> Which of the following is not a random number, but a useful mathematical constant?
I just said that it's not the numbers which are random, but rather the process which produced them. Any of those numbers could be produced by a random process, or a universal constant useful in some context. The output of a random process could (with infinitesimal probability) be equal to PI to an arbitrary number of digits, and still be random; the output of a deterministic process with known initial conditions could be completely unrecognizable without actually repeating the process, and pass all the statistical heuristics for randomness, and yet the process would still not be random.
> The larger point is that any deterministic system is "without time" in this sense, and that contrary to commonly held views Classical Mechanics is not deterministic, and QM can be deterministic.
I'm not arguing with that. True randomness does not require non-determinism, provided no one can know all the initial conditions. The difference is that, given non-determinism, a process can be random even _with_ complete knowledge of the initial conditions.
>> the question is whether the rules of the system permit _anyone_ to predict the next number with 100% certainty, given all the information which it is possible for them to know
> This is perhaps the closest to a meaningful definition of "random," but it depends upon the validity of the uncertainty principle, and who qualifies as _anyone_. It is in fact what I said earlier, that QM is how we define "random", not that QM *is* random.
First, it doesn't depend on QM at all, or the uncertainty principle. QM (whether non-local or non-deterministic) is one system which defines cases where the number cannot be predicted, but it is hardly the only one. Classical mechanics also works, if the initial conditions are unknowable. Second, "anyone" simply means "anyone"; there are no qualifications.
> In particular since physical law can never be proven complete and correct, "random" can never be proven to be correctly defined.
Whether a physical process is random does depend on the physics of the process. If we're wrong about the physical laws, we may also be wrong about whether the process is random.
That doesn't really affect the definition of "random" so much as the system to which the definition is applied: what the system permits one to know, and whether that knowledge is sufficient to predict the next result given the system's rules.