A "measurement" in the general sense (not quantum) is about amplifying or isolating some signal out of all the noise. I simply don't agree that all measurements boil down to relative displacements. Did a chemical reaction occur or not? Is it red or is it blue? Sure, in many of these measurements, something tangentially related to distance (like light wavelength) is involved, but lots of things are involved: measurements are inherently "macro" phenomena, which means many processes will be involved, not the least of which is electrochemical signaling in the brain of the measurer. It's just far too reductive to say that every quantity we measure is actually relative displacement. I could just as justifiably (or more so) say that every measurement is actually just measuring the strength of an electromagnetic field. Why is this helping? What does this have to do with which numbers "truly exist" or not?
Note that Bertrand Russel is on "team Nathan" in the sense that he thinks measurements relate to (at most) the reals.
I don't think anyone ever really measures the elecromagnetic field. We might, for example, measure the displacement of a charged object attached to a spring in an electromagnetic field. Or, if the field is changing in time we measure that displacement as a function of the position of the hands on a clock. But it is very hard for me to think of a situation where we measure something other than a distance at its most basic level. Even in a DAC we measure voltage relative to some calibrated voltage which we measured using a voltmeter which shows us our answer as a deflection in a meter.
This is particularly relevant in QM because in fact all the values we might measure are the eigenvalues of hermitian operators and they are, in fact, restricted to the real numbers.
> I don't think anyone ever really measures the elecromagnetic field. We might, for example, measure the displacement of a charged object attached to a spring in an electromagnetic field.
I don't think anyone ever really measures the displacement of a charged object attached to a spring in an electromagnetic field. We might, for example, measure the difference in strength of neuronal activation in our brain from neurotransmitters emitted in a chain traveling from the photoreceptor cells in our fovea, which are responding to differing quantities of photoisomers that have had their shape altered by the absorption of different frequencies of photons reflected differently off the needle of a gauge in the display of an instrument which is measuring the displacement of a charged object attached to a spring in an electromagnetic field.
This is what I mean when I say it's overly reductive to say a measurement is necessarily a displacement. A measurement is lots of things, and not all of them can be represented as a spatial displacement unless you really shoehorn it.
> all the values we might measure are the eigenvalues of hermitian operators and they are, in fact, restricted to the real numbers.
Is that even true? Spin is a measurable quantity, and you cannot possibly get a spin of 0.2. Most measurable quantum numbers are essentially integers (integer multiples of some conventional base). Remember this discussion is about whether complex numbers are "lower-class citizens" than real numbers in physics. If you're going for measurable quantum numbers, these are almost all counterexamples to the idea that real numbers are special, and instead hint that it's simply integers that are the only first-class citizens in physics. I also fundamentally disagree that the possible eigenvalues of hermitian operators are the sole criteria we should be using to "rank" the truthiness or realness of mathematical structures.
I deeply do not care what philosophy has to say about this, either. Philosophy as a discipline is completely incapable of determining the truth, because it is unwilling to ever reject a single idea; all it is is a giant collection of shower thoughts. Every philosophy page, including the one you linked, somewhere includes "According to Aristotle ...". If you're trying to learn evolutionary biology and you read "according to Lamarck ...", then you are not learning science, you're reading science history. Yet this is all philosophy ever can say about anything. Science curates ideas; philosophy hoards them.
