Imagination and the Unreasonable Effectiveness of Mathematics by Armin Nikkhah Shirazi

Dear Akinbo,

Thank you for your patience, I had some other issues to take care of before I could continue our debate, and I knew this post was going to be a long one.

I will get to your question towards the end of this post, but I would like to first address the issues surrounding the questions I asked of you.

First, let me at least give a sketch of an intuitive qualitative justification for my claim that the non-commutativity relations are equivalent to the uncertainty relation, and that rejecting them amounts to rejecting all of QM.

The way I think of this is that the commutator of the operators is a measure of the independence of each other of the states that result when two different operators act on a particular quantum state, in the following sense: if the resultant states are completely independent, then they can be separately specified to arbitrary precision in terms of either observable, which is mathematically signified by the fact that the resultant states are eigenstates that both operators have in common. If they are not completely independent, then acting on the particular quantum state with one operator gives a resultant state that can no longer be specified to arbitrary precision in terms of the other observable. In that case, the order of operation matters: If you have two operators, A and B, and the resultant states when they act on a particular quantum state S are not completely independent, then acting on S with A and then B gives a different resultant state than acting on S with B and then A. Thus the difference between the two coupled operations, which is just the commutator, will be non-zero.

On the other hand, lack of "arbitrary precision" in the specification of a state just means there is some uncertainty in the specification of a state, so that if the resultant states of the operation of two operators on S are not completely independent, then there will be a statistical spread of the product of the observed measurement outcomes associated with the two operators. Mathematically, this is signified by the fact that the product of the standard deviation of the outcomes associated with operations is greater than zero. The statement that it is equal to or greater than hbar/2 is just the Heisenberg uncertainty principle.

Why does rejecting these amount to rejecting quantum mechanics in its entirety? Because the entire mathematical apparatus of quantum mechanics, and in particular the mathematical identification of quantum states as rays in Hilbert space is essentially built to accommodate the consequences of these relations: The fact that any pure quantum state at all that is definite in some measurement basis can be expressed as a superposition of more than one state in some other measurement basis reflects mathematically the uncertainty, the limit in precision in being able to express the same state in terms of different observables. If all operators in quantum mechanics commuted with each other, then the whole formalism would all be completely unnecessary and could be discarded; we could just get by with a description in terms of classical physics, where we can specify the same state in terms of, say, position or in terms of momentum to arbitrary precision.

Now, let me address my reason for asking you the two questions:

1. Do you believe that the introductory textbook definition of a wavelength is exact?

2. Do you accept the non-commutativity relations?

You said "I smell a 'dialectic' bait meant to entrap me" and I confirm that you smelled correctly. I normally do not try to trap people, but your original comment "It gives me a kind of pleasure when my "opponents", wriggle and wreathe trying to explain a question in a convoluted kind of way because of a belief that they must not let go a dogma they are holding on to." made it fair game for me to give you a reality check.

The trap was that answering each question with yes, which is by far the reasonable answer in each case, leads to a contradiction. To see why, consider first how we go about measuring wavelength in an idealized sense: We measure the distance between two points, which is to say, we measure two positions in space and subtract them from each other, and then assign to each a point on one of two successive waveforms such that if the two waveforms were superimposed, the two points would be right on top of each other, and then say that the distance between the two points in space is equal to the distance between the two points on the waveform.

Notice that if this idealization were exact, we could arbitrarily re-express any distance between two points in space in terms of a wave length and vice versa, which implies that you could re-express position measurements in terms of wavelength measurements and vice versa, which implies that you can express either measurement in terms of the other to arbitrary precision, or in short, that position measurements and wavelength measurements are completely independent of each other

However, if you accept the non-commutation relations, of which the non-commutativity of position measurements and momentum measurements is the best known example, then you will run into the following problem: The commutator of x and p (where to keep things simple I am using 1D) is

[math][x,p]= i\hbar[/math]

But the DeBroglie relation says

[math] p=h/\lambda=h\lambda^{-1}[/math]

where

[math]\lambda[/math]

is the wavelength associated with any object described by QM. Dividing

[math]h=2\pi\hbar[/math]

out of both sides of the first equation gives

[math][x,\lambda^{-1}]=i/2\pi[/math]

which means that position measurements and wavelength measurements are not completely independent, in contradiction to the implication of our first answer. The factor of 2pi signifies that a given measured position could correspond to a a point anywhere within that waveform.

