The relationship between mathematics and physics is defined by a deep intimacy where mathematics provides the essential language for physical laws. Historically, this bond has been forged through necessity; for example, the development of calculus was driven by the need to describe the dynamics of motion established by Galileo and Isaac Newton. During this period, the distinction between the two was so thin that Newton regarded geometry as a branch of mechanics. This interplay continues to drive modern science, as seen when the transition to Riemannian geometry provided the specific mathematical framework Albert Einstein required to formulate general relativity.
The article discusses how calculus was developed to describe the dynamics of motion established by Newton.
Central to this relationship is the principle of simplicity, a concept emphasized in Paul Dirac's 1939 speech. Dirac suggested that the success of physical theories often stems from the fact that equations of a simple form tend to work. For a physicist, this provides a 'principle of simplicity'—a formal basis for research where the elegance of a mathematical structure serves as a guide for identifying the underlying laws of nature. This aligns with the long-standing philosophical view, shared by the Pythagoreans and later echoed by Galileo, that the 'book of nature' is written in the language of mathematics. In this view, the aesthetic quality of a mathematical expression is not merely a preference but a heuristic for truth.
However, despite their shared language, mathematics and physics remain distinct in their ultimate objectives. Mathematics seeks absolute logical certainty through rigorous proof, dealing with entities whose properties are known with certainty. Physics, by contrast, operates in the realm of 'probable reasoning,' where models are abstractions or idealizations. As noted by Georg Rasch, no physical model is ever 'true' in an absolute sense; instead, they are evaluated based on their applicability to the natural world. While a mathematician demands a proof that is universally certain, a physicist uses the 'beautiful' simplicity of mathematics to build models that are accurate enough to describe reality, even if they are not absolute truths.