I think I mention dimensionality in "integration"
(my email to grads)

Here's another application:
you can only apply -homogeneous- functions to dimensional quantities,
otherwise you scramble the dimensions
(more precisely, a homogeneous function of degree $d$ on the line
 on $L^{\otimes k}$
 yields output in
 $L^{\otimes kd}$
 ...but a non-homogeneous function outputs a sum in different
 graded components: it's not natural)

Thus you -cannot- take exp or cos of a -length-,
though of course you can of a -scalar-.
This underlies non-dimensionalization,
and is why you always need to normalize inputs to 'em.
$\cos t$ makes no sense, but
$\cos \omega t$ does (frequency = cycles/time).
