FOM: June 1 - June 24, 1999

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FOM: More on Hilbert (and Goedel)

Dear Jacques and Neil:

    I followed your interchange concerning Hilbert with great interest. 
Let me first make some general remarks concerning the "Hilbert Edition".
 William Ewald, Michael Hallett, Ulrich Majer, and I have been working
for some years on editing Hilbert's unpublished lecture notes.  These
notes have been (and are!) accessible at the Mathematical Institute
and/or the Library of the University of Goettingen. They span the period
from the early 1890's to the late 1920's, and cover roughly three
subject areas: (i) geometry, (ii) physics, and (iii) logic & arithmetic.
    As to (i), Toepell's book "Ueber die Entstehung von David Hilberts
'Grundlagen der Geometrie'", Vandenhoek & Ruprecht, 1986, contains an
informative discussion of lectures on geometry. As to (ii), I don't know
of any detailed discussions; as to (iii), there is the book by Volker
Peckhaus, "Hilbertprogramm und Kritische Philosophie", Vandenhoek &
Ruprecht, 1990.  Peckhaus uses some of the lecture notes to cast light
on Hilbert's work on foundations of arithmetic and logic roughly up to
1905. There is additional work (by a number of other people); for
information concerning such work and on Hilbert's emerging finitist
program, see my paper "Hilbert's Programs: 1917-1922", Bulletin of
Symbolic Logic 5 (1), 1999, pp. 1-44.
    The Hilbert Edition is to provide carefully edited texts of the
lectures (no translations!) with some critical apparatus and brief,
informative introductory notes.  We conceive of it as a "Working
Edition"!  In any event, we are planning to publish five volumes; one on
geometry, two on physics, and two on logic & arithmetic.  The geometry
volume will be completed by the end of this year and published next
year, as soon as possible.  The texts for the other volumes have been
TEXed to a large extent, and the first logic & arithmetic volume should
be completed by the fall of next year. 

    Secondly, let me add some more specific remarks about the Hilbert
notes on logic & arithmetic. Hilbert gave lectures on the foundations of
mathematics almost every single year in the period indicated above, thus
in particular during the period from 1905 to 1917, when -- according to
the standard view -- he was pursuing exclusively other topics, integral
equations and physics (relativity theory).  We have detailed notes for
many of these lectures.  In the fall of 1917, Hilbert hired Paul Bernays
as an assistant, and Bernays returned to Goettingen for the winter term
1917/18.  That very term, Hilbert gave lectures entitled "Prinzipien der
Mathematik".  Bernays wrote detailed notes, preserved as a typescript of
about two hundred pages.  The second, longer part of these notes is
entitled "Mathematische Logik" and is a wonderful draft of the book by
Hilbert and Ackermann that was published only in 1928.  At this point in
1917/18, Hilbert was pursuing a logicist program - clearly influenced by
Russell.  The finitist program emerged only slowly, beginning in 1920. 
It was first formulated fully in the Lecture Notes from the winter term
of 1921/22 and presented publicly in Hilbert's talk in Leipzig,
September 1922.  The text of that talk was published in 1923 as "Die
logischen Grundlagen der Mathematik".

    Finally, even more specific remarks on the 1917/18 lectures. 
Sentential logic, monadic logic, full first order logic, and ramified
analysis (with the axiom of reducibility) are developed, both
syntactically and semantically.  The systems are investigated
metamathematically in a completely novel and rigorous way: I think it is
not an exaggeration to say that the origin of modern mathematical logic
is found in these very notes!  In particular, consistency of the logics
is established, semantically. Completeness is formulated as
Post-completeness and proved for sentential logic; in a footnote,
ordinary completeness is obtained as a consequence.  Hilbert & Bernays
conjecture that first order logic is not Post-complete; the conjecture
is verified in the book by Hilbert & Ackermann.  Bernays, in his
Habilitationsschrift of 1918, formulates and proves (with a reference to
the 1917/18 Notes) ordinary completeness for sentential logic and
investigates the independence of the axioms for sentential logic as
given in "Principia Mathematica". In Hilbert & Ackermann the ordinary
completeness for first order logic is formulated in exactly the same
informal way as in Goedel's dissertation.  (Recall that Goedel used
H&A's book as providing the basic logical background.)  For detailed
support of this claim you can look at section B3 of my paper "Hilbert's
Programs: 1917-1922".  So, clearly, I do not agree with Jacques's
assessment in the following remark:

