Incompleteness Suppose L is a logic and H(T,x) is a statement in L expressing that Turing machine T halts on input x. Thus H(T,x) is true if and only if T halts on input x. Recall -- L is sound and effective. So: If H(T,x) is provable in L then it is true so T halts on input x.
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Incompleteness
Criticism of Penrose
In The Emperor's New Mind [Penrose, 1989] and especially in Shadows of the Mind [Penrose, 1994], Roger Penrose argues against the “strong artificial intelligence thesis," contending that human reasoning cannot be captured by an artificial intellect because humans detect nontermination of programs in cases where digital machines do not. Penrose thus adapts the similar argumentation of Lucas [1961] which was based on Goedel's incompleteness results to one based instead on the undecidability of the halting problem, as shown by Turing [1936]. Penrose's conclusions have been roundly critiqued, for example, in [Avron, 1998; Chalmers, 1995a; LaForte et al., 1998; Putnam, 1995].
In a nutshell, Penrose's argument runs as follows:
1. Collect all current sound human knowledge about non-termination.
2. Reduce said knowledge to a computer program.
3. Create a self-referential version of said program.
4. Derive a contradiction.
The conclusion (by reductio ad absurdum) is that the second step is invalid: A program cannot incorporate everything humans know!
(The reasoning is that humans can know that a self-referential version of this program does not halt, but the computer program cannot know this.)
Ramanujan had several extraordinary characteristics which set him apart from the majority of mathematicians. One was his lack of rigor. Very often he would simply state a result which, he would insist, had just come to him from a vague, intuitive source, far out of the realm of conscious probing. In fact, he often said that the goddess Namagiri inspired him in his dreams. This happened time and again, and what made it all the more mystifying -- perhaps even imbuing it with a certain mystical quality -- was the fact that many of his “intuition theorems” were wrong.
Mahalanobis: Now here is a problem for you.
Ramanujan: What problem, tell me?
I read out the question from the Strand Magazine.
Ramanujan: Please take down the solution. (He dictated a continued fraction.)
The first term was the solution I had obtained. Each successive term represented successive solutions for the same type of [problem] as the number of houses in the street would increase indefinitely. I was amazed.
Mahalanobis: Did you get the solution in a flash?
Ramanujan: Immediately I heard the problem, it was clear that that solution was obviously a continued fraction. I then thought, “Which continued fraction?” and the answer came to my mind. It was as simple as this.
Johann Martin Zacharias Dase, who lived from 1824 to 1861 and was employed by various European governments to perform computations, is an outstanding example. He not only could multiply two number each of 100 digits in his head; he also had an uncanny sense of quantity. That is, he could just “tell”, without counting, how many sheep were in a field, or words in a sentence, and so forth, up to about 30 … . Incidentally, Dase was not an idiot.
I(s) L
{standard integers}