FOM: June 25 - July 31, 1999

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FOM: Recent exchange on HILBERT

    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logicist tendencies in Hilbert's writings - published or unpublished,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

    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
>        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.)



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