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Saturday, September 1, 2012

Kurt Godel - Incompleteness Theorem and Human Intuition

A video clip: Kurt Godel - Incompleteness Theorem and Human Intuition (hat tip to 'JohnnyB' at Uncommon Descent)

Here is an example of a mathematical truth that cannot be proven by any machine, cannot be proven by any arithmetic logic, but can be grasped/understood by any human mind who is willing to grasp it -- 1/0='infinity' ERGO: the human mind is not a computer ... and no computer will ever be a mind.

And, by the by, while the math problem [1/0=?] cannot be recognized by a machine as being unsolvable by machine logic, the human mind can see, almost immediately, both that it is logically/mathematically unsolvable AND what the solution is. Indeed, to grasp the one is to grasp the other; the two understandings are two sides of the same coin.

This is my transcription of the very last comment of the clip --
"I think people very often, for some reason, misunderstand Gödel, certainly he intention. Gödel was deliberately trying to show that what one might call 'mathematical intuition', [I mean] he referred to what he called 'mathematical intuition' , and he was demonstrating, clearly, in my mind, demonstrated, that this is outside just following formal rules. And, I don't know ... some people picked up on what he did and said, 'Well, he's shown there are unprovable results, and therefore beyond the mind.' What he *really* showed was that for any system that you adopt, which, in a sense, the mind has been removed from it, because you ... the mind is used to lay down the system, but from there on, it [the rules so laid down] takes over, and you ask, 'What's its scope?' And what Gödel showed is that its scope is always limited, and that the mind can go beyond it."