The goals of reduction and reductionism in the natural sciences are mainly
explanatory in character, while those in mathematics are primarily foundati
onal. In contrast to global reductionist programs which aim to reduce all o
f mathematics to one supposedly "universal" system or foundational scheme,
reductive proof theory pursues local reductions of one formal system to ano
ther which is more justified in some sense. In this direction, two specific
rationales have been proposed as aims for reductive proof theory, the cons
tructive consistency-proof rationale and the foundational reduction rationa
le. However, recent advances in proof theory force one to consider the viab
ility of these rationales. Despite the genuine problems of foundational sig
nificance raised by that work, the paper concludes with a defense of reduct
ive proof theory at a minimum as one of the principal means to lay out what
rests on what in mathematics. In an extensive appendix to the paper, vario
us reduction relations between systems are explained and compared, and argu
ments against proof-theoretic reduction as a "good" reducibility relation a
re taken up and rebutted.