Dirac's prediction is from special relativity. It is already implied in E^2 = m^2c^4 (pc)^2. It's just that he took seriously the case of a particle with zero momentum and nonzero energy that relativists had dismissed as nonphysical. Symmetry demands it, however.

Not all theories are based on prior observations. Newton's dictum, hypotheses non fingo, favors correspondence of mathematical theory to physical result, and drove the whole of rational science for 300 years until scientists got into the bad habit of imposing metaphysical ideas on events they can't explain rationally.

Predicting error is useful for control systems, with arbitrary boundaries. For nature at its foundation, unboundedness is a primary property.

"It is already implied in E^2 = m^2c^4 (pc)^2." That does not even imply the existence of an electron.

I read Newton rather differently; hypotheses non fingo, simply means he did not care to speculate about the cause of gravity; "why" it was an inverse square. His theory was based on previous observations, both celestial and terrestrial.

Nature, as a whole, may be unbounded, but nothing within it appears to be.

Rob McEachern

You could be right. Maybe electron is only the name we give to a particle of fundamental negative charge. The equation, does, however imply symmetry, no matter what we call the particle.

You may also be right about Newton. If the cause of gravity is metaphysical, it will never have an explanation, so better not to deal with it as a scientific issue.

But your most interesting statement: "Nature, as a whole, may be unbounded, but nothing within it appears to be," is the crux of our dilemma. A physically real spacetime, vice a mathematical artifact we call spacetime, would solve the problem. I find myself in agreement with Petkov's essay.

Petkov stated that: "Determining which mathematical entities have counterparts in the external world is indeed quite challenging partly because it is a bit atypical task for physicists since it does not involve calculations."

I would argue that it is not "because it is a bit atypical", but because it is neither falsifiable nor verifiable; the problem is not that "it does not involve calculations", rather, it is that it does not involve observation - of the supposed cause for an observed effect.

Rob McEachern

It's verifiable, although not falsifiable.

This is the point of my essay section the correspondence principle and Popper falsifiability.

Continuous one to one correspondence without a sign change cannot be falsified. Continuous functions in Minkowski spacetime, however, are reversible. The challenge is to develop the mathematics which show where the function reverses, because the chirality is externally identical. I think the key to observation is in the quantum jump phenomenon.

Tom,

Your essay concludes with:

"If reality is objective and not observer-dependent and if theory is objectively correspondent to experimental results,..."

I would say that theory can only correspond subjectively to experimental results, since some experimental results (near the Shannon limit) can only be obtained subjectively; the subject/observer must know, a priori, what and how to observe, in order to observe anything at all, of any significance.

Rob McEachern

While that may be true, Rob, we are continually finding more sophisticated ways to observe and learn. Better mathematical models guide us, and we don't know where or if they limit out.

If we can keep extending domain and range by closed form, we can make exact -- not probabilistic --- measurements.

Rob,

Tom argued: "If reality is objective and not observer-dependent and if theory is objectively correspondent to experimental results,..."

I consider reality not observer-dependent but by definition the conjecture of something objective and the future nonetheless open to an incalculable plurality of what you are calling initial conditions. While I agree with most of your points, I would like you to comment on MP3 vs. Fourier transformation.

Eckard

Tom,

I'm not sure what you mean by an "exact measurement", except in the sense that I said - that the observer knows exactly what to measure, and what to ignore. But even then, probability can come into play, when you consider sequences of measurements, and determining/estimating the entire sequence "at once", rather than as individual measurements, rather like recognizing a word, rather than the individual letters (much less fonts) making up the word.

I think that we can indeed "keep extending domain and range by closed form", but the question remains, is the extended knowledge useful for understanding the physical world - is it about math, or physics?

Another issue is with "closed form". Wiki gives the following definition: "In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations."

While Physics may have chosen its domain to be so limited, that it can be fully characterized in a finite number of operations, I'm not so sure that the rest of reality can be. In particular, I do not think that an observer with "free-will" can be characterized in that way. That is the ultimate measurement problem. It reminds me of what an author once said about writing a book - a book is never finished, it is merely abandoned. An observer can choose to abandon additional measurement attempts, but I'm not so sure that there is any other way to render measurements, able to be completed, in a finite number of operations.

