The following three postings [pasted below] have been submitted to
the FOM discussion list [fom@math.psu.edu]. Again I've taken the
liberty of collecting these messages (slightly edited), and then
posting this small digest to [HM] for the benefit of all interested
readers. Further reactions of *historical* flavor are most welcome,
as always.
With best regards,
Julio Gonzalez Cabillon
PS Mind that this is a follow-up to my previous post to [HM] on
June 21, 1999.
===========
Date: Wed, 23 Jun 1999 17:46:37 +0000
From: Bernd Buldt <Bernd.Buldt@uni-konstanz.de>
... i've broken down my posting into several sections, each prefixed
with a reference. time doesn't permit to reply to everything that
caught my eyes, but only for a few comments that i have 'right at my
hands'.
- RE: the dubucs/tennant/tait(/sieg)-exchange on decidability.
i'm unable to present definite answers to the questions raised, but what
follows are pieces of evidence, that, i think, should be taken into account.
(i) though i'm unable to provide some early textual evidence, i.e., before
1917 as requested, i nevertheless think the following quote from bernays
should shed some light on the issue. its background is the following:
heinrich scholz, the founder of the 'muenster school of logic', used to
write letters to bernays in order to clarify the questions he or his group
had either concerning logical questions in general and concerning hilbert's
program(me) in particular. in a letter (dated zuerich, march 13th, 1939)
bernays wrote (quoted in german in a forthcoming paper, the translation
below is a little 'ad hoc'):
"it seems to me, that one cannot claim, that the assumption that every
mathematical problem is solvable, as it is mentioned in hilbert's paris'
address you cite, is refuted by goedel's incompleteness theorem. this is
so, because hilbert's conjecture does not draw on the solvability within
the frame of a particular formalized deductive system (which obeys the
conditions imposed by goedel). this is especially clear from the fact, that
the validity of an undecidable sentence of the goedel-type emerges from a
contentual reflection on its undecidability within prinicipia mathematica
or the formal system z_mu."
the second sentence, i think, makes it clear, that at least bernays thought
hilbert would subscribe to mathematical optimism only, not to math. monism,
to use notions defined by neil tennant earlier in this list. and also that,
according to bernays, hilbert had not conceived of this problem as been one
of mechanical solvability or something even close to it, as it was, i think
rightly, emphasized by mic detlefsen in the last digest. (in contrast,
bernays stresses, as von neuman did, that a general solving algorithm, even
for 1st order logic, would trivialize mathematics and hence was unprobabale
from the outset.) this point of view is further supported by the list of
paradigmatic questions, "decided by a finite number of operations", hilbert
himself gives in his "axiomatic thinking", pp. 154ff., whose solutions are
neither algorithmic, or mechanical, or even stated within a formalized
theory. that means also, that i'm unable to agree with william tait, who
proposed in the last digest that hilbert's "verfahren" (solving procedure
for diophantine equations), at least in general, has "clearly the sense of
a decison procedure in our sense" (for more evidence against this
identification, see below).
(ii) interestingly, a little later in the same letter, bernays remarks:
"also the results of church concerning the unsolvability of certain
problems do not refute hilbert's conjecture, because those [unsolvable]
problems are such that they can't be answered with "yes" or "no", since
they contain a variable numberparameter."
(iii) with respect to the general context of decidability questions at that
time let me also remark, first, that (in another letter to scholz, dated,
zuerich, october 12th, 1936) bernays distinguishes between set-theoretical
on the one hand and proof-theoretical procedures on the other hand. and
that he was not opposed the former in principle, but that they simply
failed to be in accordance with the methodology underlying the finitist
attitude (cf. Hilbert/Bernays, Grundlagen d. Math. I, 1st ed., 190-196).
second, as bernays explicitly mentions, this whole early work was done
along the line of logic initiated in germany by ernst schroeder's "algebra
of logic". (and maybe it was even done in the spirit of his program(me),
that of an "absolute algebra", which was intended to realize the leibniz
project of a characteristica universalis -- i don't know.)
