1. Hubert , “Die Grundlagen der Mathematik” (1927) This paper was presented to the Mathematical Seminar in Hamburg in July 1927; it was published in: Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6 (1928), pp. 65–85. A translation can be found in van Heijenoort (ed.), From Frege to Gödel, Cambridge, 1967, 464–479.—Incidentally, it was Hubert who provoked— by the very formulation of the Entscheidungsproblem for predicate logic—Church’s and Turing’s work on computability, work that is central in theoretical computer science.
2. Those foundational issues go back to the 19th century and were discussed, in particular, during the seventies and eighties. A brief account of this history is given in my paper “Foundations for Analysis and Proof Theory”, Synthese 60(2), 1984, 159–200.
3. Hobbes, De Corpore, in particular the section “Computatio Sive Logica”, and Leibniz, De Arte Combinatoria.
4. On the more psychological side Bernays reports in C. Reid’s Hubert biography (pp. 173–174): For Hubert’s program… experiences out of the early part of his scientific career (in fact, even out of his student days) had considerable significance; namely, his resistance to Kronecker’s tendency to restrict mathematical methods and, particularly, set theory.… In addition, two other motives were in opposition to each other—both strong tendencies in Hilbert’s way of thinking. On one side, he was convinced of the soundness of existing mathematics; on the other side, he had—philosophically—a strong scepticism.… The problem for Hubert… was to bring together these opposing tendencies, and he thought that he could do this through the method of formalizing mathematics.
5. You may ask, can this instrumentalism (radical formalism) be reconciled with Hubert’s view of the soundness (and meaningfulness) of mathematics? My answer is “YES, somewhat plausibly, if the formal theories under consideration are complete.”—The notion of completeness figures already significantly in Hubert’s earliest foundational paper “Über den Zahlbegriff” (1899). The formalisms for elementary number theory and analysis set up in the twenties were believed to be complete in the Hubert school. If a formalism allows then, as Hilbert put it, “to express the whole content of mathematics in a uniform way”, and if it provides “a picture of the whole science”, why not make a bold methodological turn and avoid all the epistemological problems connected with the (classical treatment of the) infinite?