FOM: Hilbert's Hubris
sf at Csli.Stanford.EDU
Fri Dec 5 13:16:36 EST 1997
Martin Davis was quite right to call attention to Hilbert's qualification
of his belief in the solvability of all mathematical problems. But this
is worth a closer look (cf.Felix Browder, ed., *Mathematical Developments
Arising from Hilbert Problems* for reprinting of Hilbert's 1900 address,
as well as many interesting updates of work on the Hilbert problems as of
1976). In connection with his statement that in certain cases a problem
may be "solved" by showing its impossibility of solution by prescribed
methods or "hypotheses", he mentions the parallel postulate and equations
of the fifth degree. The former is just a question of independence from
the other axioms of geometry and the latter is a question of solvability
by given means, as is the 10th problem. But Hilbert muddied the picture
in both cases by speaking of "insufficient hypotheses" or of seeking the
solution "in an incorrect sense".
In any case, it is quite clear what Hilbert's expectations were in the
case of Problem 10: "Given a diophantine equation with any number of
unknown quantities and with rational integral numerical coefficients: _To
devise a process according to which it can be determined by a finite
number of operations whether the equation is solvable by rational
At the time of formulation, this problem did not have a definite meaning.
Of course, if there were a positive solution by producing the requisite
algorithm, one would not need an analysis of the general concept of
algorithm. Some of the other specific looking problems were even less
definite in meaning. And some of the problems (e.g. #2, to prove the
consistency of real number axioms (not precisely stated) and axioms for
set theory, or #6, to axiomatize physics) were simply big programs.
For Hilbert, if it looked like a mathematical problem, if it smelled like
a mathematical problem, it could be tackled (at least HE could do it if he
simply turned his attention to it). By the way, at the end of Hilbert's
statement of #1 on CH, re the existence of a well-ordering of the
continuum, he says: "It appears to me most desirable to obtain a direct
proof of this remarkable statement of Cantor's, perhaps by actually giving
an arrangement of numbers such that in every partial system a first number
can be pointed out." One of the first applications of Cohen's forcing
method after his own, was my proof in 1963 that it is consistent with
ZFC+GCH that there is no definable well-ordering of the continuum (cf.
Fund.Math.56 (1965) 325-345).
As to Hilbert's "proof" of CH being a model for Goedel's consistency
result via the constructibles, see (_pace_ Shipman) Solovay's introduction
to Goedel's Goettingen lecture in Vol.III of the *Collected Works*,
especially pp. 120 ff.
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