Part of the consideration of the question of the connection of mathematics to the understanding of nature is the discovery of symmetries in nature, and the formulation of this symmetry in mathematics. Symmetry was established in this connection when Emily Noether proved that: with every symmetry of a system, there is a conserved quantity, or a conservation law for this quantity. The public is generally not aware of the importance that symmetry plays in our understanding of nature.
One of the earliest examples of this is Faraday’s formulation of the laws of electricity and of magnetism, with a dual or reciprocal role between them and with changes in one of the fields generating the other type of field. This was formulated mathematically by James Clerck Maxwell. This symmetry, that a changing magnetic field generates an electric field, and that the changing electrical field regenerates a magnetic field, led to the understanding of the generation and propagation of light as a coupled set of propagating electric and magnetic fields.
The next great symmetry was that the laws of physics should be formulate-able such that they are and appear the same to different observers that are only traveling with a constant velocity with respect to each other. In other words, we don’t live in a media called the ether, in which one can measure their velocity with respect to a special media rest frame, and in which the laws can depend on that velocity. This is called Lorentz invariance, and was instituted by Poincare and Einstein also. The amazing, but we now see required, fact is that Maxwell’s equations describing light had to be invariant under those uniformly moving transformations of coordinate system by a constant velocity, called Lorentz Transformations. Consequently, light moves with the same velocity called c in each of those reference frames, and in every direction with the same velocity, and with the same velocity independent of the velocity of the emitter.
What are the consequent conservation laws from this invariance under the symmetry transformations, according to Noether’s theorem? Nothing less than the conservations of momentum and energy.
When formulating gravity, Einstein extended the symmetry to that between frames that differed by acceleration. Since gravity is indistinguishable from the acceleration of a person in an elevator, both of which press you to the floor, (the Principle of Equivalence), gravity was the result of this imposed symmetry, as the field which causes the acceleration due to matter.
In accounting for the more recently discovered theories and forces of the weak and strong interactions, the symmetry of the interchange of fundamental particles with different but equivalent “charges” of new types play a key role. This occurs through the considerations of the quantum nature of the world. I leave the more complete discussion of this for books such as “The Lightness of Being” by Nobel Laureate Frank Wilczek, or his YouTube video of his talk at UC Irvine as the Frederick Reines Lecturer.