Computing the uncomputable; or, The discrete charm of second-order simulacra

We examine a case in which non-computable behavior in a model is revealed by computer simulation. This is possible due to differing notions of computability for sets in a continuous space. The argument originally given for the validity of the simulation involves a simpler simulation of the simulation, still further simulations thereof, and a universality conjecture. There are difficulties with that argument, but there are other, heuristic arguments supporting the qualitative results. It is urged, using this example, that absolute validation, while highly desirable, is overvalued. Simulations also provide valuable insights that we cannot yet (if ever) prove. If we could take as the finest allegory of simulation the Borges tale where the cartographers of the Empire draw up a map so detailed that it ends up exactly covering the territory..., this fable would then have come full circle for us, and now has nothing but the discrete charm of second-order simulacra. —Jean Baudrillard, Simulacra and Simulation