http://dx.doi.org/10.1090/psapm/052
Selected Titles in This Series
52 V. Mandrekar and P. R. Masani, editors, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, Michigan, 1994) 51 Louis H. Kauffman, editor, The interface of knots and physics (San Francisco, California, January 1995) 50 Robert Calderbank, editor, Different aspects of coding theory (San Francisco, California, January 1995) 49 Robert L. Devaney, editor, Complex dynamical systems: The mathematics behind the Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994) 48 Walter Gautschi, editor, Mathematics of Computation 1943-1993: A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brockett, editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January 1987) 36 Carl de Boor, editor, Approximation theory (New Orleans, Louisiana, January 1986) 35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and William Gewirtz, editors, Mathematics of information processing (Louisville, Kentucky, January 1984) 33 H. Peyton Young, editor, Fair allocation (Anaheim, California, January 1985) 32 R. W. McKelvey, editor, Environmental and natural resource mathematics (Eugene, Oregon, August 1984) 31 B. Gopinath, editor, Computer communications (Denver, Colorado, January 1983) 30 Simon A. Levin, editor, Population biology (Albany, New York, August 1983) 29 R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981) 28 R. Gnanadesikan, editor, Statistical data analysis (Toronto, Ontario, August 1982) 27 L. A. Shepp, editor, Computed tomography (Cincinnati, Ohio, January 1982) 26 S. A. Burr, editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981) 25 S. I. Gass, editor, Operations research: mathematics and models (Duluth, Minnesota, August 1979) 24 W. F. Lucas, editor, Game theory and its applications (Biloxi, Mississippi, January 1979) (Continued in the back of this publication) Norbert Wiener 1894-1964 Photograph taken in 1963 in Madison, Wisconsin. Proceedings of the Norbert Wiener Centenary Congress, 1994 Proceedings of Symposia in APPLIED MATHEMATICS
Volume 52
Proceedings of the Norbert Wiener Centenary Congress, 1994
Michigan State University November 27-December 3, 1994
V. Mandrekar P. R. Masani Editors
S, American Mathematical Society a Providence, Rhode Island
^NDED 1991 Mathematics Subject Classification. Primary 60H30, 42A38, 94A05, 31C15, 81P20; Secondary 60G46, 60H05, 94A15.
Library of Congress Cataloging-in-Publication Data Norbert Wiener Centenary Congress (1994 : Michigan State University) Proceedings of the Norbert Wiener Centenary Congress, 1994 : Michigan State University, November 27-December 3, 1994 / V. Mandrekar, P. R. Masani, editors. p. cm.—(Proceedings of symposia in applied mathematics, ISSN 0160-7634 ; v. 52) Includes bibliographical references. ISBN 0-8218-0452-9 (alk. paper) 1. Stochastic analysis—Congresses. 2. Fourier analysis—Congresses. I. Mandrekar, V. (Vid- yadhar), 1939- . II. Masani, Pesi Rustom. III. Title. IV. Series. QA274.2.N67 1994 519.2—dc20 96-43346 CIP
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© 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97 Norbert Wiener Centenary Congress
November 27, 1994 to December 3, 1994 Michigan State University
Department of Statistics and Probability Michigan State University East Lansing, Michigan
co-sponsored by the American Mathematical Society, International Association of Cybernetics, and World Organization of Systems and Cybernetics
Organizing Committee J. Benedetto (University of Maryland) D. L. Burkholder (University of Illinois, AMS representative) T. Kailath (Stanford University) G. Kallianpur (University of North Carolina) V. Mandrekar (Michigan State University) P. R. Masani (University of Pittsburgh and WOSC representative) S. Mitter (MIT) I. E. Segal (MIT)
Local Organizers Raoul LePage and V. Mandrekar
Supporters National Science Foundation Army Research Office Capital Area Community Foundation Institute of Mathematics and Applications Deutsche Forschungsge Meinshaft French Academy of Sciences Osterreichische Studiengesellschaft Fur Kybernetik Royal Netherlands Academy of Arts and Sciences The Royal Society The Swedish Academy of Engineering Sciences Swiss Academy of Sciences
vii Contributors
D. R. Adams, Professor of Mathematics, University of Kentucky S. Albeverio, Lehrstuhl Fakultat und Inst it ut fiir Mathematik, Ruhr-Universitat Bochum H. Bart, Professor of Mathematics, Economic Institute, Erasmus University, Rotterdam J. R. Benedetto, Professor of Mathematics, University of Maryland N. K. Bose, HRB Systems Professor, and Director, The Spatial and Temporal Processing Center, The Pennsylvania State University D. L. Burkholder, Professor of Mathematics, Center for Advanced Study, University of Illinois E. H. Carlen, Associate Professor of Mathematics, Georgia Institute of Technology, Atlanta H. A. Feichtinger, Professor of Mathematics, University of Vienna G. D. Gale, Professor of Philosophy, University of Missouri-Kansas City /. Gohberg, Professor of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University L. Gross, Professor of Mathematics, Cornell University M. A. Kaashoek, Professor of Mathematics, Vrije University G. Kallianpur, Alumni Distinguished Professor of Statistics, Center for Stochastic Processes, University of North Carolina, Chapel Hill J. R. Klauder, Professor of Mathematics, University of Florida P. Malliavan, l'Academie des Sciences, Paris R. W. Mann, Whitaker Professor of Biomedical Engineering Emeritus Massachus• etts Institute of Technology P. R. Masani, University Professor, Department of Mathematics and Statistics, University of Pittsburgh B. McMillan, Formerly Vice President, Bell Telephone Laboratories S. A. Molchanov, Professor of Mathematics, University of North Carolina, Charlotte, formerly Professor of Mathematics and Statistics, Moscow State University 0. Penrose, F.R.S., Professor of Mathematics Emeritus, Herriot-Watt University, Edinburgh K. H. Pribram, James P. and Anna King University Professor and Eminent Scholar, Commonwealth of Virginia, Radford University, and Professor Emeritus, Stanford University ix x CONTRIBUTORS
J. Rissanen, Professor of Mathematics, Technical University of Tempere, Finland, and member of Research Staff, IBM Almaden Center D. Roseman, Professor of Mathematics, University of Iowa /. E. Segal, Professor of Mathematics Emeritus, Massachusetts Institute of Technology H. A. Stapp, Senior Staff Physicist, Lawrence Berkeley Laboratories, University of California E. G. F. Thomas, Professor of Mathematics, University of Groningen R. L. Wornock, Visiting Physicist, formerly Staff Physicist, Stanford Linear Accel• erator Center S. Watanabe, Professor of Mathematics, Kyoto University Contents
Norbert Wiener Centenary Congress: Sponsors, organizing comittee and supporters vii List of contributors ix Preface: Overview of the Norbert Wiener Centenary Congress and acknowledgment by the editors xv Program xlv
I. Wiener's Concept of the Stochastic Universe.
Wiener-Kolmogorov conception of the stochastic organization of nature S. MOLCHANOV 1 The Wiener programme in statistical physics. Is it feasible in the light of recent developments? OLIVER PENROSE 37 The mathematical ramifications of Wiener's program in statistical physics LEONARD GROSS 53
II. Potential and Capacity Before and After Wiener.
Potential and capacity before and after Wiener DAVID R. ADAMS 63
III. Generalized Harmonic Analysis and Its Ramifications.
Generalized harmonic analysis and Gabor and wavelet systems JOHN J. BENEDETTO 85 Wiener-Hopf equations and linear systems H. BART, I. GOHBERG, AND M. A. KAASHOEK 115 Complex harmonic analysis in the aftermath of Paley-Wiener PAUL MALLIAVIN 129 Amalgam spaces and generalized harmonic analysis HANS G. FEICHTINGER 141 xii CONTENTS
IV. Quantum Mechanical Ramifications of Wiener's Ideas.
Wiener space and nonlinear quantum field theory IRVING SEGAL 151 Wiener and Feynman—path integrals and their applications SERGIO ALBEVERIO 163 Optical coherence before and after Wiener JOHN R. KLAUDER 195 Wiener and the problem of "Hidden Parameters" in quantum mechanics ERIC A. CARLEN 213 Finite path integrals ERIK G. F. THOMAS 225 Joint probabilities of noncommuting observables and the Einstein-Podolsky- Rosen question in Wiener-Siegel quantum theory ROBERT L. WARNOCK 233
V. Leibniz, Haldane and Wiener on Mind.
The role of Leibniz and Haldane in Wiener's cybernetics GEORGE GALE 247 Quantum mechanical coherence, resonance, and mind HENRY P. STAPP 263 What is mind that the brain may order it? KARL H. PRIBRAM 301
VI. Shannon-Wiener Information and Stochastic Complexity.
Shannon-Wiener information and stochastic complexity J. RISSANEN 331
VII. Nonlinear Stochastic Analysis.
Sharp norm comparison of Martingale maximal functions and stochastic integrals DONALD L. BURKHOLDER 343 Stochastic analysis on Wiener space SHINZO WATANABE 359 Stochastic filtering: A part of stochastic nonlinear analysis G. KALLIANPUR 371 The theory of learning before and after Wiener N. K. BOSE 387 CONTENTS xiii
VIII. Prosthesis, Ontogenetic and Phylogenetic.
Sensory and motor prostheses in the aftermath of Wiener ROBERT W. MANN 401 On Wiener's thought in the computer as an aid in visualizing higher- dimensional forms and its modern ramifications DENNIS ROSEMAN 441
IX. Wiener and the Political Economy: Automatization, Educational Decline and Unemployment (Round Table).
X. Cybernetics.
Norbert Wiener and the future of cybernetics P. R. MASANI 473
Letter to Norbert Wiener from John von Neumann 505
XI. Award of Norbert Wiener Centenary Medal.
Scientific impact of the work of C. E. Shannon BROCKWAY MCMILLAN 513 Citation and award of the Norbert Wiener Centenary Medal to Professor C. E. Shannon (in absentia) JEAN RAMAEKERS 521 Photograph: Presentation of medal to Mrs. C. E. Shannon by Jean
Ramaekers, President of the International Association of Cybernetics 523
XII. A Tribute to Wiener from Dennis Gabor, F.R.S., N.L.
Wiener and the art of communication
DENNIS GABOR 525 Academic vita of Norbert Wiener 537 Doctoral students of Norbert Wiener 541 Bibliography of Norbert Wiener 542 Defense Department documents 557 List of participants 559 Acknowledgments 565 Overview of the Norbert Wiener Centenary Congress and Acknowledgment by The Editors
The creative development of Norbert Wiener had much to do with the extremely wholesome climate that prevailed in world science during the period 1900-1930. The Principia Mathematica had secured the foundations of pure mathematics, and the work of Einstein and the new determination of Avogadro's constant by Perrin left little doubt that atomicity is a genuine aspect of the cosmos. This intellectual atmosphere, coupled with a home environment, that stressed the importance of intellectual hard work and intellectual honesty, has given us not only a great mathematician but a polymath—one of the truly great minds of this century.l In shouldering the responsibility of planning the Proceedings of the Norbert Wiener Centenary Congress, the Organizing Committee has endeavored to give priority to Wiener's intellectual evolution. First, in deciding on the sessions, we were guided by the natural contours of this evolution. Next, we chose the subjects of the lectures within a session on the basis of Wiener's own work, and finally we chose as speakers, the best available persons who could do justice to the chosen subjects. But we were of course constrained by temporal and budgetary limitations.
A. Philosophical foundations of mathematics Wiener's first research was in the philosophical foundations of mathematics. Begun at Harvard, it continued under Bertrand Russell at Cambridge University. The philosophical interest so acquired gained strength from the ongoing growth of science and technology in his midst, and lasted till the end of his life, witness his philosophically-inclined book
1 For the life of Wiener, we refer the reader to his autobiographical books [53h], [56g] and to the articles [L2], [S3], [M2] and to the book [M3]. xvi PREFACE Cybernetics [61c], first published in 1948 and his God, Golem, Inc.—A Comment on Certain Points Where Cybernetics Impinges on Religion [64c], published posthumously. It seemed best, however, not to open the Congress with a session on the philosophical viewpoint of Wiener, but to let this viewpoint, which permeated all his scientific work, seep in during the earlier sessions, then to broach an important aspect of it in Session V, and finally to attend to its other aspects in a lecture on cybernetics towards the close of the Congress.
B. The stochastic cosmos A fundamental aspect of Wiener* s philosophy was his firm faith in the stochastic organization of the cosmos—the belief that the laws of nature are probabilistic; witness the references to Josiah Williard Gibbs repeatedly in his writings. One more instance of this appears in the book Invention: The Care and Feeding of Ideas, which was published only in 1993, although composed by Wiener in the mid-1950s: Let us suppose that with the sweep of his wand a wizard could bring Newton down among a crowd of scientists of the present day . .. The first things that I can imagine him doing would be to get hold of the books of Gibbs, of Einstein, and of Heisenberg and to study these carefully. . . . What is certain, whether he accepted or rejected them, is that he would not ignore them, and that some of the work which he would do would throw light on the moderns. For a man who had the frankness and courage to say, "If I have seen further [than others] it is by standing on the shoulders of Giants," would not hesitate for a moment to climb even higher on the shoulders of our latter-day giants. [93a, pp. 109-110]
Notice that all three "latter-day giants" were firm believers in the stochasticity of the cosmos, albeit with somewhat varying shades of the term "stochastic". This stochastic viewpoint permeated especially Wiener's first major piece of mathematical research, to wit, the idealization of the Einstein- Smoluchowski theory of the Brownian motion, cf. [23d]. In the Brownian motion paper [23d], and more so in [20f], [24d], Wiener constructed a stochastic process (SP) by a method that Kolmogorov was to universalize in 1933 in his celebrated Foundations [Kl]. It thus seemed fitting to PREFACE xvn devote the first session of the Congress to Wiener's conception of the stochastic cosmos, and open with a lecture on "The Wiener Kolmogorov conception of the stochastic organization of nature" by Professor Stanislav A. Molchanov, that delves into the ongoing work in the very areas of statistical physics, in which Wiener was most interested, fluid flow, turbulence and plasma. With the advent of the Ergodic theorems of Birkhoff and von Neumann, Wiener resumed his stochastic activity, after a break of about a decade, in joint work with Paley and also Zygmund in 1933 [33a]. This was a prelude to his big paper [38a] on the Homogeneous Chaos. In this paper, Wiener laid out a clear ergodicity-based program on how, in his eyes, the statistical mechanics of continuous media ought to develop. A lot of progress has been made in this area after 1938, and it is a moot point if the Wiener program still holds water. In the second lecture "The Wiener program in statistical physics. Is it feasible in the light of recent developments?", Professor Oliver Penrose addresses this issue. The paper [38a] and the earlier papers on Brownian motion, seeded a lot of activity on the part of Cameron and Martin, S. Kakutani, K. Ito and I. E. Segal, and to attempts to re-express their contents in the setting of Hilbert spaces. In their definition of the stochastic integral Paley-Wiener had revealed the important role of absolutely continuous functions in C[a, b]. In 1944 Cameron and Martin [CM] considered the Hilbert space 9i of a suitable subclass of these functions f with f E £2 := ^ifa b], and inner product defined by (f, g)#:= (f, %')£r It then easily follows that the Paley-Wiener stochastic integration on £2 yields a linear isometry T on the
Cameron-Martin Hilbert space !H to a Gauss-Hilbert subspace § of the L2- space over Wiener's probability space C[a, b] with Wiener measure. In 1956 I. E. Segal [S2] realized that given an isometry T on any separable Hilbert space !H to a Gauss-Hilbert space, one can define a finitely additive probability measure ^i on the algebra C of Borel cylinder subsets of #as follows: For all integers n ^ 1 and all Borel subsets B of IRn, the vectors
{x\,.. ., x'n} Q!H' determine the Borel cylinder
C#- • x'n: = {x E# : ((x, xi),..., (x, x£)) e B}.
