  The consequences of adopting other definitions of the concepts of sum and
convergence of a series are discussed in the light of historical and
epistemological contexts. We show that some divergent series appearing in the
context of renormalization methods cannot be assigned finite values in a form
consistent with Hardy's axioms without at the same time equating one to zero,
thus destroying the mathematical building. We show that if the replacements for
the concept of sum of a series are required to be associative, to be invariant
under finite permutations of the terms and dilution, further restrictions
emerge. We finally discuss the epistemological costs of accepting these
practices in the name of instrumentalism.