Now, for most purposes this does not matter, and in most contexts, we can, in fact, arbitrarily interchange distance measurements in space with wavelength measurements without getting into any kind of trouble. That is why you find this definition without any further qualification in freshman physics textbooks.

But remember, we were talking about a specific situation where this distinction does matter, namely whether for light particles wavelength times frequency is the same as distance traveled in space per unit time. It turns out that the relationship is very subtle and my argument is not as strong as I would like it to be because there is a compensating uncertainty between Energy and time that hides the distinction. In particular, for photons, because

[math]E=pc[/math]

if we divide the expressions for the two uncertainty principles by each other, we get

[math]\frac{\sigma_x \sigma_p}{\sigma_t \sigma_E}=\frac{\sigma_x \sigma_{\lambda^{-1}}}{\sigma_t \sigma_f}=1[/math]

where the sigmas are the standard deviations associated with each set of measurement outcomes. If you wanted to mount an attack on my view that photons do not travel in space you might be better advised to use this as an argument against it than to accuse me of dogmatism.

I think it is true that most physicists today, along with "Roemer, Galileo, Newton, Maxwell" would disagree with my view, but there are two simple explanations for that: First, I think it is fair to say that most physicists do not spend a lot of time thinking about the ontological status of photons, and second, they have not had the opportunity to seriously examine the arguments that led me to my position. Of course, just saying "photons do not travel in space" and nothing more sounds crazy, but I believe that seriously examining the arguments that I have gathered so far would make that position far more plausible.

I admit that the arguments are not yet conclusive, but the collection of different hints from two seemingly independent theories (SR and QM) pointing to the same idea is strongly suggestive. The non-conclusiveness of my arguments leaves me open to the charge of having a prejudice towards the position I hold, and that is true, but I think this is no different from the ordinary prejudices towards speculative ideas that any scientist holds as he or she is pursuing research into that idea. You need a modicum of inductive commitment to certain tentative ideas in order to be motivated to gather evidence for or against it. At this stage, this has little to do with dogma. Dogma comes in when 1) one regards non-conlusive evidence as conclusive and/or 2) one ignores contradictory evidence.

In fact, let us see what dogma looks like in a concrete example. You successfully avoided my trap by answering yes to the first question and no to the second. But you did so at a steep cost, namely by massively damaging your credibility, as I will now show.

I cannot tell whether you really meant your "no" answer to my second question or not, so I will consider both cases.

Let me first suppose that in fact you accept the canonical commutation relations, but answered "no" to my second question only because you sensed that I was trying to trap you. I think it is instructive to relate to you my own past experience of finding myself in your situation: It has happened to me more than once that, in discussing some math/physics matter I could sense that there was a trap was a trap waiting for me, but that I willingly stepped in it just so I could find out how the opponent would "pull it off" to be able to, say, find a contradiction in my argument that I could not see. In short, getting closer to the "truth" was more important to me than winning the argument.

In contrast, if I suppose that you answered "no" to my question but in fact believe the opposite, then I have no choice but to conclude that not losing an argument is more important to you than getting closer to the truth. If we considered our discussion a debate, then that would in an of itself not be a big problem, after all the point of engaging in a debate is usually to win. But this is not how you frame your discussions with others, you always frame them as "Dialectics". A dialectic is not a debate, the point of engaging in a dialectic is not to win an argument but to get closer to "the truth".

So if I assume that you did not really mean your answer, that you answered contrary to what you really believe in order to avoid losing an argument, then I am also forced to conclude that your talk of a "dialectic" with me was not sincere.I could then not escape the conclusion that the real point of your "dialectic" with me is just to convince me of your beliefs. But the desire of convincing others of one's own beliefs while pretending otherwise presupposes that one is utterly convinced, unshakable of one's own beliefs. For some beliefs, this is still perfectly alright because there is evidence supporting the belief that can be considered, for all practical purposes, to be conclusive.

