FOM: more on Hilbert's "Axiomatishes Denken"

[dubucs]

Neil Tennant wrote:
>
>Interestingly, Hilbert had noted on p.413 that of all the questions he
>had listed about mathematics, the following is the "most frequently
>discussed, because it goes to the heart of mathematical thought":
>namely, the "problem of the decidability of any mathematical question
>by means of a finite number of operations".  As thus formulated, it is
>not quite the question of whether there is a single effectively
>decidable system of mathematics. It could instead be construed as the
>question whether every mathematical question can be decided by means
>of a finite proof in some system or other.
>
The question about Hilbert's attitude w.r.t. the Entscheidungsproblem is by
no way as easy to adjudicate.

1) The expression "in some system or other" in your last sentence is
obviously in need of some further qualification, even to be just candidate
for an explanation of Hilbert's genuine train of thoughts. For each
mathematical sentence can be decided by means of a finite (actually, of
lenght 1) proof in a system to which it belongs as an axiom. Such an
uninformative claim can't be seriously attributed to Hilbert.

2) As strange as it may appear, several indicies show that Hilbert has
pursued a long-run flirt with the idea of a general method of decision for
mathematics.

In the twenties, he has obstinately looked for a syntactic definition of
the completness of first-order logic in the followings terms: a consistent
theory T is complete iff the addition to the theorems of T of any sentence
in the language of T is enough to make the theory inconsistent. This
definition ("completness in Post' sense") obvioulsy implies that any
sentence of L(T) is provable or refutable in T, whence immediately the
decidability of T.

As late as 1928 (not very long before Goedel's right definition and theorem
of completness for 1st-order logic), his compulsion to express the
completness in syntactic terms that imply decidability is perfectly clear:

"The affirmation of the completness of the formal system of number theory
can be expressed as follows: if one adds to the axioms of the number theory
a formula which belongs to the field of number theory, but which is not
provable, then a contradiction may be derived from the enlarged axiom
system. (...) The question of the completness of the system of the logic
rules, generally considered, constitutes a problem of the theoretical
logic. Until now, we have get only by experiment (durch Probieren) the
conviction that these rules are sufficient". ("Probleme der Grundlegun der
Mathematic", Atti del Congresso Internazionale dei Matematici, Bologna,
3-10 Settembre 1928, vol. 1, p. 135-141).

Of course, you are right in emphazising as philosophically crucial the
distinction between (a) the principle of the solubility of every
mathematical question (often explicitely defended by Hilbert, e.g. in the
Paris Congress (1900)) and (b) the claim of the existence of a general
algorithmic method to solve any mathematical question. Nevertheless,
Hilbert can't by no way be considered as having been clear on this
important distinction.

JD


Jacques Dubucs
IHPST CNRS Paris I
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