For a signal $x$, the Fourier transform is defined as:
$$\mathcal{F}\{x(t)\} = F(j \omega) = \int_{- \infty}^{+ \infty} x(t) e^{-j \omega t} \,dt$$
We can see the signal $x$ as a sum of periodic signals of different frequencies. The $e^{-j \omega t}$ factor inside the integral acts as sort of an "indicator" or "scanner" so as for the Fourier transform for a certain frequency $\omega$ to reflect the weight (in physical terms, the amplitude) of the periodic constituent of that frequency in the makeup of $x$.
My question is, why make add the $-$ and make the angle vary clockwise instead of respecting the usual convention of making it vary counterclockwise? Is there any logical or historical reason for this?
The point is that to detect a "signal" $e^{j\omega t}$ that goes counterclockwise at rate $\omega$, you multiply it by $e^{-j\omega t}$ (making it constant) and integrate.