To: |
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From: |
Pat Hayes <phayes@xxxxxxx> |

Date: |
Tue, 22 Jan 2008 12:53:52 -0600 |

Message-id: |
<p06230910c3bbe561a282@[10.100.0.22]> |

At 12:19 PM -0500 1/22/08, John F. Sowa wrote:
Pat, > 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))
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.
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!)
Pat
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