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*To*: fom@math.psu.edu, Bernd Buldt <Bernd.Buldt@uni-konstanz.de>*Subject*: FOM: Recent exchange on HILBERT*From*: Wilfried Sieg <ws15+@andrew.cmu.edu>*Date*: Tue, 29 Jun 1999 09:21:33 -0400 (EDT)*In-Reply-To*: <l03130304b39511076cf6@[134.34.91.246]>*References*: <l03130304b39511076cf6@[134.34.91.246]>*Sender*: owner-fom@math.psu.edu

In his recent posting Mic Detlefsen does not find any evidence for logicist tendencies in Hilbert's writings - published or unpublished, whereas Bernd Buldt takes it in his posting to be "uncontroversial that there was a logicist stage in hilbert's development; especially so, because hilbert himself says so". I agree with Buldt on the substance of this issue and also with most of his other remarks, e.g., on solvability/decidability. However, from both an historical and systematic point of view, the remarks in "Axiomatisches Denken" and other snippets are prima facie isolated and puzzling. After all, Hilbert and Bernays both emphasize the continuity between Hilbert's considerations in the Heidelberg talk of 1904 and his finitist program of the early twenties; the intervening years are hardly discussed, the lectures on foundations of mathematics (between 1917 and 1922) not mentioned - except in Hilbert's preface to his book with Ackermann. What role Principia Mathematica played in Hilbert's development (and when!) was quite obscure. There remains ample room for improving our understanding, but at least we know now some crucial facts! Hilbert was getting acquainted with Whitehead and Russell's work definitely no later than 1913, but precisely what was studied at the time is still not clear. The bare fact comes from remarks of Russell's in early 1914 and a postcard exchange between Hilbert and Russell (from 1916 to 1919). Alasdair Urquhart informed me about these matters; a transcription of the exchange is published in Appendix B of my BSL paper "Hilbert's Programs: 1917-1922". Hilbert mentions in his first postcard of April 12, 1916 that the Mathematische Gesellschaft had intended to invite Russell (before the outbreak of the First World War) to give lectures in Goettingen and adds: "Ich hoffe, dass die Ausfuehrung dieses Planes durch den Krieg nicht aufgehoben, sondern nur aufgeschoben worden ist." - The growing acquaintance with Russell and Whitehead's work is hardly reflected in Hilbert's Lecture Notes: only some brief side remarks in Notes from 1914/15. Additional information is provided by the 1918 dissertation of Hilbert's student Behmann, that was analyzed recently in a manuscript by Paolo Mancosu. In any event, there is a radically new presentation of logic in the 1917/18 notes. To reemphasize a point I made in my note to Jacques Dubucs and Neil Tennant, these notes are a complete draft of Hilbert & Ackermann's book "Grundzuege der theoretischen Logik" (1928) and can be viewed properly as the starting point of modern mathematical logic. How difficult it was to assess the developments in Goettingen on the basis of publications can be seen, strikingly, from Warren Goldfarb's paper "Logic in the Twenties: the Nature of Quantifiers", JSL 44 (1979). Goldfarb viewed Hilbert & Ackermann's book as the endproduct of a cumulative development in Goettingen that started with Hilbert's 1922 and 1923 papers! Let me add a couple of remarks concerning the informativeness of the unpublished notes for the lectures presented between 1917/18 and 1922/23. Excerpts from mail: 23-Jun-99 FOM: recent fom-exchange on.. by Bernd Buldt@uni-konstanz > bernays confirms on p. 202 of his 1935-report on hilbert's > foundational work (enclosed to vol. III of hilbert's "gesammelte > abhandlungen"), that hilbert started out from the frege-russell-project, > trying to supply only the missing consistency proof: "thus hilbert was left > with the task of providing a consistency proof for these [i.e., frege's and > russell's unproved] assumptions." but in order to do so, he envisaged a > proof-theory, which was also designed to meet the constrcutive demands put > forward by weyl and brouwer. bernays then turns to the more 'riper' > developments of hilbert's program(me), leaving us in the dark concerning > the details. The notes lift the darkness! It is precisely the development from a logicist program through an attempted constructive presentation of mathematics to the finitist program that can be traced in reasoned detail from these notes (and that is described in my BSL paper). The essay by Bernays, mentioned in the next extract from BB's note, does give a concise "summary" of this intellectual development. Without the notes, however, one can hardly appreciate the substantive character of Bernays's remarks or, for that matter, that they refer to issues they had actually explored in detail! > looking for more details, one can consult the first report ever > given on the then on-going hilbert program(me). it is a lecture given by > bernays in september 1921, received in october that year, and published in > 1922 (jahresberichte der deutschen mathematiker-vereinigung 31 (1922), > 10-19). from this lecture it is clear, that hilbert/bernays found serious > problems in the logicist program(me) they started with, but also in trying > to meet the constructive demands. hence they settled on a new version, > later called hilbert's program(me), which, according to p. 15, was > conceived of as saving the best of both sides, but with having a heavy > constructive list (with later became the finitist attitude). The criticism of the Russellian logicist program, involving in particular a very detailed analysis of the axiom of reducibility, is presented in a stunning way in the Notes from the summer term of 1920 and winter term 1921/22. An equally penetrating and balanced discussion is, to my knowledge, only found again in Goedel's "Russell's Mathematical Logic". > turning to the influence of russell in particular, i'd like to add > that hilbert tried hard to get him for a series of lectures to goettingen. > why should a mathematician of hilbert's stature should have tried to do so, > if not for exchange and 'influence'? > See above. (Russell had been informed about the interest of the Hilbert group through Littlewood late in 1913 - and was eager to go to Goettingen, as he confided to Lady Ottoline Morrell in a letter of January 18, 1914.) Regards, WS

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