Rob McEachern

Eckard,

MP3 is a lossy, data compression technique, used to describe sequences of discrete audio samples, such that when the audio is reconstructed from the MP3 coding, the human auditory system is not likely to take much notice of the losses.

Fourier transforms are not generally "lossy", nor are they "tuned" to the representation of audio data.

There are Fourier transforms (continuous/continuous), Fourier series (continuous/discrete) and discrete Fourier transforms (discrete/discrete).

Both techniques are "after the fact", in that the data (at least a block of it), must already be "in hand", in order to compute the representation.

Rob McEachern

Robert,

Your posts to Tom and Eckard at 18:26 and 18:40 were helpful to me in getting a more general idea of your focus. I'll read you at times in various discussions but am ill equipped to follow many technicalities as you present them. Glad to have you state, more simply, what is the obvious to you.

So can we synopsize by saying that on the other side of the coin, even if we were to finitely determine a true objective reality, we could not know with any certainty that we had done so in all totality? That our free will and curiousity can both, make up things that are not, and disable us from recognizing the tangible? Not to interject, just looking on, :-) jrc

John,

I believe that an objective reality exists. I'm not so sure about it being finite, or at least having a finite information content; even if the cosmos consisted of just three "points", the ratio of distances between those points may be represented by some real number, with an infinite number of significant, incompressible digits. But more importantly, human observers can "behave" towards their own measurements and observations, in ways that are "symbolic", rather than physical; it is not the mass, or electric charge, or energy or momentum etc., of a printed text, that drives the behavior of a human capable of reading that text. In short, we "interpret" our measurements, and our theories predicting future measurements, just as we interpret texts. We believe that they are telling us something other, and more significant, than just the numerical value observed. Thus, we may as Tom has said, make "exact measurements", but that need not imply the existence of any exact or unique "interpretation". The reason we stop at red traffic lights rather than green ones, has little to do with physics. Unfortunately, the same can be said about most physical measurements - their values may be exact, but the meaning assigned to those values (it means reality is non-local!, it means there is no free-will!, etc.) is far from exact or certain.

Rob McEachern

Thanks Rob,

I appreciate the time and attention in your response. I'll 'watch and learn' if you don't mind, I'm going through some changes presently. jrc

Rob,

Why didn't you get my point? MP3 benefits from DCT which stands for discrete Cosine transformation in contrast to DFT where the F means Fourier.

You are quite right; the data on consideration must already be "in hand". Future data can definitely not be measured even if their sequence may be expected for "sure" like the Amen in church. They would belong to the realm of block-time which the physicists are still imagining to arrange even all future events along an a priori existing time scale from minus infinity to plus infinity. Heaviside managed to make FT applicable as if future date were available in advance.

Discrete representations including DCT are genuinely lossy in the sense their resolution depends on a chosen sample rate. DCT is moreover also lossy in the sense it avoids redundancy. Being a real valued Cosine instead of complex valued Fourier transformation, it avoids the mirror data that belong to Heaviside's trick of analytical continuation which is a precondition of any application of Fourier transformation on measured data, no matter whether a discrete or a continuous FT is used. This loss of unnecessary data does not mean a loss of original information but merely of the arbitrarily chosen and therefore irrelevant point of reference which is unavoidably implied in any conventional (event related) time scale. Our ear is not synchronized to midnight in Greenwich. The only objective point of temporal reference is the very moment, the now.

Decades ago I did not yet understand this. So I wrongly mistook the fact that the ear is highly phase deaf and asked myself: Why doesn't audition take advantage of so called phase information?

Fourier dealt with heat conduction within a ring. Obviously, such ring model doesn't fit to the world. Heaviside quasi squeezed the ring into two mirror-symmetrical halves, the past and its mirror picture as a substitute for the missing future. I see the Fourier-Heaviside formalism in signal processing and in physics as valuable but not always justified detours. DCT is superior to DFT.

Eckard

Eckard,

I do understand your point. But I do not see the point as being of particular significance.

For example, you said that "Heaviside managed to make FT applicable as if future date were available in advance." It has little to do with what Heaviside did. It is applicable to ALL data, past, present and future. But the data has to be "in hand". So how can data about the future be "in hand" before the future even exists? Easy - if the data is perfectly predictable. The future value of all constants are perfectly predictable. That is why physics is so concerned with "Conservation" laws. Periodic motions are also predictable. That is why physics is so concerned with periodic phenomenon. Such predictions enable integrals, like the Fourier integral, to be integrated over the "prediction in hand" (like Tom's closed-form model of the future), rather than any actual, observed data. It all works just fine, when applied to predictable behavior, like orbiting quantum particles and planets. But it does not work when applied to unpredictable behaviors, like the behaviors of observers with free-will. That is why QM has such big problems, when it claims that the wave functions ought to describe the observers, as well as the observed; the observed may have predictable futures, that can be integrated over, but the observers do not. It is easy to predict that future of Schrodinger's dead cat - it will remain dead. But it is not so easy to predict how an observer will behave when they observe the dead cat.

Coding techniques like MP3 and JPEG, do not just remove the "mirror data", they also eliminate information that the auditory and visual systems are highly insensitive to, even though they contain a great deal of information that a different type of detector, can easily exploit.

The auditory system is largely insensitive to phase information, because the system's detectors cannot respond fast enough to accurately measure it. Instead, it measures amplitude modulations, and transduces those into pitch (correlated with instantaneous frequency) modulations.

Neither the DCT nor the DFT is a good way to model sensory perception, because sensory systems are mostly sensitive to "report" only the modulations of the received signals, rather than the signals themselves. Unlike the case in physics, it is precisely the measurements that are NOT constant and NOT predicable, that conveys the information needed to keep the system "alive". That is why your visual system does not work like a spectrometer - the sun's spectrum does not change, on a human time-scale, hence, measuring it, over and over again, day after day, year after year, serves no useful survival benefit.

Rob McEachern

Rob,

Writing [FT] "is applicable to ALL data, past, present and future. But the data has to be in hand" you didn't get my point. In reality, future data are not yet available, not in hand. Tom's model is unreal. Shannon expressed my view when he distinguished between the unchangeable but known in principle past and the steerable but never completely predictable future. In contrast to Einstein and his followers, Shannon didn't disqualify himself by referring to the present in this context.

When Rao, Ahmed, and Natarajan invented DCT forty years ago, they were well advised to not reveal the fact that Fourier's ring corresponds to the metaphysical mistake of Parmenides. Perhaps they were not even aware of this background. Up to now, most experts tend to belittle cosine transformation as just a special case of the mandatory Fourier transformation.

I mentioned MP3 because this application of cosine transformation is very popular and related to obvious facts: Future sound cannot be heard in advance; Time scale of the ear is not synchronized; Complex analysis cannot be performed in cochlea where OHCs act like one-way rectifiers; FT based signal processing implies non-causalities; Etc.

In case of JPEG I don't consider this foundational background so easily to be seen. When I came up with my suggestion referring time to its natural zero, the very moment, my boss refused to judge it. He meant this is "sowas von fundamental" (utterly foundational). His boss wondered why I compared elapsed time with the likewise always positive radius.

Eckard

Eckard,

"Future sound cannot be heard in advance;" I agree. But the future value of a constant can be known. Some things can be reliably predicted. Those things are the domain of physics. Other things, like the behavior of an observer, cannot be reliably predicted. Those things are not the domain of physics. The problem is, many physicists believe that they are. Obviously observers must obey the laws of physics. But that is not quite the same as saying they do so in a predictable, deterministic way.

The auditory system does not use anything remotely like cosine transforms, anymore than it uses Fourier transforms. Hence, the characteristics of such transforms has little relevance to understanding the auditory system.

Rob McEachern

No Rob (1):

I maintain, the distinction between application of CT or FT is relevant, not although but because CT is a particular case of FT. Application of Fourier's complex transformation is based on adding an arbitrary chosen point of reference. Well, the complex-valued representation conveys the same information as does the original real-valued one. However, the unnatural arbitrary reference to the conventional zero doesn't just make calculations less efficient by adding unnecessary data. More importantly it violates causality. This parallels Einstein's Poincaré synchronization.

Eckard