Plug for implicit understanding though. Your very claim is not explicable; you can’t explicate the non existence of explicability. You can’t formally denote areas for which there’s no formal descriptions. This is the escape hatch out of logical realism (and yeah, that’s not an explicable claim, either; how fun!)
Not directly related to the above, but maybe also relevant : Wolfram's «new science» project centered around computability : https://writings.stephenwolfram.com/2017/05/a-new-kind-of-sc...
Speaking of which, might the feeling about the existence of numbers come from a similar shortcoming in our neurological structure ?
https://www.lesswrong.com/posts/mHNzpX38HkZQrdYn3/philosophy...
Not directly related, we're now surrounded by computers, which are even dumber (some forms of machine learning and analog computers aside ?) in that they have zero flexibility in treating symbols as something else...
The irony of the GGP comment is that it claims mathematical ideas are successful ones and philosophical ones aren’t, but that’s not a mathematical claim. So either it’s right and it’s wrong, or it’s wrong.
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 :)
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'.