Writing function application as a left action has been baked into mathematics since the Bernoullis in the 1700s (cf. this MSE question). Because of this tradition, a lot of mathematics notation has been chosen to complement this.
One (perhaps shocking) example is the commutator $[a,b]$ in a group. Traditionally we use left group actions (to be in line with left-function application), and so the conjugation action is $g \cdot h = ghg^{-1}$. This leads us to define $[a,b] = aba^{-1}b^{-1}$ (so that, among other things, $s \cdot [a,b] = [s \cdot a, s \cdot b]$).
It has become accepted to instead use right group actions, denoted $h^g = g^{-1}hg$, and with this convention we would instead write $[a,b] = a^{-1}b^{-1}ab$. This gives the identity $[a,b]^s = [a^s, b^s]$.
There are lots of places in mathematics where we have an arbitrary choice to make. Left/Right Cosets in algebra, Left/Right Linearity of the complex inner product, etc. I am curious if any of these conventions have to do with functions acting on the left or the right, as in the above example.
Thanks in advance!