We define ^CCgx' 1'" x'n ) to be the probability of the Gaussian n-variate, XV111 PREFACE
(T(x\) . . . T(x^) at B c Rn. A culminating point in this development was the demarcation of so-called abstract Wiener spaces by L. Gross under the guidance of Segal. Such spaces have received a good deal of attention. It thus seemed proper to close this session with a lecture on "The mathematical ramifications of Wiener's program in statistical physics" by Professor Leonard Gross, in which the ideas of [38a] are extended to Hilbert spaces and the Ito-Wiener decomposition is carried out on compact Lie groups.
C. Potential and capacity In the mid-1920s Wiener was lucky to have present at Harvard O. D. Kellogg, the great authority on the theory of the potential and capacity, the theme of our second session. Wiener quickly picked up the subject from him, and within a few years significantly advanced it in his researches. The year 1994 saw beautiful extensions of two of these Wiener results on the Laplace equation, to non-linear PDEs of order p ^ 3, at the hands of Kilpelainen and Maly. In the paper "Potential and capacity before and after Wiener", Professor David R. Adams traces the history of this subject, which began as the Newtonian theory of attractions in the 17th century, but still vigorously labors on, and on the growth which Wiener's contribution is clearly registered.
D. Harmonic analysis Our Session III on this subject calls for a preliminary comment. It squeezed in a large territory that included micro-local analysis, for instance, but our time-frame prevented us from devoting a much-needed lecture of its own to generalized harmonic analysis, and left no room whatever for anything on the cognate field of stationary processes. These omissions and the absence of a manuscript on micro-local analysis obligate us to digress a little on each topic and allude to some of the open problems. What is harmonic analysis and why is it important? The answer is given clearly in §1 of Wiener's book, The Fourier integral and certain of its applications [33i], in which he lucidly explains the significance of the characters of IR. But as early as 1925 Wiener was keenly aware of the limitations PREFACE xix inherent in the harmonic analysis of phenomena marked by very rapid changes in frequency. This "paradox of harmonic analysis", as he called it, comes from the fact that frequency is a temporal concept, and the expression "the frequency v(t) at instant t", while it can be defined as co(t)/2jt, where o>(t) is the angular velocity at instant t, makes little physical sense. Wiener lectured on this problem at Gottingen in 1925, cf. [56g, p. 106], but never published his analysis. In 1946 Dennis Gabor [Gl] in England put this paradox to constructive use. For problems marked by rapidly fluctuating frequencies Gabor proposed a new 3-parameter family of elementary functions, to replace the one parameter family of characters of R. His elementary functions are defined to be the functions that turn into an equality the fundamental inequality
(1) oo ^ J I tf(t)|2dtj I Xf(X)|2dX —00 -00
;> const. J |f (t) 12dt • J | f(k) \ 2dk, -00 -00 that relates f to its Fourier-Plancherel transform f . Furthermore, Gabor proposed partitioning the time-frequency plane into cells for each of which At • AX - the constant in (1), (say 1, after normalization), calling such cells logons. Each cell carries exactly one datum on time and one on frequency, by virtue of the restrictions imposed by (1) on their simultaneous obtainability. Unknown to Gabor, both of his key ideas had been worked out by von Neumann in the section on "Macroscopic measurement" in his celebrated book on the foundations of quantum mechanics in 1930 [V]. On p. 406 of [V], the 3-parametric family of elementary functions appears on the top, and the partitioning of the time-frequency plane, in order to discretize the parameters, is indicated in the middle.2
2 However, von Neumann's goal is to orthonormalize the denumerable number, so obtained, of elementary functions, which the modern wavelet followers of Gabor seem reluctant to do. XX PREFACE
The ideas of Gabor and von Neumann have fructified into what is now called micro-local analysis, cf. C. Fefferman [F]. The heuristic that the characters are (pseudo-) eigenfunctions of the differential operator leads one from a partial differential operator A(t, D) to the function A(t, X). In the simplest case, t eR, D = d/dt, we have X e IR, i.e. A( •, •) is a function on the time-frequency plane. To find the number of eigenvalues in a region R of this plane, one counts the number of cells in R. All this is clear from the first two pages of Fefferman's paper, on noting that "logon-partitioning" is called "packing disjoint cubes". Of course the micro-local analysts have pushed these ideas much further. They have brought in bent boxes ("bent" or "curvilinear" logons) for non- elliptic PDEs. Gabor referred to the time-frequency plane as the information plane, meaning information about the structure of time and frequency of the signal. This designation is most fitting for a sheet of music, for instance. This "information" is of course different from the Shannon-Wiener stochastic information. It is not usually appreciated that this 1946 Gabor paper, titled "The theory of communication" [Gl], antedated both Shannon's important monograph The Mathematical Theory of Communication [S4] and Wiener's Cybernetics of 1948, by about two years. Obviously we have to place Dennis Gabor alongside R. V.L. Hartley and W. Ross Ashby as a very important British pioneer in cybernetics.3 Although Wiener was aware of the above problem, and lectured on it in 1925, he contributed nothing towards its solution. His great contribution lay in a different direction, to wit, a fuller exploitation of the potentialities latent in the characters of IR. The writings of Kelvin, Rayleigh, Stokes and Schuster had strongly suggested that the subject has a scope far wider than that manifested in the classical theory of Fourier series and Fourier integrals. Wiener wanted to carry out the program of these pioneers rigorously. This extension, to so-called generalized harmonic analysis (GHA), first submitted in 1926, cf. [28a], contains nearly all the necessary ingredients. This paper prepared the way to a very definitive paper on the subject in 1930 with the title "Generalized harmonic analysis" [30a], to
3 It has been a pleasure for us to include in this volume a tribute to Wiener written by Gabor. PREFACE xxi which we now turn. Let us consider with Schuster [SI] (1900) a natural time-series f on 1R (e.g. a graph of the earth's magnetic field at a station) as the superposition of an enormous number of sinusoidals and transients. How do we find the sinusoidal frequencies X0? Schuster used the result that
iXt 2 (2) P^):= lim ^f | f f(t) e dt | = oo for X = suchXQ.
A A, for which the ratio on the RHS is very large thus indicates the presence of a sinusoidal with frequency close to A. To momentarily bypass a very short but remarkable paper of Einstein [E] (1914), the next important contribution was by G. I. Taylor [T] (1920), who saw that a useful smoothened version of the time-series f can be had by taking its autocovariance function <|), i.e. in rigorous terms,
(3) <|>(t) := lim ^ f f(t + u)f(u)du, t e IR.
To return to Einstein, he in essence independently discovered both Pf and ((), and by an ingenious heuristic argument arrived at the formula
iXt (4) Pf(X)=r iXt whence by a Fourier inversion, <|>(t) =/^ooe" Pf 4 Einstein and Wiener had a long train ride together from Leipzig to Geneva. But from Wiener's account of this conversation, relayed in a long letter to his sister Bertha in July 1925, it would appear that Einstein did not say a word about his own pioneering contribution to generalized harmonic analysis. PREFACE hindsight) to get Einstein's result (4), with suitable impositions on f, as a corollary. Wiener's result in its final form in [30a] for complex f and $ reads in our notation i / r1 rT\ eiXt i r1 eiXt~l (5) S(X) = l.i.m. -HJ +1 )^(t)H_dt+^-l *(t)£-n— dt, T-^oo 2JI J-T Jl ll 2JIJ-1 ll cf. [30a, (3.19)-(3.21)]. He showed that S exists, and has a monotonic increasing version, say Sf(-), which he called the power spectral distribution off. Obtained in all essentials about four years before the appearance of Bochner's theorem for a PD function (j), the formula (5) provides a recipe for getting Bochner's S from <|>. Soon after finishing [28a] in 1926, Wiener began to consider the possibility of obtaining an analogue of (5) for the signal function f itself (instead of for its covariance function <|)). Wiener naturally surmised that the transform s of f would emerge were the <|> in (5) to be replaced by f, i.e. 1 / r1 rT\ piXt i r1 eiXt-1 (6) s(X):= l.i.m. -H I +1 )f(t)£j-dt + — I f(t)—j—-dt, and he was hopeful that just as there is a Plancherel equality, |f |L Q^ = 11 U /JR), for f E LjCIR), so there would be one connecting suitable norms of the f and s in (6). In [30a, (5.51)] Wiener showed that the s(-) in (6) exists, and that (7) lim — I I s(u + e)-s(u-e)|2du= lim 4r f |f(t)|2dt, £->0 28 J_oo T-*oo Z1 J-T cf. [30a, (5.52), (5.53)]. His proof crucially rested on a special Tauberian equality, unknown to him but known to Hardy and Bochner, cf. [33i, (20.09M20.10)],viz. 1 fi- 1 /*° sin^E t (8) lim ± I f(t)dt = lim —I f(t)—7—dt, f (t) ^ 0. The equality (8) has become famous because it spurred Wiener to discover the general Tauberian theorem, which in turn influenced Gelfand (cf. [L2, PREFACE xxiii pp. 17-21]). GHA bears to the theory of stationary curves in Hilbert space, cf. [K3], roughly the same relation that BirkhofPs Individual Ergodic Theorem does to von Neumann's mean theorem: it is much deeper. For a stationary curve {x(t), t e R}, the counterparts, for instance, of Wiener's S and s are trivial: (9) s(k) = E(-oc,X]x(0) & S(X)= |s(X)|2, where E(-) is the spectral measure of the unitary group U(t) that propagates x(-). Its important application to stationary stochastic processes (SP) notwithstanding, the theory of stationary curves in Hilbert space is functional analytic, and not stochastic. Now GHA too is an analytic, non- stochastic theory. Can Stone's theorem be brought to bear on GHA in order to bring about the kind of simplification appearing in (9)? The difficulty is that the class of Wiener signals is not a vector space, still less a Hilbert space. Notice, however, that the class contains the (non- separable) Hilbert space of Besicovitch almost periodic functions. It has been shown that if A is any family of mutually correlated Wiener signals, such as he discusses in §9 of [30a], then the closed subspace 9(A spanned by A in the Marcinkiewicz-Banach space of functions f such that lTin Jpf |f(t)| 2dt< oo is a Hilbert space under the Besicovitch inner product, i.e. under the cross- correlation, cf. [W, Vol. II, p. 339]. For any f in H^ the function Xf(-)on R, given by ^ "Birkhoff's ergodic theory furnish the necessary point d'appui of Wiener's generalized harmonic analysis". So wrote Wiener in 1938 [W, Vol. II, p. 804]. In 1939 [39a] he showed that this Birkhoff theorem is deducible from von Neumann's mean-ergodic theorem and a maximal ergodic theorem. The question arises as to whether GHA is likewise deducible from the theory of SP. It is fairly clear from the important 1949 paper of Doob [D] that such a retrieval will be beset by serious difficulties. However, in the light of the developments mentioned in the last paragraph, a second careful look at Doob's critique may be worthwhile. As of now, the full significance of Wiener's (6), (7), have not been gauged. Unlike Wiener's function S his function s has no applications we know of. On the other hand there is no hard evidence that modern methods, which bypass Wiener's § and S, are more insightful than his own. Although GHA is a non-stochastic theory, Wiener was obliged to bring stochastic processes into his paper [30a] in order to answer the following question. Given a bounded right-continuous monotone increasing function S on R, is there anfonIR whose spectral distribution is S ? In [30a, §3] Wiener gives an affirmative answer for absolutely continuous S by proving the following remarkable theorem. Let y(t, a) := J^WCt - x)dx(x, a), where {x(t, a), t e IR, a e [0, 1]} is Wiener's Brownian motion. Then for almost all a e [0, 1], y( • , a) has the spectral density S' given by S'(k) = Vfrel W (X) I 2, where Wis the Fourier transform of W. The 5-jl" Page proof blends together three elements, the stochastic integral, the ergodicity of the flow of Brownian motion, and Birkhoff's Ergodic Theorem, each of which was to come well after [30a] was printed—an amazingly far-sighted tour de force. Since the answer to the italicized question is easy when S is a saltus function, the only case left unanswered in [30a] is when S is purely singular. Wiener settles this in a paper with Wintner [39c, §5]. But this proof is probability-free and purely analytic, and works for all S, not just singular S. It rests on Paul Levy's Continuity Theorem of 1925, [L3]. This reveals an interesting side of Wiener: the great pioneer was not a "pukka" probabilist like Levy, Kolmogorov or Doob, who kept abreast of developments. Wiener was "pukka" only in being a persistent pioneer, who reconnoitered till the end on the borders of chaos, seeking and discovering the order latent in chaos. Thus GHA remains a non-stochastic PREFACE XXV theory in that it is able to answer its questions without bringing in SPs.5 Another great Wiener contribution to harmonic analysis grew from the presence at MIT of E. Hopf of Potsdam. He got Wiener interested in the problem of radiative equilibrium in the stars. Wiener and Hopf reduced the problem to the solution of an integral equation which nowadays goes by their names. For Wiener this equation governed the transition from one regime to another across a border, such as radiant energy inside and outside the radiating core of a star, or outside or inside the human eye, and also the regime prevailing in the past of a particular instant, and that prevailing in its future. Wiener and Hopf solved the equation by a factorization which called for the extension of the Fourier transform into the upper and lower half planes. Luckily, Wiener then met Paley and their collaboration, entitled Fourier