Mathematics has proven to be "unreasonably effective" in understanding nature. The fundamental laws of physics can be captured in beautiful formulae. In this lecture, given at the Perimeter Institute for Theoretical Physics, Robbert Dijkgraaf, Director and Leon Levy Professor, argues for the reverse effect: Nature is an important source of inspiration for mathematics, even of the purest kind. In recent years, ideas from quantum field theory, elementary particle physics, and string theory have completely transformed mathematics, leading to solutions of deep problems, suggesting new invariants in geometry and topology, and, perhaps most importantly, putting modern mathematical ideas in a "natural" context.