Eckard,
I enjoyed your essay.
A few comments:
"There is a decisive advantage of digital over analog technology: Digital signals may cope with the noise-caused loss of decidability." Any actual advantage comes from choosing to "represent" only discrete symbols, like the letters of an alphabet, rather than from merely representing an analog signal via digitized samples taken from the analog signal. In other words, it is what is being represented (a discrete stream of alphabetic symbols - the most familiar being a two symbol alphabet - a bit), rather than how it is being represented (either analog or digital), that matters, when attempting to cope with any impairment, such as noise, distortion, or interference.
"However, since the laws of nature were abstracted from reality, they are no longer temporally or locally bound to concrete points on the actual scales of time or space." Too many physicists have lost sight of the fact, that the laws have only been abstracted from a small temporal fraction of reality, and from only a small spatial fraction, as well. That fact has a direct impact upon the finite information content of both the observations themselves and the laws being abstracted from them. Such laws, with only a finite (and very small) information content, can never completely represent any infinite reality, or even any finite reality, that happens to be greater than the fraction that has actually been observed. Assuming (as much of mathematical physics does) that everything that has never been observed, is going to behave in precisely the manner as everything that has been observed, is seldom a very good assumption.
"So far it is reasonable practice to tolerate so called non-causalities for instance in the current theory of signal processing for the sake of elegant calculability." It is not necessary to tolerate it - the problem you are addressing can be (and always is, in any properly functioning system) entirely avoided, by simply introducing enough delay into the processing, that everything that is being processed, already exists in the past (via a delay sampling buffer), so the future has no relevance to the only thing that is actually being processed (the previously buffered-up, past signal).
"Notice: Expansion of mathematics at will cannot expand nature... The practice to freely define axioms more or less at will... It opened the door for a considerable expansion of mathematical theories." I agree. In that regard, you may recall my 2015 essay, stating that it is precisely the different nature of their axioms, that distinguishes math from physics.
"For instance the human ear as a frequency analyzer... " Be careful. The auditory system does not analyze frequency, anymore than the visual system does. The bad assumption, that the perceptions of pitch and color, are being generated by any sort of frequency analysis, has confused scientists for generations; they are generated from ratios of amplitudes, which is why they are so insensitive to phase. I do not dispute that it may indeed seem "as if" the auditory system is analyzing frequency. I am only advising that one needs to "be careful" - there are other signal processing techniques that correlate much better with pitch perception, than any sort of frequency analysis does.
"Unfortunately, one cannot even prove the theory of quantum mechanics wrong..." Alas, it all depends on what the word "wrong" represents. It is easy to prove that QM is wrong, if it is supposed (AKA interpreted) to be describing the behavior of substances (analogous to a drug), rather than merely describing the behavior of a test (like a drug test) for the existence of some particular substance, at some particular place and some particular time. The point is, drug tests are known to exhibit "false positives"; QM only correctly predicts the likelihood that something will be detected, but makes no prediction at all, regarding the likelihood that what was in fact detected, was the same "something" that it was supposed to detect - such as actually being "up" when the detector mistaken called it "down". It is easy to show that such "false positives", occurring in the polarization tests associated with Bell's theorem, can reproduce the supposedly, impossible-to-produce-classically correlations. In short, the equations of QM do not represent (are being misinterpreted) what any of the well-known interpretations believe that they represent; The do not represent the behavior of any substance (like a photon or an electron). They only describe the behavior of a frequently "false positive" test for the substance.
Rob McEachern