What can we know of reality?

[Doctordick]

Of the math what I don’t quite get is how the shift symmetry implies differentiation. I can understand how adding any number to every entry makes no effect on the explanation but I don’t see how this implies that the derivative exists. Obviously any solution to your equation must be an explanation but how do we know that all functions that are explanations have a derivative?

We don't! That is exactly the reason I continually refer to the fact that “an interpretation” of any explanation exists which will satisfy that differential equation.

 

The issue is that any interpretation of any explanation is based upon a finite amount of information. Now I proved explicitly that, if the information in your left hand is finite then there exists an interpretation of that information (the valid ontological elements) together with a presumed set of invalid ontological elements under the rule that no two labels may be exactly the same, which is totally consistent with the presumed information (the collection of valid and invalid ontological elements) exists. I then point out that symmetry and the fact that the labeling procedure demands that

[math]\vec{\psi}(x_1+a,\tau_1,x_2+a,\tau_2,\cdots, x_n+a,\tau_n,t)=\vec{\psi}(x_1,\tau_1,x_2,\tau_2,\cdots, x_n,\tau_n,t)[/math]

 

is valid so long as the number of elements in the presumed information is finite. This means that it must be true no matter how large that finite number is. Since our “theoretical” case is, “the possibilities go to infinity”, if follows that we are interested in what that relationship looks like in the limit where the number of elements in the known information goes to infinity (the history of events we need to match are infinite). It is no more than a mathematical consequence that if the relationship being stated is always consistent (as we go to infinity) it drops directly into the fact that the definition of a derivative of [imath]\vec{\psi}[/imath] with respect to a must vanish. The rest is ordinary mathematics.

 

Have fun -- Dick

[AnssiH]

No problem at all. I am fully aware that your first priority is earning a living and I wouldn't want you to think there is anything more than enlightenment to be found here. But enlightenment can be fun. All in all, I think you are doing an excellent job of picking up the mathematics.

 

Thanks, it's just too bad that last months I've seldom had large enough chunk of free time at once to really concentrate on learning the math. The end of the year is going to be like that for me too (will be busy and then away from home)

 

None of this is a waste of time for me though, since just learning the math is useful for me. And of course it's an interesting topic :)

 

 

[unit vectors in the definitions of [imath]\vec{\alpha}[/imath] and [imath]\vec{\nabla}[/imath]]

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

Okay I think I understand that. And those partials are the partials of [math]\psi[/math], yes? I suppose that's what you mean by (from post #89):

"The term being summed is [imath]\vec{\alpha_i}\cdot \vec{\nabla_i}[/imath] and each “i” yields a different term in that sum but every term is operating on the same [imath]\vec{\psi}[/imath]."

 

 

[Expansion of [imath]\sum_{i=1}^n \vec{\alpha_i}\cdot \vec{\nabla_i}[/imath]]

Your interpretation here is erroneous. In vector relations, there is a thing called the “dot” product (or often, the “inner” product).

Oh I see, the dot product again, had forgotten all about that by now :P Thank you for the explanation of dot product. I took a look at dot product Wikipedia entry as well and now I understand that expansion of the sum.

The reason this inner or dot product is defined the way it is, is that it turns out that the result is independent of the orientation of the coordinate system.

...

which is exactly [imath]vec{A}cdotvec{B}[/imath] expressed in the (x,y) representation. The only reason I went through all of that was to convince you that the dot or inner product is indeed something defined by the vectors themselves and not by the particular coordinate system chosen to represent those vectors.

 

I didn't go through that proof very carefully since you say it is not very important at this point of the analysis yet. Nevertheless, thinking about the geometric interpretation of the dot product, I would certainly expect it to be independent of the orientation of the coordinate system.

 

So I followed the math from post #89 through and understand how you get:

[math]

\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}

[/math]

 

Next I suppose its the latter parts of posts #42 and #83 I should concentrate on?

---from post #83:---

[math]\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}[/math]

 

->

 

[math]\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}[/math]

-------

I think I understand the sign change on those terms now, but the next step on that post is little bit of a mystery so far... :I But then, I haven't had much time to stare at it yet either ;)

 

Thank you for the patience once again.

 

-Anssi

[AnssiH]

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

 

I would like to make a comment here.

I took a quick look at your home page, and while I don't understand almost any of it, I can see there are some similarities to the topic on this thread. I.e. you are looking for constraints that must be true for any valid worldview, and the consequences that such constraints would have.