But our "dialectic" is about the edge of our knowledge (and beyond) pertaining to fundamental aspects of reality, is it not? If it is, then it is an arena in which our beliefs, whatever they are, are not supported by conclusive evidence. If they were, then the arena in which they play out would not be be considered the "edge of our knowledge". But if it is the case that you hold beliefs of which you are utterly convinced, as though they were supported by conclusive evidence, when in fact, they are not (by virtue of the subject matter), then you have just satisfied the first criterion of dogmatism I gave above: one regards non-conclusive evidence as if it were conclusive.

So let me instead suppose that your "No" answer to my question was sincere, you really reject the canonical commutation relations, and, therefore, in fact, all of quantum mechanics. Here it is useful to compare what evidence you base your beliefs on vs. the belief that we should accept quantum mechanics as a fundamental description of nature.

You helpfully provided some reference articles as evidence for rejecting the non-commutation relations or, equivalently, the Heisenberg uncertainty principle, and I did look at them. Here is what I found:

Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements

Let me quote from the introduction of the article:

`Heisenberg wrote, ''At the instant of time when

the position is determined, that is, at the instant when the

photon is scattered by the electron, the electron undergoes

a discontinuous change in momentum. This change is the

greater the smaller the wavelength of the light employed,

i.e., the more exact the determination of the position'' [1].

Here, Heisenberg was following Einstein's example and

attempting to base a new physical theory only on observable

quantities, that is, on the results of measurements. The

modern version of the uncertainty principle proved in our

textbooks today, however, deals not with the precision of a

measurement and the disturbance it introduces, but with

the intrinsic uncertainty any quantum state must possess,

regardless of what measurement (if any) is performed

[2-4]. These two readings of the uncertainty principle are

typically taught side-by-side, although only the modern

one is given rigorous proof. It has been shown that the

original formulation is not only less general than the

modern one--it is in fact mathematically incorrect [5].'

In other words, this article, which you marshaled as evidence for rejecting the uncertainty principle (as commonly understood) in fact supports it. What it does is to give an argument for the incorrectness of Heisenberg's older and less general original version of his uncertainty principle. While it may be the case that, as the paragraph claims, the older version is still being taught in textbooks today (In fact, none of the textbooks that I have used mention it), I would say that among quantum physicists, the more general version is exclusively the one which is used.

Experimental realization of Popper's Experiment: Violation of the Uncertainty Principle?

This paper realizes a thought experiment first proposed by Karl Popper to take advantage of entanglement to test the limits of the uncertainty principle. I was not familiar with Popper's thought experiment, and found the paper very interesting. However, the authors recognize that, though the experiment confirms Popper's prediction, it does not support his interpretation of it due to the fact that those aspects of the entangled pair the experimented tested were not directly applicable to the HUP for a single particle.

In fact, in the conclusion they state:

"Our experimental result is

emphatically NOT a violation of the uncertainty principle which governs

the behavior of an individual quantum."[Their Capitalization]

So, this article, which you advanced as a refutation of the HUP in fact denies doing exactly that.

Particle Measurement Sidesteps the Uncertainty Principle

This interesting article discusses a new technique, called compressive sensing, which, as I understand it, provides less information in a measurement of a particle's position than usual, so that its complementary property can be measured with greater precision. The concluding paragraph begins with:

The physicists stress that they have not broken any laws of physics. "We do not violate the uncertainty principle," Howland says. "We just use it in a clever way."

So the technique described in the article does not refute the HUP, it exploits it in a novel way. Once again, an article you gave as support for your beliefs in fact supports the opposite of your beliefs (If I take your answer to my second question to correctly reflect your beliefs)

Scientists Now Uncertain About Heisenberg's Uncertainty Principle

This low-quality article discusses the same paper as above, "Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements" but does so in a highly misleading way. I already mentioned what the result of that paper is, but somehow (probably due to the author's misunderstanding) he interprets their result as refuting "one prong" of the uncertainty principle.

I will grant you that this article makes a claim of the refutation of the HUP, but this claim is the opposite of what the authors of the experiment themselves say of their result (see the above paragraph), so in choosing whom to believe I would rather go with the scientists who did the experiment than the journalist who reports on their work.

