  The well-known Eckmann-Hilton Principle may be applied to prove that
fundamental groups of $H$-spaces are commutative. In this paper, we identify an
infinitary analogue of the Eckmann-Hilton Principle that applies to fundamental
groups of all topological monoids and slightly more general objects called
pre-$\Delta$-monoids. In particular, we show that every pre-$\Delta$-monoid $M$
is "transfinitely $\pi_1$-commutative" in the sense that permutation of the
factors of any infinite loop-concatenation indexed by a countably infinite
order and based at the identity $e\in M$ is a homotopy invariant action. We
also give a detailed account of fundamental groups of James reduced products
and apply transfinite $\pi_1$-commutativity to make several computations.
