Put another way, it would be really bonkers if the scientific knowledge that has enabled us to do so many practical things, like create tiny devices like the one I’m using to tap out a message, was somehow totally wrong.
The space for functioning systems to co-exist within a wrong underlying model is huge. A Turing machine does not care about mathematics or logic. It just matches a sequence and swaps in a value and maybe ticks in a different direction. You could argue that the patterns themselves require a minimum of math/logic, but truthfully - no, they don't - for the same reason that you cut cheese without needing to know what the knife is doing at the molecular level: if the input gives an expected output, you can construct entire worlds on that alone.
You don't need to understand the composition of cheese and bread to make a grilled sandwich.
Fine, "really good instructional schematics written by the book authors themselves" on how to build a house. You get what I'm saying.