I have some further questions,
in the interest of the question, What
was Hilbert really doing in Grundlagen
der Geometrie (abbrv GG, FG)?
[1] Was there any genuinely cogent and
telling sense in which Hilbert in GG might
be said to be employing implicit rather
than explicit definition?
[1a] Bernay's seems to think not in his
article on Hilbert in the Edwards
Encyclopedia of Philosophy.
[1b] Indeed, GG does seem to be arranged
in the form of an explicit definition.
It begins with a DEFINITION. The question
is, what is the scope of this initial
DEFINITION? What immediately follows that
initial 'DEFINITION' does not look like
either an explicit or an implicit definition.
It is simply introducing terms for the
parts of the systems of object, hardly
definitions in any sense!! I thus conjecture
that the scope of that initial DEFINITION is
all of the axioms. That what we have is
an explicit definition of the system
whose principal parts are named immediately
after the initial 'DEFINITION'.
[1c] The logical form of the Appendix to
GG also called GG seems to reenforce the thought
that Hilbert is giving an explicit definition. it is
of course important to observe that Hilbert makes no
commitment to (what would amount to) first-order
logic (as Bernays observes).
[2] What hangs on explicit vs. implicit definition?
Explicit definition in geometry must draw on a
reservoir of material meaning, no?
[3] In the various quotations in Dr.Cabillon's
posting, one might conjecture that he and others
think that in GG Hilbert is carrying out investigations
of consequences of hypotheses, "wholly detached from
reality"; that his investigations are purely formal,
e.g., formal-logical consequences and independence
proofs.
But I'm wondering if this does justice to
what Hilbert intended to be doing?
Many have been struck by the modest
incoherence of GG being
frequently touted as a purely formal-logical work being
prefaced by thwe remark that geometry is the
logical investigation of our spatial Anschauung.
(Not to mention toe motto from Kant.)
As indeed remarked by Hilbert (and later Bernays),
Hilbert (in the close of the Appendix GG)
is out for bigger game, e.g., logically
clarifying the DIFFERENT ways in which our Anschauung of
continuity can be mathematically structured -- WE NOW
TEND TO BE BLIND TO THIS BECAUSE WE ARE SO IMBUED
WITH THE PERNICIOUS DOGMA OF THE correlation OF CONTINUA
AND THE DEDEKIND REALS!
(This is a point made beautifully by Hermann
Weyl on numerous occasions; of course he takes
the Bergsonian line that the continuous is
importantly irrational in the sense that our
experience of continuity by no means forces the
dfedekind reals. --Feferman's "Weyl Vindicated"
might be thought of as showing how the most
refinied physical "experience" of continuity
does not force the Dedekind reals. Thus in
part, Hilbert in GG (among other things)
might be reasonably viewed (-- and only this view
accouts for the content of the two GG's -- right??)
as investrigating, among other things the logical
geography and fluidity or freedom of our spatial
Anschauung of continuity. . .among other things.
Doesn't this make Hilbert's motives very
different from Peano, Pasch, .. . ? (An
honest question).
robert tragesser