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 1015 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