mathematical nonsense

[Chris Landauer]

this one i can't pass up - the anti-intellectualism is stunning

Mark P. Line wrote:

I wrote that ODE's were invented in order to provide analytical solubility for
otherwise intractable problems.

I say:

This is simply incorrect - ODE's were invented to provide analytic _models_ of
complicated phenomena, usually in physics at first, and only some of them were
ever analytically soluble.

Mark P. Line wrote:

I also wrote that no interesting problem formulated as ODE's or PDE's is
likely to be analytically soluble, and that therefore their rationale as tools
for analytical manipulation no longer exists.

I say:

The first part of this is correct, but the second part is not.  There are many
more analytic results that can be obtained from an ODE or PDE model than the
solution.  Indeed, many mathematicians have proved long term stability results
for systems that have no hope of analytic solution.  The power of an analytic
formulation is _not_ about solutions; it is about understanding, and there are
many different kinds of useful models.  If you naively limit your
understanding of mathematical analyses to "solutions", then you miss many of
the most interesting and exciting results.

Mark P. Line wrote:

In saying that, I was trying to imply that, if we're going to have to attack
our models numerically anyway, then we might as well go straight for an ABM
representation instead of blissfully accepting the inappropriate assumptions
(e.g. differentiability) of ODE's and PDE's.  Those assumptions were the
lesser evil when the problem at hand was otherwise intractable, period. That's
no longer the case in domains where we can build useful ABM's.

I say:

There is a grain of truth in some of this, in that the mathematical
assumptions often do not reflect the reality, but the same is true of the ABM
models also, and anyone who forgets that is likely to get less than useful
results.  It also ignores the fact that many ODE and PDE models of complex
phenomena are _much_ more easily analyzed (even without solutions) than the
corresponding collections of individual actors.

There are many people who think that modeling individual behavior in a complex
system is somehow "less modeling; more reality", but that is too often just
plain wrong, because the important behaviors are lost in the shuffle of
complex interactions.  The modeler must make choices either way; different
choices make more sense for different purposes.  For example, starting almost
100 years ago, Poincare and others studied long term stability of ODE's, such
as those derived from the 3-body problem, and Fatou and Julia studied certain
limiting sets of infinite processes in the complex plane, with no hint of an
analytic "solution".

My personal opinion is that the computational power available currently allows
much modeling laziness, and while sometimes helpful (I have on many occasions
used computer experimentation to give me insights about tricky problems), it
actually very often inhibits clear and useful thinking about complex models.

Don't throw away useful tools; we need all the help we can get.

more later,
cal

Dr. Christopher Landauer
National Systems Group, The Aerospace Corporation
The Hallmark Building, Suite 187
13873 Park Center Road, Herndon, Virginia 20171
e-mail: address@hidden
Phone: (703) 318-1666, FAX: (703) 318-5409



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