Transforms in the Complex Domain [34d], opened up a new subbranch of harmonic analysis, in which "Paley- Wiener" has become a household word. Wiener has described this subbranch, which is on the border between the theories of real and complex analysis, as follows: ... it is only recently that it has come to the attention of mathematicians that there are certain intermediate fields of work which share the methodology of both. There are certain curves which are smooth enough so that the whole course of these curves is known from any one part, but which are not smooth enough to be treated by the classical theory of the functions of a complex variable. The study of these curves is known as the theory of quasi-analytic functions. The French school of mathematics has contributed greatly to this field, and here some of the best work is that of Szolem Mandelbrojt. However, the book by Paley and myself in this field has also led to results which I was pursuing during my stay in China. [56g,p.91], 5 Is GHA a non-probabilistic theory? The answer depends on one's attitude to randomness. There has been a strong movement led by A. Church, Kolmogorov and others to return to the collectives of R. von Mises, by bringing to bear on it the concepts of mathematical logic such as "computability", "effectiveness", etc., cf. [W, Vol. II, pp. 372-377], and W. A. Gardner's recent book [G2]. In this book electrical engineering theory is done in a stochastic-free way by consistent appeal to the ideas of GHA. XXVI PREFACE The title of the Session III, "Generalized harmonic analysis and its ramifications", does not fully indicate the scope of all the topics we covered the session. We started with a paper on micro-local analysis, entitled "Wiener and the uncertainty principle in harmonic analysis" by Professor D. H. Phong, who unfortunately, did not provide us with a copy of his paper; so there is no paper on this in these Proceedings.6 The next lecture on "Generalized harmonic analysis and Gabor and wavelet systems" focusing on wavelet theory, another important offshoot of Gabor's idea, is by Professor John Benedetto. In the third lecture we turn to the Wiener-Hopf equation, today a field in itself. This is titled 'The Wiener-Hopf integral equation and linear systems". The lecturer, Professor Harm Bart, read a joint paper done with Professors I. Gohberg and M. A. Kaashoek. The closing lecture in this session entitled "Complex harmonic analysis in the aftermath of Paley-Wiener" was delivered by Professor Paul Malliavin. In this session we also had a short 20-minute communication entitled "Amalgam spaces and generalized harmonic analysis" by Professor Hans G. Feichtinger, among others. These papers show how Wiener's ideas extend and extend in the most diverse directions. E. Quantum mechanics Wiener's interests in quantum mechanics first materialized in his paper with Max Born [26d] and continued throughout his life, although intermittently, as is evident from the several papers he wrote. But even if we forget this entire episode, his much earlier work on introducing measure and integration in function space has far reaching ramifications, as M. Kac first emphasized. I. E. Segal saw significant uses for it in quantum field theory, and M. Kac pointed out the affinity of Wiener's integration with that used by Feynman for non-relativistic quantum theory. Thus there are two aspects to Wiener's quantum mechanical contribution, one stemming from Wiener's own papers on the subject, and the other from the important uses of his earlier ideas by others. 6 This omission is to a small extent mitigated by the Appendix in Professor Pribram's paper in Session V, in which the simplest ideas of Gabor needed in neurophysiology are presented. PREFACE XXVI1 It seemed convenient to begin our Session IV on the "Quantum mechanical ramifications of Wiener's ideas" with the latter, and start with a lecture on "Wiener space and non-linear quantum field theory" by Professor Irving E. Segal, and follow this up with a lecture on "Wiener and Feynman: the role of path integration in science" by Professor Sergio Albeverio. To turn to Wiener's own papers on the subject, the initial paper with Born attempted to generalize the Heisenberg matrix by an operator on the space of the highly irregular functions on IR that Wiener was encountering in generalizing harmonic analysis. This work, which was the first to reveal the nexus between quantum mechanics and operators on infinite- dimensional spaces, cf. Born [B, p. 188], became obsolete no sooner had Schrodinger's work made it plain that the operator has to be on Hilbert space. But in the wake of his paper with Born, Wiener wrote another called "Coherency matrices and quantum theory" [28d], which has turned out to be fundamentally useful in the clarification of optical coherence, an idea going back to Huygens, Fresnel and Young. Wiener's conception was rediscovered by Gabor in the 1950s, and became especially important in both classical and quantum optics in the wake of the theory of masers, lasers and holography. Accordingly, the next lecture we had was on "Optical coherence before and after Wiener" by Professor John R. Klauder. Wiener's later papers on quantum mechanics in the 1950s and 1960s, in collaboration with Armand Siegel and G. Delia Riccia, are motivated by the belief that by proper use of the Brownian motion, the Gibbsian statistical mechanics can be made to yield the strange probabilities that appear in quantum mechanics, and thereby to resolve the so-called "hidden parameter problem". Other workers interested in the hidden parameter problem (who too saw naivete rather than wisdom in the belief that cats can be in a state "neither dead, nor alive") had also found the Brownian motion useful. It therefore seemed appropriate to have a paper on "Wiener and the hidden parameter problem" by Professor Eric Carlen, in which Wiener's thought is outlined, and compared with the cognate ideas of Einstein-Podolsky-Rosen and of E. Nelson. The session on quantum mechanics also had a short contribution by Professor Erik Thomas on "Path integrals in finite sets" and by Dr. Robert L. Warnock on "Joint XXV111 PREFACE probabilities of noncommuting observables and the Einstein-Podolsky- Rosen question in Wiener-Siegel quantum theory", among others. F. Leibniz and Haldane. The mind. In a couple of his below-par papers, notably "The role of the observer" [36g], and a few stray passages elsewhere, e.g. [64c, p. 89]), Wiener's statements seem to contradict the view of reality that emerges clearly from his numerous other writings, not the least of which is his long essay on Kant's theory of space [22a]. This obligates us to recount the salient points of his conception: (i) There is a world, Nature, the laws of which (for instance, the laws of stellar radiation) are absolutely beyond human control; (ii) To discover and formulate the laws of nature, concepts emanating from the human mind, and therefore a priori, are necessary; (iii) The useful a priori conceptions (such as the "Lagrangian") hail from pure mathematics, and are by necessity ideal; (iv) The ideal concepts and ideal laws suggest the sound designing of machinery, and enhance experimental science by making available sensitive instruments such as the Michelson interferometer and the electron microscope. These principles spell out the role, both active and passive, of the investigator. Epistemologically, Wiener was close to Plato and Leibniz: the ideal is fundamental; no monad is purely passive. Wiener was not interested in the ontology of Plato or Leibniz, or of the many other philosophers he respected. Lurking in back of Wiener's mind from the days of his youth was the image of Gottfried Wilhelm Leibniz (1646-1716), whom he later was to christen the "patron saint of cybernetics". In Leibniz's calculus ratiocinator, machina ratiocinatrix and characteristica universalis was, according to Wiener, a vision of an age of automatization such as is now upon us. Wiener also found in Leibnizian monadology a philosophical framework congenial to the quantum mechanical view of the world, and to his own ideas on consciousness and cognition, i.e. on the mind. Wiener was elated when he read a paper by J.B.S. Haldane in 1934, entitled "Quantum mechanics as a basis for philosophy" [HI], in which mind is PREFACE xxix viewed as a quantum mechanical resonance, and he wrote "A letter to the Editor", pointing out the affinity of Haldane's thought to that of Leibniz. It thus seemed proper to have a Session V on "Leibniz, Haldane and Wiener on Mind", containing lectures in which the bearing of the thought of these thinkers on cybernetics, physics and neuropsychology is exposed in a way suitable for an audience for whom many of these topics would be unfamiliar. This led to the organization of three lectures, the first "On the role of Leibniz and J.B.S. Haldane in Wiener's cybernetics", by Professor George Gale, the second on "Quantum mechanical coherence, resonance and mind" by Dr. H. P. Stapp, and the third "Evidence from brain research regarding conscious processes" by Professor K. H. Pribram. Unfortunately, bad weather prevented Professor Pribram from attending, but luckily, we have a version of his lecture under the more exciting caption "What is mind that the brain may order it?". G. Information theory The notion of stochastic information in which the prior probability is absolutely continuous on IR is defined in Wiener's Cybernetics [48f], [61c] in Ch. Ill titled 'Time series, information and communication". It is also of course covered more amply in Chapter III on Continuous Information in Shannon's monogram [S4].7 Its fundamental role in communications engineering theory is well known. The first ramification of this subject outside communications engineering was in statistics. The idea of information was shown to be significant in the non-Bayesian Testing Hypothesis problem, as a measure of discrimination by Kulback and Liebler [KL], and in the Bayesian context for the comparison of experiments by Lindley [L5]. In the first case the "signal" is the null Hypothesis and the "signal + noise" is the alternative, and in the second case the difference between the entropy of the posterior and the prior, measures the information in the experiment. Also, to be noted is that Wiener's statement about R. A. Fisher's Information [61c, p. 62] is correct in the sense that the latter is a first order 7 Unlike Shannon, who introduces the definition without any ado, cf. [S4, p. 54], Wiener tries to justify its appropriateness by deriving it from considerations of the distribution of errors due to noise, cf. [61c, pp. 61- 62]. XX X PREFACE approximation to Shannon-Wiener information, cf. [KL, p. 81]. That the underlying idea of statistical entropy, the origins of which go back to Boltzmann, has essentially a combinatoric significance was emphasized in the 1960s by Kolmogorov. In his lecture at the Nice International Congress, 1970, Kolmogorov affirmed that: Information theory must precede probability theory, and not be based on it. By the very essence of this discipline the foundations of information theory have a finite combinatorial character. Kolmogorov clearly discerned the value of information theory for what he called "the algorithmic side of mathematics as a whole". This led him to what we now call Kolmogorov algorithmic complexity. This "complexity" has, however, a different designatum from the one used to describe the complexities such as turbulence and shock waves that appear in non-linear physics. It turns out that the Kolmogorov algorithmic complexity is not computable, and steps to alleviate this difficulty by Rissanen have lead to a cognate idea of stochastic complexity. Wiener wrote only a few short papers on information theory, but the notion of "message" and "noise" are absolutely central to his modus operandi, and this also applies to a lesser extent to the other types of "complexity". So we arranged a one-lecture Session VI under the rubric "Shannon-Wiener information and stochastic complexity" with a lecture under the same heading by Dr. Jorma Rissanen. H. Stochastic analysis In 1934 Paley and Wiener took what looked like a small step when they defined the stochastic integral of a function on R with respect to the Brownian motion. But at the hands of K. Ito [II] and his followers such integration has been extended to include random variable integrands and martingale integrators, and has mushroomed into the huge field of stochastic differential equations. Likewise, the seemingly specialized, classified war work of Wiener on antiaircraft fire control (1940) has had far-reaching ramifications in altering our outlook in the whole arena of the prediction and filtering of time series, and signal processing. This began PREFACE XXXI as a linear theory, but from early on Wiener saw it as just a beginning towards a theory of nonlinear filters with the capacity to learn and to reproduce themselves, i.e. to be learning filters and reproducing filters, The two topics have merged: stochastic integrals enter into nonlinear filtering. In 1951 K. Ito [12] proved the so-called "Wiener-Ito expansion", of which there is no trace [38a] but which Wiener apparently rediscovered in the mid-1950s from his interests in the response of non-linear filters to noise, cf. [58i]. In [58i] Wiener defined the so-called "multiple Wiener integral" of K(tj • • • ^) over Rn, in the way he had done in [38a], by first 8 considering K of the form <|)(t ) • • • <()(tn) and building therefrom. In 1988 Hu and Meyer [H3], not knowing of this work of Wiener, called this multiple integral the "Stratonovich integral". These multiple stochastic integrals, in conjunction with the Wiener-Ito expansion, have helped crucially in converting the stochastic PDEs of non-linear filters (or equivalently their representations as stochastic integrals) into classical, i.e. non-stochastic, PDEs. This Ito expansion has also been crucial in the implementation of Frechet's plan to develop a stochastic calculus in infinite dimensional spaces. The first steps towards such a calculus were taken by Paul Levy [L4] in 1951. But with the aid of the Ito expansion, Malliavin was able to complete the calculus on Wiener's infinite dimensional space, C[0, 1] in 1976, cf. [Ml]. Since then the calculus has been extended to