everything that goes on in the brain to control your muscles when you walk (without falling), but the brain is busy interpreting sensory data in ways that allow the muscles to compensate)

As a side note, since I started to talk about "inferring the meaning of unknown sensory data", think about the experiences of people who have had hearing, have went deaf, and later have had cochlear implants put to their skulls. The brain had already built a "worldview" with which to interpret the sounds in some meaningful ways. But the cochlear implant does not input the data in the same form as the ear did, so the interpretation of the data is initally just gibberish. The brain once again needs to learn how to interpret the data meaningful ways (albeit now it already has got a large worldview to help with the task). Take a look at the video at:

Sorry I can't post links yet, you can find the video if you google:

pbs computer in his skull. The story about Michael Chorost. Jump to 8 minutes if you just want to hear his description of what it first sounded like to have the device turned on.

Another interesting case IMO is Helen Keller. Her brain had limited amount of information to try and make sense of, and the task was much more difficult than it is for the most of us. She was a troubled child, because the brain had not figured out good meaning for the sensory data, even after having built quite substantial "worldview" (just kind of distorted one as it did not include a conception of "outside world").

When she first understood the connection between the cold sensation (water) on her hand, and the tactile pattern on the other hand (her teacher doing the sign language word "water" on her hand), she says that is the first time "she was conscious" (that is the same as realizing for the first time there is "a reality" out there, i.e. there is difference between "self" and the reality). The point of the story is, think about how "mind-bogglingly" complicated it can be to make ANY sense of HUGE amount of completely unknown information. Yet we all have done it! Our worldviews always include assumptions, but what are the logical constraints that allow us to make ANY assumptions at all? That gets us directly to:

...which leads to the conclusion that you can have consistent formal systems, and you can have "reality" but never in the twain shall they meet: any mapping between "perceived reality" and a "formal system" requires assumptions and therefore lies outside the realm defined by our eminent and irascible doctor.

Yes, and no. We can never know if our assumptions are right or wrong (I think our assumptions are just semantical boundary drawing that reality does not care or operate on), but perhaps we can know something about logical constraints that allow ourselves to build self-coherent worldviews (allows us to build "logically coherent set of assumptions")

So "what can we know about reality" is not referring to any specific ontology, but rather to what can we know about each and every possible "self-coherent" ontology that ever could be built. (The question would perhaps be more properly phrased, "what can we know about valid worldviews"?)

-Anssi

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Well, yes, I suppose it could be put as this being the limit on what can be known of ontology.

For Turtle, I think it can put simply by saying that each part of the brain is just as much as the others closed inside the skull and receiving knowledge through sensory organs and nerves. Of course even outside of these, it's all a chain of interactions; any contraption or device that can measure "something" is simply being affected by that something via some kind of an interaction with it.

Dick, if you need non-matching braces again, use the null delimiter '.' (I replaced right} with right.) so that left and right commands match up with each other but only the first one is visible.

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This of course would lead to an answer to this thread's title of "nothing," thus begging the question--asked many times previously in this thread--what are the "profound implications" if there are an infinite number of possible models whose relationship to "reality" is entirely subjective?

Once again I am confronted with someone who “knows the correct answer” without examining the problem at all.

Well of course the model:

"Person does not understand the Theory" if and only if "The Person states the Theory is Incorrect"

is a remarkably self-consistent model, whose relationship to reality is determinable solely through highly subjective interpretations! :)

But...

If you see a flaw in my logic, please point it out; otherwise, as most everyone feels compelled to inform me that nothing can be gained by thinking the thing out, it really serves no purpose to raise that battle flag again.
I didn't really call it a flaw in your theory, only a request asking for an explanation of its profundity.

Stating the seemingly obvious (to my insignificant little brain at least) statement that the conclusion of "subjectiveness is an inherent quality of perception" (i.e. "to know" is equivalent to "to assume") in conjunction with the definition of your model whose primary feature is "without assumptions" means that there is no way to conclude which model can be mapped to reality ("mapping" being an assumptive activity), and thus we can "know" precisely nothing.

That sounds internally self-consistent to me!

