October 14, 1997

Mr. George Johnson

The New York Times

Dear Mr. Johnson,

I'm writing this
to you and your colleagues, and to others who may come to read it as well.
My life is coming to an end, and I would like to review my extensive work
with M. Robert Showalter. I have enormous respect for Bob's work and his
abilities. I have been honored to work with him.

First, I'd like to
give some of my own background. I've spent my career as a professor
at Stanford, and have paid more attention to fluid mechanics than anything
else. The Japanese Society of Mechanical Engineering named me as the most
productive experimental and theoretical fluid mechanician of the twentieth
century. Most of my colleagues, I believe, would agree that I am one of
the candidates for that designation. For technical and military reasons,
fluid mechanics has been a busy, productive field. I've been involved in
breakthroughs in my field, including the central one that makes computational
fluid mechanics as we know it today possible. My work has been much involved
in the conceptual work required to make mathematically and visually understandable
models possible, including careful, step-by- step application of mathematics
to physical modelling. I took a sabbatical year at the Harvard department
of mathematics, and have been aware of mathematics, and its limitations,
throughout the years. I'm a member of the National Academy of Engineering,
and have many other awards.

I've been principal
advisor to 40 Ph.D. students, and have served on the committees of many
more. I'm proud of my students. They have gone on to distinguished careers
in the academy and in industry. One of my students is also a member of
the National Academy of Engineering. My third to last Ph.D. student is
an astronaut.

I have not been Robert
Showalter's teacher or academic advisor. I have not supervised him as a
student. I have worked with him, as a colleague, without pay, for about
the last decade. I've sometimes put aside other calls on my time to do
so. I have worked with Bob much longer, and worked with him harder, than
I ever worked with one of my Ph.D. students. I did so because I thought
his work (our work) was VERY important, in my own field and many other
fields as well.

Bob Showalter and
I have worked, together, to solve what I believe are the most essential
problems in conceptual modeling and mathematical modeling. These are problems
that had to be solved if some important kinds of progress were to become
possible in the physical and engineering sciences. These are the hardest
conceptual and mathematical problems that I've ever encountered. We have
succeeded, but the work is conceptually very difficult work to put across.

Engineers and scientists
strive to describe the systems they deal with in precise terms, that make
imaginative and mathematical sense at all the levels that matter in their
work. Even so, the processes of description have remained surprisingly
incomplete and unsatisfactory over the centuries. In my SIMILITUDE AND
APPROXIMATION THEORY (McGraw-Hill 1965, Springer-Verlag 1984) I worked
to focus the procedures involved, with considerable but still limited success.
The connection between the concrete world and an abstract model remained
in some ways unclear, and the practice of constructing such models, particularly
in complicated cases, remained obscure and unreliable.

The most difficult
and longstanding problem was the construction of models of physical systems
that went beyond the very simple - models that included coupled effects,
and that included effects at a number of coupled levels. These problems
had concerned both of us for many years. On a high stakes commercial project
Bob and I worked on together almost twenty years ago, the job of constructing
a workable mathematical model of a coupled problem defeated us, even though
we consulted the best mathematicians we could find. Bob and I both came
to see the reasons for this as fundamental and very widespread in the sciences
and engineering.

About eight years
ago, Bob came to me with a direct and very important clue to this body
of problems. The Kelvin-Rall neural transmission equation, derived by accepted
modeling techniques, had to be very wrong. There were a number of reasons
to think so. The most important reason was this: zoom FFT EEG data from
Professor David Regan showed that the inductance in neural lines had to
be higher than Kelvin-Rall by factors of 10^{15} and more. By a
simple dimensional analysis argument, Bob came up with a term of about
the right size to account for this huge inductance. The term could be found
in the derivation logic of the transmission equation. However, it was a
kind of term that had always been dismissed as infinitesimal (0). This
was a very good clue, and the first good clue we'd ever had, about the
source of difficulties in modeling coupled systems mathematically. It involved
terms that had been of concern to James Clerk Maxwell before us.

This was, to both
of us, the most important problem in mathematical modeling anyone could
possibly work on, connected to difficulties in modeling and practice that
had been problematic for centuries. Bob focused on re-evaluating the derivation
of differential equations from physical models, proceeding from first principles
and evidence. I worked with him hard on this.

I also worked on conceptual
issues in modeling, including the systems concept, hierarchy in structured
systems, complexity, and its relations to the different world views of
our academic disciplines. I set this work out in CONCEPTUAL FOUNDATIONS
FOR MULTIDISCIPLINARY THINKING (Stanford 1995). Many people helped me in
the working out of these ideas, over many discussions. Bob's help was some
of the most important.

