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 …
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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The Turing machine doesn't use an infinite tape, only an unbounded one (it adds cells as necessary). Also, as Turing himself notes when he constructs it, it is not "the model of computation", just one arbitrary convenient construction allowed by the principles of computation.
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The problem with functions is that they are a constructed meaningful interpretation. And, as Turing showed, to get meaning out of computation you do not need to bake it in at the start. So functions add much – a human that creates the concept of functions – that is not required.
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