I came across this comment:
Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more interesting: their lack of rigor is usually a reflection of the inadequacy of existing mathematical tools for their purposes. When such a theory is read by a mathematician, he is forced to invent new tools or use existing tools in a way he had never thought to before in order to make sense of it. This in turn has spurred many mathematical advances and even created new fields of study. For example, many of the most important PDEs (heat, wave, Laplace, Navier-Stokes, Schrodinger, etc.) were first encountered in physics.
I'm curious in what other ways physics (or other pursuits of physical/scientific/computational truth) have sparked the creation of new mathematical tools to understand and formalize these things. Is one example Turing's formalization of computation, or Shannon's formalization of information?
Best Answer
Fourier analysis, which has applications in fields as diverse as PDEs and algebraic number theory, has its origins in physics (which, at the time, really wasn't as distinct from mathematics as it is now). The study of waves and heat flow lead Fourier and others to the theory of trigonometric series.
For example, by working formally you can show that solutions to the heat equation in the unit disc, with boundary function $f(\theta)$, should be of the form $u(r,\theta) = \sum_{n = -\infty}^{\infty} a_{n}r^{|n|}e^{in\theta}$, where $a_{n} = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx} \ dx$. In particular, if $f$ is nice enough then it should be the case that $f(\theta) = u(1,\theta) = \sum_{n = -\infty}^{\infty} a_{n}e^{in\theta}$, which is a Fourier series. This leads to the fundamental question of Fourier analysis: is such a representation possible?