 

You are concerned with "all facts in reality are consistent, meaning no fact proves any other fact". That is what has been referred to as "self-consistent worldview" in this thread, i.e. that the mental idea you have about reality cannot contain self-contradictions for it to be taken as valid.

 

You seem to be equating a worldview/model with reality itself, and that's a stretch you should not make. Instead you should be concerned with the possibility that there are many valid ways to describe reality. For example, we are free to define different features of reality as "entities" (say they are things with temporal identity, i.e. we refer to them as the "same thing" when we see them at different times).

 

Anyway, not to get into that discussion again (you can look at my older posts for clarification if needed), what I'm getting at is that your treatment - like Doctordick's - doesn't say as much about reality as it says about our ways of describing reality. That is NOT to say it is insignificant. On the contrary, it is very significant.

 

In a nutshell, if you can derive quantum mechanics from logical constraints that are general to ANY possible valid worldview, that says something about the QM-models that we built, but not necessarily so much about reality itself.

 

Think about that

 

-Anssi

[Turtle]

Actually, I'm not so sure this is correct. If a theory (epistemology) can be derived from some kind of logic alone, then it is not possible to deny it's correct. But what kind of logic would do this. I realize this opens up a whole new can of worm, but I've been working on such a theory. I wouldn't call it completely proved yet, but I think it is close enough to merit some serious attention. ...

 

This comment stirred up my affinity for Kurt's hammers, that is Gödel's incompleteness theorems, and I don't think I or any others have yet have brought this up with the good Dr. Dick in any of our earlier discourse on his work. Consider it brought up I guess. Thoughts? :) :)

[Majik]

You seem to be equating a worldview/model with reality itself, and that's a stretch you should not make. Instead you should be concerned with the possibility that there are many valid ways to describe reality. For example, we are free to define different features of reality as "entities" (say they are things with temporal identity, i.e. we refer to them as the "same thing" when we see them at different times).

 

In a nutshell, if you can derive quantum mechanics from logical constraints that are general to ANY possible valid worldview, that says something about the QM-models that we built, but not necessarily so much about reality itself.

 

-Anssi

 

The only thing we can describe is our observations about reality. And our "observations" are not necessarily "reality" itself, or is it? And the most general truth we can abstract from all our observations is that these facts exist in conjunction. That certainly seems to me to be the most reliable starting point to derive physics. Even if we should learn later that what we thought we observed was really not correct, we should still believe that whatever that facts really do turn out to be, they all exist in conjunction and none contradicts another. This seems to suggest that the theory I(we) develope from non-contradition alone actually takes priority over actual observations. For I think we would be more willing to say we observed wrong, then to say the facts are in contradiction.

[Majik]

This comment stirred up my affinity for Kurt's hammers, that is Gödel's incompleteness theorem , and I don't think I or any others have yet have brought this up with the good Dr. Dick in any of our earlier discourse on his work. Consider it brought up I guess. Thoughts? :) :)

 

Deriving physics from first principles is NOT the same as trying to find all the mathematical statements that are true. No one is going to argue with the math used in financial accounting practices just because we don't know every mathematical statement that's true. Applying math to physical ideas is just a type of accounting scheme. Gödel's incompleteness theorem does not apply.

[wigglieverse]

I.e. you are looking for constraints that must be true for any valid worldview, and the consequences that such constraints would have.

 

You are concerned with "all facts in reality are consistent, meaning no fact proves any other fact". That is what has been referred to as "self-consistent worldview" in this thread, i.e. that the mental idea you have about reality cannot contain self-contradictions for it to be taken as valid.

 

You seem to be equating a worldview/model with reality itself, and that's a stretch you should not make.

If you look at any information source, a channel, it's a question of noise, and how well that noise is understood, or discriminated.

[Bombadil]

Dr Dick, then in a way all that the fundamental equation allows us to do is approximate an explanation to a degree of accuracy dependent on the number of known epistemological objects, and the greater the number of objects the greater the accuracy of our interpretation of an explanation to the value of a true explanation, and since we are only interested in the limiting case where the number of objects approaches infinity we are approximating an explanation to an infinite degree of accuracy?

 

When I first saw that you where making sure that no entry was repeated in the coordinate system I understood it to be a mathematical convenience but after some more reading I’m beginning to wonder if it is in fact far more then this. Is there some sort of symmetry that this is a requirement of?

