FOM: How clear was Hilbert?

William Tait wtait at
Fri Jun 18 12:30:42 EDT 1999

Dear Neil,


>In an earlier message, I claimed that Hilbert's phrase
>	"problem of the decidability of any mathematical question
>	by means of a finite number of operations"
>(from "Axiomatisches Denken", a lecture delivered to the Swiss
>Mathematical Society in September 1917) posed "not quite the question
>of whether there is a single effectively decidable system of
>mathematics", and I claimed that "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." (Call this latter view
>Mathematical Optimism.)

and later in the same posting

Does Dubucs or any other fom-er have any interesting textual evidence
from Hilbert dating from *before 1917* which would show that Hilbert had
distinguished clearly between the following claims of Mathematical
Monism and Mathematical Optimism?

Aside from what evidence there is in the Hilbert lectures on logic that
Wilfried Sieg has, note that, in the formulation of the 10th problem in
``Mathematische Probleme'', H uses the term ``Entscheidung'' and his
explanation of the problem is:

``man soll ein Verfahren angeben, nach welchem sich mittels einer endlichen
Anzahl von Operationen entscheiden l\"{a}szt, ob die Gleichung in ganzen
rationalen Zahlen l\"{o}sbar ist.''

That clearly has the sense of a decision procedure in our sense. So in 1904
this seems to the way in which he spoke of `decidability'. I don't have
available the German text of "Axiomatisches Denken"; but I am assuming that
``decidable'' is a translation of ``entscheidbar''.

Best, Bill Tait

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