I hope it isn't considered too far off-topic as I think the relation
between continuous and discrete time is very important and very difficult,
and people don't seem to think about it enough; perhaps because they regard
the issues as inconsequential, but more likely because they regard them as
solved. Once they start discussing it properly, it becomes clear that the
things they regarded as trivial, are not trivial at all, the things they
regard as solved, are not solved at all, and all sorts of difficulties
arise for the unwary.
Few people, to my knowledge, experiment with continuous time models at all.
In this respect I think the work that Duncan did in the 1980s, and which is
now being developed by a number of his former students, deserves a lot more
attention.
Incidentally, I don't see this question as divisive in relation to the
temporal/static distinction, and indeed I don't see it as the same
question. My difference with the static method is not that it fails to
understand time as continuous, but that it represents each instant in time
as the consequence of a stable equilibrium. The distinction betwen temporal
and static methods appears in both discrete, and continuous time,
representations of reality.
Nevertheless I think that anyone who makes a serious attempt to move to
continuous time, becomes obliged to confront more directly the logical
problems involving time that can be brushed aside in discrete models, but
must be faced squarely in continuous models: not least, the relation
between stocks and flows.
I think the modern view is that the theorisation of time, particularly the
relation between discrete and continuous time, hasn't been solved. It has
given rise to endless paradoxes, not least Zeno's. At the turn of the
century it was thought that this paradox had been answered but when
attempts were made to programme computers to think about time, the
practical people found that Zeno's paradox is alive and kicking. All sorts
of issues are involved, not least the following: what is an event?
In the literature, the example that is usually given is turning off a
light. Clearly, in some sense this is an event. Before it, the light is on,
and after it, the light is off. But if you wanted to take a picture of it,
when should you click the shutter? Since even a still photograph takes a
certain definite time to take, you can never hope to capture the 'turning
off' event; you can only photograph before, and after. The 'event' seems
not to be a state of affairs at all but a gap defined as the distinction
between two states.
Thus no matter how fast you run the camera, you will always find that one
frame the light is on, and on the next frame the light is off. You cannot
have a photograph of the light 'being switched off'.
This is different from the analagous situation regarding space, where we
can at least conceive (wrongly or rightly) of a point as an entity in its
own right, speak of lines as being composed of points, and so on. Taking
the arrow and the tortoise as an example, we could take a motion picture in
which, in each frame, the arrow is at a definite point, but we could not
photograph the event of the tortoise's death. We would always have one
frame with a live tortoise, and the next frame with a dead tortoise. It
doesn't seem to work to try and conceive of an event as being like a point,
as being a sort of atom of time.
There may be an underlying issue which is whether time is really granular;
whether it really is, at some quantum level, a succession of discrete
events; or whether it is 'really' continuous. Worse still, even if it is
granular, it is hard to conceive how this granularity could extend in space
without violating the relativity principle that there is no absolute frame
of reference; thus, each particle (always assuming we can define a
'particle') would have to have its own clock, and there would be no reason
for supposing that the clocks tick at the same 'moment'. We would go back
to Monads, I suppose.
If we suppose that time really is quantised, granular, a succession of
discrete moments, then an event is just some state variable taken at one of
these moments and it is quite simple; we can actually find a frame in which
the 'light is going off' or the 'tortoise is dying', unless death exists
like space for Democritus, as a gap between the atoms of time.
But if time really is continuous, then it is quite hard to conceptualise
what an event might consist of.
If time is not discrete, then no film camera, no matter how good, can ever
capture it and (in this case) this is how I would put it pedagogically;
reality is simply more complex than can be captured with one camera. So one
could explain it like this: no matter how good the camera, things can
happen while it isn't looking. No camera can hope to be adequate to the
reality and perhaps the best way to explain things pedagogically is to make
the distinction between the reality and its image.
In that case, what is a continuous time model? Is it 'the true reality'? I
would not like to be so commital. I have a sneakily heretical view that the
atomists have had a bad press on the left because of the debate about
methodological individualism; atomism was originally a heretical
alternative to the notion of the divine perfection of form.
