Dear Marc,
It is interesting to see that while you push the logical consequences of the MUH to the extreme, you still accept them, while I see them as a proof by absurdity against this hypothesis. Also I found it interesting that you have some common ideas with my view (see my essay): seeing consciousness as playing a fundamental role in giving mathematical universes a physical existence, and as you say "when we succeed at something in our universe (...), we do not change the Maxiverse in any way, we merely "visit" preexisting mathematical structures that have always been part of the Maxiverse." However we differ by the fact you consider consciousness as a particular mathematical structure, while I see it as non-mathematical.
Here are some remarks on the details:
You mentioned Occam's razor, and that the MUH is basically unfalsifiable (though you mentioned changing your mind about the latter in your comments). I consider both concepts of Occam's razor and falsifiability as different aspects of the more precise and general condition of quality for a scientific theory, as I mentioned in my essay: to give a probability law on observables which minimizes their entropy, i.e. presents the observed data in their most compressed form, where the size of the compression algorithm (or equivalently, the algorithm of probability law) is itself counted as added to the size of compressed data. Here, Occam's razor is about insisting on the need of simplicity of the algorithm.
The MUH is relatively unfalsifiable in the sense that it somehow gives a non-zero probability to every possibility, thus preventing the risk for an observation to contradict the theory : an observation which a theory excludes would mean the occurrence of a finite piece of data with infinite entropy as interpreted by the theory (theoretical probability 0 implies entropy = - ln 0 = infinity if it occurs). However, that a theory is prevented against infinite entropy, does not mean that it reaches a minimal entropy in comparison with other theories, which leaves the possibility to be relatively refuted by finding another theory doing a better job at explaining (compressing) the data, with a more precise and verified probability law. In any case, I consider that trying to avoid the issue of specifying probabilities by admitting the existence of all possibilities and qualifying the whole probability issue as a kind of mystery waiting for future elucidation, is not a very good theoretical job.
On this topic, you wrote "If you want to explain why one or only a few universes exist, you must specify the precise laws they obey and their initial conditions (at least). You must also specify and justify the rules which select these universes to be real while relegating all other possibilities to the dustbin of existence. Specifying the initial conditions alone might necessitate a mind boggling amount of information. On the other hand, to describe completely the Level IV multiverse, one short sentence is enough".
Later, you wrote: "Another argument against the Maxiverse hypothesis (in fact, against any theory which incorporates seriously the notion of a multiverse) is the belief that it critically undermines the future of theoretical physics.".
It seems to be the same character of the MUH that you first present as a quality (of satisfying Occam's razor) and then as a defect, isn't it ? I see the second view as quite amusing, as if, by principle, the Universe ought to have been well-designed for the purpose of giving jobs to physicists :) That reminds me the attitude of some climate-skeptics, which look as if the physical properties of the atmosphere had to be well-designed by God to ensure that free market structures will remain the best solution to all problems (the famous Invisible Hand) and thus for letting libertarians always remain the good guys in their defense of liberties.
In fact, a more detailed analysis of this question would let both alternatives roughly equivalent: given a piece of data that looks random as we could not find an explanation (compression), both hypothesis "It is really random" and "It only looks random but has a hidden pattern yet to be discovered" are as bad as each other at the job of actually compressing that piece of data. Chaitin's theorem ensures that the second hypothesis remains irrefutable even if it is false. However, the first hypothesis is falsifiable (by the act of finding an explanation), so that persisting failures to find any pattern (time passes without any discovery of explanation) progressively leans to the first hypothesis, while discoveries of patterns (traces of design, even if not well understood yet, as expressed in the essay of A&L.Burov) leans to the second hypothesis (if we find some patterns then there should be some metaphysical reasons for them).
You wrote : "I do not think it's possible to imagine an abstract structure which could not, in some way, be described by mathematics" I think there is, that is called feelings or qualia, discussed by the famous hard problem of consciousness. For example, what is the sensation of the red color ? It is not expressible as a mathematical structure. The physical object of red light can be described by mathematics; the sensation of the red color can't.
"Moravec explains that we observe that our universe stays lawful and predictable, even if there are many scenarios where it doesn't, because in these scenarios, our consciousness immediately ceases to exist". This reasoning is not applicable without probabilistic assumptions that beg for justifications. Namely, to imagine that just because a regular law was needed to reach some result, it will therefore continue to apply, means that just because some regularity happened, it will be more likely to happen again. But where does that law itself come from ? If I play heads and tails and happen to get 10 heads successively, will it make it more likely that I still get heads next times ? If I won at Lotto first, would it make it more likely that I will win again another time ? Is the Born rule of quantum probabilities, made more likely to be obeyed by future observations, by the fact it seemed to be obeyed in past observations ? And what other conclusion should we draw instead if it didn't seem so ?
"I expect to wake up in my bed and lead a more or less ordinary day, which must indicate that somehow (despite the measure problem), my F-clones which correspond to these ordinary scenarios greatly outnumber the other ones."
"Outnumber" : is it a matter of number ? It is well-known among specialists of the Many-worlds interpretation, such as David Wallace, that the Born rule cannot be justified as a measure of the ratio between numbers of distinct worlds with the different given states of a subsystem, because, first, there is no such a thing as a number of possibilities; second, even if there was, anyway such quantities do not fit.
Last autumn I wrote a description of the Many-worlds interpretation, with what I see as its necessary assumptions and consequences pushed to their extreme, which have a lot in common with your own description of the MUH. I do not subscribe to this interpretation but I think it is important to examine it, as part of the understanding of the one I support (mind makes collapse), because I include this many-world interpretation as part of the picture : it describes what happens in the absence conscious observation. And what remains of the physical universe when removing conscious observers, is precisely a purely mathematical world, which is ontologically equivalent to the MUH or Maxiverse (with the "small" difference that in the many-worlds interpretation of QM, the physical laws with the values of physical constants are fixed).
