  General relativity cannot be formulated as a perturbatively renormalizable
quantum field theory. An argument relying on the validity of the
Bekenstein-Hawking entropy formula aims at dismissing gravity as
non-renormalizable per se, against hopes (underlying programs such as
Asymptotic Safety) that d-dimensional GR could turn out to have a
non-perturbatively renormalizable d-dimensional quantum field theoretic
formulation. In this note we discuss various forms of highly problematic
semi-classical extrapolations assumed by both sides of the debate concerning
what we call The Entropy Argument, and show that a large class of dimensional
reduction scenarios leads to the blow-up of Bekenstein-Hawking entropy.
