Whether or not the √2 is irrational cannot be shown by measuring it.1,2
Whether or not the Church-Turing hypothesis is true cannot be shown by thinking about it.

[1] "Euclidean and Non-Euclidean Geometries", Greenberg, Marvin J., Second Edition, pg. 7: "The point is that this irrationality of length could never have been discovered by physical measurements, which always include a small experimental margin of error."
[2] This quote is partially inspired by Scott Aaronson's "
PHYS771 Lecture 9: Quantum" where he talks about the necessity of experiments.
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