Page 159 - Textos de Matemática Vol. 40

A note on the history of functional self-application 147 in fact, is directly linked to the paragraph of Hilbert’s lecture from 1905 we
study here.
Alonzo Church visited Amsterdam and G¨ottingen in 1928 and 1929 to study intuitionism and formalism with Luitzen Brouwer (1881–1966) and Hilbert as a fellow of the International Education Board, [SS01, pp. 101 and 290]. Although his first work on λ-calculus was published only in 1932 and 1933 [Chu32, Chu33], it is explicitely mentioned in [Chu32, p. 346] that the work was done by him as “a National Research Fellow in 1928–1929.” We also learn from Curry that there were already rudimentary versions of the λ-calculus available in G¨ottingen: “I recall seeing in G¨ottingen in 1928, but not comprehending, an uncirculated manuscript of his containing λ’s which was doubtless a progenitor.” [Cur80, p. 88]. Thus, also the development of the λ-calculus might have been influenced by discussions with Hilbert.49
In contrast to the former ones, the link to Johann von Neumann is less close, since we are interested here in the possibility of self-application in his computer model (and not in his contribution to set theory, which does not relate directly to self-application50). Von Neumann started his foundational work with the development of his set theory in the 1920s. He stayed in G¨ottingen during the winter term of 1926–1927 supported by the International Educational Board, cf. [SS01, p. 296], [Has06a, p. 139], and [Sie03, p. 328], (as Church later), but he apparently already visited Hilbert before, in 1924 and/or 1925. As Reid reports in her biography of Hilbert: “Von Neumann was 21 in 1924, deeply interested in Hilbert’s approach to physics and also in his ideas on proof theory. The two mathematicians, more than forty years apart in age, spent long hours together in Hilbert’s garden or in his study.” [Rei70, p. 172]. One of the topics they discussed may have been the mathematics of quantum mechanics, since the only joint paper [HvNN27] of Hilbert and von Neumann (together with Lothar Nordheim (1899–1985)) is from this area. But the foundations of mathematics will also have played a major role in these conversations. Von Neumann took an active part in the research of Hilbert’s programme, as one may see from his publications [vN27, vN29, vN31a, vN31b]51 and from the
49In [Kle03] the author discusses some papers of Russell’s Nachlaß which could be taken as evidence of an anticipation of the lambda calculus by Russell in 1903–1905. As far as we see, this relates mainly to the λ-abstraction, but does not involve the possibility of self- application. In relation to Church the author states [Kle03, p. 18]: “Certainly, Church did not simply extract the rules of the Lambda Calculus by reading Principia; Curry, Sch¨onfinkel and Hilbert surely provided more help when it came to the essential details of what was new in his innovation. If we wished to understand the actual historical development of the Lambda Calculus, a study of these figures would figure more prominently.” We hope to contribute with this paper to such an understanding. In addition, we would suggest adding the name of Bernays to the list.
50With the exception of the—excluded—use of it to model Russell’s paradox mentioned above.
51The first paper, entitled Zur Hilbertschen Beweistheorie was already written in 1925, cf. [Sie03, p. 328].