Dear Bill,
Having just read your essay, I really liked your introduction involving the deeper questions one should think to ask about unicorns (what a wonderful way to teach a child to think more deeply about things in the world, let alone adults) and was struck by how methodically you covered all the areas relevant to your argument.
Concerning your comments on probability, if you permit me, I would like to share some of my own ideas and views: I believe that, ultimately, probability is a measure of "coming into existence" relative to other possibilities. This "coming into existence" could be the outcome of a single event or many (i.e. a frequency) or, if one wishes to treat probability epistemically, the "coming into existence" of certainty of belief.
The problem you pointed out with probability, namely "What kind of mathematical axiom can distinguish the probabilities from the rest? It seems that a probability interpretation must be added separately, and lies above and beyond the definition of the mathematical structure." stems in my opinion from the fact that Kolmogorov's axioms do not distinguish this special kind of measure from other more ordinary ones, such as, say, length and mass.
In fact, I believe it cannot do this because denoting "coming into existence" is not a feature of the measure but of the sets on which the measure is taken, so in reality the problem manifests a limitation in the current foundations of mathematics, namely ZFC set theory.
I am working on developing a foundation in which such a distinction naturally appears. The key, I believe, is to extend classical logic to what is called free logic, which permits the definition of what is called an outer domain. Sets in the outer domain fail to satisfy the criterion of "existence" in the domain of discourse, which is now called the inner domain.
I believe that probability is naturally a measure over sets in the outer domain because the sets it measures cannot already "exist" in order for the interpretation of probability to be applied to that measure. If I am right, then this would allow for a natural means of formally incorporating the interpretation of probability, at least in the sense that it is distinct from other kinds of measures, into mathematics.
I am not sure how this would impact your argument against the universe being a mathematical system, but I hope that you find these ideas at least interesting.
Best wishes,
Armin
PS. I read the sample chapter of your "Relativity Made Real" book and found it to be very lucidly written. Given that you have written an entire book on relativity, I would very much like to know how you would respond if someone asked "what does it mean that for an object traveling at c, a zero proper time implies that a hypothetical observer at rest with respect to that object would measure his own duration of existence in spacetime to be precisely zero?"