In essence, I have just slightly overstated the same thing that Annsi corrects me on above:

We can never know if our assumptions are right or wrong (I think our assumptions are just semantical boundary drawing that reality does not care or operate on), but perhaps we can know something about logical constraints that allow ourselves to build self-coherent worldviews (allows us to build "logically coherent set of assumptions")

So "what can we know about reality" is not referring to any specific ontology, but rather to what can we know about each and every possible "self-coherent" ontology that ever could be built. (The question would perhaps be more properly phrased, "what can we know about valid worldviews"?)

However as I understand the direction in which you have headed here this more general description does not agree with your own.

I probably would keep my mouth shut--maybe even make some positive comments on the theory itself--if it weren't for the fact that you keep claiming "profound implications" without ever getting around to saying what they are.

To me at least there is some inconsistency between "It has profound implications" and "there's no need to raise that "battle flag" [of why there is something to be gained by thinking it out] again." I certainly don't mind discussion of subjects that don't have clear goals or even usefulness--heck, I continue to follow this one with great interest--but to claim "benefit" and then refuse to discuss it is, well, just a little bit frustrating! :)

With reference to your definition of no assumptions, it might be useful to expand upon Q's reference to Kant's a priori knowledge.

Nothing needs a purpose, but it sure makes it easier to justify the wasted effort,

Buffy

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...We both have a very strong interest in the patterns numbers can display; however, our interests are far different. From looking at your work, it seems to me that your interest is in the variety of patterns which are possible whereas my interest would be, “what do all these patterns have in common”. That is why I posted that thread ”A simple geometric proof with profound consequences”. What that proof shows is that every possible pattern in one, two or three dimensions can be seen as a projection of a rotated minimal unitary n dimensional equilateral polyhedron onto that one, two or three dimensional space. They are all patterns of exactly the same thing. From your approach, examining individual patterns, no matter how many patterns you examine, there exists another you have not yet examined. The difficulty with that attack is that the number of possibilities is infinite and there is no end to the procedure. If your purpose is merely to keep your mind busy looking at the beauty of the possible patterns, it's perhaps a nice pass time; however, if your interest is to understand those patterns, as mine is, your approach is little more than a delightful distraction. All I am saying is that our interests are evidently quite different and everyone needs to pursue their own interests.

...

...Let me stress at this point that the idea of this discussion is not so much to talk about specific models (i.e. models that attempt to explain the functions of the brain), but to investigate what is the logical procedure that allows something (like "the brain") to take a stream of completely unknown data (like "the sensory data"), and deduce some meaning from that data. (Some meaning that hopefully has got something to do with the actual reality. :)

Right then. I think Doc that you misunderstand my work as much as I do yours. :doh: And yet, we both seem to agree we are doing with our work what Anssi states above. :eek:

The 'pretty ' part of some of my work (katabataks) is more or less my engagement with what others see in it. Like you, I have yet to engage others in looking at the patterns and attempting to draw general conclusions that reveal something of the reality of the source of those patterns; folks get stuck on pretty. :) Knowing what procedure and numbers go into making a pattern, I am looking at as many patterns in a class as possible in order to backtrack my knowledge tree and arrive at "profound" numerical insights.

I think our interests are as much similar as different, and I hold out the hope that my interests may give me some insight into yours.

Having fun, Turtle :)

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Anssi, please stop me if I comment on anything which is not absolutely clear to you.

One thing right off the bat about the shift symmetry equations:

$\sum_i \frac{\partial}{\partial x_i}\vec{\psi}(x_1,\tau_1,x_2,\tau_2, \cdots , x_n, \tau_n,t) = iK_x\vec{\psi},$

I don't know what the right side of the equation means:

$iK_x\vec{\psi},$

I just know it is going to be equal to zero.

The four individual constraints are starting to get pretty clear, but I'll have to look at the latter part, where you turn it all into a single final equation, with some more time at my hands. And then I'm sure I'll have questions about it too :)

-Anssi

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Stating the seemingly obvious (to my insignificant little brain at least) statement that the conclusion of "subjectiveness is an inherent quality of perception" (i.e. "to know" is equivalent to "to assume") in conjunction with the definition of your model whose primary feature is "without assumptions" means that there is no way to conclude which model can be mapped to reality ("mapping" being an assumptive activity), and thus we can "know" precisely nothing.