I prefer to think of it like this: whether or not time is 'really
granular', the constraints on any equations we use to model reality - if
that's what we want to do, and here I really do mean 'model' - are a lot
fiercer than most people realise. This is one reason I am very sceptical
about the whole notion of economic models. Not least, since we do not know
what the granularity of time is, we have no special reason to assume any
definite period. Therefore, our equations must be equally valid, no matter
what period is chosen. Otherwise, an arbitrary assumption has been sneaked
into the equations; that there is a definite period over which things
happen. This then gives rise to all manner of ontological problems which
people just don't think about carefully enough, since it allows the
blurring of the distinction between stocks and flows, and the insertion of
all manner of confusions about when things 'really happen'.
I think of a continuous time model, oddly enough, as the best possible
approximation to reality, in the sense that one may read off the results
with any desired periodicity. I consider a period model to be correctly
stated when the period may be reduced arbitrarily and in consequence the
trajectory converges uniformly to a limit, which is then the solution of
the corresponding continuous model; in this case, the continuous version is
the lower limit of an entire family of discrete models. It is by no means
the case that this can always be done, for example there are well-known
examples from chaos theory that only exist in discrete form.
One may always *generate* a family of discrete models from any continuous
model, which is essentially what one does when one makes a numerical
approximation. Thus to every continuous equation system there corresponds
an (uncountably infinite) family of discrete systems. The continuous system
is in some sense the envelope of this family.
The reverse is not true; there are discrete systems which have no
continuous counterpart. [This is what makes me suspicious about an absolute
commitment to continous time as the true reality].
Therefore, I am inclined to think that the really important distinction is
between period-independent models, and period-dependent models. My general
approach to continuous time, which is very cautious indeed, is to ask the
following question: what can we say about reality that does not depend on
the interval of time that we choose to use?
If one wants to pursue the pedagogical example of the movie camera, we
might ask what happens when there are a large number of movie cameras all
moving at different speeds. If the results from all the cameras are in some
sense 'the same'; if there is no phenomenon that depends on the speed of
the camera, then this 'same' - the combination of information from all
possible movie cameras - is the continuous model.
Thus a movie camera synchronised to the mains frequency might always show a
fluorescent light as 'off' since these lights go on and off at mains
frequency. But we could determine that the 'real' state of affairs was
otherwise, by having a second camera synchronised to a frequency that was
prime relative to the mains frequency.
A digression: as stated above, I don't think it is quite the same question
as whether
there is temporal *succession*, an issue that exists in both discrete and
continuous time. The decisive issue for temporal succession is the causal
relation between conditions in the past and conditions in the future.
This issue exists in both discrete and continuous models, though I think
that in continuous models it takes a much simpler form.
In a discrete model, the distinction is the following. If I write down a
state variable X(t) which is a function of time, then a temporal relation
would be one like
X(t) = f(X(t-1)) (1)
where f is some non-trivial function (or more generally X(t) = f(X(t-1),
X(t-2),...))
The equivalent static relation would be
X(t) = f(X(t)) (2)
so that X is given as the fixed point of f. In the static formulation, X at
any given time cannot influence X at any other time. In the temporal
formulation, X at any given time depends on X at previous times and indeed,
cannot otherwise be defined.
In the continuous form of this system, the relation (1) is replaced by
X' = g(X)
or, more generally,
g(X, X', X'',...)=0
The equivalent static form would assert that X' is functionally independent
of X, in this case that g is identically zero.
A comparative static model singles out one or more elements of X and
'temporalises' them without temporalising the whole of X. The discrete form
is
[X(t), a(t)] = f([X(t), a(t-1)])
so that X at time t is functionally independent of X at all previous times,
instead of a genuinely temporal model
[X(t), a(t)] = f([X(t-1), a(t-1)])
The continous form of the comparative static model would be
[X, a] = f(a')
and the temporal form would be
[X, a] = f(a',X')
Thus, it is I think clearer, in a continuous formulation, what a static
ontology really means; it means that X may be determined independently of
its changes. It is tantamount to eliminating all dynamic forces, except for
the changes in the parameter a.
Alan