  Geometrical properties of spacetime are difficult to study in nonperturbative
approaches to quantum gravity like Causal Dynamical Triangulations (CDT), where
one uses simplicial manifolds to define the gravitational path integral,
instead of Riemannian manifolds. In particular, in CDT one only relies on two
mathematical tools, a distance measure and a volume measure. In this paper, we
define a notion of scalar curvature, for metric spaces endowed with a volume
measure or a random walk, without assuming nor using notions of tensor
calculus. Furthermore, we directly define the Ricci scalar, without the need of
defining and computing the Riemann or the Ricci tensor a priori. For this, we
make use of quantities, like the surface of a geodesic sphere, or the return
probability of scalar diffusion processes, that can be computed in these metric
spaces, as in a Riemannian manifold, where they receive scalar curvature
contributions. Our definitions recover the classical results of scalar
curvature when the sets are Riemannian manifolds. We propose seven methods to
compute the scalar curvature in these spaces, and we compare their features in
natural implementations in discrete spaces. The defined generalized scalar
curvatures are easily implemented on discrete spaces, like graphs. We present
the results of our definitions on random triangulations of a 2D sphere and
plane. Additionally, we show the results of our generalized scalar curvatures
on the quantum geometries of 2D CDT, where we find that all our definitions
indicate a flat ground state of the gravitational path integral.
