  Over the past few decades the notion of symmetry has played a major role in
physics and in the philosophy of physics. Philosophers have used symmetry to
discuss the ontology and seeming objectivity of the laws of physics. We
introduce several notions of symmetry in mathematics and explain how they can
also be used in resolving different problems in the philosophy of mathematics.
We use symmetry to discuss the objectivity of mathematics, the role of
mathematical objects, the unreasonable effectiveness of mathematics and the
relationship of mathematics to physics.