abstract Wiener spaces. Thus it was fitting to have a Session VII on "Nonlinear stochastic analysis" with several papers ranging from the purely mathematical to the applied. The lectures were: (i) "Nonlinearity and martingales", by Professor Donald L. Burkholder, (ii) "Stochastic analysis on Wiener space", by Professor Shinto Watanabe, (iii) "Nonlinear prediction and filtering'" by Professor G. Kallianpur, (iv) "Uncertainty, feedback and Wiener's vision of cybernetics", by Professor Sanjoy K. Mitter, (iv) "The theory of learning before and after Wiener", by Professor 8 There are lacunae in Wiener's treatment. XXXI1 PREFACE N. K. Bose. Unfortunately, Professor Mitter was not able to attend for reasons of health, and gave us no manuscript for publication. In this session, as elsewhere, want of time prevented us from saying more, for instance, on Wiener's own unique "engineer-friendly" 9 approach to non-linear filters in [58i]. I. Prosthesis Arturo Rosenblueth's circle at the Harvard Medical School, which Wiener began attending from 1934, revived his early biological curiosities. In the mid-1940s and the 1950s, Wiener collaborated in papers on neurophysiology with Rosenblueth and his Mexican colleagues. An abiding interest of his was in prosthesis, in interpreting which Wiener used the word "handicapped" in a very broad sense. Microscopes and telescopes are prosthetic devices for the human species no less than eye glasses are for individuals. The electronic computer was also prosthetic, and Wiener saw in it an aid that might enable us to visualize geometrical forms in 4 and higher dimensions [53d]. Wiener was struck by the remarks of Stephen Bergman that progress in the theory of holomorphic functions of two complex variables was held up by our inability to visualize the distinguished boundaries of their domains of holomorphy. This was in 1953. Earlier Wiener had written short notes on sound communication with the deaf, and had ideas on the iron lung and artificial limbs. His thoughts on the latter fructified. The first lecture in this session on "Muscular and sensory prosthesis in the aftermath of Wiener" was by Professor Robert W. Mann, who has been involved with Wiener's "Boston arm" from the start. The second on "Wiener's thoughts on computers as an aid in visualizing higher dimensional forms" was by Professor Dennis Roseman, whose vocation seems to be to do knot-theory in 4-dimensions, exploiting the computer to the fullest. With his unbounded faith in man-machine concatenation, Wiener would have liked this venture to do "analysis situs" by computer. In 1915 (during his philosophical doldrums under John Dewey at Columbia), 9 Term used by Dr. Poduska in his short lecture. PREFACE XXXlll Wiener began working on analysis situs, and in 1917 joined J. W. Alexander and Veblen in using the computer to prepare ballistic tables for the U.S. Army. Unfortunately, Wiener lost his manuscript on analysis situs (cf.[56g, pp. 26-27]). J. The political economy By 1942 Wiener had clearly surmised the coming of the automatic factory run by high-speed electronic computers, and soon began making efforts to alert American labor to the dangers of the dislocation and unemployment that loomed on the horizon. In Wiener's view the only authentic solution to the problem lay in seeing education and automatization as faces of the same evolving coin—each step in automatization to be matched by a corresponding step in the intellectual upgrading of the work force. The actuality during the last 35 years, however, is one of advancing automatization amidst steady and sometimes sharp educational decline. During and after his visit to the Indian Statistical Institute in 1955- 1956, and conversations with the eminent Dutch economist Jan Tinbergen, Wiener gave more thought to the political economy, and formed more definitive ideas on the allocation of resources, and on the apportionment of automation in the producer goods sector and in the consumer goods sector of the economy. In the light of this, it became clear early on in our planning that there ought to be a session in the Congress on "Wiener and the political economy: automatization, educational decline and unemployment". We expected difficulties in organizing this session, but nowhere near the amount we actually encountered. All the 15 or so economists we invited to be speakers turned us down no sooner they learned the purpose of the session. Only with much difficulty and the good offices of Professor Kallianpur were we able to get the acceptance of Professor Leonid Hurwicz and with his help of Professor Lawrence Klein. The third speaker we counted on was Professor Stafford Beer, President of the World Organisation of Systems and Cybernetics. But he backed out rather late, and we were fortunate in getting Professor Dennis Murphy, to step into his shoes. Our Economics Session with Round Table discussion got off on a bumpy start. The late discovery of a conflict in Professor Klein's calendar XXXIV PREFACE caused us to postpone the session, and one of the speaker's lectured for thrice the stipulated time. The round table was thereby thrown into confusion, and we were unable to benefit from the comments of Professor Kenneth Stunkel on education and the economy, a subject he was especially well versed with, and for which we had invited him. We were unable to persuade the speakers to submit manuscripts for publication in this volume. We submit our apologies for this lack of control over the session on our part. We compensate by giving our readers a bird's eye view of Wiener's economic thought in the next few paragraphs. It should be clear from the first two paragraphs of this section that Wiener conceived political economics, the way Aristotle had conceived it, i.e. as a branch of ethics: economics has to teach us how to build a good economy much as aeronautics has to teach us how to build good airplanes. This created a fundamental breach between Wiener and the academic establishment on the subject, according to whom economics is a "natural science" like geology or aerodynamics. This breach was widened by Wiener's outspoken honesty in voicing his skepticism of most of the mathematical economics in vogue in his day: Just as primitive peoples adopt the western modes of denationalized clothing and parliamentarism out of a vague feeling that these magic rites and vestments will at once put them abreast of modern culture and technique, so the economists have developed the habit of dressing up their rather imprecise ideas in the language of the infinitesimal calculus. In doing this, they show scarcely more discrimination than some of the emerging African nations in the assertion of their rights. An econometrician will develop an elaborate and ingenious theory of demand and supply, inventories and unemployment, and the like, with a relative or total indifference to the methods by which these elusive quantities are observed or measured . . . To assign what purports to be precise values to such essentially vague quantities is neither useful nor honest, and any pretense of applying precise formulae to these loosely defined quantities is a sham and a waste of time. [64e, pp. 90, 91] Did this imply that Wiener had doubts as to the efficacy of mathematics in matters economic? Definitely not, for Wiener wrote "One of the most potent tools in reanimating a science is mathematics" [93a, p. PREFACE xxxv 25]. Just as aeronautics has to draw on the most advanced parts of the pure sciences of aerodynamics, so economics must draw on whatever ancillary pure science it needs. Wiener wanted the right mathematics and right sciences that fitted the subject, such as the thermodynamics and geological aspects of manufacturing, and the actuarial theory of calculable risk. He saw promise in the directions of research pursued by Benoit Mandelbrot and John von Neumann. Wiener wrote: He [Mandelbrot] has shown that the intimate way in which the commodity market is both theoretically and practically subject to random fluctuations arriving from the very contemplation of its own irregularities is something much wilder and much deeper than has been supposed, and that the usual continuous approximations to the dynamics of the market must be applied with much more caution than has usually been the case, or not at all. [64e, p. 92] As for von Neumann, Wiener repeatedly refers to his "important joint book on games" as an attempt to study social organization by methods close to his own. This book faces the reality that all except the simplest markets are arenas of contest, in which the successful accomplishment of one's own goals requires the prevention of the second party from attaining its own. Thus the teleological climate in these markets is unlike that conceived by Adam Smith, according to whom, the competition would redown to the good of both parties. Von Neumann, however, conceived of this contest as one between perfectly rational players whose moves are based on the changing "board" before them. On this Wiener could not let his profound respect for von Neumann (whom he classed with Herman Weyl as "two of the greatest forces in modern mathematics", cf. [56g, p. 182]), get the better of his honesty and criticality. He pointed out that: It is rare to find a large number of thoroughly clever and unprincipled persons playing a game together. Where the knaves assemble, there will always be fools; and where the fools are present in sufficient numbers, they offer a more profitable object of exploitation for the knaves. The psychology of the fool has become a subject well worth the serious attention of the knaves. Instead of looking out for his own ultimate interest, after the fashion of von Neumann's gamesters, the fool operates in a manner which, by and large, is XXXVI PREFACE as predictable as the struggles of a rat in a maze. [61c, p. 169] And again: The individual players are compelled by their own cupidity to form coalitions; but these coalitions do not generally establish themselves in any single, determinate way, and usually terminate in a welter of betrayal, turncoatism, and deception, which is only too true a picture of the higher business life, or the closely related lives of politics, diplomacy, and war. [61c, p. 159] Thus for completion, the von-Neumann-Morgenstern theory needed, according to Wiener, a "noise analysis", the noise not being of thermodynamic origin and purposeless as in engineering, but of human origin and purposeful. The attributes of such noise are greed, deception, naivete and the like, what theologians classify as sin or evil. This presupposes that such "teleological noise" is also subject to laws. Confronted by such wisdom, far outside the confines of their own abstractions, the economists took the easy way out, by proceeding as though the wisdom did not exist. Wiener was especially against attempts to use his prediction theory for investment decisions on the stock exchange. "I could not look upon the proposed commercialization of my own ideas with a clear conscience" are his words, [93a, p. 115]. Would Wiener's thinking on this matter have changed after the rediscovery of L. Bachelier's thesis (1900), and the subsequent outflow of papers in so-called "financial risk management", using rigorous mathematics close to his own? For the answer, we must turn to the chapter on "Noncalculable risks and the economic environment of invention" in [93a], where a line is drawn between "calculable" and "noncalculable" risk. He describes an incident that occurred in the early 1950s: At this time, a scion of a great industrial family visited me in my office. He was eager for me to go ahead with the problem of stock-market prediction. His notion was that even a one percent improvement in stock-market prediction would regulate the investment of so many million dollars that it would become a major factor in economic life. When we brought this to the attention of von Neumann, he pronounced an opinion with which I concurred completely. For an improvement in stock-market prediction to be of any PREFACE XXXVll commercial value at all, it would not be enough for it to be serviceable in some uncheckable way to the tune of millions of dollars. Those who employ it must have some way to be aware of its proved serviceability. [93a, p. 115] We cannot pursue the argument that follows, except to say that Wiener was concerned about what happens when the acceptable risk is exceeded. For disasters, "acts of God", the insurance companies, operating on market forces, are unable to deliver, and charity (based on religious non-pecuniary concerns) comes to the rescue [93a, p. 121]. Where is the rescue from financial disaster to come from? The complementarities in the quantum world are a manifestation of cosmic orderliness, witness Planck's universal constant of nature. The complementarities in the business world are different: on the beautiful, orderly, stochastic chaos of the cosmos is superimposed an ugly confusion stemming from human frailty: "Where the knaves assemble, there will always be fools." The sharp lines that Wiener always drew, make it hard to hazard a guess as to what he would have thought about the current research on the mathematics of financial risk management. K. Cybernetics The term xvPepvr|TiKT] occurs in Plato and the terms "cybernetique" and "cybernetics" were coined by Ampere and Wiener in 1840 and 1948, respectively. The word refers to much more than the electronic computer—it definitely carries a connotation of control. Today, however, the prefix "cyber", with inane hyphenations, refers primarily to telecommunication via the electronic digital computer. Wiener would have been saddened to see the term so severely misused, commercialized and degraded, but not at all surprised. He had warned that "this new development [the computer] has unbounded possibilities for good as well as for evil" [61c, p. 27]. There are, however, scientists who, even after dismissing such obvious abuses of the term "cybernetics", view it with suspicion, because of its frequent use to cover up fuzzy thinking, and some even dismiss Wiener's cybernetics, as "Wiener noise" rather than Wiener wisdom. They overlook the fact that Wiener was himself aware of what he called the "charlatan fringe" of the cybernetical movement. XXXV111 PREFACE The object of the one lecture on cybernetics in the Congress