We always have some “map”, so technically all we can ever check is whether “map + experimentalData(territory)” is consistent. The trick is to use the map in weird ways, which can exaggerate possible incosistencies.
In given example, let’s suppose that the exponent is not 2, but for example 2.001 -- then yes, we would be measuring it using imprecise electronic equipment, but there is a small chance that all those differences would exactly cancel each other out.
As a metaphor, imagine that we have a function “f” that we believe returns f(x)=x^2, but in reality it returns f(x)=x^2.001. Unfortunately, we can never inspect numbers directly, only their f-values. Trivial checks like “f(5) = f(5)?” would not help us discover the problem. Some more complicated checks like “f(2×3) = f(2) × f(3)?” would still give the expected answer. But for example check “f(2+2) = f(2) + f(2) + f(2) + f(2)?” would fail. A complicated test like “f(2×3+2×3) = f(2)×f(3) + f(2)×f(3) + f(2)×f(3) + f(2)×f(3)” will more probably fail that appear correct. -- Using electronic devices seems to me like using these complicated tests; there is very small chance they would fail in exactly the necessary way to make the error in theory invisible.
We always have some “map”, so technically all we can ever check is whether “map + experimentalData(territory)” is consistent. The trick is to use the map in weird ways, which can exaggerate possible incosistencies.
In given example, let’s suppose that the exponent is not 2, but for example 2.001 -- then yes, we would be measuring it using imprecise electronic equipment, but there is a small chance that all those differences would exactly cancel each other out.
As a metaphor, imagine that we have a function “f” that we believe returns f(x)=x^2, but in reality it returns f(x)=x^2.001. Unfortunately, we can never inspect numbers directly, only their f-values. Trivial checks like “f(5) = f(5)?” would not help us discover the problem. Some more complicated checks like “f(2×3) = f(2) × f(3)?” would still give the expected answer. But for example check “f(2+2) = f(2) + f(2) + f(2) + f(2)?” would fail. A complicated test like “f(2×3+2×3) = f(2)×f(3) + f(2)×f(3) + f(2)×f(3) + f(2)×f(3)” will more probably fail that appear correct. -- Using electronic devices seems to me like using these complicated tests; there is very small chance they would fail in exactly the necessary way to make the error in theory invisible.