Dear Roger,
I found your essay to be one of the rare ones having significant quality (among those I reviewed for now).
I agree about the difference of view of randomness between maths and physics, as I also expressed in my essay (that is, coexistence of possibilities with weights in maths, vs. realization of only one possibility in physics). I agree that randomness in physics does not make absolute sense as opposed to unpredictability, which may depend on the range of considerations to which the considered event may be related. However we do have a law of quantum physics that gives some probabilities quite successfully.
You wrote: "There are deterministic interpretations of quantum mechanics, so not everyone agrees with quantum randomness."
Indeed it is interesting to wonder what is behind quantum randomness, since randomness does not make absolute sense. The main interpretation of quantum physics that actually tries to be deterministic is Bohmian mechanics and other hidden variable theories, which I criticize (one of the reasons being that I see their claim of determinism as meaningless). As I explained in my essay, I see the source of quantum randomness as metaphysical, which, depending on circumstances, may either be the expression of free will by individual minds (diverging from the quantum probability law), or of a hidden, impersonal aspect of universal consciousness (usually obeying the quantum probability law, as far as "obeying a probability law" is relatively meaningful). So I disagree with the argument that "quantum mechanics adds only randomness to decision making processes, not freedom".
"Statisticians have taken the task of interpreting probability theory so it can be applied to the real world. Interpretations range from probability being a physical entity like temperature and called propensity, to being just a subjective measure of how persuaded you are of something. The major interpretations are the frequentist and Bayesian."
I see no sense of "interpreting probability theory" in the absolute. I can agree that it is an intrinsically unclear concept (a defect which can be seen as a quality, that is how I see mathematical probability laws as the perfect formalism by which mathematical laws of physics can leave a door open to the non-mathematical laws of free will), but I still consider it as a primitive concept in physics, i.e. I do not see a sense in trying to reduce it to something else. For example, frequentism suggests to set a limit to how far may frequency depart from average expectations. The fact is that probability laws induce values of the probabilities for different frequencies to occur in any given sample, so that frequencies away from the probable average just have a very low probability to occur, but a very low probability is not an impossibility, so that we still need an a priori concept of probability to interpret the frequentist interpretation of probability.
Instead, I see more sense to discuss more specifically about interpreting quantum theory: you wrote "A physicist might say that [some things] are random if current theory cannot predict them."
Sorry, I do not see things so vague as this. Lack of prediction does not mean randomness because randomness is a precise concept that refers to a specific probability law, which needs a theory to be specifically formulated. We are not in generic unknown universes with long series of successive speculative theories that might more or less be tried to fit, but in a specific universe where the specific probability law of quantum theory was found extremely successful. This law is the reference by which physicists can meaningfully speak about randomness, instead of simple ignorance. So, I consider the need to distinguish whether a considered probability law comes from quantum theory or not. And I actually do not see well where can serious probability laws come from, if not as particular cases of quantum probabilities. I mean, I see most probability laws that emerge in practice in this world, as actually particular cases of quantum probabilities, even if they do not directly look like such. I commented this further in my page on the problems with Bohmian mechanics.
"To the patient, a cancer seems like bad luck whether it came from genes, smoking, or mutation. All three factors presumably have completely causal explanations, if cancer were better understood. The reason they call the mutation bad luck is just that they have no good way of modeling it as a causal factor."
For most phenomena where microscopic physics or the butterfly effect is involved, deterministic explanations cannot exist in a non-deterministic world (do you mean anything else by "causal" ?), but in guise of causality, only probabilistic laws ultimately apply. In particular, I see mutations as directly coming from quantum randomness. Smoking is a conscious choice, which comes from free will, which uses quantum "randomness" as a medium of expression but departs from quantum probability laws. But I guess that genetic mutations are not subject to psychological bias away from quantum probabilities, since otherwise I would not understand why it is not also manifested in the form of intelligent design.
"A common view is that quantum mechanics describes true randomness, while all the other sciences have a pseudo randomness that results from complexity or uncertainty. Physics also has pseudo randomness, such as in the study of statistical mechanics."
There are good reasons to consider quantum randomness as fundamental in physics, since alternatives (hidden variable theories) have lots of troubles as I said. Statistical mechanics and other sciences express a practical randomness, and can happily remain agnostic about the nature of their randomness. This practical randomness can be logically compatible with explanations by pseudo randomness from complexity or uncertainty, but there is no point for positively claiming it to be pseudo randomness, so this is usually not claimed either ; and most cases, especially statistical mechanics, actually have their source or randomness in quantum randomness, so that this conceivable difference between the nature of randomness of diverse sciences is usually not real.