is to show that cybernetics is a fundamental part of Wiener wisdom, and to separate this cybernetics, which is a part and parcel of science at its best, from the other brands. Another purpose is to draw attention to a very important letter in 1946 from John von Neumann to Wiener, wherein is suggested a sure way to always keep the cybernetical movement on the scientific path. L. Contributed papers and session on Cybernetical organizations The Congress benefited from the 14 twenty-minute papers that were read in the course of the week. In the program following this preface, these papers are listed in the sessions to which they belong. Unfortunately, space limitation has allowed us to include only three of the papers presented in these Proceedings. A special short session captioned "Cybernetical Organizations" was held during the Congress. There is a needlessly large number of such organizations in the world. The speakers were Jean Ramaekers, on behalf of the International Association of Cybernetics, Masani, on behalf of the World Organisation of Systems and Cybernetics, and L. Volkov on behalf of the International Academy of Informatization in Russia. Unfortunately, Professor V. I. Astaafyev, the delegate from this Academy, was unable to attend. M. Award of Wiener Centenary medal to C. E. Shannon Our session on the last day of the Congress, presided over by Professor Jean Ramaekers, President of the International Association of Cybernetics, was especially memorable. It was our acknowledgment of the indispensability of the creative work of C. E. Shannon for the comprehendibility of Wiener's own thought. We were delighted to have in our midst, Mrs. Elizabeth Shannon. Dr. Robert Schiller, Superintendent of Education of the State of Michigan, opened the meeting and welcomed us to Michigan, the birth place of Dr. Shannon. Dr. Brockway McMillan, close colleague of Shannon, and former associate of Wiener, then lectured on "The scientific impact of the work of C. E. Shannon". Professor Ramaekers then read the citation and awarded the Norbert Wiener Centenary Medal to Mrs. Shannon, who accepted it on behalf of her PREFACE XXXIX husband. Mrs. Shannon then took the podium and made a few very amusing remarks about Wiener's quirks on one of the few occasions she met him, and of her husband's amusing side, and fascination with designing the strangest automata. So far in this Overview we have kept to regions of Wiener's work covered in our sessions, and have attended to some of the issues that did not get into the lectures. But Wiener's domain of research was much wider. It included electrical network theory. Carried out in the 1930s with his then student Professor Y. W. Lee, this work revealed the important role of the Hilbert transform, and of Laguerre functions in network design, cf. [LI]. His 1940 proposal to Dr. Vannevar Bush for the design of a special purpose computer [85b] was very far reaching in its general implications, cf. [MRFS]. In physiology, in collaboration with his Mexican colleagues, Drs. Arturo Rosenblueth and J. Garcia Ramos, Wiener worked on the cardiac muscle, the spike potential of axons and muscular clonus. He also wrote papers on the self-organization of systems by the non-linear entrainment of frequencies, and its possible bearing on brain-wave encephalography, in which he had an abiding interest. Also left out is the difficult pioneering work with Bigelow on the antiaircraft fire control problem, cf. [W, Vol. IV, pp. 141-179], and its mathematical ramifications into prediction theory [49g], the work of Helson and Lowdenslager [H2], and its impact on the operator theoretical work of Nagy and Foais [N], and on the Odessa School of M. G. Krein. Acknowledgment Our very sincere thanks go to all our speakers, but here we would like to single out those who were good enough to read short papers: Madjid Amir, S. Balaji, James E. Brassert, Hans G. Feichtinger, A. Himonas, Vladimir V. Kisil, Shlomo Levental, John W. Poduska, J. Schafer, Erik F.G. Thomas, Lev N. Volkov, Quoc-Phong Vu, Robert L. Warnock, and Georg Zimmerman. We also sincerely thank the participants who, though not speakers, contributed less formally to the intellectual life of the Congress. Our thanks go to Professors Anders Martin-L6f, Wellington Wo, Sheldon Axler, Tom Kaplan, S. D. Mahanti, Dennis Roseman, Anatolii V. xl PREFACE Skorokhod, Anil K. Jain, Nicoli V. Krylov, John Eulenberg, P. R. Masani and Jean Ramaekers for chairing the sessions. We were much honored to have had the support of the following European learned Societies and Foundations, who accepted co- sponsorship of the Congress, and were good enough to send one or more delegates, and who assisted us more concretely by providing travel grants to their delegates. Deutsche Forschungsgemeinschaft: Professor S. Albeverio and Dr. J. Schafer, Bochum French Academy of Sciences: Professor Paul Malliavin, Paris Osterreichische Studiengesellschaft fur Kybernetik: Professor Hans G. Feichtinger, University of Vienna Royal Netherlands Academy of Arts and Sciences: Professor Erik G.F. Thomas, Groningen The Royal Society: Professor Oliver Penrose, Heriot-Watt University The Swedish Academy of Engineering Sciences: Professor A. Martin-Lof, Stockholm University Swiss Academy of Sciences: Professor H. Carnal, Berne. All but two of the delegates spoke at the Congress. Thus, we are extremely grateful to these learned Societies for their assistance. The materialization of the Congress itself of course required major funding from American foundations. In this regard, we gratefully acknowledge generous grants from the following Foundations: The National Science Foundation, Army Research Office Capital Area Community Foundation Institute of Mathematics and Applications Anonymous Donor Finally, the materialization of the Congress depended on an academic institution that was willing to host it. In this regard, our heart felt gratitude goes to the Michigan State University and especially to the Department of Statistics and Probability, and The College of Natural Science, and Vice President Percy Pierre, for their constant PREFACE xli encouragement, financial and infrastructural support. We are indebted to the local organizing committee under Professor R. LePage, and the graduate students Hao Zhang, Alex White, Rudy Blzek, Srikarith and others for their enthusiastic help with the local arrangements. Last but not the least we thank the excellent supporting staff consisting of Mary Gowans, Cathy Sparks and Laurie Secord whose devotion to their work made the conference pleasurable for both the participants and the organizers. Would posterity benefit from the Norbert Wiener Centenary Congress? The answer depended upon our ability to have the Proceedings printed. For this we are most grateful to the American Mathematical Society for agreeing to accept this volume in their hard-cover series on Symposia of Applied Mathematics. In this regard, our thanks go to the officers and staff of the American Mathematical Society, Dr. Robert M. Fossum, Dr. Jane E. Kister, Dr. Sergie Gelfand, Mrs. Donna Harmon, Mrs. Victoria Ancona, Mr. Thomas Costa, and no doubt others. For the organization of this volume, and a great deal of retyping of the manuscripts, both Editors are especially grateful to Mrs. Naomi Rhodes. References [B] M. Born, The Natural Philosophy of Cause and Chance, Clarendon Press, Oxford, 1949. [CM] R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under translations, Ann. Math. 45 (1944), 386-3%. [D] J. L. Doob, Time series and harmonic analysis, Proc. Berkeley Symposium on Mathematical Statistics and Probability, Univ. of Calif. Press, Berkeley, CA, 1949,303-343. [E] A. Einstein, Methode pour la determination de valeurs statistiques dfobservations concernant des grandeurs soumises a des fluctuations irregulieres. Archives des Sciences Physiques et Naturelles, t. 37 (1914), pp. 254-255. [F] Charles L. Fefferman, The Uncertainty Principle, Bull. Amer. Math. Soc. 9 (2) (1983), 129-206. [Gl] D. Gabor, Theory of Communications, J. IEEE London, Part III B (November issue) (1946), 429-457. [G2] William A. Gardner, Statistical Spectral Analysis: A Nonprobabilistic Theory, Prentice-Hall, Englewood Cliffs, NJ, 1988. [HI] J.B.S. Haldane, Quantum mechanics as a basis for philosophy, Philosophy Sci. 1 (1934), 78^-98. xlii PREFACE [H2] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Parti, Acta Math. 99(1959), 165-202; Part II, 106(1962), 175-213. [H3] Y. Z. Hu and P. A. Meyer, Sur Les Integrales multiples de Stratonovitch, Seminaire de Prob XXII, Lecture Notes in Mathematics #1321, Springer-Verlag, 1988,72-81. [II] K. Ito, Stochastic integral, Proc. Imp. Acad. Tokyo 20 (1944), 519-524. [12] K. Ito, Multiple Wiener integral, J. of the Math. Soc. of Japan 3, No. 1 (1951), 157-169. [Kl] A. N. Koimogorov, Foundations of the Theory of Probability, (1933), Chelsea Publishing Company, New York, 1950. [K2] A. N. Koimogorov, Curves in Hilbert space, which are invariant with respect to a one-parameter group of motions, DAN SSSR 26 (1940), 6-9. (In Russian.) [K3] A. N. Koimogorov, Stationary sequences in Hilbert space (Russian), Bull. Math. Univ., Moscow 2, No. 6 (1941), 40 pp. (English translation by Natasha Artin.) [KL] S. Kulback and R. A. Liebler, Information and Sufficiency, Ann. Math. Stat. 22, (1951), 79-36. [LI] Y. W. Lee, Synthesis of electric networks by means of Fourier transforms of Laguerre'sfunctions, J. Math. Phy. 11 (1932), 83-113. [L2] N. Levinson, Wiener's Life, in: American Mathematical Society Bulletin 73 (No. 1, Part II) (1966), dedicated to Norbert Wiener, pp. 1-32. [L3] P. Levy, Calcul des Probabilites, Gauthier-Villars, Paris, 1925. \LA] P. Levy, Problemes concrets dyanalyse fonctionnelle, Gauthier-Villars, Paris, 1951. [L5 ] D. V. Lindley, On a measure of information provided by an experiment, Ann. Math. Stat 27; (1956), 986-1005. [Ml] Paul Malliavin, Stochastic Calculus of variation and hypoelliptic operators, Proc. Int. Symp. on Stochastic Differential Equations, Kyoto (1976), Wiley, New York, 1978,195-164. [M2] P. R. Masani, Norbert Wiener: a survey of a fragment of his life and work, in: A Century of Mathematics in America, Part III (Peter Duren, ed.), Amer. Math. Soc., Providence, RI, 1989, pp. 299-341. [M3] P. R. Masani, Norbert Wiener: 1894-1964, Vita Mathematica Series 5, Birkhauser, Basel, 1990. [MRKS] P. R. Masani, B. Randell, D. K. Ferry and R. Saeks, The Wiener memorandum on the mechanical solution of partial differential equations, Annals of the History of Computing 9, No. 2 (1987), 183-197. [N] B. Sz. Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, 1970. [SI] A. R. Schuster, The periodo gram of magnetic declination, Camb. Phil. Trans. 18 (1900), 107-135. [S2] I. E. Segal, Tensor Algebras Over Hilbert Space, Trans. Amer. Math. Soc. 81 (1956), 106-134. [S3] I. E. Segal, Norbert Wiener, Biographical Memoirs, 61, Natl. Acad. Press, 1992, pp. 388-436. [S4] C. E. Shannon, The Mathematical Theory of Communication, The University of Illinois Press, 1949. (Reprint from Bell Syst. Tech. Journ. 27 (1948), 379-423; 623-656.) [T] G. I. Taylor, Diffusion by continuous movements, Proc. Lond. Math. Soc. 20 (1920), 196-212. PREFACE xliii [V] J. von Neumann, Mathematical Foundations of Quantum Mechanics, (1930), Princeton University Press, Princeton, NJ, 1955. [W] N. Wiener, Collected Works (P. Masani, ed.), I, II, III, IV, MIT Press, Cambridge, MA, 1976, 1976,1979, 1981, 1985. Norbert Wiener References [20f] N. Wiener, The mean of afunctional of arbitrary elements, Ann. of Math. (2) 22 (1920), 66-72. [22a] N. Wiener, The relation of space and geometry to experience, Monist 32 (1922), 12-60,200-247,364-394. [23d] N. Wiener, Differential-space, J. Math, and Phys. 2 (1923), 131-174. [24d] N. Wiener, The average value of a functional, Proc. London Math. Soc. 22 (1924), 454-467. [28a] N. Wiener, A new method in Tauberian theorems, J. Math, and Phys. 7 (1928), 151-184. [28d] N. Wiener, Coherency matrices and quantum theory, J. Math, and Phys. 7 (1928), 109-125. [30a] N. Wiener, Generalized harmonic analysis, Acta Math. 55 (1930), 117-258. [33a] N. Wiener (with R.E.A.C. Paley and Z. Zygmund), Notes on random Junctions, Math. Z. 37 (1933), 647-668. [33i] N. Wiener, The Fourier Integral and Certain of Its Applications, Cambridge University Press, New York, 1933. [34d] N. Wiener (with R.E.A.C. Paley), Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ. 19, Amer. Math. Soc., Providence, RI, 1934. [36g] N. Wiener, The role of the observer, Philos. Sci. 3 (1936), 307-319. [38a] N. Wiener, The homogeneous chaos, Amer. J. Math. 60 (1938), 897-936. [38c] N. Wiener (with H.R. Pitt), On absolutely convergent Fourier-Stieltjes transforms, Duke Math. J. 4 (1938), 420-440. [39a] N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939), 1-18. [39c] N. Wiener (with A. Wintner), On singular distributions, J. Math, and Phys. 17 (1939), 233-246. [48f] N. Wiener, Cybernetics, or Control and Communication in the Animal and the Machine, Actualites Sci. Ind., no. 1053; Hermann et Cie., Paris; The MIT Press, Cambridge, MA, and Wiley, New York, 1948. [49g] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications, The MIT Press, Cambridge, MA; Wiley, New York; Chapman & Hall, London, 1949; paperback edition with the title Time Series, The MIT Press, 1964. [53d] N. Wiener, Les machines acalculer et la forme (Gestalt), Les machines acalculer et la pensee humaine, Colloques Internationaux du Centre National de la Recherche Scientifique, Paris, 1953,461-463. [53h] N. Wiener, Ex-prodigy: My Childhood and Youth, Simon and Schuster, New York, 1953: The MIT Press, Cambridge, MA, 1965 (paperback edition, The MIT Press). [56g] N. Wiener, / Am a Mathematician. The Later Life of a Prodigy, Doubleday, Garden City, New York, 1956; paperback edition by The MIT Press, 1964. [58i] N. Wiener, Nonlinear Problems in Random Theory, The MIT Press, Cambridge, xliv PREFACE Mass., and Wiley, New York, 1958; paperback edition, The MIT Press, 1966. [61c] N. Wiener, Cybernetics, Second edition of [48f] (revisions and two additional chapters), The MIT Press and Wiley, New York, 1961; paperback edition, The MIT Press, 1965. [64c] N. Wiener, Machines smarter than men? (Interview with N. Wiener), U.S. News and World Rept. 56 (1964), 84-86; abbreviated in Reader's Digest 84 (1964), 121-124. [64e] N. Wiener, God, Golem, Inc.