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I think your argument doesn't stand. Take this paradox:

1) the next sentence is false
2) the previous sentence is false

I can conceive a word where one only one of the two is correct, and indeed it will be consistent, but it is the conjunction of the two that creates the paradox.
Extrapolating from both the branches of the paradox is obviously not a problem, but this doesn't mean that a conjunction is always the trivial, as your example seems to entail.

This is an old debate: can God create logical inconsistencies? If not, can it be really said that it is omnipotent?

Word thinking is better at getting at weird things. Weird or not, it’s there.

Another way I think of it is, imagine a function f(x), that is unique for all values, which returns the same value for x=3 and x=10.

That's sort of like... Okay... So there is some function with two properties that imply the other cannot exist. What does it mean?!

I think lots of these philosophical problems were very reasonable to discuss when they were first brought up. I can't prove this, since it's based on lots of general concepts I've found in old books or academic papers, but those ideas seem related to the same idea of the human brain as mystic, and language an expression of that mysticism. Rather than an evolved mechanism to filter patterns from reality and communicate them to one another. In that anti-mystical view, which views words and math as the same subset of pattern classification and information, poorly defined thought experiments are as interesting as "what if 2+2=5".

Also, I want to speculate on an idea I have with near zero proof that I half stole from/half conceived from reading "Godel, Escher, Bach:" We know computers cannot process undefined statements. When humans try to process undefined statements we don't crash. Instead we conceive something that seems like it could be meaningful, even if it's inconceivable. I can 'conceive' of 2+2=5, or it feels like I can, but maybe that's just the error-catching subroutine of my brain. We then sort of interpret this inconsistency as meaningful and call it a 'thought experiment' -- when the inconsistency is just our brain's way of saying "does not compute."

When humans try to process undefined statements we don't crash. Instead we conceive something that seems like it could be meaningful, even if it's inconceivable.

Physics is consistent. Humans are physics. "What a human would predict from this inconsistent set of statements" is often consistent and unambiguous as most communication. I think this is a necessary tool for thinking about law.

"You can deduce that verbally. But I bet you can’t predict it from visualizing the scenario and asking what you’d be suprised or not to see."

I like this.

In my mind, this plugs into Eliezer's recent facebook post regarding thinking about the world in mundane terms or in terms of what is merely-real or in terms of how you personally would go and fix a sink or how you go and buy groceries at the store VS. the way you think about everything else in the world. I think these methods of thought in which you are visualizing actual objects and physics in the real world, thinking of them in terms of bets, and checking your surprise at what you internally simulate all point at a mindset that is extremely important to learn and possess as a skill.

I don't quite see the connection between the title and first sentence and the rest of the post you have there; logically inconsistent is not the same as inconceivable

Logically inconsistent implies inconceivable, right? Falsely attributing a paradox is a way of issuing a false positive for inconceivability. The rest of the post is an example of falsely attributing a paradox and concluding inconceivability, when someone already has the concept.

I think I can conceive of things that are logically inconsistent. I might just be ignoring the details that make it inconsistent when I do, but other cases where I conceive of a concept but don't keep every detail in mind at once don't seem examples of inconceivability.

Wouldn't the ability to have a false positive for a paradox itself be a sign that people can conceive of things that are paradoxical?