 

This is starting to sound to me like we are using an epistemological construct to approximate something composed of ontological items. The biggest problem that I have with this idea is that it seems that I am suggesting exactly what you have been warning people against doing and that is starting to make epistemological constructs, although, this would be a very general one.

[Turtle]

Deriving physics from first principles is NOT the same as trying to find all the mathematical statements that are true. No one is going to argue with the math used in financial accounting practices just because we don't know every mathematical statement that's true. Applying math to physical ideas is just a type of accounting scheme. Gödel's incompleteness theorem does not apply.

 

I had no physics principles in mind, and I'm primarily interested in Doc's take on whether Gödel's theorems apply to his work inasmuch as he is talking about internally consistent systems. It just so happened your turn of phrase prompted my post. :)

[wigglieverse]

internally consistent systems.

Have you looked at any information-theoretic models, esp computational algebras?

[Turtle]

Have you looked at any information-theoretic models, esp computational algebras?

 

Yes. Cellular automata for example. Can you clarify the context of your retort? :)

[wigglieverse]

So, you have papers or you've built models on real computers? Do you use any FP languages?

[Buffy]

Have you looked at any information-theoretic models, esp computational algebras?

 

So, you have papers or you've built models on real computers? Do you use any FP languages?

 

If you'd bother to read this thread, you'd know that this is completely irrelevant to to the topic being discussed. Do not disrupt it.

 

Thank you for your cooperation,

Buffy

[Doctordick]

Hi Anssi, it's nice to hear from you again and I am sorry that we need to cover so much mathematics. I read in a recent article that Finland is number one in the world in mathematics education; too bad your teachers failed to perk your interest in the subject when you were young and picked things up like a sponge. On the other hand, perhaps they are number one for the very same reson your interest was destroyed. In my grandfathers day, mathematics here was taught more by rote memorization than by explanation and we used to be pretty high on the list :confused: :shrug:

And those partials are the partials of [math]psi[/math], yes? I suppose that's what you mean by (from post #89):

"The term being summed is [imath]vec{alpha_i}cdot vec{nabla_i}[/imath] and each “i” yields a different term in that sum but every term is operating on the same [imath]vec{psi}[/imath]."

Yes and another term for such a thing is “an operator”. In mathematics, “an operator” is a thing which performs some mathematical operation on a mathematical object. The performance of the act is called “an operation”. For example, addition is an operation performed by “an operator” called a “plus sign” and the defined operation requires two mathematical objects. The result of the operation is called a sum. It is nothing more than a general way of referring to what is going on in English.

So I followed the math from post #89 through and understand how you get: [...] I think I understand the sign change on those terms now, but the next step on that post is little bit of a mystery so far... :I But then, I haven't had much time to stare at it yet either ;)

This is nothing but straight forward algebra taking into account the more complex nature of the mathematical objects (non-commutation in particular). As long as one does exactly the same thing to both sides of an equation, the equation remains true; that is the essence of algebra itself. Originally I multiplied both sides of the equation from the left (since commutation is now important, the order of the terms is important) by the mathematical object [imath]\alpha_{qx}[/imath].

 

At that point each side of the equation consists of two factors: the left side consists of [imath]\alpha_{qx}[/imath] operating on (multiplying) a sum of terms and the right side consists of [imath]\alpha_{qx}[/imath] operating on a single term. I then move [imath]\alpha_{qx}[/imath] inside the Sum sign and distribute it across the entire sum. At that point, I can examine the consequences of commuting it through the alpha and beta objects inside that sum, obtaining the expression which I think you understand:

[math]\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}[/math]

 

As I said, commutation merely changes the sign and adds one additional term (that partial with respect to [imath]x_q[/imath] sans alpha or beta) which occurs exactly once: i.e., when i happens to be q (remember, the sum is over all i so the case must occur exactly once and no more and stands by itself outside the sum). At this point, the term [imath]\alpha_{qx}[/imath] occurs in every term in the sum (just as it did when we first distributed it out over all the terms in the sum). Thus it can again be factored out of that sum; however, at this point it is on the other side of the alpha and beta objects in that sum. That fact brings it directly against [imath]\vec{\Psi}[/imath] with no intervening terms (remember, it commutes with all the other mathematical objects except for [imath]\vec{\Psi}[/imath]). At that point, we have:

[math]\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\}\alpha_{qx}\vec{\Psi} +\frac{\partial}{\partial x_q}\vec{\Psi}= iKm\alpha_{qx}\vec{\Psi}.[/math]

 

Since we have factored the terms involving q such that they are no longer in the explicit sums over i and j, the sums themselves can be seen as mere mathematical objects (their definitions do not require knowing what q is). If we sum both sides of that equation over q, the entire sum still stands as factored and we have the expression:

[math]\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}.[/math]

 

That equation still must be satisfied by [imath]\vec{\Psi}[/imath]; however, when we perform that sum we get some interesting results.

