Now about your page 7.
You wrote : "the answer to Wigner's question is that mathematics is reasonably effective in physics, which is to say that, where ever it is effective, there is reason for it". This claims comes as logically deduced from your philosophy, in the traditional way of philosophers, that is, as a pure theoretical (but not even so carefully logical) blind guess, that proudly comes as self-sufficient reasoning with no need to check it against any review of how things were observed to be : here, the measure of how mathematics was found to be effective in reality. Indeed, where is your review of these observations ? Instead of observing or checking anything, you satisfy yourself to prophesy: "There will never be discovered a mathematical object whose study can render unnecessary the experimental study of nature". Still you are coming with philosophical principles whose study seem to suffice for you to deduce in the abstract how effective should math be to the study of nature. Just like usual (bad) philosophers, your confidence in your principles makes you see unnecessary not only to abstain concluding and humbly consider to wait and see what future discoveries may show (maybe giving your claims a status of falsifiable predictions to be tested and eventually refuted), but you also see it unnecessary to check their compatibility with the present record of the state of things actually found by modern science: whether the effectiveness of mathematics that was actually "observed" by the development of modern science fits these expectations of effectiveness you are presenting. Does the self-evidence of your principles and prophecies carry sufficient logical or metaphysical reliability to give you such a faith in their truth that this confidence can legitimately supersede for a rational mind, any concern for experimental check, any verification against any past or future research, such as a search for a counter example to your claims (some mathematical object that might be successful enough to make some experiments unnecessary) ?
Actually, theoretical physics happened to be so successful that, well of course there is still some place for experiments, but this place is now quite reduced either to very complex (macroscopic) systems (where computations would be too complex for our supercomputers, so that the studied properties are only consequences of known laws in principle but not in practically computable ways), or to the case of extreme conditions that are very hard to explore (with particle accelerators, some subtle aspects of astronomy and cosmology to analyze the properties of dark matter... not mentioning the mind/brain interaction that I expect, as I explained in my essay, to involve subtle processes, linked to the nature of quantum measurement, beyond established mathematical physics, that have not yet been well investigated); in many other cases, such as gravitation, theory suffices. Fortunately indeed we do not need to send hundreds of probes in space all over again for each space exploration mission until finding out by chance which trajectory may actually lead to the desired destination.
After this, in guise of illustration of your belief, you give examples from modern physics, so as to make it look as if your principles were not pure abstract principles disconnected from modern science, but compatible with it, or even supported by it. I am deeply amazed at what a badly distorted report you manage to make of how things go in modern physics, so as to make it look as if it supports your philosophy. This is so ridiculous, and just the same style of absurd distortions and misinterpretations of modern physics as what is usual from the part of cranks who claim to refute Special Relativity by criticizing Einstein's book and finding a "new explanation" for the Michelson-Morley experiment (or rather an old one, always the same : a "mechanically explained" Lorentz contraction of moving things and absolute slowdown of clocks with respect to an absolutely still ether), or who similarly "explain" quantum physics by classical waves, or who claim there must be a local realistic deterministic explanation of quantum randomness because they believe that any randomness must hide such a determination (assuming that physicists just did not try to look for one but lazily and dogmatically preferred to "shut up and calculate") and they did not learn about the logical and experimental arguments against it.
You see "a large degree of arbitrariness" in mathematical physics. Of course there is some arbitrariness in the list of particles in the Standard Model and the values of all constants there as we know them (about 20), but this is nevertheless often qualified by many physicists as quite elegant as compared to the amount of observations this theory explains, far from "a large degree of arbitrariness" as you say. The Higgs boson, like many other particles (such as antiparticles), was predicted before being observed.
You wrote "In most cases the equation describing the law could be complicated by the addition of extra terms, consistent with the symmetries and principles expressed, whose effects are merely too small to measure [by] given state of the art technology. These "correction terms" may be ignored because they don't measurably affect the predictions, but only complicate the analysis". Sorry I do not see well what kind of example you are thinking about here.
On the contrary, I see in most cases that such "correction terms" you mention, such as "correcting" classical mechanics by Special Relativity and then General Relativity, indeed complicate the work of numerical computation of results with "additional terms" from the viewpoint of numerical analysis, however the corresponding theoretical picture is simplified instead. What they actually reflect is a more unified, simple and elegant theory. They are not arbitrarily added for complications, but they come as more or less theoretical necessities. Indeed I explained in my web site how, for example, Special Relativity is simpler as a theory than Galilean space-time. Consequently, Relativistic mechanics is also simpler than classical mechanics, as it comes from a simple principle (the least action principle, more elegantly applied to the space-time of Minkowski than it was to the Galilean space-time) and unifies all conserved quantities (mass, energy, momentum, angular momentum, center of mass) in a unique mathematical object (an antisymmetric tensor in the 5-dimensional vector space associated with the 4-dimensional affine space-time). General Relativity is very elegant too, should I develop this point ?
"every one of the famous equations we use is merely the simplest of a bundle of possible forms of the laws". Please list 10 possible non-equivalent theories of speed and movement that behave approximately the same in many practical cases of experiments, and among which Galilean space-time and Special relativity are non-remarkable particular possibilities. If for any reason you do not like this example, please do a similar thing for other problems such as electromagnetism, gravitation, quantum physics or whatever. You say the only advantage of admitted versions is their simplicity just because it is convenient for us ? But how to explain that in so many cases of theories, among all possible alternatives, there happens to be one that is both extremely simpler than any alternative that can be thought of, and extremely well-verified by observation, with no need of correction by any alternative (no arbitrary complication that we may naturally think of for the sake of complication rather than for the sake of elegance, ever turns out to be better verified, as far as I know) ? Or do you claim this is not the case ?
So I'm sorry but this is bullshit : "Often we assert that the right one is the simplest, evoking a necessarily mystical faith in "the simplicity of nature." The problem is that it never turns out to be the case that the simplest version of a law is the right one". First, we do not assert by faith that the right one is the simplest we think of. Instead, we conclude it as we verified it by observation. Second, when we had a seemingly simple equation which worked (such as Newton's law of gravitation), the new one that turns out to be more correct to replace it (General Relativity), turns out to be conceptually simpler (more elegant) than the first one; only it was not thought of at first because it is a more subtle, sublime kind of mathematics that requires some familiarity with high mathematics to be grasped. Finally thus, it remains true that the right one is the simplest, except only that we did not know at first the theory which turned out to be both simplest and better verified.
You gave another example : "Maxwell's equations received corrections that describe light scattering from light-a quantum effect that could have been modelled-but never anticipated-by Maxwell". This example is supposed to illustrate your claim of possible complications and "under-determination" of laws among multiple possibilities. It doesn't. The truth is that these corrections by light scattering from light are not an option among alternative possibilities, but a logically necessary consequence of inserting electromagnetism in the framework of quantum field theory. Of course Maxwell could never anticipate it because quantum theory was not known at that time, but this impossibility to anticipate it before the birth of quantum physics is completely irrelevant here. It does not change the fact that this effect is a necessary consequence of quantum physics. This quantum physics had to be introduced for very different (and necessary) reasons than looking for corrections to electromagnetism. There is no logical possibility for this "correction" of electromagnetism to be not there with its exact necessary amplitude as soon as we live in a quantum world with all its other, more direct consequences (such as the stability of atoms). There is no trace of any "radical under-determinacy" here. To take a related example, consider the measure of the anomalous magnetic moment of the electron, where the calculation as logically determined from theory was verified by observation to an amazing degree of accuracy. We did not need to adjust anything in the theory to put it in agreement with this observation.
