I would say that the solution to sqrt(2) is the expansion (to however many finite number of iterations you run it for) of the bits produced by the Turing machin

I would say that the solution to sqrt(2) is the expansion (to however many finite number of iterations you run it for) of the bits produced by the Turing machine that computes sqrt(2). Sqrt(2) is a computable number, so it exists both in my system and in yours.