  Mathematical challenges punctuate the history of early modern mathematics.
While cultural historians have attempted to contextualize these challenges
among contemporary practices, in particular duels or advertisements in a
competitive market, thus emphasizing their interpersonal and social dimensions,
historians of mathematics have generally treated them as somewhat childish
remnants of a pre-scientific age, that the advent of modern science and its
Baconian ideal of efficient collaboration would soon bring to an end. However,
the number of challenges did not decrease but rather multiplied inside one of
the first scientific organizations aiming at cooperative work--Marin Mersenne's
network. This paradox has suggested the focus of this article : to examine the
role of challenges in the economy of mathematical exchange (and mathematical
creation) in early modern France. Through examples of successful, but also of
unsuccessful challenges, we shall see how challenges operated, not only as
"mises en sc\`ene" of methodological oppositions, but also, and primarily, as
links in a mathematical environment structured inside correspondences around
the resolution of problems. This situation also exemplifies how controversies
may have been a constitutive part of normal scientific activities--and not
their disruption--while shaping their development in specific, limited
directions.
