What can we learn by observing nature? How can we understand and predict natural phenomena? This talk is on the mathematics of precision measurement. How can we solve for the input that generated the output of some measurement apparatus? Our starting point is an information theoretic prior of sparsity. We investigate sparse inverse problems where we assume the input can be described by a small number of parameters. We introduce some of our recent theoretical results in superresolution and in spectral clustering. In particular, we show how to solve infinite dimensional deconvolution problems with finite dimensional convex optimization. And we show why dimensionality reduction can be such a useful preprocessing step for mixture models.