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.
So, it’s like if you use a map of the territory to build a road from A to B, and then you later realize that the map that you used to build the road was wrong. But the road still takes you from from A to B! So it doesn’t matter if you built it with the wrong map, it still works.
I think this summary omits a key point: How we know that the road still takes us from A to B? As far as I can tell, the answer is "by experiment" --- we know from repeated use how the instruments behave, and therefore it doesn't matter what our map of the world was when we initially built them. It could, of course, be that they are off --- but we know (by experiment) their precision, and so we can use that to check precision on other things.
What do you think?
I was reading the introduction to a textbook on electrodynamics (J.D. Jackson), and in there was a description of an experiment designed to measure the exponent of the inverse square force law that governs forces between charges (Coulomb’s law). So, the experiment was designed to detect how close the exponent is to 2, and put error bars on the value of the exponent. The experiment was performed in 1971 and established that if the exponent was not 2, then the error had to be in the 15th or greater decimal place! So far, so good...
So, the experimental design seemed to use all kinds of fancy modern electronic equipment. Here’s what bugged me: they were testing the oldest law in electrodynamics, using all this technology, which was based on over a century of development in the theory of electromagnetism. It reeked of circularity to me. You build an instrument using a set of laws, and you use those instruments to test one of the laws? It’s almost like you use a map to build a territory and use that territory to check the map. What’s going on here?
So, trying to exercise the virtue of scholarship, I went on the Stanford Encyclopedia of Philosophy, and read the article on Experiment in Physics (highly recommended). And I found the resolution in this line:
Instruments create an invariant relationship between their operations and the world...When our theories change, we may conceive of the significance of the instrument and the world with which it is interacting differently, and the datum of an instrument may change in significance, but the datum can nonetheless stay the same, and will typically be expected to do so.
So, it’s like if you use a map of the territory to build a road from A to B, and then you later realize that the map that you used to build the road was wrong. But the road still takes you from from A to B! So it doesn’t matter if you built it with the wrong map, it still works. So in the context of the Coulomb’s law measurement, as long as you’ve secured the input-output characteristics of the fancy equipment you’re using, it doesn’t matter if they were built using a wrong theory. So, there is no circularity in testing the theory which you used to build the instrument, since the instrument is going to follow reality no matter what.
So, experimental physics is very epistemically secure. What about other fields? For sure, other fields dealing with understanding reality have one reality to deal with. But the aspect of reality they’re interested in maybe much more fragile. And for an truly accurate map of the territory, you need to take into account how just your presence is changing the territory. Goodhart’s law in economics is the perfect example.