icient for empirical adequacy, we assume that the set $\Omega$ of all possible configurations of distance relations between $N\in\mathbb{N}$ matter points can be represented as follows:

\[def:dist\] Let $\mathcal M=\{1,2,\dots,N\}$ and $\mathcal E=\{(i,j)\,|\,i,j\in\mathcal
    M, i\neq j\}$. The set $\Omega$ comprises elements $\Delta=(\Delta_{ij})_{(i,j)\in\mathcal E}$ that can be represented by numerical assignments fulfilling the following requirements:

(i) \[one\] $\Delta=(\Delta_{ij})_{(i,j)\in\mathcal E}$ is a $\frac{N}{2}(N-1)$-tuple of positive values $\Delta_{ij}\in\mathbb R^{+}$ for each $(i,j)\in\mathcal E$.

(ii) \[two\] For all $(i,j)\in\mathcal E$ one has $\Delta_{ij}=\Delta_{ji}$.

(iii) \[three\] For all $i,j,k\in \mathcal M$, it is the case that $\Delta_{ij}\leq\Delta_{ik}+\Delta_{kj}$.

(iv) \[four\] For all $i \neq j$ there is a $k \neq i,j$ such that $\Delta_{ik} \neq \Delta_{jk}$.

Due to requirements (\[one\]) and (\[two\]), the distance relation is irreflexive, symmetric, and connex. Requirement (\[three\]) is the triangle inequality in virtue of which the relation is a distance relation. Requirement (\[four\]) states that the distance relations individuate the matter points: if matter point $i$ is distinct from matter point $j$, there exists at least one other matter point $k$ such that matter points $i$ and $j$ are distinguished by their relation to $k$. Formulating these requirements in terms of numerical assignments is a means to express the features in virtue of which a relation is a distance relation. However, the numerical assignments do not belong to the ontology, let alone the notion of absolute scale that comes with them; they are just introduced for representational purposes. Consequently, the fact that the values assignable to $\Delta_{ij}$ are real numbers does not smuggle in any infinity in this ontology: there is a finite number of $N$ matter points and, hence, finitely many distance relations.

Nonetheless, by introducing a labelling $\mathcal M$ of the matter points, thi