There is a beautiful tension in topology between positive classification theorems and negative “no-go” theorems. The positive results come from geometry, and often derive ultimately from analysis. The negative results, by contrast, come from undecidability results in logic. I’ll give a survey of the history of this tension, and mention the highlight theorems — examples include Markov’s theorem that 4-manifolds cannot be classified (on the negative side), and Perelman’s Geometrization Theorem in dimension 3 (on the positive side). I’ll then go on to describe some recent undecidability results, which limit possible computations in matrix groups. This is joint work with Martin Bridson.