I've found a few articles that write the ring of formal Laurent series in $t$ as $R((1/t))$, but what's the underlying meaning of $\cdot ((\cdot))$?

A mathematician of my acquaintance swears that $R((t))$, not $R((1/t))$, should be used to denote the ring of formal Laurent series in $t$. We can't decide who's right without knowing what $\cdot((\cdot))$ means. (We both agree that $R[[t]]$ denotes the ring of formal power series in $t$ with coefficients in $R$.)