Real algebraic geometry, at least to start with, traditionally studies the zero-sets of real polynomials in a given set of variables. But treating, say, the Euclidean plane as an uncoordinatized metric space, one may still consider the ring of functions generated by all functions $D_p(\cdot)=d(\cdot,p)^2$, and also constant functions. Then one has really lost nothing, because (bring back coordinates $x,y$ just for the moment), $x=(D_{(0,0)}-D_{(1,0)} +1)/2$ and $y=(D_{(0,0)}-D_{(0,1)} +1)/2$.