Suppose a human cognition apparatus (loosely, "mind") is modeled as a Turing Machine. In that case, if that TM "knows" (has available from memory) its Description Number, then it can compute its design code -- its particular rules of computation.
But, such a computation is equivalent to the derivation or proof of axioms from the theorem, "This TM's DN is x."
Yet, we know from Goedel's results that such a TM has a low level of sophistication, that the TM must be equivalent to the level of sophistication needed to do the arithmetic that lacks the division operation. But since humans routinely do arithmetical calculations that include the division operation, it follows that no human can know the equivalent of her or his DN. This consequence further implies that there is a component of human cognition that transcends human understanding. That is, as Roger Penrose has argued at length, computation is insufficient to account for all human cognition.
Hence Penrose is correct to say that a scientific explanation must be found in some novel approach, possibly via quantum physics. Also correct are those who say that human cognition implies a "noumenal" or hidden means of "intuition" (for want of a more precise English word).
Turing's halting problem result is a variant of Goedel's theorem, but that is not my thrust here.
But, such a computation is equivalent to the derivation or proof of axioms from the theorem, "This TM's DN is x."
Yet, we know from Goedel's results that such a TM has a low level of sophistication, that the TM must be equivalent to the level of sophistication needed to do the arithmetic that lacks the division operation. But since humans routinely do arithmetical calculations that include the division operation, it follows that no human can know the equivalent of her or his DN. This consequence further implies that there is a component of human cognition that transcends human understanding. That is, as Roger Penrose has argued at length, computation is insufficient to account for all human cognition.
Hence Penrose is correct to say that a scientific explanation must be found in some novel approach, possibly via quantum physics. Also correct are those who say that human cognition implies a "noumenal" or hidden means of "intuition" (for want of a more precise English word).
Turing's halting problem result is a variant of Goedel's theorem, but that is not my thrust here.
