Re: Why "geometric"?

From: George Cox (george_coxanti_at_spambtinternet.com.invalid)
Date: 11/10/04 

Date: Wed, 10 Nov 2004 19:13:57 +0000 (UTC)

KRamsay wrote:
>
> In article <4185690D.E1476ACB@spambtinternet.com.invalid>, George Cox
> <george_coxanti@spambtinternet.com.invalid> writes:
> |A first order formula is called "geometric" if it is built up from
> |atomic formulae using only conjunction, disjunction and existential
> |quantification. But why "geometric"?
>
> Perhaps you have heard of topos theory. The category of sheaves
> over a topological space X is a topos. Given a continuous function
> X->Y we get an adjoint pair of functors between the sheaves over
> X and the sheaves over Y. The notion of "geometric morphism" from
> one topos to another is a generalization. The geometric formulas
> are preserved under geometric morphisms in a certain sense. You
> can read about this in Mac Lane and Moerdijk, _Sheaves in Geometry
> and Logic_.

Thank you. So now I wish to know why some morphisms are called
"geometric". May one hope that there is some obvious connection with
geometry--geometry in the layman's sense? Maybe I have to read Mac Lane
and Moerdijk, oh dear!

>
> I believe this is related to what they call "geometric logic". When
> I did a web search for "geometric logic", one of the first pages to
> turn up was praising the acting ability of Humphrey Bogart, ;-)
> in his role as Queeg in "The Caine Mutiny":
>
> Ahh, but the strawberries that's... that's where I had them. They
> laughed at me and made jokes but I proved beyond the shadow of a
> doubt and with... geometric logic... that a duplicate key to the
> wardroom icebox DID exist, and I'd have produced that key if they
> hadn't of pulled the Caine out of action. I, I, I know now they were
> only trying to protect some fellow officers...
>
> Keith Ramsay