So, to summarize, if I take your "no" answer to reflect a genuine rejection of the canonical communication relations and thereby, all of quantum mechanics, then, at least going by the evidence you have provided me, your belief is based on essentially no evidence. There are of course beliefs that are held in the absence of any evidence, they are usually identified as "faith". What takes your purported belief in the present situation far beyond faith is the fact that the majority of technological innovations over the last several generations provide a massive body, a mountain of evidence that contradicts your belief. By choosing to ignore it, you satisfy the second criterion for dogmatism in a way that could hardly be satisfied any more strongly:one ignores contradictory evidence.

So no matter whether I suppose that your "no" answer was genuine or not, I am forced to conclude that whatever dogmatic beliefs I may hold, yours are manifold greater. By evading my trap, you won the battle but lost the war.

Let me now finally get to your question. I repeat it below:

"Given a light source, e.g. a pulsar say 10^3 light seconds away, and sending out pulses once every 60 seconds, such that the moment a pulse is detected, another is already emitted and on on its way and would be detected also after 60 seconds. So we have detections every 60 seconds. Now if, on detecting a pulse, the observer moves towards the next incoming pulse, can he reduce the detection time to 59 seconds? Again, if on detection, the observer moves away from the already incoming and in-flight photon, can he delay the detection time to 61 seconds?"

There are two factors to consider here: one is the fact that if the observer moves relative to the source, then successive waves crests travel different distances before they reach the observer (notice that here I am adopting the standard language of waves traveling in space in order to avoid the much more cumbersome language I would have to use if I wanted to express my beliefs about the existence of photons in spacetime. What licenses me to do this is the fact that the ontological status of photons is irrelevant to the present question). The other is the fact that if the observer moves relative to the source, then he will observe the emission events at the source to occur spaced out more due to the time dilation effect.

The first factor implies that the inverse relation between frequency of emission and emission period must be corrected by a factor derived from the relative speed. The corrected relation if the relative speed is such that the source and observer are moving towards each other is

[math]f=\frac{c}{(c-v)T}[/math]

where f is the frequency, T is the period, v is the relative speed. The second factor implies that we must substitute T_0, the period in the rest frame by

[math]T=\gamma T_0=\frac{T_0}{\sqrt{1-v^2/c^2}}[/math]

where gamma is the Lorentz factor. Substituting this into the first equation gives

[math]f=\frac{c}{(c-v)\gamma T_0}=\frac{c\sqrt{1-v^2/c^2}}{(c-v) T_0}=\frac{\sqrt{c^2-v^2}}{(c-v)T_0}=\frac{\sqrt{(c-v)(c+v)}}{(c-v)T_0}=\sqrt{\frac{(c+v)}{(c-v)}}\frac{1}{T_0}[/math]

but since in the frame of the source the inverse relationship between frequency and period still holds, we can substitute

[math]f_0=\frac{1}{T_0}[/math]

into the last equation to get

[math]f=\sqrt{\frac{(c+v)}{(c-v)}}f_0[/math]

If the source and observer are moving away relative to each other, the above relation becomes

[math]f=\sqrt{\frac{(c-v)}{(c+v)}}f_0[/math]

The reason I went through the derivation is because I suspect that you want to point out what you perceive as another "illogicality" of SR, and having the derivation handy will make it easier to pinpoint if what you wish to argue is based on a misunderstanding of some aspect of the situation.

Ok, now to your question. What you call the "Detection time" is in fact the inverse of the frequency of detections in the frame of of the observer (i.e. f), but remember the inverse of the frequency is NOT the period associated with the emission event in the frame of the moving observer because each successive wave travels a shorter distance to get to the observer than the previous one. So plugging in 1/59 for f and 1/60 for f_0 into the second last equation and using Wolfram Alpha, I get the answer that yes, it is possible if the relative speed is on the order of 5*10^6 km/s toward each other. Similarly, if I plug in 1/61 for f and 1/60 for f_0 into the last equation, then using WA again I get a solution if the relative speed is on the order of 4.9*10^6 away from each other.

