Discrete versus continuous -- Ive been thinking about this for
several days after a private communication with Gene Bellinger,
and I think I might finally have some admittedly esoteric ideas
as to why he remains somewhat confused.
First of all, let me say that this topic comes up all the time
in electrical engineering (EE), where it goes by the name,
"digital versus analog" -- different names, same problem in that
smart people continue to disagree about the "best" way to solve
engineering problems. When I studied EE, I came down on the
analog side of the fence as, essentially, all EE problems
are analog. If you look at the response of a transistor at
a small enough time scale, it is indeed nonlinear, hence analog.
Digital circuits are thus a simplification of the underlying
analog nature of their constituent parts. Keep aggregating
up and computers are the same things, digital simplifications
of their analog silicon substrate.
The ability of digital computers to simulate continuous response
has been recently discussed here, and so I wont go into it again,
but what I do what to discuss is how really easy it is to go back
and forth between digital and analog (discrete and continuous) as
each is good at different things. The question remains, what are
these things?
Weve already highlighted two dichotomies -- discrete vs.
continuous and digital vs. analog -- so let me bring up some more,
left vs. right brain, or "logic" versus "pattern matching."
The brain actually encompasses both types of cognitive processes,
discrete and continuous, although they go by the names, "logic" and
"pattern matching." As most of us probably already know, Russell and
Whitehead tried to put the finishing touches on logic but was
thwarted by Kurt Godel and math types have since been trying to
interpret just what Godels theorem means ever since with some
holding this is the most important theorem ever and others holding
that its just a proof with little application outside mathematics.
Nevertheless, mathematician Rudy Rucker writes,
On the face of it, the application of formal logic
might be expected to resolve all kinds of disagreements.
As it turns out though, the known lows of logic are too
few in number to be of any great help.
Where does this leave us? Is logic, and therefore discrete
simulation, useless? Of course not. What it does mean though
is that logic (zeros and ones, ands, ors, and nots), is a simplification
of the types of situations describable through differential equations.
This has a cognitive component as well. The left brain contains the
sections that process logic and language, language being reducible
to logic. These are the simplifications that allow people to
communicate with one another. The right brain is able to "pattern
match," that is to perform much more sophisticated computations in
real time. For instance, fighter pilots and artists both primarily
use the right part of their brains. So we are left with the idea
that there is a complex reality out there that is more complex than
be communicated through language. One need only think of being at
the receiving end of a complex description that all of a sudden
becomes clear and exclaiming, "NOW I see what youre saying!"
This phrase captures the interface between left and right brain.
(as an aside, let me note that people are indeed language creatures
as Chomsky argues, but this logical process takes place on an
continusous substrate, neurons. One might want to read William
Calvins "Cerebral Symphonies" to appreciate the continuous aspects
of brains.)
Now let me return to the topic of this discussion group, system
dynamics. While system dynamics has always appealed to me as an
old analog EE type (but younger than Jay whos also an analog EE
type), Ive always wanted to know why system dynamics is right.
My current opinion is that system dynamics specifically and
continuous simulation generally allows one to make a more right
brain, pattern oriented argument than differential equations or
logic would allow. Moreover, system dynamics allows a richer
description of "reality" and all that implies.
Also, now that Im on a roll and have probably lost all my audience,
the scenario analysis of system dynamics allows one to explore the
entropy space inherent in any continuous system. I am not going to
explain this here as it would take to long and draw on knowledge
learned many years ago, but the key point is that a logical,
discrete analysis doesnt have the mathematical subtlety to do the
same.
Although theres much more to be said on this topic, I think Ill end
the argument on this note: continuous simulation is richer and harder
to comunicate while discrete is simpler and easier to communicate.
-------------------------------------------------
| Corey L. Lofdahl -*-
lofdahl@sobek.colorado.edu |
| Institute of Behavioral Science |
| University of Colorado at Boulder |
| "there is an ethic to design, interact" |
-------------------------------------------------