Just a quick comment that may or may not be helpful;

Think about what I might have meant by "this discussion is not about a theory, but about a logical construction that is simply either valid or invalid, and it's validity doesn't say anything about what exists in reality"

You can think of the results of the procedure as loosely similar to statements like:

"If you choose to describe the components of a physical system in 'such and such ways', that will force you to describe other things in your worldview in certain way".

The point being, while you may hold in your head right now a completely internally coherent worldview that gives you explanation of the reality around you (i.e. you can predict things), you could always just about anything in that worldview without losing any prediction ability, as long as you changed other things in it appropriately too.

The different QM interpretations are all specific examples of this, and so are different ontologies of relativity (static spacetime or something else). Note that the math of relativity does not force you to assume static spacetime, and not only that, it doesn't even force you to assume relativity of simultaneity!

That is rather simple issue really, although it tends to strike some people as "obviously invalid" because they do not recognize what are the undefendable assumptions in their worldview that lead them to think that the only valid way to describe reality (our observations) is to assume simultaneity is relative. (If you describe everything with a spacetime diagram, the way you draw "simultaneity planes" on top of it has no effect on anything inside it. You can draw them in S shape or draw one that everyone shares, who cares?)

dammit, that was not such a quick comment after all -Anssi

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That is rather simple issue really, although it tends to strike some people as "obviously invalid" because they do not recognize what are the undefendable assumptions in their worldview that lead them to think that the only valid way to describe reality (our observations) is to assume simultaneity is relative.
Actually I'm pretty sure I do get what you're trying to say here, and I'm not at all trying to say that any of it is "obviously invalid!" I was drawn to math and computer science precisely because the notion of abstract systems that have no analogue to reality are easy to build and manipulate.

And that is precisely why I'm drawn to your restatement of the goal of this exercise insofar as it leaves the mapping to reality as an implementation detail instead of it being a central goal while simultaneously insisting that this implementation is assumptionless.

And yes, I don't have any problems with having multiple self-consistent and inter-contradictory ontologies (yes, I realize that is a half-redundant phrase and its on purpose for...clarity ) simultaneously in play: they're all wonderfully valid on their own (and I'm kinda like Turtle in that I think that makes them pretty!).

dammit, that was not such a quick comment after all :eek:
In this kind of discussion, they never are! And that's okay! :friday:

My group is running rings around the field,

Buffy

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I probably would keep my mouth shut--maybe even make some positive comments on the theory itself--if it weren't for the fact that you keep claiming "profound implications" without ever getting around to saying what they are.
I think that for now the thing is to check the arguments from which Dick derives his fundamental equation. It would be interesting to understand them, as he claims to give a very general logical grounding for many things, including Dirac's equation. I'd like to understand the logical basis better, but the effort has been somewhat lengthy.

With reference to your definition of no assumptions, it might be useful to expand upon Q's reference to Kant's a priori knowledge.
Kant, in his Transcendental Aesthetics, discusses what knowledge we may consider to be "a priori" of any logical argument, IOW directly as perceived through our senses. That is where things get to be tricky. However, I really only meant to distinguish "what we perceive" from "what exists" or "das ding an sich" (the thing in itself); the assumptions aren't so much about your perception of things but rather about the things themselves.

Just after having banged your head against a stone wall, you might be inclined to think you have an excellent idea of what those stones are. While you're quite sure of what the bang felt like to you, and you're attributing various qualities to them, do you really know what the stones actually are?

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Sorry it took me so long to respond; we had house guests and I had no real opportunity to look at the internet. Thanks Qfwfq for fixing that expression, (the null replacement). I went ahead and removed my “red” complaint.

