[T]he relationship between cognition and neurobiological processes might be similar to the one between a car engine and locomotion. ...[A] car's loco… - Joscha Bach

What Gödel could show is using classical mathematical semantics you run into contradictions, and because Gödel strongly believed in these semantics... he was shocked. It... shook his world to the core, because in some sense he felt that the world has to be implemented in classical mathematics.

For Turing it wasn't quite so bad. ...[T]uring could see that the solution is to understand that mathematics was computational all along. ...For instance pi in classical mathematics is a value. It's also a function, but it's the same thing. In computation, a function is only a value when you can compute it, and if you cannot compute the last digit of pi, you only have a function. You can plug this function into your local sun, let it run until the sun burns out... This is it. This is the last digit of pi you will know. But it also means that there can be no process in the physical universe, or in any physically realized computer that depends on having known the last digit of pi. ...Which means that there are parts of physics that are defined in such a way that cannot strictly be true, because, assuming that this could be true leads into contradictions.

[A] functionalist model of cognitive processes might be implemented as a computer program (computationalism)... hand in hand with... Artificial Intelligence.