Misinterpreting Reality: Confusing Mathematics for Physics by Robert H McEachern

Dear Robert McEachern,

I think this part of your abstract is deeply mistaken:

"Equations contain very little information. This fact is what makes it possible to symbolically represent them, in a computer memory, by a very small number of bits. As a direct result of this fact, we can conclude, contrary to the fervent belief of most physicists, that equations cannot describe anything other than the most trivial physical phenomenon; those nearly devoid of all information."

This would be exactly true if the terms of an equation did not vary in their numerical values. You have treated a bit of algebra as if it was a single set of numbers. But because the equation is a kind of shorthand, it actually contains a lot of information. When you add this information up, you need to take into account every possible combination of numbers that equation can represent. This shows the limitations of looking at things that way - trying to assess the information content of something is sometimes a futile game, because information is a human idea that is more mental than fundamental.

I hope this information is useful...

Anonymous,

"information is a human idea that is more mental than fundamental." ?? I would like to support Robert.

Moreover there is definitely an objective difference between past and future data. The distinction between them is not a futile game but overdue. See also here and the subsequent correction.

Eckard

Rob McEachern,

"future signals can, in fact, be analyzed, if they can be perfectly predicted" ?? I see this a typical unnecessarily misleading confusion. In reality there is nothing that can be predicted for sure. You are referring to a model of reality, and in this case the decisive border between the still open future and the finally determined past is shifted. A predicted signal is not really a future signal. Analyzing a predicted signal is equivalent to analyzing the past.

What about your filter bank, you wrote: "I mean a mathematical filter; the equation for a Fourier Transform can, and usually is, interpreted as a superposition. But by simply reordering the computations, it can easily be interpreted as a filter-bank."

The FT of a signal yields a complex spectral decomposition, also called a superposition of (by the way positive and negative) frequencies. In order to interpret this complex spectrum as the output of a filter bank, you must omit some parts of it. Perhaps you did not learn the fundamentals from the very beginning. EEs like me are e.g. aware of the non-causality of so called ideal filters. We would also not question the principle of superposition in general. FT and CR are integral transformations, and the symbol for integration is a "s". Integration means limit of an infinite superposition. DFT and DCT even more obviously summations. I do not understand your intention when you are trying to "explain" for instance "the integration as a lowpass filter".

My intention is to demonstrate foundational shortcomings in the theory of signal processing and in QM. I infer from the restriction in reality to only positive elapsed time and only positive frequency that the shift properties must also be considered restricted. A distance cannot be shifted into negativity. Moreover, representation of a function of positive real values includes after FT in complex domain three redundant copies. They must not be carelessly omitted but correctly interpreted.

I hope you got this and will support me against those like Feynman, Schulman, and Ken Wharton.

Eckard

An equation often contains information about a very wide range of auxillary conditions. In one way of looking at it, that's a lot of information. One of the things we find beautiful in physics is that this high information content can be expressed in such an elegant and economical way.

The amount of information depends on how you look at it, because information is a mental idea, not fundamental to the universe. When we get over the novelty of computing, it will go down in importance. Your statement that "equations cannot describe anything other than the most trivial physical phenomenon" attempts to diminish equations, and physics itself. But you may have stumbled onto a way of showing that information is a thin concept, because of the nonsense that comes out of your approach to it.

Eckard,

"A predicted signal is not really a future signal. Analyzing a predicted signal is equivalent to analyzing the past." True, but irrelevant. Comparing a predicted signal to a "future signal", is very different from comparing it to a past signal. Making such predictions and then comparing them to future observations is a central part of modern physics.

"In order to interpret this complex spectrum as the output of a filter bank, you must omit some parts of it. Perhaps you did not learn the fundamentals from the very beginning. EEs like me are e.g. aware of the non-causality of so called ideal filters." Nothing needs to be omitted. It is a mathematical IDENTITY. Perhaps you did not learn that Fourier Transforms are non-causal representations.