I am not sure where you are going with this example, but my suspicion, based on the fact that you used the word "detection time" instead of frequency is that you wish to mount an objection based on the assumption that in the observer's frame the inverse association between frequency and period holds. But as I pointed out above and in the derivation, this assumption is false.

Already, this got to be a very long post, so I'll cut it off here.

Best,

Armin

Dear Vladimir,

Thank you for your comments. It is too bad that you got lost after the word "modal", if anything I would have expected that as an artist you might actually appreciate it because there is a neat analogy to color and painting:

Working within classical logic is like being able to paint only using black and white as available colors. But because the only limit on finding interpretations for modalities is one's imagination, working within modal logic is like painting using many more different colors. The fact that modalities capture subtle nuances in meaning is like having access to a rich palette of colors to paint subtle nuances that you see in your mind's eye.

"I also empathized with your words "pseudo-nonlocality" because in BU angular momentum is transmitted node to ether node as in a slippery gear train and nothing is lost or hidden"

Well, the "pseudo" in pseudo-nonlocality in my theory comes out of something that most opponents of non-locality would probably not find palatable, namely a radical non-realism according to which there is "nothing" (i.e. no spacetime object) in between quantum measurements. This ensures that there are no "beables" in Bell's sense (The absence of a "beable" is what marks the ontological significance of the incomplete spacetime vector), and so there is nothing that could receive a non-local influence. In my view, Bell's arguments for non-locality are in and of themselves correct, but they simply fail to apply because they require "beables" in order to do so.

"Did you read Klingman's essay debunking Bell's theorem in this contest?"

Yes, I read his essay and found that he did no such thing. What he did is to point out the possibility that a particular experiment (the SG experiment) that could be used to test (really, I should say check its applicability to) Bell's theorem might not be appropriately modeled by Bell's theorem. While I doubt that he is correct I don't think it is impossible, but even if he is right then this has nothing to do with the correctness of Bell's theorem, because the theorem stands independently of any experiment.

Bell's theorem is a mathematical theorem, and trying to debunk it is like trying to debunk the theorem that the product of the number 1 and the number -1 is something other than -1. The only way this could be done is by changing some axioms in the foundations of mathematics, but that would have the highly unwelcome side effect that a whole host of other intuitive commonsense results would become mathematically invalid.

Also, I am a bit disturbed by how quickly you were willing to jump on his "debunking" claim, presumably because it agrees with your metaphysical preferences. Let me just mention that the mark of good science is applying one's skepticism to new claims equally and without regard to whether they agree with one's philosophical predilections or not. Few people may achieve perfection in this regard, but it is a goal that should be aspired to.

"You lost me in your next section about the path integral because you take quantum probability as a given, which I think is only a mathematical analogy."

It's too bad that I lost you in section 6 because that is the heart of my paper. It shows directly, by means of a theorem, the connection between the novel objects definable under an expanded foundation for mathematics and quantum mechanics.

Also, it is not the case that I took probability as a given. Rather, I showed (although omitting a lot of details) that the concept of probability is one of the thing that arises from the the application of axiom D, and then applied the example at the end of section 3, which pertains to an incomplete space vector, in section 6 to an incomplete spacetime vector, thereby setting the stage for deriving the Feynman path integral.

"Time to listen to music- I hope you are still composing!"

Well, thank you, yes as a matter of fact I still am. Last year I composed nearly 25 new pieces, so that my total is now up to almost 140. This year, I have not taken the time to compose because I am trying to focus filling in all the details of my theory.

Thank you again for your comments, I will read your essay shortly and leave some comments.

All the best,

Armin

Dear Akinbo,

I suppose nothing I say is going to change your perception that I am desperately clinging to SR in the face of puzzles like the existence paradox (and even the other ones I pointed out in my existence of photons paper), which should have dissuaded me from accepting the theory.

