The Emperor?s New Gödel
Chris Homan
By overstating the power of human intelligence, Penrose is able to incorrectly use Gödel?s incompleteness theorems to claim that human intelligence is not strictly logical. This slight knock on computational intelligence mars an otherwise thought-provoking article. His argument rests on the assumption that humans can recognize incompleteness in any algorithm, and thus must have a reasoning power beyond that of logic. As a counter-example to this assumption, we will describe a model of human intelligence that can neither prove a so-called Gödel sentence nor recognize its inherent truth. Such a model can exhibit many characteristics, including Platonic existence, which Penrose sees as beneath the power of logic to describe.
Our first concern is to disprove Penrose?s above-mentioned assumption. Could there exist a Gödel sentence in human intelligence? Such a sentence would have to be inherently true but also unproveable by the logic system in question. In human terms, this means that the sentence must be true (in some perhaps archetypal sense), but that a human could not recognize it as such.
Let us try to build a consistent logical representation of my consciousness. Consider the following sentence,
My consciousness cannot prove this statement.
If this statement is a theorem of my consciousness, and my consciousness is logically consistent, then it can?t be proven, therefore it is not a theorem. If it isn?t a theorem, then it can?t be proven, and so it must be true. Have we found a Godelian sentence for my consciousness?
No; if I can recognize the above sentence as true, and my consciousness is strictly logical, then there must exist some proof of the above statement. This would imply that the statement is false, which leads to a contradiction. Presumably, Penrose would stop here. However, it is also conceivable that my original representation was ill formed, in which case I expand it into a greater one that includes logic to handle the above sentence. But now, a new Gödel sentence exists in this representation. We can continue in this manner until the finite capacity of my brain is exhausted, and we will still have a logical representation that is incomplete. But now we can no longer prove the resulting representation?s Gödel sentence nor can we recognize it as true (even though it is, in some archetypal sense). Let us call this final representation "human logic." Note that human logic may be a set of representations.
Now, in essence, Gödel?s second incompleteness theorem states,
No logical representation exists which can prove its own constancy.
Thus we will never know if adding new axioms to our system in this way results in a consistent new representation. This is really what Penrose?s argument boils down to. If no "axiomatic expansion" of any conscious system can be made consistent whenever it is below the capacitive limits of any human brain, then logic is not sufficient for conscious intelligence. We shall leave this question unresolved.
Penrose also claims that logical systems cannot account for much of the mystery of the physical world in general and of human intelligence in particular. Let us continue to assume that conscious intelligence is strictly logical. By the second incompleteness theorem, there is no way of knowing whether we are logically consistent. If we can?t determine the constancy of our own reasoning, then we can never prove anything (including whether a particular logical system is intelligent, which will be important later on), at least not in a rigorous, mathematical way. Intuitively this makes sense; for example, there is no way of proving that I am not the only human in existence, or that everything I perceive is not an illusion. In this way, logic ? human logic in particular - can actually account for a great deal of mystery in the "real" world by simply resolving that it is inherently unproveable. Thus, in the "real" world, human logic is of little use unless we limit its scope to "provable" problems by assuming new axioms. Note that Penrose seems to recognize only these subsets of the greater human logic.
Now consider the example of human cloning (which is, in a sense, a crude form of artificial intelligence). Certainly, a human clone would be intelligent, but how do we prove this fact? Do we assert that all humans are intelligent? If so, what axioms are needed to prove this? Because we don?t understand the mechanism that makes humans intelligent, we accept human intelligence on faith. The construction of artificial intelligence through cloning is actually a variation of Searle?s Chinese Room argument, where the clone plays the part of the Chinese dialect. Although we cannot prove through logical inference that the clone is intelligent, we do know that if we follow the rules of cloning, we will end up with an intelligent system. Penrose finds Searle?s arguments convincing in the case of simple acts, but unreasonable when one considers "more complicated putative" procedures, which might "conjure up actual consciousness". Yet here we see a case where a very complicated procedure, the generation of human consciousness, is made into a merely functional procedure. Perhaps Penrose?s concept of consciousness needs refinement.
Likewise, for any algorithm we construct, we can never prove that it is intelligent; we simply have to believe that it is. Our belief in the algorithm?s intelligence would have to come from interacting with it and forming opinions -- the mechanics of which are (by Gödel?s second incompleteness theorem) beyond our capacity to understand logically -- about it?s intelligence. That human intelligence is strictly logical ? and therefore necessarily limited -- could be one reason why no one has yet invented a test for intelligence better than the Turing test.
Many philosophical arguments also suggest that human intelligence is incomplete. Nietsche?s abyss and Derrida?s diffrènce both represent a reality that is beyond the grasp of human comprehension. It could even be said that Plato?s archetypes are in fact truths that are beyond the ability of conscious intelligence to prove. This assertion sounds very much like a textbook definition of Platonic truth, which, of course, contravenes Penrose?s belief that Platonism is beyond logic. Indeed, maybe Platonism is human logic.
By establishing an adversarial relationship between logic and his alternate models, Penrose misses some very intriguing descriptions of intelligence. For example, what if human intelligence changes -- in a mathematically significant way -- as our perception of it changes? For instance, what if human intelligence is truthfully strictly logical, but only as long as we believe it is so. Such a view (which I don?t necessarily hold) seems to resonate strongly with quantum physics, and presents even more possibilities than Penrose imagines in his article.
Penrose deserves praise for searching for alternate models of conscious intelligence. It is too bad that he didn?t give logic more of a chance.