Issue 98038 | Editor: Erik Sandewall | 22.4.1998 |
Today |
Today: interventions by John McCarthy and Erik Sandewall concerning the ontology of time, and an answer by Sergio Brandano to Jixin Ma and Pat Hayes in the same debate.
Debates |
Pat Hayes wrote
In my view, axioms are tools which we can manipulate at will; they are not
set in stone or somehow inevitable. Different formal accounts of time might
be appropriate for different purposes or to capture different intuitions.
(I agree with Jixin that it is useful to seek a common 'core' theory which
can be extended in various ways to describe various possible more complex
temporal structures; and that this theory will have to be rather weak.)
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A more mundane example is given by temporal databases, which usually assume
in their basic ontology that time is discrete: for example, they routinely
describe times as integers representing the number of milliseconds since
the birth of Christ. (Of course, one can always insist that these are to be
understood as being embedded in a continuum, but then what use is an axiom
whose sole purpose is to insist that times exist which have no name and
about which nothing can be asserted, other than that they exist?)
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As to the rhetorical "what use", suppose the theory is to tolerate the elaboration that two successive events, shooting Pat and his falling to the ground, occurred between successive ticks of the clock. If you guarantee that no such elaborations will be required or that you are willing to do major surgery on your theory should elaboration be required, then you are ok with a weak theory even if it is unextendable.
In reply to Pat and Jixin.
I apologize for the length of this message, although it mainly consists of quoted text. As ``skin perception'', it seems to me my critique hits the target. The arguments of reply I received, in fact, are not as convincing as they were supposed to be. The details follow.
To Jixin
First of all, what do you mean "the classical one"? (the classical
continuous time structure)? Does it refer to the classical physical
model of time, where the structure is a set of points which is
isomorphic to the real line?
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Here in Pisa, we write ``continuity'' and we read ``axiom of
completeness'', which is what everyone commonly means when speaking
about (the founding notion of) continuity.
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At the ontological level, the notion of continuous time vs.
discrete time is closely related to questions "Is the set of
time elements dense or not?", and " Are there really time atoms?".
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For a point-based model, the continuity is usually characterized as
"Between any two points, there is a third"; while for an
interval-based model (like that of Allen), it is characterized as
"Every interval can be decomposed into two adjacent sub-intervals".
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Let be
Now, the set
As for general treatments, the Basic Time Structure does not
have to impose the axiom of density or discreteness (Similar
arquements apply to issues such as linear/non-linear,
bounded/un-bounded). Therefore, the time structure as a whole may be
continuous or discrete, or neither continuous nor discrete.
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Now, "why an alternative notion of continuous structure is needed at
all"? It has been noted that temporal knowledge in the domain of
aritifical intelligence, including "temporal reasoning about actions
and change", is usually imcomplete, and using time intervals in
many cases is more convenient and more in-keeping with common
sense of temporal concepts than to use the classical abstraction of
points. In fact, the notion of time intervals (or periods) has been
introduced for a long time in the literature. In addition, in order
to overcome/bypass the annoying question of if intervals are open or
closed, various approaches have been proposed. An example is Allen's
interval-based time theory. As for these time theories, the old
(classical?) notion of continuity no longer simply applies.
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Let me ask you a more stringent question.
Premise: It is evident that if you assume the axiom of completeness,
the domain
Question: Suppose that you define your neither continuous nor discrete
Temporal Structure. What is your domain
To Pat
Why cannot time be continuous in some places but discontinuous at others?
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There is no mathematical objection to such a structure, and it has been
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You talk about a 'founding notion' of continuity as being that captured by
the axiom of completeness. Here, in my view, you commit a philosophical
error (especially in Pisa!) There are intuitions about continuity which
one can try to capture in various formal ways, but there is no 'founding
notion' of continuity other than those intuitions. ...
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Concerning the intuition, let me remind that the student who discovered the square root of 2 was killed (down the cliff), and no one was allowed to speak about ... ``the fault of the god'' for long time. Humans' common sense, to me, is something we shall not call too much.
century, famous mathematicians objected strongly to the view of the
continuum as consisting of a set of points, for example. This modern
perspective, now taught in high schools, is a modern invention, not a
'founding' notion. It is more recent than the gasoline engine, yet people
have had intuitions about smoothness, instantaneity and continuity for
eons. (Whether or not one agrees with me on this admittedly controversial
point, it seems unwise to identify a mathematical property such as
continuity with any kind of axiom until one has verified that no other
axiom will do as well; and as I am sure Sergio knows, there are many
alternative ways to axiomatize continuity.)
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If another axiom exists, which does as well, then it is surely equivalent to the axiom of completeness, just because it does as well. Alternative notions are clearly equivalent, until we speak about continuous domains. The point here, instead, was whether one can have a continuous domain with exceptions, that is the claim I originally criticized.
In my view, axioms are tools which we can manipulate at will; they are not
set in stone or somehow inevitable. Different formal accounts of time might
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be appropriate for different purposes or to capture different intuitions.
(I agree with Jixin that it is useful to seek a common 'core' theory which
can be extended in various ways to describe various possible more complex
temporal structures; and that this theory will have to be rather weak.)
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Finally, concerning your examples:
The 'dividing point' problem which gave rise to this discussion would do.
According to the modern account of the continuum, this point must exist,
and since all intervals consist of points, the light is therefore either on
or off at it. But it seems more natural, as well as formally simpler, to
just say that the question is meaningless; perhaps (though this is no
longer my own preference) because that point doesn't exist.
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A more mundane example is given by temporal databases, which usually assume
in their basic ontology that time is discrete: for example, they routinely
describe times as integers representing the number of milliseconds since
the birth of Christ. (Of course, one can always insist that these are to be
understood as being embedded in a continuum, but then what use is an axiom
whose sole purpose is to insist that times exist which have no name and
about which nothing can be asserted, other than that they exist?)
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Best Regards Sergio
Pat,
In answer to Sergio, you wrote
Why cannot time be continuous in some places but discontinuous in
others?
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The first problem is with respect to motivation. For what reasons would Time suddenly skip over potential timepoints? If the reason is, as you wrote, that
The 'dividing point' problem which gave rise to this discussion would do.
According to the modern account of the continuum, this point must exist,
and since all intervals consist of points, the light is therefore either on
or off at it. But it seems more natural, as well as formally simpler, to
just say that the question is meaningless; perhaps (though this is no
longer my own preference) because that point doesn't exist.
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The other problem is with respect to the axiomatizations. Since your
article "A catalog of temporal theories" characterizes the various
theories through axiomatizations, I thought I'd go back to that article
and check how you had done this formally. However I was not able to
find it; the closest I got was the denseness axiom on page 15. If the
intuitive notion is that time itself is continuous in some places but
not in others, wouldn't it be natural to start with an axiomatization
of continuous time (such as the real numbers) and then to proceed from
there? For example, a domain of piecewise continuous time could be
represented as a twotuple
Maybe I'm missing something - are constructs of this kind subsumed by the axioms in your report, or can they be inferred as theorems? Or why is this not the natural way of doing things?
Erik