Excerpts from mail: 14-Jun-99 FOM: Re: more on Hilbert, M.. by 
> I suspect that there is no such possible clue, for a simple and well-known
> reason. As lenthly and convincingly explained by Goedel in several places,
> the very notions in terms of which your (3) is expressed were not available
> to Hilbert, who was always very reluctant (and, by the way, unable) to
> formulate the completness problems in such semantical terms. Moreover, one
> can wonder if Hilbert's views favouring decidability were not a simple
> by-product of his propensity of thinking the foundational problems in
> exclusively syntactical terms (cf my posting of Jun 11 22:34:58 1999).

In his next remark, Jacques views this as making the "enigma about
completeness" --  why was completeness established in Vienna and not in
Goettingen -- even more acute. (In the remark, "notes by Ackermann"
should be replaced by "notes by Bernays".)

Excerpts from mail: 14-Jun-99 FOM: Re: more on Hilbert an.. by 
> I'm very curious and impatient to cast a glance at the lectures notes by
> Ackermann that have been found by Sieg, for they seem able to change the
> current view (at least my own!) concerning the completeness problem during
> the twenties.
> Note that, in a sense, such textual evidence make the historical enigma
> about completeness more acute, rather than less: *why* in that case the
> completeness theorem has not been proved before 1929, viewing that all the
> mathematical pieces of the puzzle were available much earlier (cf Goedel's
> letter to Wang from December 7, 1967, according which the theorem was an
> "almost trivial consequence of Skolem 1922" (quoted in Wang's "From
> Mathematics to Philosophy", p. 8)). Goedel's claim (of course not founded
> on the evidence of the lack of any textual evidence of the contrary !) had
> at least the merit of proposing a prima facie acceptable explanation of
> this enigma, in terms of Hilbert's finitistic "prejudices" against
> semantical notions.

Consistency was the fundamental problem for the Hilbert school, and it
was to be established -- for excellent programmatic reasons -- by
restricted mathematical means. If there was a "blindspot", it did not
result from a finitist prejudice, but rather from a lack of focus on the
completeness problem: they did not turn their attention to it, being
preoccupied with other, for them presumably more central issues. In
addition and contra Goedel, I don't think that Goedel's proof is so
trivially extractable from the various pieces available in the
literature before 1929.  

In the Bologna lecture of 1928, to which Jacques pointed earlier on,
Hilbert gives also a perfectly correct (though interestingly different)
definition of semantic completeness for first order logic; he
conjectures that elementary number theory is (syntactically) complete,
but considers it as possible that "higher areas" of mathematics,
presumably set theory, might be incomplete.  

Best regards,


concerning the remark:

Excerpts from mail: 14-Jun-99 FOM: Re: more on Hilbert, M.. by 
> Incidentally, Johannes von Neumann, who was one of the most lucid members
> of the Hilbertian School (and, anyway, the first to grasp the significance
> of Goedel's incompletness results), found the kind of *morphological*
> consistency proof that Hilbert attempted in 1904 very dubious, just because
> its non-trivial implications concerning mechanizability (cf *Zur
> hilbertschen Beweistheorie* (1927), in Collected Works, Pergamon Press,
> 1961, vol. 1, p. 276).

The solvability of the decision problem (not the character of the 1904
consistency proof) had, in von Neumann's view, a direct implication on
the mechanizability of mathematics.  Indeed, he used the "morphological"
kind of argument in his own consistency proof and described it most
lucidly in his contribution to the 1930 discussion with Heyting and
Carnap (in Koenigsberg).  The texts that arose from this discusssion
have been translated into English and are published in Benacerraf and
Putnam's "Philosophy of Mathematics".  It was also at this meeting that
von Neumann learned about the first incompleteness theorem; Goedel
presented it there.  As to the direct impact and von Neumann's
subsequent discussions with Herbrand, see the Appendix of my paper
"Mechanical Procedures and Mathematical Experience", in: "Mathematics
and Mind", A. George (ed.), Oxford University Press, 1994, pp. 91-117. 
There is also some highly interesting correspondence between von Neumann
and Goedel from that time; that is to be published in volume IV of
Goedel's "Collected Works".

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