Re: Multisets and 3VL

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.

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

> 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."

Cheers,
David.

-- 
David Fetter david_at_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