Why should I believe that the real numbers model distances along a line? Taking the real numbers to be a complete ordered field, why do we believe that they model distances along a line? How do we know (or why do we believe) that any length that can be drawn is a real number multiple of some unit length?
Completeness often feels a bit technical at first: we show that there is exactly one complete ordered field up to isomorphism, but why should that confluence of properties correspond to line-ness?
I think it's more intuitive to focus instead on connectedness. This is really the same thing in our context, but is a priori phrased in more convenient language: the idea is that if I &quot;cut&quot; the line into a &quot;lower piece&quot; and an &quot;upper piece,&quot; then there is some point which captures this cut. Specifically:

The line is connected, in the sense that it cannot be written as the disjoint union of two nonempty sets $A$ and $B$ where $A$ is downwards closed, $B$ is upwards closed, $A$ has no greatest element, and $B$ has no least element.

This is - for me at least - a pretty fundamental piece of the intuition I have about the line. (Connectedness is equivalent to completeness in our context, but is in my opinion more obviously a fundamental line property.) The fact that $\mathbb{R}$ is the unique connected ordered field then tells me that $\mathbb{R}$ is the only possible way we could faithfully &quot;model&quot; the line by a field, that is, the only way we could sensibly add/subtract/multiply/divide lengths.
This whole line of attack was developed by Dedekind in his Essays on the Theory of Numbers.

That said, we can critique this idea.
The line has three fundamental properties (to me anyways): it is a linear order without endpoints (duh), it is connected, and it is dense (= between any two elements is a third). These three properties are not enough to pin down a single linear order up to isomorphism: there are lots of connected dense linear orders without endpoints much larger than $(\mathbb{R};&lt;)$. This does not contradict the above, since these &quot;unreal&quot; linear orders do not support a field structure.
So one way I could argue that the ordered field $(\mathbb{R};+,\times)$ does not faithfully model the intuitive line is if I gave up the assumption that lengths do in fact form a field! This raises a natural question:

What sort of algebraic structure can a connected dense linear order without endpoints other than $(\mathbb{R};&lt;)$ support?

It turns out that the answer is very little: any two (nontrivial) connected ordered  groups are in fact isomorphic,$^*$ and in particular up to isomorphism the only (nontrivial) connected ordered group is $(\mathbb{R};+)$. So assuming we want to be able to add and subtract lengths in a reasonable way, we're stuck with $\mathbb{R}$.
This is pretty much case-closed for me: the idea of &quot;non-additive intervals&quot; is sort of a non-starter as far as my own intuition is concerned.

$^*$Here's a proof sketch:
Suppose $G$ is a nontrivial connected ordered group. First, note that by connectedness $G$ must be divisible: e.g. for each $x$ there must be some $y$ such that $y+y=x$, since otherwise we could partition $G$ into $\{y: y+y&lt;x\}$ and $\{y: y+y&gt;x\}$ contradicting connectedness.
Now - by nontriviality - fix some positive group element $g$. By divisibility we get an embedding $\eta$ of $(\mathbb{Q};+)$ into $G$ generated by sending $1$ to $g$. By connectedness, $ran(\eta)$ is cofinal in $G$ (otherwise look at the downwards closure of $ran(\eta)$); this gives us a homomorphism $h:G\rightarrow \mathbb{R}$ (think about sending $x\in G$ to the least upper bound in the sense of $\mathbb{R}$ of $\eta^{-1}(\{y: y&lt;x\})$). And we can show that $h$ is a bijection - hence an isomorphism - by applying connectedness again.
I think steven gregory's answer is probably the best so far: we simply choose to believe that real numbers describe distances - it has turned out to be a useful notion, but it is simply a choice we made, it isn't absolute truth.
This really about the fundamental nature of mathematics: there are certain things we choose to accept as true witout proof (called axioms), and when we that maths represents absolute truth, it means simply that the logical statements of the form &quot;if the axioms are true, then ...&quot; are true.