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shminux comments on Open Thread, October 1-15, 2012 - Less Wrong Discussion
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I'd work backwards. As I said, "finite integers as the most "real", given that they correspond to discrete objects we can see, count and calculate". If you accept that, instrumentally, we can only ever see a count of something, like the number of notches on a ruler, or marks to the left of the needle of some gauge, or the number of clock ticks before a certain event, then you can extrapolate backwards in your logic to include negative numbers, fractions, rationals, reals and complex numbers as convenient abstractions. But you have to leave a path forward in your models to eventually end up with the number of notches on a dial. If your observables are not Hermitian, you cut off this path from models to counts.
If that's your view, though, I don't see why you think real expectation values are better than complex ones. If our measurements have the structure of integers, and our expectation values are real, then in our model of measurement we're going to have a map from the real line (or some segment thereof) representing possible expectation values to the integers representing notches on your dial. This map will presumably associate an element of some partition of the real line into finite regions with each integer. So this map must be part of your model of measurement in order to connect your integer measurements with the continuous space of expectation values allowed by the theory.
But if you're relying on some such map anyway, why can't the expectation values be complex? You could construct a map from a partition of the complex plane to the integers. More specifically, your claim that the ultimate measurement results must be integers doesn't rule out the possibility that each measurement result gives a pair of integers (instead of a dial moving along a single dimension, think of one that can move in two dimensions). For such measurements, it may even be more natural to think of a map from the complex plane to the measurement outcome, with one of the integers associated with the real part and the other with the imaginary part. As long as you allow for this possibility, I don't see the motivation for restricting observables to Hermitian operators. Surely it isn't a requirement of instrumentalism that the measurements we make must only be one-dimensional.
Yeah, that would be a dangling requirement.
That would be an interesting counterfactual universe, where you need at least two numbers to make sense of anything. I cannot quite imagine how this would work, but maybe this is a limit of my imagination. In the Universe we are currently stuck with, with the classical world described by classical mechanics based on real numbers, and the quantum world manifesting itself only through classical measurements, any useful model must ultimately lead to a set of readings off some dial, each of which should make sense separately.