TITLE: How does one actually show from associativity that one can drop parentheses?
QUESTION [28 upvotes]: I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?
$$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$$
Do you just say that the grouping of the parentheses now corresponds to just multiplying straight through? Thanks.

REPLY [2 votes]: Here is a geometric version of a proof:
First, note that parenthecizings of a product of length n  are in bijection with triangulations of convex (n+1)-gon $A$. This is illustrated in the picture below, copied from p. 240 of the book “Discriminants, Resultants, and Multidimensional Determinants” of Gelfand Kapranov and Zelvinski.

In words, to go from triangulations of $A$ to  parenthecizings of $x_1x_2\ldots x_n$, label all but one edges of  $A$ in order as $x_1, x_2, \ldots, x_n$, leaving the last edge as “output edge”. Then a triangulation of $A$ gives a parenthesizing in an inductive manner:  Take the triangle $T$  incident on the output edge; then $A$ with $T$ removed is a union of two polygons $A_l$ and $A_r$ (one of which could be degenerate, i.e. a segment), each of which is incidents on one “non-output” edge of $T$. Then assign to our triangulations the product of parenthecizings assigned to $A_l$ and $A_r$.
One can easily go back in a similarly inductive manner, by thinking of a parentecising $p$ as a product of two subparenthesizings $p_l$ and $p_r$, and correspondingly  getting the triangulation associated to $p$ by “inserting a triangle $T$” into the triangulations of smaller polygons associated to these $p_l$ and $p_r$.
Now that we have a geometric interpretation of parenthecizings, we can give geometric interpretation of single application of associativity law. This is known as a flip — in a given triangulation, one takes two triangles sharing an edge, thus forming a quadrilateral,  and replaces this edge with the other diagonal of this quadrilateral. 
Thus the question becomes: can we connect any two triangulations of a convex polygon by a sequence of “diagonal flips”? The answer is yes, and again is best illustrated by a picture, this time stolen from page 7 of book “Triangulations” by Jesús A. De Loera, Jörg Rambau, and
Francisco Santos.  

In words, pick any vertex  $i$ and make triangulation by drawing all diagonal from that vertex. To connect any triangulation to that “i-standard” one, make flips that increase the degree of $i$: as long as you are not at  “i-standard” triangulation, there exists a triangle $T$ with vertex $i$ and opposite edge not an edge of $A$ (but rather a diagonal); complete $T$ to a guadrilateral, and flip the diagonal.
This shows that “the graph of
flips of the triangulations of the n-gon” is connected (thus establishing what we want). 
QED!
In fact much more is true. The graph of flips is just the union of 1 dimensional faces of a convex polytope, all of whose faces correspond to incomplete triangulations/incomplete parenthecisings, the one open face corresponding to incomplete triangulation with no triangles/incomplete parenthecising with no parenthesis.
What we proved is simply that one can get from any vertex of this polytope to any vertex by following some edges. This is of course true for any convex polytope (take the straight path and push it to faces of lower and lower dimension, for example by projecting "radially" from a point of the face not on the path; this is an easy version of cellular approximation theorem in algebraic topology).
The polytope in question is called an associahedron, or Stasheff polytope, and its existence follows from various direct constructions, notably it is constructed by Gelfand-Kapranov-Zelevinsky as the secondary polytope of a convex n-gone in the plane (this is described in the same book where the first picture came from). 

On a related and more ridiculous note, existence of a permutohedron (secondary polytope of the prism $I\times \Delta^n$ on the simplex $\Delta^n$) shows that commutativity law allows one to write products of any number of elements in any order (this is equivalent to showing that the permutation group is generated by transpositions).