—A Comment on Certain Points Where Cybernetics Impinges on Religion, The MIT Press, Cambridge, MA, 1964, paperback edition, The MIT Press, 1966. [85b] N. Wiener, Memorandum on mechanical solution of partial differential equations, Coll. Works, IV, 124-134. [93a] N. Wiener, Invention: The Care and Feeding of Ideas (Introduction by Steve Joshua Heims), MIT Press, Cambridge, MA, 1993. V. Mandrekar P. R. Masani 1996 Program* I. Wiener's Concept of the Stochastic Universe Wiener-Kolmogorov conception of the stochastic organization of nature S. Molchanov, University of North Carolina, Charlotte The Wiener Programme in Statistical Physics. Is it feasible in the light of recent developments? Oliver Penrose, Herriot-Watt University, Edinburgh The mathematical ramifications of Wiener's program in statistical physics Leonard Gross, Cornell University Contributed Paper Wiener's random theory applied to turbulence John Poduska, Advanced Visual Systems II. Potential and Capacity Before and After Wiener Potential and capacity before and after Wiener David R. Adams, University of Kentucky III. Generalized Harmonic Analysis and its Ramifications Wiener and the uncertainty principle in harmonic analysis D. H. Phong, Columbia University Generalized harmonic analysis and Gabor and wavelet systems John J. Benedetto, University of Maryland, College Park Wiener-Hopf equations and linear systems H. Bart, Erasmus University, Rotterdam Complex harmonic analysis in the aftermath of Paley-Wiener Paul Malliavin, Acad. Sci, Paris, France Contributed Papers Amalgam spaces and generalized harmonic analysis Hans G. Feichtinger, University of Vienna The general Wiener Tauberian theorem and asymptotic behavior of C0-semigroups Quoc-Phong Vu, Ohio University *In the actual time-table, the sessions could not be organized as indicated here. Some had to be broken up. xlv xlvi PROGRAM Double projective approximations for the construction of wavelet basis G. Zimmerman, University of Maryland On the Regularity of solutions to degenerate elliptic PDEs on the torus Alexandrou Himonas, University of Notre Dame IV. Quantum Mechanical Ramifications of Wiener's Ideas Quantum field theory and functional integration Irving Segal, MIT Wiener and Feynman—path integrals and their applications Sergio Albeverio, Ruhr-University, Bochum Optical coherence before and after Wiener John R. Klauder, University of Florida Wiener and the problem of "hidden parameters" in quantum mechanics Eric A. Carlen, Georgia Institute of Technology Contributed Papers Relative quantization and improved Dirac equations Vladimir V. Kisil, Cinvestav del I.P.N., Mexico Path integrals on finite sets Erik G. F. Thomas, University of Groningen, The Netherlands Joint probabilities of noncommuting observables and the Einstein-Podolsky-Rosen question in Wiener-Siegel quantum theory Robert L. Warnock, Stanford University, Accelerator Center Markov property and cokernels of local operators J. Schafer, Ruhr-University, Bochum V. Leibniz, Haldane and Wiener on Mind The role of Leibniz and Haldane in Wiener's cybernetics George Gale, University of Missouri, Kansas City, MO Quantum mechanical coherence, resonance, and mind Henry P. Stapp, Lawrence Laboratory, Berkeley, CA Evidence from brain research regarding conscious processes Karl H. Pribram, Radford University VI. Shannon-Wiener Information and Stochastic Complexity Shannon-Wiener information and stochastic complexity J. Rissanen, IBM Research Division, San Jose, CA VII. Nonlinear Stochastic analysis Sharp norm comparison of Martingale maximal functions and stochastic integrals Donald L. Burkholder, University of Illinois Stochastic analysis on Wiener space Shinzo Watanabe, Kyoto University, Japan Stochastic filtering: Non-linear prediction and filtering G. Kallianpur, University of North Carolina, Chapel Hill PROGRAM xlvii Uncertainty, feedback and Wiener's vision of cybernetics S. K. Mitter, MIT The theory of learning before and after Wiener N. K. Bose, Pennsylvania State University Contributed Papers From random walks to sticky Brownian motion Madjid Amir, University of Mannheim, Germany Recurrence and transience of reflecting diffusion in half space S. Balaji, Indian Statistical Institute, Bangalore A necessary and sufficient condition for absense of arbitrage with tame portfolios Shlomo Levental, Michigan University VIII. Prosthesis, Ontogenetic and Phylogenetic Sensory and motor prostheses in the aftermath of Wiener Robert W. Mann, MIT On Wiener's thought on the computer as an aid in visualizing higher dimensional forms and its modern ramifications Dennis Roseman, University of Iowa IX. Cybernetics Norbert Wiener and the Future of cybernetics P. R. Masani, University of Pittsburgh Contributed Papers On formal and real connection of cybernetics and philosophy J. E. Brassert, Tubingen Norbert Wiener and the new branches in cybernetics development (with emphasis on homeostatics) Lev N. Volkov, Siberian Institute of Energetics, Irkutsk X. Wiener and the Political Economy: Automatization, Education Decline and Unemployment (Round Table) The relationship of cybernetics and automation to the economy L. Klein, University of Pennsylvania The role of information in economic processes L. Hurwicz, University of Minnesota A cybernetical approach to unemployment: the role of ethics and education D. Murphy, Concordia University, Montreal Round Table Kenneth L. Stunkel, Monmouth College, New Jersey Gary Wolfram, Hillsdale College, Michigan xlviii PROGRAM XI. Special Short Session on Cybernetical Organizations Speakers: J. Ramaekers: The International Association of Cybernetics P. R. Masani: World Organization of Systems and Cybernetics L. Volkov: International Academy of Informatization XII. Award of Norbert Wiener Centenary Medal Welcome Dr. Robert Schiller, Superintendent of Education, State of Michigan Scientific Impact of the Work of C. E. Shannon Brockway McMillan, Bell Laboratories (retired) Award of the Norbert Wiener Centenary Medal to Professor C. E. Shannon (in absentia) by Jean Ramaekers, International Association for Cybernetics, Namur, Belgium, and its presentation to Mrs. Shannon. Proceedings of Symposia in Applied Mathematics Volume 52, 1997 Academic Vita of Norbert Wiener1 1894 Born on November 26 in Columbia, Missouri, to Bertha Kahn Wiener and Leo Wiener, a professor of foreign languages at the University of Missouri. 1895 The family moved to Cambridge, Massachusetts, where Leo Wiener became a professor of Slavic languages at Harvard. 1901 Entered the third grade at the Peabody School, but was removed shortly and taught by his father until 1903. 1903 Entered Ayer High School. 1906 Graduated from Ayer High School and entered Tufts College where he studied mathematics and biology. 1909 Received an A.B. degree, cum laude, from Tufts, and entered the Harvard Graduate School to study zoology. 1910 Entered the Sage School of Philosophy at Cornell University with a scholarship, and studied with Frank Thilly, Walter A. Hammond, and Ernest Albee. 1911 Transferred to the Harvard Graduate School to study philosophy, and studied with E.V.Huntington, Josiah Royce, G.H. Palmer, Karl Schmidt, and George Santayana. 1912 Received an M.A. degree from Harvard. 1913 Received a Ph.D. degree from Harvard; dissertation under J. Royce, but supervised by K. Schmidt of Tufts College. Appointed a John Thornton Kirkland Fellow by Harvard, and entered Cambridge University. Studied logic and philosophy with Bertrand Russell, G.E. Moore, and J.M.E. McTaggart, and mathematics with G.H. Hardy and J.E. Littlewood. 1914 Joined the University of Gottingen and took the courses of David Hilbert, Edmund Husserl, and Edmund Landau. Appointed a Frederick Sheldon Fellow by Harvard; returned to Cam• bridge University to study mathematics and philosophy. Received the Bowdoin Prize from Harvard. 1915 Studied philosophy under John Dewey at Columbia University. Appointed an assistant and a docent lecturer in Harvard's Philosophy Department for 1915-1916, and lectured on the logic of geometry. 1916 Served with Harvard's reserve regiment at the Officer's Training Camp in Plattsburg, N.Y. Appointed instructor of mathematics at the University of Maine in Orono for 1916-1917. 1 This data is extracted from the Chronology in the spiral-bound publication entitled the "Inventory of Norbert Wiener, 1894-1964", processed by Mary Jane McCavitt, September 1980. Reprinted from Academic Vita of Norbert Wiener. Norbert Wiener: 1894-1964, Vita Mathematica Series 5, Birkhauser Basel. 1990. 537 538 ACADEMIC VITA OF NORBERT WIENER 1917 Served with the Cambridge ROTC; briefly worked as an apprentice engineer in the turbine department of the General Electric Corp. in Lynn, Massachusetts. Appointed a staff writer for the Encyclopedia Americana in Albany, N.Y. 1918 Joined the Aberdeen, Proving Grounds of the U.S. Army under O. Veblen, and worked on computations of ballistic tables. Joined the American Mathematical Society. 1919 Served as an Army private at the Aberdeen Proving Ground, Mary• land. Worked as a journalist with The Boston Herald. Appointed instructor of mathematics at MIT. 1920 Attended the International Mathematical Congress in Strasbourg as MIT's representative and presented a paper on Brownian motion. He also visited Cambridge and Paris. 1924 Appointed assistant professor of mathematics at MIT. 1925 Attended the International Mathematical Congress in Grenoble and the British Association for the Advancement of Science meeting in Southampton; visited Gottingen University. 1926 Elected fellow of the American Academy of Arts and Sciences. Married Marguerite Engemann. Received a Guggenheim Fellowship to study in Gottingen and in Copenhagen during 1926-1927. Collaborated with Harald Bohr, and taught a course of general trigonometric developments at Gottingen. 1928 Addressed the Symposium on Analysis Situs of the American Math• ematical Society. 1929 Appointed associate professor of mathematics at MIT. Lectured at Brown University as exchange professor during 1929-1930. 1931-1932 Visiting lecturer at Cambridge University; lectured on the Fourier integral and its applications at Trinity College. Appointed professor of mathematics at MIT. Attended the International Congress of Mathematics, Zurich, as MIT's representative. 1933 Awarded Bocher Prize by the American Mathematical Society. Elected to the National Academy of Sciences. Began participation in the interdisciplinary seminar at Harvard Medical School under Arturo Rosenblueth. Collaborated with REAC. Paley. 1934 Delivered the AMS Colloquium Lectures at Williamston, Massa• chusetts. 1935 Patented electrical network systems with Yuk Wing Lee. (Two more patents were issued in 1938.) Lectured at Stanford University and in Japan on his way to China. Visiting professor at Tsing Hua University in Peiping, China, during 1935-1936. 1936 Attended the International Congress of Mathematicians in Oslo, Nor• way, and lectured on Tauberian gap theorems. Collaborated with Harry Ray Pitt at MIT during 1936-1937. 1937 Delivered the Dohme lecture at Johns Hopkins University on Tauberian theorems. 1938 Lectured on analysis at the semicentennial of the AMS. 1940 Appointed chief consultant in the field of mechanical and electrical aids to computation for the National Defense Research Committee. ACADEMIC VITA OF NORBERT WIENER 539 Consultant with the NDRC's Office of Scientific Research and De• velopment, Statistical Research Group and Operational Research Lab• oratory at Columbia University. Consultant to the War-Preparedness Committee of the American Mathematical Society. Joined a team at MIT under S.H. Caldwell to study the guidance and control of antiaircraft fire. Worked on the theory and design of fire control apparatus for anti• aircraft guns with Julian Bigelow, under NDRC Project. 1941 Resigned from the National Academy of Sciences. 1945 Participated in a study group set up by John von Neumann, and attended a meeting on communication theory in Princeton. Collaborated with Arturo Rosenblueth at the Instituto National Car- diologia in Mexico, and attended the Mexican Mathematical Society's Conference held in Guadalajara. 1946-1950 With Arturo Rosenblueth received a five-year Rockefeller Foundation grant that allowed them to collaborate in Mexico and at MIT on alternating years. 1946 Received an honorary Sc.D. degree from Tufts College. Attended the first three Josiah Macy, Jr. Foundation Conferences and the Conference on Teleological Mechanisms sponsored by the New York Academy of Sciences. Lectured at the National University of Mexico. 1947 Visited England and France, and gave lectures on harmonic analysis in Nancy, France. 1948 Spoke at the AMS's Second Symposium on Applied Mathematics. 1949 Received the Lord & Taylor American Design Award. Delivered the AMS's Josiah Willard Gibbs Lecture at the annual meeting. 1950 Attended the seventh Macy conference. Lectured at the International Congress of Mathematicians at Harvard University. 1951 Lectured at the University of Paris, College de France, under a Ful- bright Teaching Fellowship, and also lectured in Madrid. Received an honorary Sc.D. degree from the University of Mexico. 1952 Received the Alvarega Prize from the College of Physicians in Philadelphia. Delivered the Forbes-Hawks Lectures at the University of Miami. 1953 Lectured on the theory of prediction at the University of California at Los Angeles. Taught a summer school course with Claude Shannon and Robert Fano on the mathematical problems of communications theory. 1953-1954 Lectured at the Tata Institute of Fundamental Research, Bombay, attended the All-India Science Congress, and visited research centers. 1955-1956 Visiting professor at the Indian Statistical Institute in Calcutta. 1956 Lectured in Japan and gave a summer school course at UCLA. 1957 Received an honorary Sc.D. degree from Grinnell College. Awarded the Virchow Medal from the Rudolf Virchow Medical Soci• ety. 1959 Gave a summer school course at UCLA. Appointed institute professor at MIT. 1960 Lectured at the University of Naples in Italy, and visited the USSR. 540 ACADEMIC VITA OF NORBERT WIENER Received the ASTME Research Medal. Retired from MIT, and appointed institute professor emeritus. 1961 Gave a summer school course at UCLA. 1962 Lectured at the Institute of Theoretical Physics, University of Naples, Italy. Delivered the Terry Lectures at Yale University, titled "Prolegomena to Theology". 