Re: Multisets and 3VL

From: vc <boston103_at_hotmail.com>
Date: 21 Jan 2006 21:10:09 -0800
Message-ID: <1137906609.713381.129280@o13g2000cwo.googlegroups.com>

David Fetter wrote:
> vc <boston103_at_hotmail.com> wrote:
> > David Fetter wrote:
> >> vc <boston103_at_hotmail.com> wrote:
> >> >> neutral geometry has sharper limits on what it can prove than
> >> >> Euclidean geometry does.
> >> >
> >> > That does not make any obvious sense. What "sharper limits" do you
> >> > have in mind ?
> >>
> >> Well, at this stage, it's just fuzzy intuition, but if I had to
> >> assign a reason, it would be that I've noticed that when you "know
> >> extra stuff" about a problem domain, for example, that every
> >> multiset has multiplicity one, or that truth values will only be in
> >> {T,F}, you can then use that knowlege to get to places you couldn't
> >> have gotten to if you hadn't have it.
> >
> > I still do not understand your analogy. Say, in neutral geometry,
> > one can deduce that the angle sum of any triangle is not more than
> > 180 degrees.

>

> Actually, one can't. Neutral geometry includes spaces with positive
> (aka spherical), negative (aka hyperbolic) and zero (aka Euclidean)
> curvature.

That is not true. Elliptic geometry is not neutral, whereas Euclidean and hyperbolic are.
According to the Saccheri-Legendre theorem, in neutral geometry the sum of the three angles in any triangle is less than or equal to 180 (equal in Eucleadian geometry and less in hyperbolic).

>


> > In Euclidian geometry,  one can prove that the angle sum is exactly

> > 180 degrees thanks to the fifth postulate. So,  it's the Eucleadian

> > geometry that "has sharper limits", not neutral, unless you redefine
> > the word "sharper".

>

> With the parallel postulate, you can prove things that you simply
> can't prove without it. In this sense, when you're using neutral
> geometry, you have to "stop short" in places where you could go
> further with the parallel postulate in any of the above formulations.
> "Sharper limits" == "Sharper limits on what particular things you can
> prove using the more general theory."

So saying that something is less or equal 180 degrees is "sharper" (or more precise) than saying that something is exactly 180 degrees ? A very strange notion indeed.

By the same token you'd claim that rings are "sharper" than fields and integers (being an example of the former) are "sharper" than rationals that are an example of the latter ?

>

> Cheers,
> David.
> --
> David Fetter david@fetter.org http://fetter.org/
> phone: +1 510 893 6100 mobile: +1 415 235 3778

>

> Whenever a theory appears to you as the only possible one, take
> this as a sign that you have neither understood the theory nor
> the problem which it was intended to solve.
> Karl Popper
Received on Sat Jan 21 2006 - 23:10:09 CST