Intuitionistic mathematics is basically always asking "show me the money", nothing is accepted before we build it. Intuitionistic math may be a much better foundation to foundational physics than classical math. http://math.andrej.com/wp-content/uploads/2014/03/real-world-realizability.pdf …
Yes, that is all largely uncomputable stuff, which means that it is not the language that reality is written in.
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Turing showed that a computational reality need not be "written" in any language (a language is a post hoc human interpretation) and so, by Occam's razor, it isn't. For Turing, there was no "true" math, only useful math (he rejected both classicism and intuitionism as "correct")
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Since the Turing Machine uses an infinite tape, it is not the best metaphor to capture computation. The set of computable functions is the set of mappings between finite integers.
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