The Role of Mathematics in Fundamental Physics

In this chapter, I argue that the unreasonable effectiveness of mathematics in fundamental physics has become evident through our growing understanding of the significance of symmetry principles in nature and the ever-expanding repertoire of mathematical concepts and tools developed to extend that understanding. Symmetry principles are not inherently mathematical—they have formed the conceptual basis of philosophical arguments since earliest times. However, in the last 100 years or so, they have been taken to a new level of sophistication, enabling us to explore unobservable physical realms where the concepts are, of necessity, physico-mathematical, in the sense that the unobservable physical entities are directly represented by the mathematical forms and our only knowledge of them is implicit in the mathematics. For example, consider the modern gauge theory of QCD which describes the dynamics of quarks and gluons inside nucleons. The forces and particles which it contains are determined by the invariance of its equations under the local symmetry group SU(3). Wilczek, one of the discoverers of QCD, writes: