  Part of the title of this article is taken from writings of Einstein, which
argue that we need to exercise our ability to analyse familiar concepts, to
demonstrate the conditions on which their justification and usefulness depend,
and the way in which these developed, little by little $\ldots$. My aim is to
do this for the initial negative reactions to the seminar by E. Cech on higher
homotopy groups to the ICM meeting in Z\" urich in 1932; then the subsequent
work of Hurewicz, the use of groupoids and so the use of many base points, and
how J.H.C. Whitehead's use of free crossed modules gave rise to a successful
search for higher dimensional versions of the fundamental group and of the
theorem of Van Kampen.
