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This paper presents a systematic study of the prehistory of the traditionalsubsystems of second-order arithmetic that feature prominently in the reversemathematics program of Friedman and Simpson. We look in particular at: (i) thelong arc from Poincar\'e to Feferman as concerns arithmetic definability andprovability, (ii) the interplay between finitism and the formalization ofanalysis in the lecture notes and publications of Hilbert and Bernays, (iii)the uncertainty as to the constructive status of principles equivalent to WeakK\"onig's Lemma, and (iv) the large-scale intellectual backdrop to arithmeticaltransfinite recursion in descriptive set theory and its effectivization byBorel, Lusin, Addison, and others.

The Prehistory of the Subsystems of Second-Order Arithmetic