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At 10:23 AM +0000 1/24/08, Barker, Sean (UK) wrote:
This mail is publicly posted to a
distribution list as part of a process
of public discussion, any automatically generated statements to
the
contrary non-withstanding. It is the opinion of the author, and does
not
represent an official company view.
The problem with "points" is that one cannot exactly
explicitly
represent most points - or any non-rational number.
Why does one need to explicitly represent (I think you mean
numerically represent) something in order for it to be useful in one's
ontology? One does not. A physics ontology might talk of atoms, even
though they are too small to see and impossible to capture or name
singly.
And in any case, you are presuming that time is the real line.
Maybe it isn't. Intuitive time doesn't seem to be.
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.
:-) I almost went to graduate school to study topology.
Fascinating subject. Fuzzy intervals aren't the way to go, IMO
tolerance spaces are much more 'natural' and don't need real numbers.
They also have a natural metric structure.
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".
Old story, but I fail to see the relevance. The, er, point of
points is that they are useful. And, by the way, you can't not have
them, in a sense. One can construct the points from the
intervals. So in a sense they have to be there, whether you talk about
them or not.
Sean Barker
BAE SYSTEMS - Advanced Technology Centre
Bristol, UK
+44(0) 117 302 8184
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
________________________________
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Pat
Hayes
Sent: 22 January 2008
18:54
To: [ontolog-forum]
Subject: Re: [ontolog-forum]
Time representation
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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
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