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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 1second grain) the point 02:13:01 and the moment
> (irreducible interval) 2:13:012:13:02. See section 3.4 (page 21) of
> http://www.ihmc.us/users/phayes/TimeCatalog.pdf, especially the discussion of
> 'models' on pp 234.
>
> 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 "justindistinguishable" 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 workedout 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
>>
>>
>>
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> (04)

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