Abstract: I will answer the following problem: Let I be a differential invariant, no matter how it was computed, determine systematically the corresponding approximating discrete invariant. Our approach takes advantage of the new geometrical construction of multi-spaces introduced by P. Olver where we can switch at our convenience from/to normal derivatives to/from divided differences. Multi-spaces are a far reaching generalization of the classical blow-up construction in algebraic geometry. New examples that heretofore were intractable are now easily solved.