[FOM] Hilbert and conservativeness

praatika@mappi.helsinki.fi 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?



Panu Raatikainen

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

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