The global dynamics of nonlinear systems can exhibit structures over wide varieties of temporal and spatial scales and the same phenomenon is true with respect to parameters. However, from an experimental or computational perspective only a finite amount of information can be collected and measurements can only be made up to a certain level of precision. With this in mind I will argue that we need an alternative description to the qualitative theory of dynamics, preferably based on principles that are easily accessed computationally. I will present some results that represent efforts to develop a theory of dynamical systems that is based on combinatorics and algebraic topology.