Suppose I suggested you can have a distinguishable solid subsystem of a fuzzy system. Distinguishing objects in the environment is fuzzy, assigning meaning to "three" is fuzzy, but once counting and naming is done, the arithmetic subsystem is locally quite solid.
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Yes! Rationality works, when it does, because somehow inferences within the mathematical system turn out to be true-enough in the real world.
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Suppose I then suggested that adopting any number of purposes and object distinguishers will nail down this core subsystem surprisingly hard. To the point that alien systems probably agree with ours not only about 2 + 2, but about formulae with no practical use like 85378^397642.
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I agree that math is a domain in which absolute truth applies. (Modulo maybe stuff like the independence of the continuum hypothesis, but let’s ignore that.)
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Okay. What about "absolute" systems that cast nonabsolute prescriptive recipes as shadows, like "try counting digits and adding to estimate the product to within a couple of orders of magnitude"?
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That particular example is just true, isn’t it? (Arithmetic isn’t my strong suit)
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It'll actually end up being an overestimate. 20 * 20 * 20 ~ 10000 not 100000.
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Oh, I was thinking about multiplying just two numbers. I guess I don’t follow your question about shadows and so on. If you’re talking about products of any number of numbers, this rule is useless. Maybe we can think of a better example of a roughly-right heuristic.
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It's in fact useful for eye-checking many sets of numbers we multiply in practice. It tells us ordering 37 of a $16 product should not cost $100,000 like the computer says, even if you can't do the arithmetic in your head.
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Replying to @ESYudkowsky @Meaningness and
You need to understand the right context in which to use this tool, and its limits, and not take the recipe as an absolute, and check the results against common sense and reality.
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OK, this is great—that’s exactly what I’m calling “meta-rational judgement”! My thesis is that you always need to do that when applying rational methods, and there’s skillsets for doing that, and those tend to be neglected, and it would be good to help people learn them.
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Threading is a nightmare here. See my continuing reply. "Decision theory" isn't the adding-digits recipe, it's the more abstract idea of Peano arithmetic that helps us understand when the recipe might fail or succeed, even though applying the Peano axioms is way too laborious.
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OK, another attempt at finding a crux. For you, decision theory is THE TRUE framework, according to which any practical method must be judged. For me, it’s just one bit of math among many, with no special value.
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