(iv) in the text book "grundzuege der theoretischen logik" by
hilbert/ackermann, published in 1928 (for whose precursor, see sieg's
posting), the decidability problem apparently is understood as an
algorithm: "the decision problem is solved when one knows a procedure, that
allows for deciding the validity resp. the satisfiability of any given
expression with a finite number of operations" (p. 73). but further, it is
called -- in italics -- the "main problem of math. logic" (1st. ed. p. 77,
cf. also the whole of p.III§11, or 2nd. ed. (1938), p. 90, p.III§12).
taken this literally, btw, means, that even though the consistency
problem might have been considered as the most pressing one, as sieg
stressed, but only from a practical point of view; but considered from a
systematical point of view, it, apparently, was not considered as the most
fundamental one.
but maybe more interestingly is, that in the 2nd. ed. from 1938
ackermann doesn't say that the church-turing results provide a fatal blow
for hilbert's optimism. instead he points out that they do not provide
instances of 'absolute undecidable' sentences, which, in fact, would lead
to a contradiction. while ackermann prepared this 2nd. edition he was only
a 'corresponding member' of the hilbert school, i.e., he discussed the
revision with bernays in letters. but at least in his view (just like
bernays did, see above), recursive unsolvability doesn't undercut
hilbertian optimism.
thus, some kind of solvability/decidability-request runs through all of
hilbert's writings: from hilbert's 'problem address' at paris (1900), over
'axiomatic thinking' (1917/18) and his text book (1928) up to one of his
last lectures "naturerkennen und logik" from 1930, where, towards the end,
hilbert takes up again the ignorabimus-question.
gathering all these evidences, what remains unclear to me, is,
then, what was the exact relationship between hilbert's "axiom of
solvability", his demand for "decidability with a finite number
operations", both stressed in his more popular lectures from 1900
throughout to 1930, and the "main problem of math. logic" from his more
technical writings. i hesitate to subscribe to dubucs conclusion that
hilbert was simply confused on this matters, rather, i'm inclined to think
that hilbert himself considered this as an open problem, since he lists on
p. 153 of his "axiomatic thinking" the "solvability in principle of every
math. question" and the "decidability of a math. question in a finite
number of steps" as two separate problems; two problems btw, that should be
solved with the help of his proof theory.
- RE: the dubucs on post-completeness.
this is only a short note to call back to memory, that to think of
completeness in terms of post-completeness was kind of 'natural' for
hilbert, since he stated the completeness of his axioms for geometry (from
the second ed. onwards, missing in the first ed. of his "foundations of
geometry") and for the real number field in 1900 via a kind of
post-completeness, i.e., that any addition would render the system
inconsistent.
but what strikes me as odd is that initially the hilbert circle
apparently searched for this kind of completeness also for 1st order
predicate logic. but suppose it were, then we couldn't have formalized
axiomatic systems, since additions of axioms (unprovable within 1st order
logic) could render the system inconsistent because of post-completeness,
which, in my view, would have partly threatened hilbert's program(me).
(e.g., think of an axiom system for some entities x containing an axiom,
expressable within 'pure logic', to the effect that there are, say, n
entities of sort x.) maybe someone on the list has a clue on this. (for
more on completeness see the next section.)
- RE: tennant's request for the bernays quotation
first, one finds a comprehensive though incomplete bernays bibliograhy in
the book "sets and classes. on the work of paul bernays", ed. by g.h.
mueller, amsterdam 1976.
second, the requested quotation is from "die philosophie der
mathematik und die hilbertsche beweistheorie" (philosophy of mathematics
and hilbert's proof-theory), in: blaetter fuer deutsche philosophie 4
(1930), 326-367, reprinted, with a postscript, in p. bernays, "abhandlugen
zur philosophie der mathematik", darmstadt 1976, 17-61. on p. 58 bernays
denies the possibility that an end to the creation of new concepts in
mathematics is within sight. nevertheless, he states on p. 59, formalized
theories can enjoy this possibility, viz., that no new concepts furnish new
results. he then defines a theory to be deductively closed iff the theory
is post-complete (see above), or, as some would prefer to say, iff it is
maximal. before he goes on to conjecture that the then current
formalization of arithmetic of the hilbert school, i.e., the peano axioms
with recursive definitions, is deductively closed, he adds footnote nr. 19:
"please note that the demand for deductively closedness is weaker then
[geht nicht so weit wie] the demand for decidability of each question of
the theory in question, viz., that there should be a procedure to decide of
any pair of contradictory assertions belonging to this theory, which one is
provable (true)."
this is the first mentioning of this modern notion of completeness
(i.e., being closed under ...) i'm aware of in either hilbert or bernays.
possibly, bernays learned it from herz or zermelo, who both used similar
closing conditions in papers that were published in 1929. (whereas our
modern notion, i think, comes from tarski's papers on the 'methododology of
the deductive sciences', that appeared around the same time.)
