[FOM] Hilbert and conservativeness
praatika at mappi.helsinki.fi
Sun Sep 4 03:44:47 EDT 2005
Robert Black <Mongre at gmx.de> wrote:
> Panu Raatikainen wrote:
> >My hypothesis is the following: I think that Hilbert simply assumed
> >that finitistic mathematics is deductively complete with respect to the
> >real sentences (i.e., is real-complete)
> No doubt Hilbert (and Bernays) believed that finitary reasoning was
> complete for real sentences. And this would follow anyway from other
> things they believed...
>... However, to have assumed this in argument would have been a serious
> mistake (and not one I think we should attribute to them), since the
> Enemy was Brouwer, and Brouwer would (rightly, as it turns out) not
> have conceded it.
Well, I am inclined to think that Hilbert just did not think that it was a
controversial assumption but (wrongly) expected that it was a
mathematically provable fact (acceptable also for Brouwer). Anyway, this
wouldn't be the only case where Hilbert misinterpreted the dialectical
situation between him and Brouwer...
> The quote from Bernays just doesn't entail real completeness.
I must say that I can't easily recall how I interpreted (some 5 years ago)
this passage, and I don't have the paper at hand here now. I should think
more about it before any comment. But even if I was wrong about Bernays
here, it does not really affect my main point.
The question in my paper was:
Why was Hilbert so convinced, and why did he insist repeatedly with no
further argument (from early on), that a finitistic consistency proof
guarantees real-soundness and real-conservativity?
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
P.O. Box 9
FIN-00014 University of Helsinki
E-mail: panu.raatikainen at helsinki.fi
More information about the FOM