This example isn't analogous but I'm curious how you'd reply to a student saying, "The notion of 'objectively true sentences' is wrong and can't be rescued, because words don't have culturally independent meanings and there's no Objective Teacher to grade answers as correct."
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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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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