> Now suppose that we construct the Gödel sentence for this formal system. Since the Gödel sentence cannot be proven in the system, the machine will be unable to produce this sentence as a truth of arithmetic. However, a human can look and see that the Gödel sentence is true. In other words, there is at least one thing that a human mind can do that no machine can. Therefore, “a machine cannot be a complete and adequate model of the mind” (Lucas 1961: 113). In short, the human mind is not a machine.
That is not a difference between human and machine. It's a rhetorical trick at best, and an unreasonable appeal to mysticism at worst.
A human mind looking at the Gödel sentence while strictly adhering to the same formal structure could also not see that the Gödel sentence is true. Conversely, a machine that is allowed to operate outside of this postulated formalism could perfectly well make an inference that the Gödel sentence is true.
The Lucas-Penrose argument boils down to a questionable attempt at a logic bomb. So by first tying the machine's hands, they argue that the machine's hand's are indeed tied, and therefore humans - who in this example have been given free rein - can do something that you explicitly disallowed machines from doing.
That is not a difference between human and machine. It's a rhetorical trick at best, and an unreasonable appeal to mysticism at worst.
A human mind looking at the Gödel sentence while strictly adhering to the same formal structure could also not see that the Gödel sentence is true. Conversely, a machine that is allowed to operate outside of this postulated formalism could perfectly well make an inference that the Gödel sentence is true.
The Lucas-Penrose argument boils down to a questionable attempt at a logic bomb. So by first tying the machine's hands, they argue that the machine's hand's are indeed tied, and therefore humans - who in this example have been given free rein - can do something that you explicitly disallowed machines from doing.