The idea that there is some "real" mathematical structure, and statements about other structures are nonsensical because "numbers don't work like that!"
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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You're arguing that the absence of knowledg of a perfect model, is evidence for the correctness *untestable* implicit assumptions in your admittedly incomplete current model? That's certainly not Popperian or Bayesian reasoning.
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You seem to be arguing that because we can never make a measurement of a real number, there can't *be* any real numbers. I think the fact that physics does so well by assuming space and time are continuous is evidence that space and time are, in fact, continuous.
End of conversation
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