But, being the optimist that I am, let me just emphasize one last time that the reason I accept the special theory of relativity is that I find the totality of evidence compelling, and the reason that these puzzles do not only weaken my acceptance of SR but actually strengthen it is because I see these as fundamental implications of the theory which have at present not been noticed to give clues about how to understand reality at a more fundamental level and how special relativity can be derived from a common framework that also leads to quantum theory. These implications of SR, being consistent with a picture of quantum theory according to which quantum objects really "come into existence" only when they are measured point to a picture in which spacetime itself, our repository of existence, emerges from something more fundamental. I don't know if you know of my prior work on the dimensional theory, but that "something more fundamental" is, I believe, a 2+1 analog of spacetime which I call areatime. Once you understand this, then (coupled with an understanding of the dual role of metric intervals as "distance" and as "proper time" in spacetimes with Lorentzian metrics) these puzzles make perfect sense. There really is no reason for me to take these puzzles as difficulties for the theory.

"I get the impression that we reach some form of agreement that 'detection times', 'observed frequency', 'observed period' or 'observed light arrival times' can be influenced by the relative velocity. This if you crosscheck is against the principle of Lorentz invariance, according to which the velocity of the observer towards or away from the source has no effect on these observed parameters."

Well, I gave the derivation of the relativistic doppler shift in frequency earlier, what do you think is wrong with it? Or better yet, what do you think the correct expression for the relativistic doppler shift should be?

"Now what is the v in the equation? If it is relative velocity between source and receiver/observer, then in the Michelson-Morley interferometer since this is zero, then no length contraction or time dilation mechanism should occur or be applicable to explain the null findings."

I don't understand this assertion, it seems to me that you are mixing the length contraction phenomenon with the relativistic doppler shift. They are separate phenomena.

"On the other hand, if v is the observer's velocity in space towards the incoming light travelling at speed, c, then there is a dilemma of which velocities of the earthly observer in space to apply. v could simultaneously have three different velocities as it moves in space towards the light. In the M-M experiment, the v considered was the orbital velocity 30km/s about the sun. However, today, we now know that the sun as well as the Milky Way are moving with the result that in the Lorentz factor, the magnitude of v for a moving earth in space can be 30km/s, 225km/s (solar system motion in space) or 370km/s (earth motion relative to CMB). So which v applies in the Lorentz factor during the M-M experiment?"

It seems to me that you are arguing from a point of view which assumes that there is such a thing as an absolute velocity. From that point of view, I agree, that the fact that the LT could change depending on the fact that earth is moving at different velocities relative to different objects seems egregiously wrong.

But, the central lesson of special relativity is *there is no absolute frame*. All three velocities are on an equal footing, and which v you use depends on which frame is relevant for the problem you are trying to solve. So if you accept the central lesson of relativity, there is no problem at all.

"What is the - sign in the Lorentz factor? If it applies to the direction of the observer relative to the incoming light, then it means we can also have length dilation for a + sign and time contraction for a + sign apart from the contradiction that the observer's frame of motion then becomes capable of influencing whether it is contraction or dilation that happens."

No, you have misunderstood the significance of the sign. It cannot refer to the direction of relative motion because the sign stands in front of a squared quantity, so whether the quantity itself is positive or negative, the square will always be positive. If you want to understand the fundamental significance of the minus sign, I can only refer you to my old paper in which I derived the invariance of the speed of light.

http://deepblue.lib.umich.edu/handle/2027.42/83152

Look at Axiom I and divide by c. What you should hopefully realize is that the significance of the minus sign is that every spacetime object is associated with c, which depending on the frame is distributed over motion in space and "motion in time", better known as "aging".

To sum up, what I perceive here is negative feedback loop. There are aspects of SR that are distasteful to your philosophical prejudices, and which prevent you from understanding what the theory is really saying. This leads to misunderstandings which in turn seem to confirm your prior prejudices. I really don't know how this cycle can be broken except if you really take the time to really understand the theory, even if only for the sake of trying to mount stronger arguments against it.

Hope this helps.

Best,

Armin

Thank you for letting me know, I will leave a comment there shortly. Also, please note: My last name is Nikkhah Shirazi, not Shirazi.

Thanks,

Armin

Hi Jon,

"Am I the only one who thinks maybe HAL could be looked at in a positive light?"

No, I'm sure all the peers of HAL would also look at his actions in a positive light, presumably because they would be sufficiently advanced to have something like empathy for HAL, but the not the astronauts.

