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Ezekiel comments on Godel's Completeness and Incompleteness Theorems - Less Wrong
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Comments (85)
So... not everyone. In Godel, Escher, Bach, Hofstadter presents the second-order explanation of Godel's Incompleteness Theorem, and then goes on to discuss the "human-superiority" crowd. Granted, he doesn't give it much weight - but for reasons that have nothing to do with first- versus second-order logic.
Don't bash a camp just because some of their arguments are bad. Bash them because their strongest argument is bad, or shut up.
(To avoid misunderstanding: I think said camp is in fact wrong.)
I think it's worth addressing that kind of argument because it is fairly well known. Penrose, for example, makes a huge deal over it. Although mostly I think of Penrose as a case study in how being a great mathematician doesn't make you a great philosopher, he's still fairly visible.
The reason it's not random-strawman is that the human-superiority crowd claims we have a mystical ability to see implications that machines can't. If some of them, while making this claim, actually fail at basic logic, the irony is not irrelevant - it illustrates the point, "No, humans really aren't better at Godelian reasoning than machines would be."
Why is the mystical ability to see implications that machines can't mutually exclusive with the ability to fail at basic logic?
It's not logically exclusive. It's just that the only evidence for the existence of this ability comes from logical reasoning done by people. Which contains failures at basic logic.
I didn't evaluate the strongest arguments for the human-superior crowd, because I find the question irrelevant. If some evidence comes from arguments which have not been shown to be flawed, then there is reason to believe that some humans are better at Godelian reasoning than machines can be.
The response wasn't "All of the evidence is logically flawed". The response was
(emphasis added)
I disagree with EY. I think all of them, while making this claim, fail at basic logic, although their failures come in several kinds.
This is based on arguments I have seen (all flawed) and my inability to come up myself with a non-flawed argument for that position. So if you think I am wrong, please point to evidence for human-only mystical powers, which is not logically flawed.
Suppose that humans had the ability to correctly intuit things in the presence of inadequate or misleading evidence. That ability would require that humans not follow first-order logic in drawing all of their conclusions. Therefore, if did not follow perfect logic it would be (very weak) evidence that they have superior ability to draw correct conclusions from inadequate or misleading evidence.
Humans do not always follow perfect logic.
I don't have good evidence, but I haven't searched the available space yet.
This is negligibly weak evidence, not even strong enough to raise the hypothesis to the level of conscious consideration. (Good evidence would be e.g. humans being actually observed to deduce things better than the evidence available to them would seem to allow.)
Consider that there are much much better reasons for humans not to follow logic perfectly. The stronger these are, the less evidence your approach generates, because the fact humans are not logical does not require additional explanation.
Logic is hard (and unlikely to be perfect when evolving an existing complex brain). Logic is expensive (in time taken to think, calories used, maybe brain size, etc.) Existing human adaptations interfere with logic (e.g. use of opinions as signalling; the difficulty of lying without coming to believe the lie; various biases). Existing human adaptations which are less good than perfect logic would be, but are good enough to make the development of perfect logic a bad value proposition. There are others.
Ever known someone to jump to the correct conclusion? Ever tried to determine how likely it is, given that someone is jumping to a conclusion with the available evidence, that the conclusion that they reach is correct?
Consider that several people have asserted, basically, that they have done the math, and more people than expected do better than expected at reaching correct conclusions with inadequate information. I haven't gathered empirical data, so I neither support nor refute their empirical claim about the world; do your empirical data agree, or disagree?
In my personal experience I can't think offhand of people who guessed a correct answer when a random guess, given available evidence, would have been very unlikely to be correct.
Sometimes people do guess correctly; far more often they guess wrong, and I expect the two to be balanced appropriately, but I haven't done studies to check this.
Can you please point me to these people who have done the math?