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shminux comments on Logical Pinpointing - Less Wrong
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How come we never see anything physical that behaves like any of of the non-standard models of first order PA? Given that's the case, it seems like we can communicate the idea of numbers to other humans or even aliens by saying "the only model of first order PA that ever shows up in reality", so we don't need second order logic (or the other logical ideas mentioned in the comments) just to talk about the natural numbers?
Umm... wouldn't they be considered "standard" in this case? I.e. matching some real-world experience?
Let's imagine a counterfactual world in which some of our "standard" models appear non-standard. For example, in a purely discrete world (like the one consisting solely of causal chains, as EY once suggested), continuity would be a non-standard object invented by mathematicians. What makes continuity "standard" in our world is, disappointingly, our limited visual acuity.
Another example: in a world simulated on a 32-bit integer machine, natural numbers would be considered non-standard, given how all actual numbers wrap around after 2^32-1.
Exercise for the reader: imagine a world where a certain non-standard model of first order PA would be viewed as standard.