- RE: the sieg/detlefsen-exchange on logicism in hilbert
mic detlefsen states he hasn't found any evidence of hilbert's logicism
around 1917-1920, as claimed by wilfried sieg in his recent paper in the
bsl. well, even before learning of sieg's papers two years ago or so, i
thought it uncontroversial that there was a logicist stage in hilbert's
development; especially so, because hilbert himself says so.
in his "axiomatic thinking" (again p. 153) hilbert says: "... hence
it seems necessary, to axiomatize logic itself und to establish, that
number theory and set theory are only parts of logic. this path, prepared
since long -- not in the least through the penetrating investigations of
frege -- was finally and most successfully followed by the brilliant
mathematician and logician russell." thus we see hilbert -- contrary to
earlier remarks of his own -- taking an explicit logicist stance (number
and set theory being simply logic). he then goes on with acknowledging,
that in order to bring this frege-russell-project to a happy ending, there
still needs a lot of work to be done, which he plans to pursue in the years
to come (a first step of which was to hire bernays right away from zuerich,
where he gave that lecture).
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.
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).
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'? (the mathematical circle at goettingen
also studied frege's foundational works, as is evident from the reports of
sessions held there, published in the "jahresberichte der deutschen
mathematiker-vereinigung".)
wilfried sieg has corrected and improved much on what was extractable from
the hitherto published sources like those mentioned. and my reason to hint
at these things is only, that even without sieg's recent paper, and even
without drawing on nachlass-material, one find clues in both, hilbert's as
well as bernays papers, that there had been some serious logicist
flirtations until hilbert's program(me) emerged into the shape we all know
it.
- RE: detlefsen/tennant/dubucs on rationalistic optimism
i won't deal here in extenso with mic detlefsen's claim that hilbert was a
kantian, but focus instead on one point only, that is connected also to
rationalistic optimism, brought up earlier by neil tennant and jacques
dubucs.
mic detlefsen suggests that we should place hilbert's axiom of
solvability in a kantian context, because of striking parallels from kant's
critique of pure reason. there are surely better quotation from kant's
critique then the one he gave, because there (p. A vii) kant states that
there are UNanswerable questions ("questions ...[that] ... reason ... is
also not able to answer"). one good example is found at B 508, where kant
says: "it is not so extraordinary as it first seems the case, that a
science be in a position to demand and expect none but assured answers to
all questions within its domain ... although up to the present they have
perhaps not been found." kant goes then on to say (and prove), that pure
reason (i.e., metaphysics), pure mathematics, and pure ethics are examples
of such sciences.
is it wise to interpret this as a kantianism? jacques dubucs has
mentioned in digest #216, that this solving optimism can also be found in
goedel. but goedel, as we know, was a student of leibniz. so what we have
here, is, i think, some kind of a 'common cause', namely, both, kant and
leibniz, were within the rationalistic line of western philosophy. and that
the products of the mind are transparent to itself, i.e., that there are no
undecidable questions in this area, has been a prejudice common to (more or
less) all members of that rationalistic line. so what we find in goedel or
in hilbert is no peculiar "-ism", but simply that they belong to this
rationalistic line of western thought. in addition, whether they studied
philosophy (like goedel did) or not (like i assume hilbert did) doesn't --
imho -- matter; mathematicians are anyway, by default, so to speak, members
of this rationalistic tradition and hence there no need to explain it by
assigning them any '-ism'.
*******************************************************
Bernd Buldt
FG Philosophie
Universitaet Konstanz
D-78457 Konstanz
Germany
Fon: [+49] (0)7531/88-2794 (Office)
[+49] (0)7531/88-2750 (Secretary)
Fax: [+49] (0)7531/88-4121
Email: bernd.buldt@uni-konstanz.de
Fon: [+49] (0)7731/25486 (Private)
Fon/Fax:[+49] (0)7731/24116 (Study)
*******************************************************
Date: Fri, 25 Jun 1999 12:23:30 -0500
From: Michael Detlefsen <Detlefsen.1@nd.edu>
In a posting of 22 June 1999 I expressed reservations regarding a
suggestion by Wilfried Sieg (in a posting of 18 June 1999) that Hilbert
adopted a form of logicism sometime around 1917. I wrote:
"By the way, I would like to register my reservations regarding Wilfried's
suggestion that H was pursuing a logicist project around the time of the
1917 essay and that he was, in that, heavily influenced by Russell. I think
there's little in the way of evidence to support such a view. He expressly
rejected logicism in either its Fregean, its Dedekindean or its Cantorian
forms in the 1904 essay. This rejection appeared again, in expanded form,
in his work in the 20s. The chief influence of Russell was, I believe, in
his (Russell's) having provided the vehicle of formalization which was
needed to give the third option of H's 'kein ignorabimus' view (i.e. the
option of 'unsolvable under such-and-such conditions') a precise form. I've
looked at the lecture notes from this period and seen the parts of the
material of H's student Behmann that are supposed to 'evidence' the
connection with Russell and see nothing in any of this that gives much
credence to the idea that H was influenced by the substantive logicist
views of Russell."