Well of course the model:

"Person does not understand the Theory" if and only if "The Person states the Theory is Incorrect"

is a remarkably self-consistent model, whose relationship to reality is determinable solely through highly subjective interpretations! :)

Your problem is that you think I am presenting a theory. I am not. I am showing some relationships which can be deduced from my definition of an explanation (and a few other very specific definitions).
But...I didn't really call it a flaw in your theory, only a request asking for an explanation of its profundity.
Before such an explanation can exist, you must first understand the deduction. So long as you do not understand the definitions I have put forward, I am quite confident that you are completely missing the issues covered in the deduction and it is that critical issue which leads you to invariably misidentify the opus as a theory.
Stating the seemingly obvious (to my insignificant little brain at least) statement that the conclusion of "subjectiveness is an inherent quality of perception" (i.e. "to know" is equivalent to "to assume") in conjunction with the definition of your model whose primary feature is "without assumptions" means that there is no way to conclude which model can be mapped to reality ("mapping" being an assumptive activity), and thus we can "know" precisely nothing.
The problem here is that you don't comprehend the basis of my attack on explanation. English words possess rather vague definitions and, if we want to do mathematically precise deductions, we need mathematically precise definitions. I have used (over the years) different English terms for what I am talking about and, in all cases people invariably choose to work with what what they have grasped the common words to mean. At the moment, I am using the terms “valid” ontological elements and “invalid” ontological elements (“valid” meaning they are truly elements of reality and “invalid” meaning they have been created in order to facilitate an explanation). When I first started, forty years ago, I referred to the same division as “knowable data” and “unknowable data” (knowable meaning actual data about reality and unknowable being presumed data in defense of the explanation). I mean something very specific here and apparently the people I talk to find it very difficult to pick up on what is intended almost always presuming valid/invalid has something to do with experimental results or thinking knowable has something to do with what can be proved real: i.e., “to know” is equivalent to “to assume”. I am totally at a loss as to how to get my readers to comprehend that, if they want to follow my deductions, they have to use my definitions.
Nothing needs a purpose, but it sure makes it easier to justify the wasted effort
And I agree with you and. yes, I said there are profound consequences, but those consequences can no more be seen without understanding my proof than the consequences of calculus can be understood without understanding calculus.
And yet, we both seem to agree we are doing with our work what Anssi states above. :eek:
Not really, I suspect you are concerned with understanding the patterns themselves, whereas I am concerned with the constraints on understanding itself. That issue actually has nothing to do with reality and everything to do with exactly what “an explanation” is. You are more concerned with “numerical insights” (which could be seen as constraints on the patterns. My interest can be seen more as “constraints on insights”.
I think our interests are as much similar as different, and I hold out the hope that my interests may give me some insight into yours.
I would agree that they are certainly related and, that understanding my work will yield insights into yours.
One thing right off the bat about the shift symmetry equations:

$sum_i frac{partial}{partial x_i}vec{psi}(x_1,tau_1,x_2,tau_2, cdots , x_n, tau_n,t) = iK_xvec{psi},$

I don't know what the right side of the equation means:

$iK_xvec{psi},$

I just know it is going to be equal to zero.

Again, it seems that the new LaTex implementation does not implement within quotes. The two equations above are:

$\sum_i \frac{\partial}{\partial x_i}\vec{\psi}(x_1,\tau_1,x_2,\tau_2, \cdots , x_n, \tau_n,t) = iK_x \vec{\psi},$

and

$iK_x \vec{\psi},$

Sorry Anssi, but you happen to be wrong on this point. It is no more than the fact that your math background is limited. Let me do the following algebra for you. The definition of a derivative is

$\frac{d}{dx}f(x)=\lim_{\Delta x \rightarrow 0}{\frac{f(x+\Delta x)- f(x)}{\Delta x}}.$

The question is then, given that definition, what is the derivative of f(x) multiplied by g(x). It should be clear from the above that the correct expression is:

$\frac{d}{dx}f(x)g(x)=\lim_{\Delta x \rightarrow 0}{\frac{f(x+\Delta x)g(x+\Delta x) - f(x)g(x)}{\Delta x}}.$

