"Mathematicians don't use square brackets for functions"

Not true at all:

1) Operators acting on a distribution: E[X], Var[X], Cov[X,Y]

2) Square brackets are used in nested parentheses to avoid confusion: [1(2 + 3) + 4]/5

3) Functionals: L[f] for lagrangian

The main reason why [] isn't used as frequently as () is because most domains interpret the square brackets to mean something special. For example:

1) [x] is often used to denote floor(x)

2) [i=j] is used to denote the kronecker delta