On the mathematical
work, I was in an advisory role, as Bob did the work that, we both felt,
only he could do. His job was to reassess old, ad hoc patterns for deriving
differential equations from coupled models, and create patterns that made
it possible to go, in a careful step-by- step fashion, from physical model
to mathematical representation, and back again. To do this, he had to work
out new schema, some in conflict with his own old ideas and mine. It was
VERY hard work. He kept at it (and sometimes I kept him at it) because
the work was so important. This matter of mathematical modeling is a matter
of life and death in some medical applications, and much else. We both
felt that, if Bob turned away from it, the problem might not be solved
for generations.

In my judgement, Bob is one of
the two or three most creative, most effective applied mathematicians I
have ever known, or known enough about to judge. My judgement covers mathematics
that can be applied to practical problems. I have some basis for comparison.
I've known mathematicians at Stanford, Harvard, and elsewhere. Bob Showalter
has excellent vision, to sense what problems need to be addressed, and
a good judgement of the relative importance of different problems. He is
brilliant. He works to be productive. He is flashy when he has to be, and
as conservative as he can be - good traits for a mathematical engineer.

Bob's quantitative skills
are linked to a strong and meticulously schooled sense of physical reality.

I believe that the mathematical
and modeling work that we have done together, set out in a number of pieces
we have co- authored, and that Bob has authored, will be of vital importance
in the sciences.

We've found out how
to derive equations so that they MATCH THE GEOMETRY OF THE PHYSICAL MODEL
THEY REPRESENT and so that they MAKE DIMENSIONAL SENSE at finite scale,
BEFORE the equations are mapped into abstract differential equations.

The mechanics of
our discovery is simple, but a challenge to the imagination of those with
conventional training, which is to say, everyone today. The work yields
an "unthinkable" result.

We are saying that
there can be PHYSICAL interactions between several kinds of physical laws,
that occur over a length or over an area, or over a volume, or over time,
that can be represented in terms in differential equations. That is, these
coupled effects can be represented at POINTS in valid terms in differential
equations.

We are saying that terms
that people have called infinitesimal (called 0) have finite values.

We are saying that terms
that people have called infinite have finite values.

We have had enormous difficulty
getting people to accept this, and the difficulty continues.

Even so, we have no reason
to doubt the result, on the basis of either theory or data. There are strong
reasons to believe it.

Strong reasons to believe
our results are embedded in our experience in fluid dynamics. In fluids,
the existence of the new crossterms permits us to organize our data conceptually.
Perhaps the clearest way to get a sense for fluid motion
is AN ALBUM OF FLUID MOTION assembled by Van Dyke. Again and again, as
the pictures show, flow patterns change mode as the value of the flow parameters
change. The number of different modes and patterns is now very, very large.
Shifts in patterns are COMMONPLACE all through our flow data. Experience
in fluid mechanics shows that, when values of the parameters are very different,
very different patterns are to be EXPECTED. The existence
of the crosseffects that our math shows makes such shifts expected.

In neurophysiology such
a shift is very important, and Bob has explained that shift in detail.
The Kelvin-Rall neural conduction equation corresponds to the conduction
equations that are dominant in electrical engineering. But in neurophysiology,
the values of the parameters R and C are enormously different from the
values we experience in ordinary electrical engineering practice. And our
experience in fluid mechanics repeats. The conduction pattern is VERY different
under the new conditions, with new terms becoming dominant. The
conduction pattern we see seems ideal for neural logic: opening or closing
membrane channels switches neural conduction between very different conduction
modes. Fits with data give us good reason to trust our neural conduction
equation, that we've taken to calling the Showalter-Kline (S- K) equation.
With the S-K equation, a whole new view of brain logic opens up. This work
should be important, both medically and scientifically.

I hope that anything
that can be done will be done to see that our results are tested, and discussed
under observed circumstances with "experts" who now reject it.
The results are straightforward. They yield straightforward, testable conclusions.

Perhaps some aspect
of the work will be shown wrong when it is tested further, theoretically
and empirically. I don't expect this to happen,
and if it does, expect the problem can be fixed. I believe that too much
fits for this work to be very wrong. Even so, the most surprising result
of the work - that a whole class of terms, never considered to be finite
before, can be finite and even large, is hard for people. The
difficulty of getting the work considered gives some index of how difficult
(and how important) the work is.

I hope that THE NEW YORK
TIMES, or anyone else who comes to read this, will do what they can to
get this testing done.

Let me close
with this. Bob Showalter is my colleague, not my student.
I've worked with him, just because I thought the work of vital importance,
for a long time. I haven't signed a Ph.D. thesis for Bob Showalter, but
I wish I could have done so. I respect his work as much as any work I've
ever been involved with. Bob Showalter deserves the respect and support
a productive scholar gets. I believe that he will continue
to do work of value to the nation and to the world at large.

Sincerely yours.

Stephen J. Kline

cc: M.R. Showalter