This talk is a case-study about the utility of probabilistic formulations for numerical mathematics. I present a recent result showing that quasi-Newton methods can be interpreted as performing Gaussian (least-squares) regression on the Hessian of the objective function, using a particular noise process to keep uncertainty constant, and a non-obvious structured prior which ignores the duality between vectors and co-vectors. This insight connects these numerical methods to important areas of machine learning (regression) and control (Kalman filters). It allows cross-fertilization: Better numerical algorithms can be built using existing knowledge from machine learning, and machine learning can benefit from a new structured prior model allowing linear-cost inference on matrix-valued operators. Arguing for more and closer interaction between the fields of learning and numerical mathematics, I also point out some challenges arising from cultural differences between these communities.