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

"Not just an idle game":(examining some historical conceptual arguments in homotopy theory)