To this, Bernd Buldt replied (in a posting of 23 June 1999):
"mic detlefsen states he hasn't found any evidence of hilbert's logicism
around 1917-1920, as claimed by wilfried sieg in his recent paper in the
bsl. well, even before learning of sieg's papers two years ago or so, i
thought it uncontroversial that there was a logicist stage in hilbert's
development; especially so, because hilbert himself says so.
in his "axiomatic thinking" (again p. 153) hilbert says: "... hence
it seems necessary, to axiomatize logic itself und to establish, that
number theory and set theory are only parts of logic. this path, prepared
since long -- not in the least through the penetrating investigations of
frege -- was finally and most successfully followed by the brilliant
mathematician and logician russell." thus we see hilbert -- contrary to
earlier remarks of his own -- taking an explicit logicist stance (number
and set theory being simply logic). he then goes on with acknowledging,
that in order to bring this frege-russell-project to a happy ending, there
still needs a lot of work to be done, which he plans to pursue in the years
to come (a first step of which was to hire bernays right away from zuerich,
where he gave that lecture)."
I was, of course, well aware of this statement of Hilbert's. Unlike Buldt,
however, I do not take it to be a clear declaration of logicism on the part
of Hilbert--not, at any rate, if by 'logicism' one means anything very
close to the philosophical positions of Frege, Dedekind, Cantor and/or
Russell. There are a number of reasons for this.
One is that Hilbert makes clear in the 1917 paper that he views our
understanding of Russell's axiomatization of logic as far from adequate.
One point of concern is how to understand the distinction between the
contentual and the formal in Russell's axiomatization. The
content/formalism distinction and the relationship between the two in
Russell's axiomatization is, I believe, a foreshadowing of the real/ideal
distinction that would figure so prominently in Hilbert's later work. Put
most plainly, the problem is this: to tell which of the axioms of Russell's
axiomatization are contentual propositions (propositions having a
truth-value) and which are mere 'formalisms' serving some instrumental
purpose of simplification or efficiency in our thinking. In his 1904
lecture, Hilbert characterized the axiomatic method as marked by creative
freedom. He wrote: "Once arrived at a certain stage in the development of a
theory, I may say that a further proposition is true as soon as we
recognize that no contradiction results if it is added as an axiom to the
propositions previously found true." (van Heinenoort, 135). He refers to
this as the 'creative principle' ... a principle, Hilbert said, "which
justifies us in forming ever new notions, with the sole restriction that we
avoid contradiction" (l.c., 136).
Here, then, is a problem: Did Hilbert understand Russell's axiomatization
of logic in this way? That is, did he see certain of its axioms as
justified solely because they were useful instruments that could be
consistently added to other axioms justified previously (and perhaps in
other ways)? If so, then his position would surely not have been a
'logicist' position in anything like the sense in which Frege, Russell, and
others were logicists. (N.B. I do not mean to suggest by this that Frege,
Russell, Dedekind and Cantor all held the same or closely similar
positions. They didn't, as I have argued at length in my essay 'Philosophy
of mathematics in the 20th century'.) The logicists required their axioms
to be true ... true in a radically different sense from that described in
Hilbert's 1904 address. I see no evidence to suggest that Hilbert saw
things this way. Rather, it seems he may have viewed Russell's
axiomatization as calling for justification itself ... justification in the
form of a consistency proof.
Buldt cites Bernays' 1935 essay 'Hilberts Untersuchungen ueber die
Grundlagen der Arithmetik' as providing support for his 'logicist'
interpretation of H 1917.
"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.""
In fact, this provides anything but confirmation of a logicist reading of H
1917. If one is serious about calling the axioms of a system laws of logic,
then one does not need to justify the system by prving its consistency.
Yet, as Bernays stressed, the only grounds that Whitehead and Russell had
for belief in the consistency of their system was one based on experience,
and one which therefore provided 'keine voellige Sicherheit' (Bernays,
201). But, as Bernays goes on to emphasize, the complete certainty of the
consistency of their axioms is something that, even in the 1917 essay,
Hilbert judged to be 'a requirement of mathematical rigour' (l.c., 202).
