Proof-theoretic reduction as a philosopher's tool (Q1841003)

The philosophical significance of proof-theoretic reductions in the sense of \textit{S. Feferman} [J. Symb. Logic 53, 364-384 (1988; Zbl 0656.03002)] is discussed. According to the author, these reductions are usable neither as theory reduction nor as ontological reductions. Proof-theoretic reductions, however, have their uses for refuting indispensability arguments, because they may show that certain mathematical theories are dispensable; they provide also foundational reductions, as pointed out by Feferman (loc. cit.) and relative consistency proofs. The latter are especially important if some theory is assumed to be primary and true, to which then others can be reduced. The author takes arithmetic as such a base theory and endorses an instrumentalist view to mathematical theories that can be reduced to arithmetic. How proof-theoretic reductions might be useful in a general instrumentalist account of large portions of mathematics is only outlined.