In the interest of stirring the pot, I'll make a few bold (but not really
original) assertions about derivatives:
Generally I find it pointless (though admittedly fun) to debate whether
differential or integral equations are a better mental model or description
of reality given that the models we're talking about can be described
either way. The practical point boils down to just two things: don't
confuse levels and rates, and don't connect rates to other rates. It is the
second of these principles that's really at stake here.
Forrester asserted long ago that all measurement devices available to us
report averaged rates (which are levels), not instantaneous rates. I have
yet to see a counter-example that proves him wrong. Admittedly, sometimes
our measurement devices have short time constants, so that it is useful to
treat them as if they were instantaneous. And sometimes (as with coflows)
it is convenient to connect flows to flows rather than to their common
determining levels, but I think this does not undermine the argument.
As to the more general assertion that ""nature never takes a derivative"" I
also have yet to hear a convincing counter-example, though I believe that
Ray and Bruce may be correct to suggest that one could exist, e.g. at the
quantum level (if that's what they meant). Since this theoretical problem
is remote from the practical problem of policy makers presuming that the
earth will cool off the moment energy efficiency technology improves, I'm
content to wallow in my ignorance and bumble along with my rates neatly
segregated.
>From: ""Weaver, Elise A"" <
eweaver@WPI.EDU>
>That was a good answer. But then, integrals don't exist in nature either,
>because they also depend on our choice of scale.
Are you suggesting that the volume of water in a bathtub or some other
quantity we would normally think of as a stock/level/state in SD is
indeterminate at a given point in time (neglecting Heisenberg)? Or merely
that the process of integration becomes as elusive as the derivative when
you start to look at it at finer and finer scales? Or ... ?
>From: ""Jim Thompson"" <
james.thompson@strath.ac.uk>
>We might start with the position that we can only access the real world
>through our senses or instrumental extensions of our senses. In other
>words, we can take the question of whether there is an absolute or real
>world off the table.
Great idea ... then we can skip the discrete/continuous debate as well.
>Fiddaman observes that our ""perception of slope is not independent of
>scale and becomes uselessly noisy on fine scales."" I take that to mean
>that our mental model or computer model or physical model does not produce
>useful understanding of our experiences when the scale is wrong for the
>problem at hand. That is, I understand Fiddaman to be saying that there
>are practical limits on the use of derivatives as part of one's mental
>model and the limitation on the usefulness of derivatives can be estimated
>by comparing the granularity produced by the derivative to the maximum
>value of another construct of thought, the integral.
Again, since SD models written as integral equations can easily be
rewritten as differential equations I'd hesitate to say that derivatives
are less useful than integrals as a mental model, though integrals work for
me personally. I'd simply suggest that, whether or not derivatives (slopes
in time) exist, we (or, more to the point, the decision makers we model)
have little if any practical ability to make direct use of them.
>Fiddaman implies that we cannot experience the future now and that we can
>accurately recall our past experiences. As Yogi Berra said about past
>experience, ""You can look it up."" Or was it Casey Stengel who said
>that? In any case, if you have had some experience with mountains and see
>one some distance away from you, your organised experience can give you a
>sense of what that pile of earth might mean to your travels. But that
>mountain can only be in your future. If your mental model includes
>constructs -- knowledge -- of integrals and derivatives, you might make
>some observations and useful predictions about any mountain in your future.
In the spirit of Forrester's caution against rate-to-rate connections, I
would merely suggest that slope, as perceived by a mountaineer wearing a
large sombrero and neck brace (""You can't look up"" in this case), is likely
to be biased towards the downhill terrain. In heavy fog, it is likely only
the recent downhill terrain that will be salient.
Tom
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Tom Fiddaman
Ventana Systems, Inc.
http://www.vensim.com
PO Box 153 Tel (406) 578 2168
Wilsall MT 59086 Fax (406) 578 2254
Tom@Vensim.com http://www.sd3.info
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