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> Any definition of explicit to make your proposition work is equivalent to “computable”. any decent formalism of mathematics is computable.

I am not sure that this is the case actually. Many results in mathematics only relate to the (non) existence of certain objects, without explicit recipes for finding (i.e., computing) them. And obviously philosophers started arguing whether an object that cannot be found actually exists or not :)

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My favourite example this is Mark Braverman's demonstration that all Quadratic Julia sets are computable (as in computable real numbers: meaning that you can produce arbitrary fine approximations).

However, as Mark notes, this proof proceeds by deriving 5 alternative programs to do the computation. For any given Julia set parameter 'c' (given as a computable real number) one of the 5 programs will compute the Julia set. Which one? Who knows. It depends discontinuously on the properties of 'c'.

Proving non existence uses some type of law of excluded middle. It’s a very concrete and computable style of argument, although its subjects may not be.

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