1963 Gave a summer school course at UCLA. 1964 Received the National Medal of Science from President Johnson. 1964 Visiting professor and honorary head of neurocybernetics at Nether• lands Central Institute for Brain Research, Amsterdam. Lectured in Norway and Sweden. Died on March 18 in Stockholm, Sweden. Proceedings of Symposia in Applied Mathematics Volume 52, 1997 Doctoral Students of Norbert Wiener1 Shikao Ikehara Ph.D. 1930 Sebastian Littauer Sc.D. 1930 Dorothy W. Weeks Ph.D. 1930 James G. Estes Ph.D. 1933 Norman Levinson Sc.D. 1935 Henry Malin Ph.D. 1935 Bernard Friedman Ph.D. 1936 Brockway McMillan Ph.D. 1939 Abe M. Gelbart Ph.D. 1940 Donald G. Brennan Ph.D. 1959 1 Reprinted with the kind permission of Professor Irving Ezra Segal of MIT. Reprinted from Doctoral Students of Norbert Wiener, Norbert Wiener: 1894-1964, Vita Matnematica Series 5, Birkhauser Basel, 1990. 541 NOTE In the accompanying Bibliography, Wiener's publications are indexed by the official year of their appearance. The internal ordering of the publications appearing in a given year is not chronological but according to the categories laid down in the Norbert Wiener Collected Works, Editor P. Masani, Volumes I, II, III, IV (1976, 1979, 1981, 1985), MIT Press, Cambridge, Massachusetts. For instance, [14c] means "the cth paper, according to category, which appeared in a journal marked 1914". The Roman numeral IV in the right hand column indicates that this paper is reprinted in Volume IV of the Collected Works. Proceedings of Symposia in Applied Mathematics Volume 52, 1997 Bibliography of Norbert Wiener Volume* [13a] On a method of rearranging the positive integers in a series I of ordinal numbers greater than that of any given funda• mental sequence of omegas, Messenger of Math. 43 (1913), 97-105. [14a] A simplification of the logic of relations, Proc. Cambridge Philos. I Soc. 17(1914), 387-390. [14b] A contribution to the theory of relative position, Proc. Cambridge I Philos. Soc. 17 (1914), 441-449. [14c] The highest good, J. Phil. Psych, and Sci. Method 11 (1914), IV 512-520. [14d] Relativism, J. Phil. Psych, and Sci. Method 11 (1914), 561-577. IV [ 15a] Studies in synthetic logic, Proc. Cambridge Philos. Soc. 18(1915), I 14-28. [15b] Is mathematical certainty absolute?, J. Phil. Psych, and Sci. I Method 12(1915), 568-574.1A [16a] Mr. Lewis and implication, J. Phil. Psych, and Sci. Method 13 I (1916), 656-662. [16b] The shortest line dividing an area in a given ratio, Proc. Cambridge III Philos. Soc. 18 (1916), 56-58. [16c] Review ofCassius J. Keyser, Science and Region: the Rational and IV the Super rational, J. Phil. Psych, and Sci. Method 13 (1916), 273-277. [16d] Review of A. A. Robb, A Theory of Time and Space,}. Phil. Psych. IV and Sci. Method 13 (1916), 611-613. [17a] Certain formal invariances in Boolean algebras, Trans. Amer. I Math. Soc. 18(1917), 65-72. [17b] Review of C.J. Keyser, The Human Worth of Rigorous Thinking, IV J. Phil. Psych, and Sci. Method 14 (1917), 356-361. [18a] Review of Edward V. Huntington. The Continuum and Other IV Types of Serial Order, J. Phil. Psych, and Sci. Method 15 (1918), 78-80. [18b] Aesthetics, in Encyclopedia Americana, 1918-20 edition, vol. I, IV 198-203. [18c] Algebra, definitions and fundamental concepts, in Encyclopedia III Americana, 1918-20 edition, vol. L 381-385. [18d] Alphabet, in Encyclopedia Americana, 1918 20 edition, vol. I, IV 435-438. [18e] Animals, chemical sense, in, in Encyclopedia Americana, 1918-20 IV edition, vol. I, 704. 18f] Apperception, in Encyclopedia Americana, 1918-20 edition, IV vol. II, 82-83. Reprinted from Bibliography of Norbert Wiener, N. Wiener, Collected Works (P. Masani, ed.), I, II, III, IV, MIT Press, Cambridge, MA 1976, 1979, 1981, 1985. 543 544 BIBLIOGRAPHY OF NORBERT WIENER [18g] Category, in Encyclopedia Americana, 1918-20 edition, vol. VI, IV 49. [18h] Dualism, in Encyclopedia Americana, 1918-20 edition, vol. IX, IV 367. [18i] Duty, in Encyclopedia Americana, 1918-20 edition, vol. IX, IV 440-441. [18j] Ecstasy, in Encyclopedia Americana, 1918-20 edition, vol. IX, IV 570. [19a] Geometry, non-euclidean, in Encyclopedia Americana, 1918-20 III edition, vol. XII, 463-467. [19b] Induction, in logic, in Encyclopedia Americana, 1918-20 edition, IV vol. XV, 70-73. [19c] Infinity, in Encyclopedia Americana, 1918-20 edition, vol. XV, IV 120-122. [19d] Meaning, in Encyclopedia Americana, 1918-20 edition, vol. IV XVIII, 478-479. [19e] Mechanism and vitalism, in Encyclopedia Americana, 1918-20 IV edition, vol. XVIII, 527-528. [19f] Metaphysics, in Encyclopedia Americana, 1918-20 edition, vol. IV XVIII, 707-710. [19g] Pessimism, in Encyclopedia Americana, 1918-20 edition, vol. IV XXI, 654. [19h] Postulates, in Encyclopedia Americana, 1918-20 edition, vol. IV XXII, 437-438. [20a] Bilinear operations generating all operations rational in a domain I Q, Ann. of Math. 21 (1920), 157-165. [20b] A set of postulates for fields, Trans. Amer. Math. Soc. 21 (1920), I 237-246. [20c] Certain iterative characteristics of bilinear operations, Bull. Amer. I Math. Soc. 27 (1920), 6-10. [20d] Certain iterative properties of bilinear operations, G. R. Strasbourg I Math. Congress, 1920, 176-178. [20e] On the theory of sets of points in terms of continuous transforma- I tions, G.R. Strasbourg Math. Congress, 1920, 312-315. [20f] The mean of a functional of arbitrary elements, Ann. of Math. (2) I 22 (1920), 66-72. [20g] Review of C.I. Lewis, A Survey of Symbolic Logic, J. Phil. Psych. IV and Sci. Method 17 (1920), 78-79. [20h] Soul, in Encyclopedia Americana, 1918-20 edition, vol. XXV, IV 268-271. [20i] Substance, in Encyclopedia Americana, 1918-20 edition, vol. IV XXV, 775-776. [20j] Universals, in Encyclopedia Americana, 1918-20 edition, vol. IV XXVII, 572-573. [21a] A new theory of measurement: A study in the logic of mathematics, I Proc. London Math. Soc. 19 (1921), 181-205. [21b] The isomorphisms of complex algebra, Bull. Amer. Math. Soc. 27 I (1921), 443-445. [21c] The average of an analytic functional, Proc. Nat. Acad. Sci. U.S.A. I 7 (1921), 253-260. [2 Id] The average of an analytic functional and the Brownian movement, I Proc. Nat. Acad. Sci. U.S.A. 7 (1921), 294-298. BIBLIOGRAPHY OF NORBERT WIENER 545 [21e] A new vector method in integral equations (with F. L. Hitchcock), HI J. Math, and Phys. 1 (1921), 1-20. [22a] The relation of space and geometry to experience, Monist 32 I (1922), 12-60, 200-247, 364-394. [22b] The group of the linear continuum, Proc. London Math. Soc. 20 I (1922), 329-346. [22c] Limit in terms of continuous transformation, Bull. Soc. Math. I France 50(1922), 119-134. [22d] The equivalence of expansions in terms of orthogonal functions II (with J.L. Walsh), J. Math, and Phys. 1 (1922), 103-122. [22e] A new type of integral expansion, J. Math, and Phys. 1 (1922), III 167-176. [23a] On the nature of mathematical thinking, Austral. J. Psych, and I Phil. 1 (1923), 268-272. [23b] Nets and the Dirichlet problem (with H. B. Phillips), J. Math, and I Phys. 2 (1923), 105-124. (Reprinted in [64f].) [23c] Discontinuous boundary conditions and the Dirichlet problem, I Trans. Amer. Math. Soc. 25 (1923), 307-314. [23d] Differential-space, J. Math, and Phys. 2 (1923), 131-174. I (Reprinted in [64f].) [23e] O szeregach Ej°(± \/n)—Note on the series £j°(± 1/rc), Bull. I Acad. Polon. Ser. A, 13 (1923), 87-90. [23f] Note on a new type of summability, Amer. J. Math. 45 (1923), II 83-86. [23g] Note on a paper of M. Banach, Fund. Math. 4 (1923), 136-143. Ill [24a] Certain notions in potential theory, J. Math, and Phys. 3 (1924), I 24-51. [24b] The Dirichlet problem, J. Math, and Phys. 3 (1924), 127-146. I (Reprinted in [64f].) [24c] Une condition necessaire et suffisante de possibilite pour le probleme I de Dirichlet, C.R. Acad. Sci. Paris 178 (1924), 1050-1054. [24d] The average value of a functional, Proc. London Math. Soc. 22 I (1924), 454^467. [24e] Un probleme de probabilites denombrables, Bull. Soc. Math. I France li (1924), 569-578. [24f] The quadratic variation of a function and its Fourier coefficients, J. II Math, and Phys. 3 (1924), 72-94. [24g] Review of four books on space, Rudolf Carnap's Der Raum: Ein IV Beitrag zur Wissenschaftslehre, E. Study's Mathematik und Phy~ sik: eine erkenntnistheoretische Untersuchung and Die realistische Weltansicht und die Lehre vom Raume: zweite Auflage; erster Teil, and Hermann WeyPs Mathematische Analyse des Raum-problems. Vorlesungen gehalten in Barcelona und Madrid, Bull. Amer. Math. Soc. 30 (1924), 258-262. [24h] Review of E. Study, Denken und Darstellung: Logik und Werte; IV Dingliches und Menschliches in Mathematik und Naturwissenschaf ten, Bull. Amer. Math. Soc. 30 (1924), 277. [24i] In memory of Joseph Lipka, J. Math, and Phys. 3 (1924), 63-65. IV [25a] Note on a paper of O. Perron, J. Math, and Phys. 4 (1925), I 21-32. [25b] The solution of a difference equation by trigonometrical integrals, J. II Math, and Phys. 4 (1925), 153-163. 546 BIBLIOGRAPHY OF NORBERT WIENER [25c] On the representation of functions by trigonometrical integrals, II Math. Z. 24 (1925), 575-616. [25d] Verallgemeinerte trigonometrische Entwicklungen, Gottingen II Nachr. (1925), 151-158. [25e] Note on quasi-analytic function, J. Math, and Phys. 4 (1925), II 193-199. [25f] A contribution to the theory of interpolation, Ann. of Math. (2) 26 III (1925), 212-216. [26a] The harmonic analysis of irregular motion, J. Math, and Phys. 5 II (1926), 99-121. [26b] The harmonic analysis of irregular motion (Second Paper), J. II Math, and Phys. 5 (1926), 158-189. [26c] The operational calculus, Math. Ann. 95 (1926), 557-584. II [26d] A new formulation of the laws of quantization of periodic and III aperiodic phenomena (with M. Born), J. Math, and Phys. 5 (1926), 84-98. [26e] Eine neue Formulierung der Quantengesetze fur periodische und III nichtperiodische Vorgange (with M. Born), Z. Physik 36 (1926), 174-187. [26f] Analytic approximations to topological transformations (with P. Ill Franklin), Trans. Amer. Math. Soc. 28 (1926), 762-785. 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Ill Sci. Paris 185(1927)42-44. [27i] Sur la theorie relativiste des quanta (Note), C. R. Acad. Sci. Paris III 185 (1927), 184-185. [27j] Laplacians and continuous linear functional, Acta Sci. Math. Ill (Szeged) 3 (1927), 7-16. [27k] Une generalisation des fonctions a variation bornee, C.R. Acad. Ill Sci. Paris 185 (1927), 65-67. [28a] The spectrum of an arbitrary function, Proc. London Math. Soc. II (2) 27 (1928), 483-*96. [28b] A new method in Tauberian theorems, J. Math, and Phys. 7 (1928), II 161-184. [28c] The fifth dimension in relativistic quantum theory (with D.J. Ill Struik), Proc. Nat. Acad. Sci. U.S.A. 14 (1928), 262-268. BIBLIOGRAPHY OF NORBERT WIENER 547 [28d] Coherency matrices and quantum theory, J. Math, and Phys. 7 III (1928), 109-125. [29a] Harmonic analysis and group theory, J. Math, and Phys. 8 (1929), II 148-154. [29b] A type of Tauberian theorem applying to Fourier series, Proc. II London Math. Soc. (20) 30 (1929), 1-8. [29c] Fourier analysis and asymptotic series. Appendix to V. Bush, III Operational Circuit Analysis, New York, John Wiley, 1929, 366-379. [29d] Hermitian polynomials and Fourier analysis, J. Math, and Phys. 8 II (1929), 70-73' [29e] Harmonic analysis and the quantum theory, J. Franklin Inst. 207 III (1929), 525-534. [29f] On the spherically symmetrical statical field in Einstein's unified III theory of electricity and gravitation (with M.S. Vallarta), Proc. Nat. Acad. Sci. U.S.A. 15 (1929), 353-356. [29g] On the spherically symmetrical statical field in Einstein's unified III theory: A correction (with M.S. Vallarta), Proc. Nat. Acad. Sci. U.S.A. 15(1929), 802-804. [29h] Mathematics and art (Fundamental identities in the emotional IV aspects of each), Tech. Rev. 32 (1929), 129-132, 160, 162. [29i] Einsteiniana (Facts and fancies about Dr. Einstein's famous IV theory), Tech. Rev. 32 (1929), 403^104. [29j] Murder and mathematics, Tech. Rev. 32 (1929), 271-272. IV [30a] Generalized harmonic analysis, Acta Math. 55 (1930), 117-258. II (Reprinted in [64f] and [66b].) [30b] Review of A. Eddington's Science and the Unseen World, Tech. IV Rev. 33(1930), 150. [31a] Uber eine Klasse singularer Integralgleichungen (with E. Hopf), HI Sitzber. Preuss. Akad. Wiss. Berlin, Kl. Math. Phys. Tech., 1931, pp. 696-706. (Reprinted in [641].) [31b] A new deduction of the Gaussian distribution, J. Math, and Phys. HI 10(1931), 284-288. [31c] Reports from Cambridge—1931, Tech. Rev. 34 (1931), 82-83, 131, IV 218, 220. [32a] Tauberian theorems, Ann. of Math. 33 (1932), 1-100 (Reprinted in II [64f] and [66b].) [32b] A note on Tauberian theorems, Ann. of Math. 33 (1932), 787. II [32c] Back to Leibniz! (Physics reoccupies an abandoned position), Tech. IV Rev. 34 (1932), 201-203, 222, 224. [32d] Reports from Cambridge—1932, Tech. Rev. 34 (1932), 62, 74. IV [32e] Review of A.S. Besicovitch, Almost Periodic Functions, Math. IV Gaz. 16 (1932), 275-277. [32f] Analytic properties of the characters of infinite Abelian groups (with II R.E. A.C. Paley), Abstract, Int'l. Math. Congr., Zurich, 1932, 95. [33a] Notes on random functions (with R.E. A.C. Paley and A. Zyg- I mund), Math. Z. 37 (1933), 647-668. [33b] A one-sided Tauberian theorem, Math. Z. 36 (1933), 787-789. II [33c] Characters of Abelian groups (with R.E. A.C. Paley), Proc. Nat. II Acad. Sci. U.S.A. 19 (1933), 253-257. [33d] The total variation of g(x + h) - g(x) (with R.C. Young, Trans. II Amer. Math. Soc. 35 (1933), 327-340. 