[Corpora-List] PS:minimal changes in a paragraph (based on a corpus it appeared) ... (2nd attempt (after first one was deleted))

[Graham White]

This may be straying a bit off topic (well, it already has strayed
off topic), but there are two things that tell against this line of
argument. 
i) Goedel was extremely interested in our own cognitive capacity: he had
a very strong doctrine of mathematical intuition, for example. He wasn't
just interested in formal systems for themselves, but as products of
human cognitive abilities, because (I think) he was deeply interested 
in human cognitive abilities.
ii) Furthermore, if you consider formal systems (to be specific, 
formalisations of areas of mathematics which are important in 
mathematical practice) you find there are two big classes. Firstly, you
have the ones which can't express arithmetic, for example formalisations
of Euclidean geometry, large parts of algebraic geometry, and so on.
These have sound and complete axiomatisations, have other good
properties (for example, quantifier elimination). Goedel's theorem
doesn't apply to these, precisely because you can't express arithmetic
in them. Secondly, you have the ones which can express arithmetic, and
Goedel's theorem does apply to them. Systems of both types are
significant in mathematical practice: Tarski, for example, worked on
both of them (he wrote a lot on quantifier elimination in systems of the
first type as well as his famous paper on truth in formalised
languages). By the way, I'm not saying that any logical system at all 
belongs to either the first type or the second type: if you want a 
better characterisation of the first type, then the theory of
o-minimality does that quite well. 

Graham

On Wed, Aug 10, 2011 at 01:14:05AM +0000, Albretch Mueller wrote:
> > Godel showed that a formal system capable of doing arithmetic perfectly cannot be both complete and consistent …
> ~
>  I agree with you. I think abusing Goedel's theory has become some
> sort of new-age pastime, but I think you may be getting bit off course
> with your colloquial explanations first the adverb "perfectly" was not
> part of the statement in Goedel's theory and second and foremost, it
> is not about us "humans" not being able to do arithmetic "perfectly",
> not being "consistent" … "complete" (as we have limitations like
> finite attention spans and finite lifetimes) …
> ~
>  I would simply say that Goedel's never had in his mind (not even in
> his wildest dreams) our own cognitive capacity he, very explicitly
> indeed, was just referring to -formal systems- and to explain what
> those beasts are and why Mathematicians fancied them so crazily (some
> still do), I would first going through Frege's Begriffsschrift and
> Hilbert (sort of trivialization of Euclid's work (my opinion more as a
> semiotician than a Mathematician)), go as early as Euclid's's element
> and explain what a proof is and why Mathematicians thought we could
> make such a machine (syntactic device) capable to exhaustively -and
> progressively- prove things for us.
> ~
>  lbrtchx
> 
> 
> On 8/9/11, Patrick Juola <juola at mathcs.duq.edu> wrote:
> > On Tue, Aug 9, 2011 at 12:11 PM, Angus Grieve-Smith
> > <grvsmth at panix.com>wrote:
> >
> >>   Russell continues:'IT IS A LANGUAGE THAT HAS ONLY SYNTAX AND NO
> >> VOCABULARY WHATSOEVER.(My emphasis). Barring the omission of a vocabulary
> >> I
> >> maintain that it is quite a nice language. It aims at being the sort of
> >> language that, if you add a vocabulary, would be a logically perfect
> >> language.' He adds that actual languages are not logically perfect in this
> >> sense...
> >>
> >>
> >>     Then it's not a language, it's a model of a language.  I'm sure it's a
> >> very nice model, and useful for all sorts of purposes.  But Gödel showed
> >> that in general models can never completely capture reality,
> >>
> >
> > No, he didn't.
> >
> >  |  One of the biggest mistakes a scientist can make is to reify their own
> > model.  It is a dangerous form of hubris that can lead science off course
> > for generations.
> >
> > And an even bigger mistake that scientists can make is to misunderstand what
> > they read and then misapply it in a completely inappropriate way in a
> > totally wrong discipline.
> >
> > Godel showed that a formal system capable of doing arithmetic perfectly
> > cannot be both complete and consistent.  Since human beings are not formal
> > systems, this is of limited application.   In particular, we know for
> > independent reasons that a) humans can't do arithmetic perfectly, b) humans
> > aren't consistent, and c) humans aren't "complete" (as they have limitations
> > like finite attention spans and finite lifetimes).
> >
> > So, Godel's theorem only shows that if human beings were something we know
> > they're not, they would have properties that we have already known them to
> > have.
> >
> 
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