The idea that there is some "real" mathematical structure, and statements about other structures are nonsensical because "numbers don't work like that!"
Just because our measurements of reality can't be real numbers doesn't mean the state of the universe can't involve them. And even if we discover the universe is a complicated cellular automaton, current physics is so accurate we'll go on using it for most purposes anyway.
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So there *is* something that the real numbers map to in reality; our current model of physics, which is absurdly accurate and useful even if it turns out to be wrong. I don't see the "but it's not the base level" argument as having a point for the "reality" of real numbers.
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You're arguing that because X allows Y, where Y is known to be a simplified model that doesn't capture everything, therefore X is real? Yes, "real" numbers are conceptually useful and consistent. That doesn't imply they exist in the territory, it means they help with mapmaking.
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I'm not arguing they are not useful, I'm arguing that they are unnecessary, and that, AFAICT, reality cannot contain them.
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I'm confused about your assertion that they are unnecessary. Has someone invented a fully discrete version of physics that I don't know about?
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