  We introduce an algebraic system which can be used as a model for spaces with
geodesic paths between any two of their points. This new algebraic structure is
based on the notion of mobility algebra which has recently been introduced as a
model for the unit interval of real numbers. Mobility algebras consist on a set
$A$ together with three constants and a ternary operation. In the case of the
closed unit interval $A=[0,1]$, the three constants are 0, 1 and 1/2 while the
ternary operation is $p(x,y,z)=x-yx+yz$. A mobility space is a set $X$ together
with a map $q\colon{X\times A\times X\to X}$ with the meaning that $q(x,t,y)$
indicates the position of a particle moving from point $x$ to point $y$ at the
instant $t\in A$, along a geodesic path within the space $X$. A mobility space
is thus defined with respect to a mobility algebra, in the same way as a module
is defined over a ring. We introduce the axioms for mobility spaces,
investigate the main properties and give examples. We also establish the
connection between the algebraic context and the one of spaces with geodesic
paths. The connection with affine spaces is briefly mentioned.
