Is there a demonstration that a physics based on the computables is more complex than a physics based on the reals?
This is a complicated question. In practice, it is difficult in this particular context to measure what we mean by more or less complicated. A Blum-Shub-Smale machine which is essentially the equivalent of a Turing machine but for real numbers can do anything a regular Turing machine can do. This would suggest that physics based on the real is in general capable of doing more. But in terms of describing rules, it seems that physics based on the reals is simpler. For example, trying to talk about points in space is a lot easier when one can have any real coordinate rather than any computable coordinate. If one wants to prove something about some sort of space that only has computable coordinates the easiest thing is generally to embed it in the corresponding real manifold or the like.
This is a complicated question. In practice, it is difficult in this particular context to measure what we mean by more or less complicated. A Blum-Shub-Smale machine which is essentially the equivalent of a Turing machine but for real numbers can do anything a regular Turing machine can do. This would suggest that physics based on the real is in general capable of doing more. But in terms of describing rules, it seems that physics based on the reals is simpler. For example, trying to talk about points in space is a lot easier when one can have any real coordinate rather than any computable coordinate. If one wants to prove something about some sort of space that only has computable coordinates the easiest thing is generally to embed it in the corresponding real manifold or the like.