What are the implications of Gödel's Incompleteness Theorem?
For those not familiar with it, Gödel’s Incompleteness Theorem says that for any set of mathematical axioms that includes arithmetic, there are statements that can be formulated that are not provable in the system. What I know about the theorem, I got from the book Gödel, Escher, Bach
On one extreme, there are those who say that since the true (outside the axiom system) but unprovable (within the system) statement that Gödel came up with was so convoluted, that there is no serious implication. On the other end, there are those who say that the theorem shows that there are limitations to any computational system and therefore true artificial intelligence is impossible.
I go along with Hofstadter’s middle position. There seems to be an implication that it is sometimes necessary to think outside the box and to be able to think about the thought process. This does not preclude thinking robots, but it supports the idea that these robots might have to feel emotions and to respond to the world more like humans than Mr. Spock.
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I had an emotional reaction to the theorem when I encountered it. It seems, in an almost poetic way, to demand from science a deeper wisdom than that found by mathematical proof.
I’m not sure why it would mean artificial intelligence is impossible. by the same logic, any intelligence would be impossible, including our own.
Have you read “I am A Strange Loop”? I haven’t read GEB but I’m led to believe that the former goes into much more detail about the Incompleteness THeorem and it sort of blew my mind as an explanation for how Godel’s work can point the way to how consciousness might work (the theorem itself works with an example of a strange loop.)
GEB is all about Godel’s Theorem, but I will make a point of looking into I Am A Strange Loop. I can see how there might be a connection with consciousness, since the theorem is all about self-reference. It is in effect a grand effort to get a statement to say, in effect, “I am unprovable.”
I think Godel’s Theorem is troubling only to theoreticians. It says there are limits to what any logical system of thought can prove but it doesn’t say what those limits are. It doesn’t really change anything; it just says logic will always be limited.
It proves artificial intelligence can never be complete, that we can’t make a machine that understands and explains everything, or even explain itself to itself, but this is true of human beings also. Again it doesn’t set a boundary as to how intelligent artificial intelligence can be it just says there is a limit and that isn’t really much of a constraint.
Einstein’s quote is just as profound “As far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
@Qingu GA!
@LostInParadise I Am a Strange Loop is a fascinating book. I highly recommend it.
@flutherother Good synopsis. I agree exactly with that. No wonder it took so long to just get to 42.
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