It seems like your conception is isomorphic to mine to the extent that the ultimate ensemble can be considered as a set of counterfactual (with respect to local physical law) universes. You'll see below that I brought up Zeno machines, and noted that they seem to follow this paradigm. Does this seem accurate?

The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.

Even the most abstract symbol manipulation can turn out to be functionally equivalent to a more concrete mode of reasoning and hence be quite useful. It is still an exploration of some sort of algorithmic reasoning, and coming from the view that mathematics is something like the art of predicting the outcome of various manipulations of structured data, such abstract games are not totally invalid.

My intuition is currently along the lines of thinking that uncomputable mathematics is disguised computable math, since the manipulation and creation of the system itself is computable (again, given the assumption that the human mind is computable), and so it is perfectly valid but misinterpreted in some sense.

I feel that I should say that my primary motivation for this line of thought is a desire to think about how a machine might create mathematics from scratch. I've started reading Simon Colton's work, it seems quite interesting. Perhaps you could offer some insight into this sort of issue?