"Regardless of the branch of science, attributing causality to randomness does not explain it at all. Randomness is just a buzz word to replace an admission that the some causal factors are unidentified."
Hilbert spaces, at the basis of the formulation of quantum physics, are not a buzz word, but a very high, rigorous and elegant mathematical concept that is amazingly effective in explaining physical observations with a high degree of accuracy. Quantum probabilities are their direct logical consequences. Quantum theory is a well identified theory, leading to well identified mathematical evidences that any try to "explain" its randomness by hidden variables would lead to a lot of troubles. And, as I showed in my essay, we can find a well-identified logical connection between the paradoxes of quantum theory and what, in my view, can be naturally expected for the universe to allow the presence of consciousness, which I see as strongly suggesting a well-identified source for quantum randomness which is a metaphysical one (consciousness, which is non-physical and non-mathematical).
"When a uranium atom decays, it is often assumed that it is a spontaneous event that is independent of the history of the atom. But we have no proof of that. We can collect a large number of apparently similar atoms, and notice that they decay at much different times, but we do not truly know that the atoms are all in the same quantum state."
We do know it, not just assume, as the proof is found from the mathematical formalism of quantum physics: a system of rigorously identical particles behaves very differently in how they mix together (by the interferences between them), than would behave a system of different particles with only superficially similar properties that we would fail to distinguish by measurement.
I did not notice that "Physicists believe that causality is fundamental to physics, and philosophers reject that idea." On the contrary, I once saw a philosophical work followed in academic philosophy, which insists that causality is fundamental in physics, while I notice that the notion of causality is only an indirect one in the fundamental equations of physics which are time-symmetric (and, in many cases do not even fundamentally distinguish time from space), while causality is a philosophical concept which is fundamentally bound to the time dimension and is time-asymmetric; still I see a sort of concept of causality as important to describe the quality of quantum physics as "causal" in the sense of letting the probabilities of measurement results only depend on a "finite number of causes" from the state of what is there at a previous time. Otherwise, as people, physicists may talk about "causality" but I would say this is only a philosophical interpretation from outside the equations of physics in a strict sense.
It seems to me that the Bayesian/psi-epistemic view is actually a position of agnosticism with respect to the idea of hidden causes, and thus a non-interpretation, so I would not really classify this in the list of views, except if taken as a case of logical positivism.
You point out the diversity of views on quantum physics, but these divergences are internal to the small part of the community of physicists who care, and thus does not point to any idea of opposition between maths and physics which you took as the main topic of your essay, or does it ? (We might also find "controversies" in the foundations of maths if we search well)
I agree about the difference between math and physics on the role of infinity. I also commented this in my essay : how quantum physics, despite being a mathematical theory formally involving infinite sets, avoids any effective dependence with respect to actual infinity and its undecidabilities that can be found in other branches of mathematics.
We have the same interests in the foundations of mathematics and its paradoxes :)
Since the fine structure constant is only the macroscopic limit of the effective value of the charge, while some possible determination, if it exists, would rather come from the Planck scale, there is no surprise that it is not algebraic ; its deduction from the renormalization process would make it algorithmic but not algebraic (even though it would be very hard to write down the relevant algorithm). But I'm not even convinced that mathematicians would be troubled at the idea that physical constants are just arbitrary real numbers without any specific definitions (and thus possibly escaping countable models of set theory). After all, that would leave the axiomatic system of physical laws incomplete, just in a similar way as axiomatic systems of arithmetic or set theory are. Do you know, the most naturally "constructed" countable models of ZF also have wrong arithmetical properties (with a nonstandard model of arithmetic). The failure of countable models of ZF to contain the correct values of physical constants would be quite a similar phenomenon.
I also (moderately but for many practical purposes of truth research) agree with logical positivism. Your position seems unclear as you seem to promote logical positivism at the end, yet you argued about the impurity of the concept of randomness (the difference between pure randomness and pseudo-randomness) at the beginning, a difference which logical positivism would dismiss as senseless metaphysics. Some physicists do argue about it, as they look for interpretations (other than the logical positivist non-interpretation of Copenhagen and Bayesianism), but they are a minority. Having pointed them out, suggests that you give it importance yourself, in seeming contradiction with your final expression of support for logical positivism.
I guess your 2012 essay should be interesting as well, but for now I wish to focus on reviewing other essays of this year :-/