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*To*: Neil Tennant <neilt@mercutio.cohums.ohio-state.edu>, dubucs@idf.ext.jussieu.fr, fom@math.psu.edu*Subject*: Re: FOM: How clear was Hilbert?*From*: William Tait <wtait@ix.netcom.com>*Date*: Fri, 18 Jun 1999 11:30:42 -0500*In-Reply-To*: <199906121622.MAA27320@mercutio.cohums.ohio-state.edu>*Sender*: owner-fom@math.psu.edu

Dear Neil, Regarding >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

**References**:**Neil Tennant**- FOM: How clear was Hilbert?

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