FOM: Hilbert's logicism

Michael Detlefsen Detlefsen.1 at
Fri Jun 25 13:23:30 EDT 1999

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'

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;

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
exagerrate, 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

Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
e-mail:  Detlefsen.1 at
FAX:  219-631-8609
Office phones: 219-631-7534
Home phone: 219-273-2744

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