The impact of Godel's Incompleteness Theorem, which was the subject of the last essay, was felt beyond the bounds of mathematical logic and, indeed, beyond the bounds of mathematics itself. The essay by Lucas entitled Minds, Machines and Goedel tries to make the case that the Incompleteness Theorem has as a consequence "that Mechanism is false, that is, that minds cannot be explained as machines". This, of course, was the topic of debate at C. P. Snow's dinner table in The Cambridge Quintet. Had Lucas been invited to dinner that evening he would have found himself at odds with Turing, but his arguments would have been entirely different from those of Wittgenstein. Lucas does not take the humanist approach that minds can not be mechainzed beacuse they are embedded in society. Rather, he uses Turing's own weapons, Godel's Incompletenss Theorem in particular, against him.
In order to understand Lucas' argument, it is necessary to review the proof of Godel's Incompleteness Theorem. To begin, the theorem is a result about arithmetic which says that given any "understandable" axiomatization of arithmetic it is always possible to find a true theorem of arithmetic which can not be proved from the axioms. One of the keys to the proof is that givena an "understandable" axiomatization it is possible to code statements and proofs of arithmetic as natural numbers and then define an arithmetic function which takes the code for a potential proof as input an tells us whether not it is a valid proof using the allowed axioms and rules of logic. Using this fucntion is it is possible to define the Godel statement: "I have no proof". The reading from the last topic show that this must be a true statement which cannot be proved within the given system of axioms.
But, what does this have to do with "mechanized minds"? Ironically, it was Turing who provided the essential step required in Lucas' chain of reasoning, the step which established a connection between Godel's Incompleteness heorem and machines. Turing machines, in spite of various outrageous and largely non-sensical claims made about them, were invented by Turing to serve as simple models of computation. They key facts one neds to know about them are:
Lucas uses to this fact to conclude Godels Incompleteness Theorem is also a theorem about machines and hence can be applied to the problem of whether or not human thought is a machine-like process. His conclusion is that it is not. If human thought follows the rules of some Turing machine (as it must if it is mechanical in nature) then it would never be able to see that its Godel Statement is true. The fact that humans can see this leads Lucas to conclude that out thinking could never be accomplished by a mechanical system.
This is essentially the same argument as that put foreward by Penrose, although he does add some refinements of his own. For example, he elaborates on Lucas' discussion of the possible objections to his argument that Godels' theorem applies only to consistent formal systems and human beings display inconsistency.
A more difficult argument for a supporter of Lucas to argue against is that Lucas's argument only shows that if a human mind knows completely the rules a machine is following then it can construct a Godel sentence which the mind sees to be true but the machine cannot prove. In order to apply this process to itself the human mind/machine would have to be able to understand completely all the rules it is following. Each time it adds a Godel sentence to its set of axioms is class of rule becomes more complicated. Adding too many such rules will eventually lead to a situation in which the mind/machine can not keep track of the rules it "sees clearly" to be true. This is the point of the Achilles-Turtle dialogue by Hofstadter.
The philosopher C. H. Whitley has given an interesting twist to Lucas' argument. Consider the statement, "Lucas cannot consistently assert this sentence." This is a true statement because if Lucas were to assert it he would be contradicting himself. Does this mean that Whitley can prove things that Lucas cannot? Does it mean that Lucas' thinking is machine-like and Whitely's is not?
The topic of the essay concerns the fourth reading, the short story by Christopher Cherniak. There are hints that this is related to Godel's sentence but the assignment for this topic is to write an essay providing a clearer hypothetical explanation of what is happening in the story based on the ideas of Lucas' argument. As a starting point, your essay should include a clear explanation of Lucas' argument and how itr relies on Godel's Incompleteness Theorem. Your exlpanation of the fictional phenomenon wil probably be based on at least one of the objections to Lucas' argument. (Why?) A discussion of the story in this context is provided in the last reading, but other interpretations are possible, and I will be more interested in seeing these than a restatement of D. Hofstadter's views.