I am not sure if this is the most appropriate place to post this but here goes nothing:
Assume we were trying to come up with system of numbers $S$ to model our intuition of length. We want $S$ to have these properties intuitively at least:
- S is an abelian group under some operation (say +). Abelian because we want it count stuff.
- It should be a $\mathbb{Q}$ - vector space. This corresponds to our notion of divisibility of length, given a ruler, we can imagine $1/k$ th of that ruler.
- It should be archimedean. No observation contradicts it.
- We should have a notion of limits in it through something like the intuition behind Zeno's paradoxes. Essentially, we are asking for completeness.
I think these are all our intuitions in mathematical language. Of course, these are also enough to force our system to be the familiar reals. I am not sure if anyone has done something similar to this before and I have a few questions.
Does the other stuff we model by the reals(temperature, probabilities, entropy etc) also follow the same/similar intuitions? Is there any reason that all our measurements have these properties?
If not, do we measure other properties in physics by other systems? The only one I can think of is complex numbers in Quantum Mechanics but I don't know anything about that.
Finally, is it coincidence that we have a uniqueness theorem for exactly those properties that model our intuitions about the world?
I am sorry that my questions are vague/philosophical/physical. This seemed like an interesting enough phenomenon to post anyway.
In my opinion any physical measure process is a finite set of operations, so, if this process gives a number as result, this number must be a rational number. If the process gives a set of numbers as result, than we can represent this set as an element of a more complex structure, as a vector, a complex number or a matrix ecc.. But every number in the set, if it is genuinely the result of a measure process, is always a rational. The field of rational numbers fits yours first three requests, but not the fourth. For completeness we need reals, but do we really need completeness ? This is an open question (in my opinion), see: Do we really need reals?.