“All the Resources of Pure Mathematics”: Mathematical Physics, and Mathematics Against Physics

This chapter argues for an asymmetry in the reciprocal impact of modern mathematics and modern physics on each other, an impact commonly seen more symmetrically. By modern mathematics I refer to mathematics that, from its emergence around 1800 to our own time, was defined, as abstract mathematics, by separating, abstracting, itself from natural objects and thus from physics, as well as from the objects of daily thinking. Modern physics, which emerged two centuries earlier, along with what we call modernity itself, and has continued to our own time as well, is a mathematical-experimental science, with mathematics defining this conjunction. As such it has not and could have not separated itself from mathematics. Indeed, as this chapter argues, twentieth-century physics, especially relativity and quantum theory, was able to use modern mathematical theories born from the separation of modern mathematics from physics. Reciprocally, throughout its history, beginning with the invention of calculus, modern physics has had a major influence on mathematics, including modern mathematics and, with relativity and quantum theory, on the twentieth- and twenty-first century of mathematics. This chapter argues for an asymmetry in this reciprocal impact, in favor of the impact of modern mathematics on modern physics, while this mutual impact is commonly seen more symmetrically, and sometimes by way of a reverse asymmetry, in favor of theoretical physics in helping to resolve some outstanding problems of pure mathematics. This chapter will challenge and qualify the latter view by arguing that theoretical physics cannot do so short of becoming mathematics, by, as modern mathematics itself has done, separating itself from physics. This chapter sees all modern physics as defined primarily by the invention of new theories, comprised of mathematized conceptual conglomerates, capable of predicting and, in some cases, representing the physical reality responsible for the phenomena considered, which are part of this reality. The invention of new mathematics, at least new for physics (this mathematics can be borrowed from already existing mathematics), is, the chapter argues, what has most fundamentally defined modern physics from its emergence until our own time.