 

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

[math]\sum_i \vec{\alpha}_i \vec{\Psi} = \sum_{i \neq j}\beta_{ij} \vec{\Psi} = 0.[/math]

 

Just as an aside, it is this portion of the definition of the alpha and beta objects which disallow commutation with [imath]\vec{\Psi}[/imath] for the simple reason that [imath]\vec{\Psi}\sum_i \vec{\alpha}_i[/imath] is undefined. Please notice also that the index i here is nothing except a summation index: i.e., if you are going to sum over all of them, it makes no difference what letter you use to indicate the index (so long as you use different indices when you have more than one sum in an expression). At any rate, since the sum over [imath]\alpha_{qx}\vec{\Psi}[/imath] vanishes by definition, only one term in the above expression fails to vanish and the equation reduces to:

[math]\sum_q\frac{\partial}{\partial x_q}\vec{\Psi}=0[/math]

 

and it follows that our solution [imath]\vec{\Psi}[/imath] obeys the shift symmetry constraint on the [imath]x_i[/imath] which was originally deduced as necessary.

 

There is nothing really subtle going on here. The real issue is that these alpha and beta objects have been defined such that they are actually operators which separate out what appears to be one coherent equation

[math]\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} = iKm\vec{\Psi}.[/math]

 

into four separate equations. Just as a vector equation is, in reality, a number of independent equations, one equation for each component of the vector, these alpha and beta objects perform exactly the same kind of mathematical effect. We like to use vector equations because they simplify the notation and make relationships appear simpler. In the same fashion, the alpha and beta objects I have defined yield an apparently simple equation; however, in actual fact, it is little more than a notational convenience. So long as we stick with the definitions of these objects and their respective operations, we can perform ordinary algebra as if we were dealing with a single coherent equation. It makes life much more convenient than trying to handle the logic of each constraint on its own.

 

Qfwfq and Buffy, if you have any complaints with what I have just said, please let me know. Thank you.

This comment stirred up my affinity for Kurt's hammers, that is Gödel's incompleteness theorems, and I don't think I or any others have yet have brought this up with the good Dr. Dick in any of our earlier discourse on his work. Consider it brought up I guess. Thoughts? :cup:

I am afraid I have to agree with Majik on this:

Applying math to physical ideas is just a type of accounting scheme. Gödel's incompleteness theorem does not apply.

And Majik, what you have to remember is that our “observations” are themselves an interpretation of reality (that is why I have defined reality to be a “valid ontology”). Contradiction is an aspect of our explanation of reality, not an aspect of reality itself.

Dr Dick, then in a way all that the fundamental equation allows us to do is approximate an explanation to a degree of accuracy dependent on the number of known epistemological objects, and the greater the number of objects the greater the accuracy of our interpretation of an explanation to the value of a true explanation, and since we are only interested in the limiting case where the number of objects approaches infinity we are approximating an explanation to an infinite degree of accuracy?

I think you have missed the entire point of my presentation. Anyone who understands the comment that you cannot disprove solipsism has to know is that there exists no way of knowing what is illusion and what is reality. That is the very issue upon which my analysis is built. I define reality to be a “valid ontology” and point out that explanations are facilitated through the invention of “invalid ontological elements”. It is our very freedom to do this that allows us come up with an explanation of that “valid ontology”. Recognizing the power of that freedom, I laid out a specific procedure for yielding exactly the “valid ontological elements” no matter what they were. Accuracy does not even become an issue here.

When I first saw that you where making sure that no entry was repeated in the coordinate system I understood it to be a mathematical convenience but after some more reading I’m beginning to wonder if it is in fact far more then this. Is there some sort of symmetry that this is a requirement of?