Against that we can be confident that [imath]f(x+\Delta x)g(x+\Delta x) - f(x+\Delta x)g(x+\Delta x)[/imath] is exactly zero and furthermore, in the limit as [imath]\Delta x[/imath] goes to zero, [imath]f(x)g(x+\Delta x) - f(x)g(x+\Delta x)[/imath] is also exactly zero: i.e., in that particular limit, f(x)g(x)-f(x)g(x)=0. This means that

$\frac{d}{dx}f(x)g(x)=\lim_{\Delta x \rightarrow 0}{\frac{f(x+\Delta x)g(x+\Delta x) - f(x)g(x)+f(x)g(x+\Delta x) - f(x)g(x+\Delta x)}{\Delta x}}.$

or, reordering terms,

$\frac{d}{dx}f(x)g(x)=\lim_{\Delta x \rightarrow 0}\left\{{\frac{f(x+\Delta x)g(x+\Delta x)- f(x)g(x+\Delta x)}{\Delta x}+\frac{f(x)g(x+\Delta x) - f(x)g(x)}{\Delta x}}\right\}.$

Which is identical to

$\frac{d}{dx}f(x)g(x)=\lim_{\Delta x \rightarrow 0}\left\{\left\{\frac{d}{dx}f(x)\right\}g(x+\Delta x)+f(x)\left\{\frac{d}{dx}g(x)\right\}\right\}.$

That limit should be clear to you. It is exactly

$\frac{d}{dx}f(x)g(x)=\left\{\frac{d}{dx}f(x)\right\}g(x)+f(x)\left\{\frac{d}{dx}g(x)\right\}.$

Now, if we add the fact that [imath]i=\sqrt{-1}[/imath] and that the conjugate (the dagger on [imath]\vec{\psi}[/imath]) means change the sign of all imaginary components, then the fact that [imath]P=\vec{\psi}^\dagger \cdot \vec{\psi}[/imath] implies that the derivative of P with respect to a (the shift parameter) must be

$\sum_i \frac{\partial}{\partial x_i}P = \left\{\sum_i \frac{\partial}{\partial x_i}\vec{\psi}^\dagger \right\}\cdot \vec{\psi} + \vec{\psi}^\dagger \cdot \left\{\sum_i \frac{\partial}{\partial x_i}\vec{\psi}\right\}$

If [imath]\vec{\psi}[/imath] satisfies the equation,

$\sum_i \frac{\partial}{\partial x_i}\vec{\psi}=iK\vec{\psi},$

where K is a simple constant, then [imath]\vec{\psi}^\dagger[/imath] must satisfy the equation

$\sum_i \frac{\partial}{\partial x_i}\vec{\psi}^\dagger=(iK\vec{\psi})^\dagger= -iK\vec{\psi}^\dagger.$

Substitution of this result into the expression for the derivative of P in terms of [imath]\vec{\psi}[/imath] simply allows the iK term to factor out and (because of the change in sign due to complex conjugation) we will get (iK - iK) which is exactly zero. Thus the implied constraint on the derivative of [imath] \vec{\psi}[/imath] is not that it must be zero but rather that it must be a constant. Of course, it could be zero as zero is indeed a constant; however, it corresponds to a conserved quantity which allows mathematical transformation consistent with the existence of that shift symmetry.

My perspective on symmetry is quite alien to the norm and it would be valuable for anyone interested to take a quick look at my post to savior machine (post #696 on the “Is time just an illusion?” thread on physicsforums), selfAdjoint’s response to it (post number 697 immediately below that post) and my response to selfAdjoint’s (post number 703 on that same page). These three posts should be read very carefully as they clarify my contention that all proofs are tautological in nature: i.e., what is proved must be embedded in the axioms themselves or the proof could not be accomplished. What is significant here is that mathematical deduction can carry tautological consequences far beyond what can be comprehended by the human mind.

I find it quite sad that the only person with sufficient understanding of the problem I have attacked to have made an attempt to follow my deduction has such a limited education in mathematics. This is going to slow things down greatly particularly when we get to solving that fundamental equation. I also noticed that no one took the trouble to explain the differential issue to Anssi though I am sure most of you were well aware of where that “K” came from.

Have fun -- Dick

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