Piero Scaruffi(Copyright © 2013 Piero Scaruffi | Legal restrictions )
These are excerpts and elaborations from my book "The Nature of Consciousness"
Following the British philosopher John Randolph Lucas (“Minds, Machines and Gödel”, 1959), the British physicist Roger Penrose (following the British philosopher John Lucas) resorted to Goedel's theorem to undermine the very foundations of Artificial Intelligence. Goedel’s theorem states that every formal system (which is bigger than Arithmetic) contains a statement that cannot be proven true or false. Indirectly, Goedel's theorem states the preeminence of the human mind over the machine: some mathematical operations are not computable, nonetheless the human mind can treat them (at least to prove that they are not computable). Humans can realize what Goedel’s theorem states, whereas a machine, limited to mathematical reasoning, would never realize what it states. We can intuitively comprehend a truth that the computer can only try (and, in this case, fail) to prove. Therefore a computer will never be equal to a mind. And, in general, no mathematical system can fully express the way my mind thinks.
Again, countless replies, have been provided.
First of all (Hilary Putnam), a computer can observe the failure of “another” computer’s formal system, just like a human mind can observe it. A computer can easily prove the proposition “if the theory is consistent, then the proposition that there is at least one undecidable proposition is true”. Which is exactly all the human mind is capable of doing.
Second, even if Goedel’s theorem sets a limit, it is not a limit of the machine, it is a limit of the human mind: the human mind will never be capable of building a machine that can think. This does not prove that machines cannot think.
Third, Penrose’s demonstration can be used to prove that a machine cannot prove the validity of a mathematical demonstration, a fact that is contradicted by our experience.
Fourth, Goedel’s theorem is false in some nonstandard mathematical systems, as, for example the Zimbabwe-born mathematician Aaron Sloman pointed out (“The Emperor's Real Mind“, 1992). One of Goedel’s conditions is that the mathematical system must be consistent (i.e., not contain a contradiction), but that can only be if the undecidable statement is added to the system, assuming either true or false. Nonstandard models assume that it is false. Goedel’s theorem, because of the way Goedel carried it out (by employing infinite sets of formulas), leaves the illusion of proving a truth which in reality is never proved, cannot be proven and must be arbitrarily decided.
Fifth, Rudy Rucker believes that conscious machines could be built, following an observation of Goedel himself, that we cannot build a machine that has our mathematical intuition but such a machine can exist and can be discovered by humans. If such a machine exists, humans cannot understand its functioning. Such a machine cannot be built by humans, but could be built by Darwinian evolutionary steps starting from a man-made machine. If a machine can be built that exhibits a behavior completely similar to that of humans, then a machine can be built that is as conscious as humans. What Goedel's theorem asserts is that "the human mind is not capable of formulating all of its mathematical intuitions" (quoting Goedel himself).
Sixth, the British physicist Stephen Hawking notes that the behavior of earthworms can probably be simulated adequately with a computer, because they do not worry about Goedel sentences. Darwinian evolution can generate human intelligence from earthworm intelligence through a process (natural selection) for which Goedel's theorem is also irrelevant. Therefore, Goedel's theorem does not forbid the birth of an intelligent computer.
What Goedel’s theorem proves, if anything, is an intrinsic limit to “any” form of intelligence, including the machine’s but also Penrose’s...
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