"My intention is to demonstrate foundational shortcomings in the theory of signal processing..." What "shortcomings" are you talking about? How do you propose to algebraically describe the directions of anti-parallel vectors, if not via the use of a negative number for one and a positive number for the other?

"A distance cannot be shifted into negativity." True, but "distance" is not "location". Distance is the magnitude of a vector, and so it is always positive. But location is the vector itself, not just its magnitude.

Rob McEachern

The "nonsense" was only where you say that the physical phenomena we have described with equations are the most trivial ones. Gravity equations, for instance, describe what the relationship between the terms will be in a very wide range of situations, so telling us a lot about what gravity does, and therefore (indirectly) telling us something about many "auxillary conditions". If this is short on "information", then it contains a lot of something else that is important, which you haven't defined, but which makes the phenomenon better than trivial.

Rob Mc Eachern,

Comparing predictions to future observations can only be performed after the observation, i.e. in what is then already the past. The future cannot be observed. My Fig. 1 is correct.

So called ideal filters are non-causal. Why? Do not confuse mathematics with reality. I reiterate: "In order to interpret this complex spectrum as the output of a filter bank, you must omit some parts of it." Even Feynman did not devote the due attention to the steps from reality to the mathematical model and return. My Fig. 1 shows necessary steps before FT: first abstraction and then analytic continuation (Heaviside's trick).

Is the FT a mathematical IDENTITY that relates a f(t larger than zero) to a complex F(omega)? No. The equation for the kernel of FT is still an identity: exp(iwt) = cos(wt) i sin(wt) with w=omega. However, it adds an imaginary part with in principle arbitrarily chosen sign to a real also in the sense of realistic real part. You can alternatively consider this relationship an omission: 2cos(wt) = 2ch(iwt) = exp(iwt) exp(-iwt) mutates into a complex exp-function by omission of either the clockwise or the anticlockwise rotating phasor.

Feynman correctly wrote in vol.1, 23-1 (my reinterpretation from a translation into German): "We will speak of the "force" F_0 exp(iwt). Of course, the true force is the real part of this expression." He merely ignored the steps that prepared f(t larger than zero) for integration from minus infinity to plus infinity - and arrived at backwards running time.

EEs like me love non-causal filters while being aware of them as dirty tricks. Do not confuse mathematics with reality.

I wrote: "My intention is to demonstrate foundational shortcomings in the theory of signal processing..." You: "What "shortcomings" are you talking about?"

A lot of: For instance, the failure of spectrograms to efficiently and accurately mimic the function of cochlea. If the cochlea did perform a complex analysis then the one-way rectification by inner hair cells were impossible.

It is also not true that one needs a complex cepstrum. Cosine transformation works well. Redundancies can be avoided.

Apparent symmetry in QM is still mysterious unless one obeys the consequences of my Figs. 1 and 2. ...

You: "How do you propose to algebraically describe the directions of anti-parallel vectors, if not via the use of a negative number for one and a positive number for the other?"

I do not deny the benefits of using negative and complex numbers. However, there are quantities like absolute temperature, pressure, probability, frequency, wave number, distance, and elapsed time that are always positive.

I: "A distance cannot be shifted into negativity." You: "True, but "distance" is not "location". Distance is the magnitude of a vector, and so it is always positive. But location is the vector itself, not just its magnitude."

Again: Do not confuse mathematics with reality. A vector may be useful to mathematically describe reality. It is not reality.

Eckard

The fact that the law tells us so much with so little of what is defined as "information" shows how thin the concept of information is.

You say this point makes what the law describes trivial, I say it makes the idea of information trivial (certainly compared with anything fundamental). Let's leave it there, each to his own.

Bob,

In spite of the quality of a few of the top esssays, yours should have been at or near the very top. It's been a pleasure reading your essay and your comments, and I hope that you participate again.

Edwin Eugene Klingman