"But if the axioms are enumerable, couldn't a computer just systematically go through them, computing say the first 1000 "theorems" of each. If the goal of physics is to compress complex physical phenomenon into simple equations, then if you were to take this idea to the extreme, could you just systematically (without imagination) just looking through the space of all simple axiom systems (with simple updating rules/rules of inference) with some romantic hope that our universe might happen to be describable with a very simple system/program? Do you think that this is a worthwhile endeavor or just some pipe-dream?"

I think being able to answer these questions intelligently requires a background in both complexity theory and in computability theory, neither of which I posses, unfortunately. I would like to clarify that by using imagination in this sort of situation I had in mind something like the following: You pick some axiom system and think about what requirements it fails to satisfy, then use your imagination to add/subtract or modify the axioms and now you can take advantage of the ability to have computer system check the thousands of theorems that follow from the tweaked system.

"If equations didn't have closed-formed solutions that harnessed infinities and we saw things as a computation do you think the concept of time would stand out more in the equations/programs? Could time just be where you are in the calculation?"

I have a hard time understanding what it means for something physical to be a computation, so I am afraid I can't really answer that question. However, I suspect that computation conceptualized broadly in this way pre-supposes the existence of time, and if that it is true, then you cannot just say that time is defined by where you are in the calculation on risk of circularity.

"Do you think nature, evolution, or the universe is creating this way, with some high-level purpose in mind?"

No, I do not think there is such a high-level purpose. In my view, whatever adaptions you see are the result of environmental constraints, which though obeying regular patterns, do not serve to advance any particular purpose.

"How do I know that the truncated version of the one you provided is some particular noncomputable real and not another? How do you even know which one you had in mind that you are approximating?"

I would say that if you have to ask these questions, then you have not yet approximated the number to sufficient precision and need to approximate further. It seems doubtful to me that you would run into a realistic situation in which you need an arbitrary uncomputable number to infinite precision. What would you do with it? Any manipulation involving it to infinite precision is also going to be uncomputable, unless it happens to be expressible in simple form like e and pi.

"In the case of you flipping a coin, we could say that the outcome has already been logically determined, but we just don't know what the outcome is because the universe hasn't done the computation yet. In certain situations the calculation might be compressible, and these would be the cases in which we can make precise predictions. In the cases where the computation the universe is doing isn't compressible, we would resort to probabilities."

Again, because I have a hard time wrapping my mind around the idea of physical processes as computations, I'm afraid I can't say much. But if what you are saying is right, it seems that this would imply a novel interpretation of probability.

"...maybe some of Chaitin's youtube lectures might interest you if the Wikipedia page didn't offer too much insight on his Omega constant."

Thank you for the suggestion, I'll look into it.

I hope I was able to at least answer some of your questions, though I felt that they were somewhat outside of what I consider to be my area of knowledge.

Best,

Armin

Dear Akinbo,

"To form an all encompassing opinion, and to know where any conspiracy could be coming from..."

Are you seriously considering the possibility of such a worldwide conspiracy?!?

All those physicists working on the Manhattan project generations before GPS must have done an outstanding job not only maintaining the secrecy of the conspiracy but of giving terrifying fake evidence that the theory works.

Or wait...maybe you think Hiroshima and Nagasaki was not A-bombed, either?

My point is that if your really believe such things, then it seems that such beliefs, in order to be consistent with other beliefs, entail beliefs which become ever more absurd. And citing one person in one particular field is not going to help much.

I see that Ronald Hatch has written a lot of critical material on SR. I find it curious that, at least upon a cursory search, I was not able to find any critique of his work except for one particular blog.

"By any stretch of imagination going by his Bio above, Ron Hatch cannot be said to be someone with a high school background who misunderstands relativity. Why, my friend Armin would feel such a person should be disregarded along with other references like Prof. Herbert Dingle is what looks more like conspiracy, probably unintentional or well meaning, I can't really say."

This is called a straw-man, a fallacy which only further undermines your credibility. Anyone reading my comment knows that my "high school background" comment was a reference to Harry's claim that "anyone with a high school math background can see that relativity is mathematically inconsistent".