Bernays may, of course, have been wrong in his description of Hilbert's
views. All I am saying is that, contrary to what Buldt suggests, one does
not find support for a serious 'logicist' reading of H1917 in Bernays'
essay.
Indeed, Bernays took essentially the same view I offered: namely, that what
H prized in Russell's logicist work was not his logicism but his production
of a a comprehensive or synoptic formalization of mathematics (and ne which
therefore captures arithmetic) ... something which facilitated H's
incipient metamathematical interests which he described in the 1917 essay
as 'the study of the essence of mathematical proof' (155) or the 'making of
the concept of the specific mathematical proof itself the object of
research' (ibid.). Bernays thus wrote (201):
"In fact, the standpoint of Principia Mathematica contains an unsolved
problematic. What is accomplished by this work is the working out of a
synoptic (uebersichtlichen) system of postulates (Voraussetzungen) for a
common deductive construction of logic and mathematics and so thus a proof
that this construction actually can be given."
I thus remain unconvinced that there was commitment to any serious or
genuine form of logicism in H 1917 ... or elsewhere in Hilbert's work.
Indeed, it seems to me that Axiomatisches Denken is rather another
testimony to the Kantian character of Hilbert's foundational program. What
Russell did was something that Kant thought could not be done ... namely
to find a 'topmost' layer of axioms in the ever-deepening layers of
axiomatization. Kant believed that we can always move from one major
premise to a higher one in our drive for ever-increasing unity in our
understanding, but that there is no 'highest' major premise beyond which
our application of the regulative use of reason will not take us. Reason
therefore launches us on a (potential) infinity of ever greater
developments towards unity. This, I believe, is what H thought Russell
may have changed. He produced a last or highest (or deepest) layer of
axiomatization ... and one which, therefore, should make it possible for a
final justification of the regulative use of reason in mathematics. That is
why it deserved to be called the 'crowning achievement' of the axiomatic
method. As Hilbert himself said (156):
"Through progressing to ever deeper layers of axioms ... we obtain ever
deeper insight into the nature or essence of scientific thought itself, and
this will make us ever more conscious of the unity of our knowledge."
Compare with Kant's description of the proper use of pure reason (A: 305;
B:361).
I want now briefly to turn to Buldt's criticism of my use of Kant's general
critical epistemology as providing the right model for understanding
Hilbert. He begins with this:
"mic detlefsen suggests that we should place hilbert's axiom of
solvability in a kantian context, because of striking parallels from kant's
critique of pure reason. there are surely better quotation from kant's
critique then the one he gave, because there (p. A vii) kant states that
there are UNanswerable questions ("questions ...[that] ... reason ... is
also not able to answer"). one good example is found at B 508, where kant
says: "it is not so extraordinary as it first seems the case, that a
science be in a position to demand and expect none but assured answers to
all questions within its domain ... although up to the present they have
perhaps not been found." kant goes then on to say (and prove), that pure
reason (i.e., metaphysics), pure mathematics, and pure ethics are examples
of such sciences."
I certainly agree that there are other passages that support my view, but
the one I chose had this unique characteristic ... and for exactly the
reason Buldt identifies as making it inappropriate.Kant here seems to
combine a statement of the completeness of reason with (an apparently
conflicting) statement that reason can nonetheless sometimes reveal that
it has no answer to a question. This is exactly the type of 'third option'
of solvability that Hilbert stressed in his 1900 address: namely, that in
addition to positive and negative solutions, another type of solution is
a proof of unsolvability. As Buldt notes, the passage I cited seems to
include this possibility, and that, it seems to me makes the parallel
between Kant and Hilbert even more thorough-going. I don't want to
exagerate, though. I see no evidence that Hilbert was a Kant scholar.
And I am not saying that his characterization of solvability in the 1900
address was consciously patterned after that laid down in the preface to
the 1st ed. of the CPR that I cited. I make only the more modest (but still
very strong ... indeed, some would say, outrageously strong) claim that
Kant's ideas (whether or not Hilbert actually got them from reading
Kant)--or, more specifically, the general structure of Kant's critical
epistemology--is the most important determinant of Hilbert's philosophical
views.
**************************
Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
U.S.A.
e-mail: Detlefsen.1@nd.edu
FAX: 219-631-8609
Office phones: 219-631-7534
219-631-3024
Home phone: 219-273-2744
**************************
Date: Tue, 29 Jun 1999 09:21:33 -0400 (EDT)
From: Wilfried Sieg <ws15+@andrew.cmu.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 by Bernd.Buldt@uni-konstanz.de:
> ... 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