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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:

  1. S is an abelian group under some operation (say +). Abelian because we want it count stuff.
  2. 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.
  3. It should be archimedean. No observation contradicts it.
  4. 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.

Asvin
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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?.

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Emilio Novati
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  • My thinking was along the lines that completeness makes things easy and it certainly seems as though space is complete from our vantage point(don't know about QM). Therefore, any observations about space should be consistent with it being complete. – Asvin Mar 04 '15 at 17:48