TITLE: Eckmann-Hilton and higher homotopy groups
QUESTION [17 upvotes]: How does the Eckmann-Hilton argument show that higher homotopy groups are commutative? 

I can easily follow the proof on Wikipedia, but I have no good mental picture of the higher homotopy groups, and I can't see how to apply it. Wikipedia mentions this application in one sentence, with no explanation. 
(Motivation: I'm thinking of giving a talk on algebraic topology for a general audience of math majors. The purpose would be to try to explain how purely "algebraic" methods can be used to gain serious insight into purely "geometric" problems. This would be a great example, if I could do the proof!)
Thanks in advance!

REPLY [14 votes]: The following is relevant to the point made by Qiaochu Yuan.
What is usually called the Eckmann-Hilton argument is actually a special application of the interchange law, whose quite general setting is for double categories, i.e. sets with two distinct category structures such that, and this is the interchange law, each is a morphism for the other. However, even when each of the structures is a groupoid, i.e. all arrows are isomorphisms, the interchange law does not lead to a commutative structure but only that it contains a family of abelian groups. Also the interchange law implies that double groupoids contain the structures known as crossed modules, which occur in homotopy theory and the cohomology of groups. In fact $n$-fold groupoids become more complicated with increasing $n$.
The existence of such double structure, first formulated by C. Ehresmann in the 1960s, raised the question of their potential use in homotopy theory, a question relevant to the history of homotopy groups, and to their abelian nature.
Topologists of the early 20th century were aware that the non commutative nature of the fundamental group was useful in geometry and analysis;
that the first homology group was,  for a connected space, the fundamental group made abelian; and that the homology groups were defined in all dimensions. Consequently,
there was a desire to find higher dimensional versions of the fundamental group, keeping its nonabelian nature.
In 1932, E. Cech submitted to the ICM at Zurich a paper on higher homotopy groups; however, Alexandroff and Hopf, the kings of topology at the time, objected to the fact that they were abelian  for $n \geq 2$, and on these grounds persuaded Czech to withdraw his paper, so that only a small paragraph appeared in the ICM Proceedings.
It seems that Hurewicz attended this ICM. In 1935, the first of his notes on homotopy groups was published, and from then the concerns about the abelian nature of higher homotopy groups were regarded  as  a failure to accept a basic fact of life.
In a publication of 1978  R. Brown and P.J. Higgins  defined the homotopy double groupoid $\rho(X,A,x)$ of a pointed pair of spaces, which consisted of homotopy classes of maps of a square $I^2$ into $X$ which mapped the edges to $A$ and the vertices to the base point $x$. This enabled the proof of a 2-dimensional van Kampen theorem, which had  as a special case a result of Whitehead on free crossed modules.
This work was in fact inspired by that  work by J.H. C. Whitehead who in the 1940s  developed a deeper understanding of the nonabelian second relative homotopy group of a pair $(X,A)$, with an operation of $\pi_1(A,x)$,  and introduced the notion of crossed module for the structure of the boundary map $\delta: \pi_2(X,A,x) \to \pi_1(A,x)$. He also gave a deep determination of $\pi_2(A \cup \{e^2_\lambda\}_{\lambda \in \Lambda}, A,x)$ as a free crossed $\pi_1(A,x)$-module on the characteristic maps of the $2$-cells.
One advantage of this result is that allows for an expression that we very much want, namely that in the standard representation of a Klein bottle as a square $\sigma$ with boundary we would like to write $$\delta \sigma= a+b-a+b,$$
which is a noncommutative formula, and this is exactly possible in the context of crossed modules, with $\sigma$ as a free generator. In the usual chain complex system we can of course write only
$$\partial \sigma= 2b,$$
thus giving a loss of information.
The interchange law is thus central to  the use of higher groupoids in homotopy theory and other areas. There are lots of fun things, with an extra structure of connections to enable such things as the notions of commutative cubes and of rotations.
For more details on maths and history, see the pdf of this  book, and also this paper, available there  in a special issue of Indagationes Math. in honor of L.E.J. Brouwer.