Here is the same email non-HTML formatted for readability.  (01)
D  (02)
On 1/22/08 10:53 AM, "Pat Hayes" <phayes@xxxxxxx> wrote:  (03)
> At 12:19 PM -0500 1/22/08, John F. Sowa wrote:
>> Pat,
>> 
>> That statement is true of the standard model:
>>> if you have intervals, you have the points at their ends.
>
> Its true of all models, standard or not. One can mathematically construct the
> points from the intervals (they are maximal filters on the space of all
> meeting pairs of intervals.) See p 32 et. seq. of
> http://www.ihmc.us/users/phayes/TimeCatalog.pdf
>
>> 
>> Suggestion: use the word 'instant' instead of 'point':
>
> No, even instants have endpoints, and they may not be the same (though they
> can be: one gets very different meeting algebras in the two cases))
>
>> 
>> 1. That allows instants to be infinitesimally small (i.e.,
>> mathematical points).
>
> Points are not quite the same as infinitesimally small intervals. Intuitively,
> the latter are the limits of intervals, but the former are the limits of
> places where intervals meet.
>
>> 2. But it leaves open the question of finite granularity.
>
> Even in a discrete granularity model, there is a necessary distinction between
> (for example, assuming a 1-second grain) the point 02:13:01 and the moment
> (irreducible interval) 2:13:01-2:13:02. See section 3.4 (page 21) of
> http://www.ihmc.us/users/phayes/TimeCatalog.pdf, especially the discussion of
> 'models' on pp 23-4.
>
> By the way, you have to be very careful when combining discreteness
> assumptions with 'limit' notions such as infinitesimal. I found that many
> apparently intuitive axioms about discreteness in fact have models in the real
> line when limits are allowed. See the discussion on page 44.
>
>> 
>> 3. It also avoids the question of whether the grain is
>> a sharply delimited interval or a distribution, such
>> as a quantum mechanical wave packet that fades away
>> without any sharply defined boundary.
>
> Hah. Good luck with giving axioms for that model.
>
>> 
>> 4. It also leaves open the nature of an interval, which
>> could be defined with instants at the ends that might
>> themselves be have fuzzy boundaries.
>
> Again, Ive never seen a coherent axiomatization of the idea of a fuzzy
> boundary. One related idea which is fully formalized is that of 'tolerance
> spaces', which are defined in terms of a "just-indistinguishable" relation on
> a set of points. That seems like a good approach to formalizing notions of
> approximation: but again, I have yet to see a fully worked-out ontology for
> this. And I wonder, in fact, if it is really necessary in order to do almost
> all practical temporal reasoning
>
>> 
>> By using the word 'instant', we can state generalizations
>> that are true of a wide range of models without making a
>> firm commitment to the nature of the granularity.
>
> We can do that already: the 'catalog' has a very wide range of options.
> Nevertheless, it is always necessary to make at last a conceptual distinction
> between intervals and points, or else to face up to the sometimes unintuitive
> consequences of conflating them (see section 5 of the 'catalog'. I actually
> find this 'vector continuum' theory quite elegant and intuitive, but it
> certainly is not the traditional real line!)
>
> Pat
>
>> 
>> John
>> 
>> 
>> 
>> _________________________________________________________________
>> Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
>> Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
>> Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
>> Shared Files: http://ontolog.cim3.net/file/
>> Community Wiki: http://ontolog.cim3.net/wiki/
>> To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
>> 
>  (04) 
-- 
**********************************************************************
"Speaking only for myself"
Senior Technical Evangelist - Adobe Systems, Inc.
Blog - http://technoracle.blogspot.com
Community Music - http://www.mix2r.com
My Band - http://www.myspace.com/22ndcentury
Adobe MAX 2008 - http://technoracle.blogspot.com/2007/08/adobe-max-2008.html
**********************************************************************  (05)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ 
Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ 
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/ 
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx  (06)