I'm reading Adams' book Infinite Loop Spaces. He explains that the product map on a loop space isn't associative, but it is associative up to coherent homotopy. I'm confused about the coherent part of this though. In his explanation he uses a pentagon where each edge is a homotopy between the various ways to place parentheses on a product of four elements (as in (ab)(cd), a(b(cd)) and so on). He says that those five homotopies give a map on $S^1 \times X^4 \rightarrow X$, where we're thinking of $S^1$ as the boundary of a pentagon. Then he says that sometimes it can be extended to a map $D^2 \times X^4 \rightarrow X$ and sometimes it can't, and this ability to be extended is the coherence we're looking for.
But what does being able to extend the map to the whole disk do for us. Why isn't associativity up to homotopy sufficient?