Justifying the use of real numbers for measuring length

Question

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.

Questions:

1. 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?

2. 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.

3. 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. This seemed like an interesting enough phenomenon to post anyway.

x-posted from math.stackexchange.

I have also seen a few other questions similar to this one but none of them carried out the analysis to an axiomatic level as far as I am aware.

Answer 1

One wants to be able to talk about paths --- paths in physical space, paths in configuration space, etc. etc. etc.

And one wants to be able to talk about the concatenation of paths. Which means that one wants to be able to glue the starting point of one path to the ending point of another and have a new path.

This suggests that a "path" should be modeled as the image of some space we might as well call $I$ with a distinguished "starting point" (which we might as well call $0$), a distinguished "ending point" (which we might as well call $1$), and a distinguished homeomorphism from $I$ to $I^*$, where $I^*$ is two copies of $I$, with the first copy's $1$ glued to the second copy's $0$.

One candidate for $I$ is the unit interval with the usual $0$, the usual $1$, and the obvious homeomorphism.

Now suppose you've got another candidate, called $J$. Then one can prove that there exists a unique structure-preserving map from $J$ to $I$. So there's a sense in which $I$ is the universal candidate for the domain-space of all paths.

So the unit interval is almost forced on you. From there, it's of course only a tiny step to the real numbers.

Answer 2

In my experimental physicist's view, it depends on how many dimensions the variable under study has. Length has one dimension and can be mapped one to one with real numbers.

The classical waves has fields that have a magnitude and a phase, and there, complex numbers were found to map the waves one to one with the measurements. The algebra of complex numbers matches the behavior of the fields and waves in general. It simplifies the formulae.

It is similar with complex numbers used in quantum mechanics whose basic model depends on differential wave equations. It is just that the complex conjugate squared of the wave function gives a measurable value, a probability for the variable under study.

Answer 3

Real numbers contradict the Church–Turing–Deutsch principle and are therefore unphysical. As explained in this article, real numbers lead to problems that from a purely mathematical perspective may be just about tolerable, but it would be crazy to assume that real numbers should play an essential role in physics.

Real numbers are an artifact of taking limits in the wrong order. If you first take the classical limit and then want to model the continuum as described by classical physics, then you are led to real numbers. This is how mathematicians in the late 19th century were led to the axioms of set theory and real numbers.