548 BIBLIOGRAPHY OF NORBERT WIENER [33e] Notes on the theory and application of Fourier transforms (with II R.E.A.C Paley) I, II, Trans. Amer. Math. Soc. 35 (1933), 348-355; III, IV, V, VI, VII, Trans. Amer. Math. Soc. 35 (1933), 761-791. [33f] Putting matter to work ( The search for chapter power), Tech. Rev. IV 35 (1933), 47-49, 70, 72. [33g] Review of Harald Bohr, Fastperiodische Funktionen, Math. Gaz. IV 17 (1933), 54. [33h] R.E.A.C. Paley—In Memoriam, Jan. 7, 1907—Apr. 7, 1933, Bull. IV Amer. Math. Soc. 39 (1933), 476. [33i] The Fourier Integral and Certain of Its Applications, Cambridge University Press, New York, 1933; reprint, Dover, New York, 1959; review by E.C. Titchmarsh in Math. Gaz. 17 (1933), 129. [34a] Random functions, J. Math, and Phys. 14 (1934), 17-23. I [34b] A class of gap theorems, Ann. Scuola Norm. Sup. Pisa, E II (1934-1936), 1-6. [34c] Quantum mechanics, Haldane, and Leibniz, Philos. Sci. 1 (1934), IV 479-482. [34d] Fourier Transforms in the Complex Domain (with R.E.A.C. Paley), Amer. Math. Soc. Colloq. Publ. 19, Amer. Math. Soc, Providence, R.I., 1934. [34e] Aid for German-refugee scholars must come from non-academic IV sources, Jewish Advocate (December 1934). [35a] Fabry's gap theorem, Sci. Repts. of Nat'l. Tsing Hua Univ., Ser. II A, 3 (1935), 239-245. [35b] Limitations of science (The holiday fallacy and a response to the IV suggestion that scientists become sociologists), Tech. Rev. 37 (1935), 255-256, 268, 270, 272. [35c] The student agitator (Is he accepting radicalism as an opiate?) IV (with Carl Bridenbaugh), Tech. Rev. 37 (1935), 310-312, 344, 346. [35d] Mathematics in American secondary schools. J. Math. Assoc. IV Japan for Secondary Education (Tokyo) 17 (1935), 1-5. [35e] The closure of BesseI functions, Abstract 66, Bull. Amer. Math. IV Soc. 41 (1935), 35. [351] Once more ... the refugee problem abroad, Jewish Advocate IV (February 5, 1935). [36a] A theorem ofCarleman, Sci. Repts. of Nat'l. Tsing Hua Univ., Ser. II A, 3 (1936), 291-298. [36b] Sur les series de Fourier lacunaires. Theoremes directs (with S. II Mandelbrojt), C.R. Acad. Sci. Paris 203 (1936), 34-36. [36c] Series de Fourier lacunaires. Theoremes inverses (with S. Man- II delbrojt), C.R. Acad. Sci. Paris 203 (1936), 233-234. [36d] Gap theorems, C.R. de Congr. Int'l. des Math., 1936, 284-296. II [36e] A Tauberian gap theorem of Hardy and Littlewood, J. Chinese II Math. Soc. 1 (1936) 15-22. [36f] Notes of the Kron theory of tensors in electrical machinery, J. Ill Electr. Engrg., China 7 (1936), 277-291. [36g] The role of the observer, Philos. Sci. 3 (1936), 307-319. IV [37a] Taylor's series of entire functions of smooth growth (with W.T. II Martin), Duke Math. J. 3 (1937), 213-223. [37b] Random Waring's theorems, Abstract (with N. Levinson), Science 85 (1937), 439. BIBLIOGRAPHY OF NORBERT WIENER 549 [38a] The homogeneous chaos, Amer. J. Math. 60 (1938), 897-936. I (Reprinted in [64f].) [38b] On absolutely convergent Fourier-Stieltjes transforms (with H.R. II Pitt), Duke Math. J. 4 (1938), 420-440. [38c] Fourier-Stieltjes transforms and singular infinite convolutions (with II A. Wintner), Amer. J. Math. 60 (1938), 513-522. [38d] Taylor's series of functions of smooth growth in the unit circle (with II W.T. Martin), Duke Math. J. 4 (1938), 384-392. [38e] The historical background of harmonic analysis, Amer. Math. Soc. II Semicentennial Publications Vol. II, Semicentennial Addresses, Amer. Math. Soc, Providence, R.L, 1938, 513-522. [38f] Remarks on the classical inversion formula for the Laplace II integral (with D.V. Widder), Bull. Amer. Math. Soc. 44 (1938), 573-575. [38g] The decline of cookbook engineering, Tech. Rev. 40 (1938), 23. IV [38h] Review of L. Hogben, Science for the Citizen, Tech. Rev. 40 IV (1938), 66-67. [39a] The ergodic theorem, Duke Math. J. 5 (1939), 1-18. (Reprinted in I [64f].) [39b] The use of statistical theory in the study of turbulence, Proc. 5th I Int'l. Congr. of Applied Mechanics, Sept. 12-16, 1938, Wiley, New York, 1939, 356-358. [39c] On singular distributions (with A. Wintner), J. Math, and Phys. 17 II (1939), 233-246. [39d] Convergence properties of analytic functions of Fourier-Stieltjes II transforms (with R.H. Cameron), Trans. Amer. Math. Soc. 46 (1939), 97-109; Math. Rev. 1 (1940), 13; rev. 400. [39e] A generalization of Ikehara's theorem (with H.R. Pitt), J. Math. II and Phys. 17 (1939), 247-258. [39f] Review of Roger Burlingame, March of the Iron Men, Tech. Rev. IV 41 (1939), 115. [39g] Review of W. George, The Scientist in Action, Tech. Rev. 41 IV (1939), 202. [39h] A new method in statistical mechanics, Abstract 133 (with B. I McMillan), Bull. Amer. Math. Soc. 45 (1939), 234; Science 90 (1939), 410-411. [40a] Review of M. Fukamiya, On dominated ergodic theorems in IV Lp(p>l), Math. Rev. 1 (1940), 148. [40b] Review of M. Fukamiya, The Lipschitz condition of random func- IV tion, Math. Rev. 1 (1940), 149. [40c] Review of Th. De Donder, L'energetique deduite de la mecanique IV statistique generate, Math. Rev. 1 (1940), 192. [40d] A canonical series for symmetric functions in statistical mechanics, I Abstract 133, Bull. Amer. Math. Soc. 46 (1940), 57. [41a] Harmonic analysis and ergodic theory (with A. Wintner), Amer. J. I Math. 63 (1941), 415-426; Math. Rev. 2 (1941), 319. [41b] On the ergodic dynamics of almost periodic systems (with A. Wint- I ner), Amer. J. Math. 63 (1941), 794^824; Math. Rev. 4 (1943), 15. [42a] On the oscillation of the derivatives of a periodic function (with II G. Polya), Trans. Amer. Math. Soc. 52 (1942), 249-256. [43a] The discrete chaos (with A. Wintner), Amer. J. Math. 65 (1943), I 279-298; Math. Rev. 4 (1943), 220. 550 BIBLIOGRAPHY OF NORBERT WIENER [43b] Behavior, purpose, and teleology (with A. Rosenblueth and J. IV Bigelow), Philos. Sci. 10 (1943), 18-24. [44a] Review of Hugh Gray Lieber and Lillian R. Lieber, The Education IV of T. C. Mits; What Modern Methematics Means to You, Tech. Rev. 46 (1944), 390, 392. [45a] La teoria de la estrapolacion estadistica, Bol. Soc. Mat. Mexicana III 2 (1945), 37-45; Math. Rev. 7 (1946), 461. [45b] The role of models in science (with A. Rosenblueth), Philos. Sci. 12 IV (1945), 316-322. [46a] A generalization of the Wiener-Hopf integral equation (with A. E. Ill Heins), Proc. Nat. Acad. Sci. U.S.A. 32 (1946), 98-101; Math. Rev. 8 (1947), 29. 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Atomic Scientists 4 IV (1948), 338-339. [48e] Review of L. Infeld, Whom the Gods Love. The Story of Evariste IV Galois, Scripta Math. 14 (1948), 273-274. [48f] Cybernetics, or Control and Communication in the Animal and the Machine, Actualites Sci. Ind., no. 1053; Hermann et Cie.. Paris; The MIT Press, Cambridge. Mass., and Wiley, New York. 1948; Math. Rev. 9 (1948), 598. Partly reprinted as Rigidity and learn• ing: ants and men, in Classics in Biology (A Course of Selected Reading by Authorities), Philosophical Library, New York, 1960, pp. 205-213. [49a] Sur la theorie de la prevision statistique et du filtrage des ondes, HI Analyse Harmonique, Colloques Internationaux du CNRS, No. 15, pp. 61-1 A. Centre National de la Recherche Scientifique, Paris, 1949; Math. Rev. 11 (1950), 376. [49b] A statistical analysis of synaptic excitation (with A. Rosenblueth, IV W. Pitts, and J. Garcia Ramos), J. of Cellular and Comparative Physiol. 34(1949), 173-205. [49c] A new concept of communication engineering, Electronics 22 IV (1949), 74-77. [49d] Sound communication with the deaf, Philos. Sci. 16 (1949), IV 260-262. [49e] Some problems in sensory prosynthesis (with J. Wiesner and L. IV Levine), Science 110 (1949), 512. BIBLIOGRAPHY OF NORBERT WIENER 551 [49f] Obituary—Godfrey Harold Hardy, 1877-1947, Bull. Amer. Math. IV Soc. 55 (1949), 72-77. [49g] Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications, The MIT Press, Cambridge, Mass., Wiley, New York; Chapman & Hall, London, 1949; paper• back edition with the title Time Series, The MIT Press, 1964; Math. Rev. 11 (1950), 118. [49h] Review of Philipp Frank, Modern Science and its Philosophy, New IV York Times, Book Review, August 14, 1949, sec. 7, p. 3. [50a] Some prime-number consequences of the Ikehara theorem (with L. II Geller); Acta. Sci. Math. (Szeged) 12 (1950), 25-28, Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B; Math. Rev. 11 (1950), 644; Math. Rev. 12 (1951), 1002. [50b] Comprehensive view of prediction theory, Proceedings of the Inter- III national Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pp. 308-321; Amer. Math. Soc, Providence, R.L, 1952, Expository lecture; Math. Rev. 13 (1952), 477. [50c] Some maxims for biologists and psychologists, Dialectica 4 (1950), IV 186-191. [50d] Purposeful and non-purposeful behavior (with A. Rosenblueth), IV Philos. Sci. 17(1950), 318-326. [50e] Cybernetics, Bull. Amer. Acad. Arts and Sci. 3 (1950), 2-4. IV [50f] Speech, language, and learning, J. Acoust. Soc. Amer. 22 (1950), IV 696-697. [50g] Entropy and information, Proc. Sympos. Appl. Math., vol. 2, IV Amer. Math. Soc, Providence, R.L, 1950, p. 89; Math. Rev. 11 (1950), 305. [50h] Too big for private enterprise, Nation 170 (1950), 496^97. IV [50i] Too damn close, Atlantic 186 (1950), 50-52. [50j] The Human Use of Human Beings, Houghton Mifflin, Boston, 1950; paperback edition by Doubleday, Anchor, Garden City, N.Y., 1954. Chapter 3 reprinted in Classics in Biology (A Course of Selected Reading by Authorities), Philosophical Library, New York, 1960, pp. 205-213. [50k] The brain (short story), Tech. Eng. News 31 (1950), 14-15, 33-34, 44, 50. (reprinted in paperback anthology, Cross-roads in Time, ed. Groff Conklin, Doubleday, Garden City. N.Y., 1953.) [51a] Problems of sensory prosthesis, Bull. Amer. Math. Soc. 57 (1951), IV 27-35. (Reprinted in [64f].) [51b] Homeostasis in the individual and society, J. Franklin Inst. 251 IV (1951), 65-68. (Reprinted in [64f].) [51c] Mathematical relationships of possible significance in the study of IV human leukemia (with P.F. Hahn), Federation Proc. 10 (1951). [52a] Cybernetics (Light and Maxwell's demon), Scientia (Italy) 87 IV (1952), 233-235. [52b] The miracle of the broom closet (short story), Tech. Eng. News 33 (1952), 18-19, 50. 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[53e] The concept of homeostasis in medicine, Transactions and Studies IV of the College of Physicians of Philadelphia (4) 20 (1953), No. 3, 87-93. [53f] Problems of organization, Bull. Menninger Clinic 17 (1953), IV 130-138. [53g] The future of automatic machinery, Mech. Engrg. 75 (1953), IV 130-132. [53h] Ex-prodigy: My Childhood and Youth, Simon and Schuster, New York, 1953; The MIT Press, Cambridge, Mass., 1965 (paperback edition, The MIT Press); Math Rev. 15 (1954), 277. [53i] The electronic brain and the next industrial revolution, Cleveland IV Athletic Club Journal (January, 1953). [53j] The machine as threat and promise, St. Louis Post-Dispatch IV (December, 1953). [53k] We can V attain truth without risk of error (from This I Believe IV radio show), Minneapolis Tribune (November, 1953). [54a] Men, machines, and the world about, in Medicine and Science, IV New York Academy of Medicine and Science, ed. I. Galderston, International Universities Press, New York, 1954, pp. 13-28. [54b] Conspiracy of conformists, Nation 178 (1954), 375. IV [54c] Automatization (with Donald Campbell), St. Louis Post-Dispatch IV (December, 1954). [55a] Nonlinear prediction and dynamics, Proc. Third Berkeley Sym- HI posium on Mathematical Statistics and Probability, University of California Press, Berkeley, Calif., 1954/5, pp. 247-252; Math. Rev. 18 (1957), 949. [55b] On the factorization of matrices, Comment, Math. Helv. 29 (1955), HI 97-111; Math. Rev. 16 (1955), 921. [55c] The differential-space theory of quantum systems (with A. Siegel), HI Nuovo Cimento (10) 2 (1955), 982-1003, No. 4, Suppl. [55d] Thermodynamics of the message, in Neurochemistry, ed. K.E.C. IV Elliott, Thomas, Springfield, 1955, pp. 844-849. [55e] Time and organization, Second Fawley Foundation Lecture, Uni- IV versity of Southampton, 1955, pp. 1-16. [56a] On a local L--variant oflkehara's theorem (with A. Wintner), Rev. II Math. Cuyana 2 (1956), 53-59. [56b] The theory of prediction, in Modern Mathematics for the Engineer, HI ed. E.F, Beckenbach, McGraw-Hill, New York, 1956, pp. 165-187. [56c] "Theory of Measurement" in differential-space quantum theory III (with A. Siegel), Phys. Rev. 101 (1956), 429-432. BIBLIOGRAPHY OF NORBERT WIENER 553 [56d] Pure patterns in a natural world, in The New Landscape in Art and IV Science, ed. G. Kepes, Paul Theobald and Co., Chicago, 1956, pp. 274-276. [56e] Brain waves and the interferometer, J. Phys. Soc. Japan 18 (1956), IV 499-507. [56f] Moral reflections of a mathematician, Bull. Atomic Scientists 12 IV (1956), 53-57. (Reprinted from [56g].) [56g] / Am a Mathematician. The Later Life of a Prodigy, Doubleday, Garden City, New York, 1956; paperback edition by The MIT Press, 1964; Math. Rev. 17 (1956), 1037. [57a] The definition and ergodic properties of the stochastic adjoint of a I unitary transformation (with E.J. Akutowicz), Rend. Circ. Mat. Palermo (2) 6 (1957), 205-217, Addendum, 349; Math. Rev. 20 (1959), rev. 4328. [57b] Notes on Polya's and Turdn's hypotheses concerning Liouville's III factor (with A. Wintner), Rend. Circ. Mat. Palermo (2) 6 (1957), 240-248; Math. Rev. 20 (1959), rev. 5759. [57c] On the non-vanishing of Euler products (with A. Wintner), Amer. II J. Math. 79 (1957), 801-808. [57d] The prediction theory of multivariate stochastic processes, Part I III (with P. Masani), Acta Math. 98 (1957), 111-150; Math. Rev. 20 (1959), rev. 4323. [57e] Rhythms in physiology with particular reference to encephalography, IV Proceedings of the Rudolf Virchow Medical Society in the City of New York, vol. 16, 1957, pp. 109-124. [57f] The role of the mathematician in a materialistic culture (A IV scientist's dilemma in a materialistic world), Columbia Engineer• ing Quarterly, Proceedings of the Second Combined Plan Confer• ence, Arden House, October 6-9, 1957, pp. 22-24. [57g] The role of the small cultural college in education of the scientists; IV a speech given at Wabash College, Indiana, October 10, 1957. [57h] Cybernetics, in The Universal Standard Encyclopedia (abridgment IV of The New Funk and Wagnail's Encyclopedia), Standard Refer• ence Works Publishing Co., New York, 1957, p. 180. [58a] Logique, probability et methode des sciences physiques, in La HI Methode dans les Sciences Modernes, Editions Science et Indus• trie, ed. Francois Le Lionnais, Paris, 1958, pp. 111-112. [58b] The prediction theory of multivariate stochastic processes, Part II III (with P. Masani), Acta Math. 99 (1958), 93-137; Math. Rev. 20 (1959), rev. 4325. [58c] Random time (with A. Wintner), Nature 181 (1958), 561-562. IV [58d] Sur la prevision lineaire des processus stochastiques vector iels a III densite spectrale bornee. I (with P. Masani), C. R. Acad. Aci. Paris 246 (1958), 1492-1495; Math. Rev. 20 (1959), rev. 4324a. [58e] Sur la prevision lineaire des processus stochastiques vector iels a III densite spectrale bornee. II (with P. Masani), C. R. Acad. