|To:||"[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>|
|From:||Pat Hayes <phayes@xxxxxxx>|
|Date:||Tue, 22 Jan 2008 12:53:52 -0600|
At 12:19 PM -0500 1/22/08, John F. Sowa wrote:
> 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
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))
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.
without any sharply defined boundary.
Hah. Good luck with giving axioms for that model.
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
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!)
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