It is not a convenience, it is a requirement. In order to make sure that no possibility is eliminated, every valid ontological element must be represented as individual. Now theories (or epistemological constructs) invariably identify some elements as being the same. If we are to allow such labeling, we must add an “invalid” orthogonal axis in order to represent our valid ontological elements as points in a Euclidean space.

This is starting to sound to me like we are using an epistemological construct to approximate something composed of ontological items. The biggest problem that I have with this idea is that it seems that I am suggesting exactly what you have been warning people against doing and that is starting to make epistemological constructs, although, this would be a very general one.

I have been warning them against creating an epistemological construct which defines what is and is not real. I have explicitly kept the two separate without defining what they are: i.e., my construct presents no constraints whatsoever on the possible explanation.

 

The only constraint on my construct is due to symmetry alone and, if you go and read those three post I have referenced a number of times, you should be able to comprehend why the constrains imposed by fundamental symmetry are required (see the three posts on “physicsforums.com”: my post on the what symmetries are, selfAdjoint's response to that post and my response to selfAdjoint). Your solution to a problem cannot produce information which is not contained in the presentation of that problem.

 

Have fun -- Dick

[wigglieverse]

OK, is this something more relevant, or germane, or non-perturbative?:

 

A serious problem is how to select a microstructure where each state is a priori "equally likely". This is not a trivial, especially if you try to understand this in a context where you want all constructs to be induced from real observations. Taking arbitrary prior distributions into account, one is led to various relative entropies (K-L divergence, or information divergence)

[AnssiH]

The only thing we can describe is our observations about reality. And our "observations" are not necessarily "reality" itself, or is it?

 

Yeah, the way I would put it is, our "observations" are an interpretation of the "noumena". There are always many valid ways to interpret them (~to "understand" reality).

 

When you see that an "apple" is "falling" "down" the "tree", that's an interpretation of a situation. A useful one since it also implies something about the future of that "apple". All those words in italics are concepts that we formed, and their definitions exist as part of our worldview. They are not so much part of reality, as they are a way to see reality. Look at QM as a set of defined entities and behaviours just the same way.

 

So by "understand" I mean, one has got some sort of idea about what is happening, going to happen, or has happened, and by "valid" I mean the model/worldview that is used for the interpretations does not contradict the noumena, nor itself. You can think of a worldview just as a set of facts, and many self-coherent worldviews can be constructed.

 

And the most general truth we can abstract from all our observations is that these facts exist in conjunction. That certainly seems to me to be the most reliable starting point to derive physics. Even if we should learn later that what we thought we observed was really not correct, we should still believe that whatever that facts really do turn out to be, they all exist in conjunction and none contradicts another. This seems to suggest that the theory I(we) develope from non-contradition alone actually takes priority over actual observations.

 

Yes, that is not so different from what Doctordick is talking about. I think we think somewhat alike, what I just wish to stress is that having a self-coherent worldview that explains all the noumena you have come across, does not mean it is the only possible worldview you could have built.

 

To bring it closer to your terminology, you can think of a physics model more as a set of "made up facts" (albeit "fact" can be a confusing word here :lol:, in that you are free to make up any "facts" as long as they are both consistent with the noumena (there are many ways to be consistent with them!) and with the other facts that belong to the same worldview. The validity of one set of "facts" cannot be investigated from within another set, as they are different self-coherent sets... In one paradigm a falling apple is considered to be accelerating, in another it is the tree that is accelerating, and all that depends on how you define things like "acceleration")

 

Anyhow, that perspective is important for this thread; that a self-coherent worldview is a set of made up facts rather than an unavoidable platonistic set of "real ontological facts". When you draw unavoidable conclusions from self-coherence, that doesn't mean we have found the ontological reality from logic alone. It kind of means those conclusions can be considered valid even in the absence of ontological knowledge. Or in other words, they are valid for any self-coherent description of any possible reality.

 

-Anssi

[Bombadil]

I’m somewhat confused just what is the difference between an explanation and an interpretation of an explanation?

 

What I understand the explanation function to be is that we input the entire set of ontological elements that fall under any particular “t” coordinate and the function then gives us a one (1) if it is a set of valid ontological elements or a zero (0) if it is not. Is this correct?

 

If this is correct, then it seems that the explanation is not a function of the coordinates but rather a function of the order in which the elements appear in the coordinate system, so that wherever the origin of the coordinate system is it still gives the same result.