Explore a 46-minute conference talk from the Workshop on "Non-regular Spacetime Geometry" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into Connes's spectral distance, an extended metric on the state space of a C*-algebra that generalizes Kantorovich's dual formula of the Wasserstein distance of order 1 from optimal transport. Discover a new dual formula, presented as an infimum, which generalizes Beckmann's "dual of the dual" formulation of the Wasserstein distance. Gain insights into noncommutative geometry and its applications in mathematical physics through this advanced presentation on the dual of the dual formula for distance.