Godel proved unquestionably that there were truths which could never be arrived at via logical processes, or reason and yet, theoretically, they could be known, but how? According to Godel, the answer was that a machine, for example, would always be limited to arriving at the answer by progressing through a successive series of steps, while, alternatively, a conscious mind could tap into an infinite and transcendental aspect of itself and thereby arrive directly at the truth by way of a direct knowing, or intuition.
"It is interesting to note that although Alan Turing believed humans were merely machines, much like the computers he had envisioned, failed to realize that his idea for computers came to him suddenly, 'in a vision', thus confirming Godel's contention that humans had access to the 'divine spark of intuition'. A divine spark which enables humans to transcend the limits he, and Turing, had found in his incompleteness theorem for computers, mathematics, and even for all material reality generally..."
"Gifted people being able to instantaneously know answers to complex problems, as Turing himself did with his vision of a computer, is something that argues forcefully against the notion that our minds are merely the 'emergent' products of molecules in motion in our brain..."
From the BBC, Dangerous Knowledge series.