The Fourier Integral

Question: What aspect of nature is responsible for the pervasive importance of Fourier analysis?

Answer: Translation invariance. Suppose a linear system is invariant under time or space translations. Then that system's behaviour becomes particularly perspicuous, physically and mathematically, when it is described in terms of translation eigenfunctions, i.e., in terms of exponentials which oscillate under time or space translations. (Nota bene: real exponentials are also translation eigenfunctions, but they won't do because they blow up at $-\infty$ or $+\infty$ .) In other words, it is the translation invariance in nature which makes Fourier analysis possible and profitable.