The idea that there is some "real" mathematical structure, and statements about other structures are nonsensical because "numbers don't work like that!"
As a physicist I use the real numbers to describe reality all the time. You can't do physics without them.
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Yes, you use a convenient mapping of states to the real number system. But accepted physics implies there literally cannot be evidence that there are arbitrary precision numbers for position/velocity, much less infinite information density, as the real numbers would imply.
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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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