Michaek Spivak advocates that "One of the dogmas of modern mathematics is that for any object that is invariant in any sort of way, a definition must be found that exhibits this invariance directly (even if such a definition is harder to understand than the original!)"
His main example is the fruitful development of Riemannian geometry and in particular the curvature tensor motivated by the search of an intrinsic definition of Gauss curvature, which was shown by Gauss's Theorema Egregium to be invariant under local isometry. Another example he gave is the basis independent definition of determinant based on exterior algebra.
In our paper, we show that the proximity condition is invariant under change of coordinates. However, it does not mean that we have a coordinate independent definition of the proximity condition. It remains open to find such an invariant definition.
(Thomas Yu, May 9, 2012.)