This mail is publicly posted to a distribution list as part of a process
of public discussion, any automatically generated statements to the
contrary nonwithstanding. It is the opinion of the author, and does not
represent an official company view. (01)
The problem with "points" is that one cannot exactly explicitly
represent most points  or any nonrational number. Computers make
matters worse, as they only explicitly represent a small subset of the
rationals. I did start looking at formalization of fuzzy edged intervals
as part of a PhD in formal definitions of computational geometry, but
when I realised I'd have to start by rewriting topology, I got on with
my life. (02)
The discussion reminds me of the story of a man looking for his keys by
a street lamp. When asked where he had dropped them, he pointed to a
place a little way off  "but the light is much better over here". (03)
Sean Barker
BAE SYSTEMS  Advanced Technology Centre
Bristol, UK
+44(0) 117 302 8184 (04)
BAE Systems (Operations) Limited
Registered Office: Warwick House, PO Box 87, Farnborough Aerospace
Centre, Farnborough, Hants, GU14 6YU, UK
Registered in England & Wales No: 1996687 (05)
________________________________ (06)
From: ontologforumbounces@xxxxxxxxxxxxxxxx
[mailto:ontologforumbounces@xxxxxxxxxxxxxxxx] On Behalf Of Pat Hayes
Sent: 22 January 2008 18:54
To: [ontologforum]
Subject: Re: [ontologforum] Time representation (07)
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Keep this in mind if you answer this message. (09)
At 12:19 PM 0500 1/22/08, John F. Sowa wrote: (010)
Pat, (011)
That statement is true of the standard model: (012)
> if you have intervals, you have the points at their
ends. (013)
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 (014)
Suggestion: use the word 'instant' instead of 'point': (015)
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)) (016)
1. That allows instants to be infinitesimally small
(i.e., (017)
mathematical points). (018)
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. (019)
2. But it leaves open the question of finite
granularity. (020)
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. (021)
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. (022)
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 (023)
without any sharply defined boundary. (024)
Hah. Good luck with giving axioms for that model. (025)
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. (026)
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 (027)
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. (028)
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!) (029)
Pat (030)
John (031)
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