e := v \;|\; \lambda v.e \;|\; (e\,e)

He also very explicitly told us that we were never to leave off parentheses, as this would change the meaning. We also did several examples in this notation; here's our formulation of Church numerals:

\begin{eqnarray} \lambda f.\lambda x.x & \equiv & 0 \\

\lambda f.\lambda x. (f\,x) & \equiv & 1 \\

\lambda f.\lambda x. (f\,(f\,x)) & \equiv & 2 \\

& \vdots &

\end{eqnarray}

\lambda f.\lambda x. (f\,x) & \equiv & 1 \\

\lambda f.\lambda x. (f\,(f\,x)) & \equiv & 2 \\

& \vdots &

\end{eqnarray}

On the other hand, Wikipedia's formulation of lambda calculus is a little different:

e := v \;|\; (\lambda v.e) \;|\; (e\,e)

Wikipedia also states that the outermost parentheses can be omitted "to keep the notation... uncluttered" and does a number of other minor abbreviations.

Both sources present their side as definitive and do not acknowledge alternative formulations; Wikipedia does state that dropping the outer parentheses is a "usual convention" and presumably not universal.

I guess what I'm asking is whether there is any one "definitive" notation here, and if not, which one is awesomer.

Incidentally, one of our assignments we're to build a Scheme parser, which appears to be closer to the Wikipedia notation; the professor did not mention this and I think it may cause considerable confusion once other people start looking at it.

Honestly, the first notation makes more sense to me since a beta-reduction won't have extra internal parentheses to deal with; maybe it's just because I was introduced to it first.