The hydrodynamic limit of the Kawasaki dynamics states that a certain stochastic evolution of a lattice system converges macroscopically to a deterministic non-linear heat equation. We will discuss how the statement of the hydrodynamic limit can be made quantitative. The key step is to introduce an additional evolution on a mesoscopic scale that emerges from projecting the microscopic observables onto splines. The hydrodynamic limit is then deduced in two steps. In the first step one shows the convergence of the microscopic to the mesoscopic evolution and in the second step one deduces the convergence of the mesoscopic to the macroscopic evolution. The talk is about a joint work with Deniz Dizdar, Felix Otto and Tianqi Wu.