It is quite obvious to me that your reason for holding your physics beliefs is not because you are guided by evidence to your beliefs, but because you have already decided what you want to believe. This leads you to grasp every straw you can find, regardless of its merits, to support the position which you already had decided beforehand you wanted to hold. This became especially clear with the articles you cited on the purported invalidity of the Heisenberg uncertainty principle, all of which, except for the one in which the author himself suffered a misunderstanding, contradicted your position.

I have not investigated Hatch and Dingle's works, and so I cannot judge their merits. However, based on my understanding of special relativity and my own research in it, I think if their findings were so revolutionary, we would have heard of it by now. Until that happens, it is much more reasonable to work with the theory for which there are mountains of evidence.

Best,

Armin

Dear Marcel,

Thank you for your comments. As for scientific publication, yes that is a goal, but I omitted a lot of details that still need to be filled in.

Best wishes,

Armin

Dear Lloyd,

Thank you for your kind remarks.

Yes, as you can see, most essays here try to explain the relationship between physics and mathematics solely in terms of those fields (and perhaps some philosophy). But it is easy to ignore the fact that we do both activities in our minds, and therefore its role on the connection is less likely to be appreciated. My essay was in part an effort to bring attention to an important factor that might be easily overlooked.

Best wishes,

Armin

Dear Jon,

"Seth Lloyd has a view of the universe being a quantum computer, ..."

First, whenever you have someone proclaiming that the universe "is" a giant version of some contemporary piece of technology, you should always take that with (more than) a grain of salt . As you may know, a couple centuries ago, during the enlightenment, people proclaimed the universe to be a giant clock, and who knows what people will proclaim the universe is in a couple centuries. To me, the universe is just the universe.

I watched the short video and found Seth LLoyd to be a bit fast and loose with words (For example, he claimed that we already have quantum computers, but I am more skeptical (My view is informed by Scott Aaronson's blog posts on the subject matter).

"...which may be a little more appealing to you..."

No, you know (I think) enough about my ideas to understand that in my view our contemporary conception of the universe, which equates it with spacetime, is too undifferentiated.

If you want to talk just about spacetime, then I definitely disagree with him, because I conceive of quantum theory (well QM, QED, and EW interactions) as the physics of spacetime objects emerging out of areatime. If you already have spacetime to begin with (which is the reasonable interpretation of his use of "universe"), then the appropriate theory is Einstein's General Relativity, not quantum theory.

"Stephen Wolfram, Ed Fredkin, Jurgen Schmidhuber, and some other scientists talk about the universe being a (classical) computation, but these ideas aren't quite as accepted."

Do they give concrete examples for how some physical process can be reframed as a computation?

"From a purely theoretical perspective, no matter how far an uncomputable real number is specified, it will always be ambiguous, since there are an infinite number of real numbers that start with the specified sequence of numbers, and there is no way to refer to or conceive of one specific uncomputable real since they have an infinite amount of information that can't be compressed into a formula like those that represent/generate pi or e. If you don't think this point is relevant when it comes to physics and your ideas regarding ZFCD, then that's probably a slightly different discussion."

Yes, I interpreted you previous question as a physicist, from the point of view of mathematics I agree. Yes, I am uncertain about the relevance of AC. The only reason at this point that I chose ZFCD vs. ZFD (not to be confused with Zermelo-Fraenkel with axiom of Determinacy) is that ZFC is the standard set theory. This does not mean that subtle deep connections may not be there, in fact, more likely than not, I think there may well be.

"I don't think computation broadly defined pre-supposes time, although I do think it would imply a sequential relationship...which I think is slightly different."

You may be right. As mentioned, the notion of a physical process as a computation is not intuitive to me, so my intuitions are more likely to lead me astray than in other areas.

"I tried to raise the question that maybe some of the statistical laws of physics are actually modeling pseudorandom processes, as opposed to truly random processes. If the computation that the universe was doing was too complex, a statistical approach analogous to the PNT might be are only practical approach to making predictions. "

Can you tell me which statistical laws of physics you had in mind?

"I hope some of this made sense."

Rest assured that you did:)

Best,

Armin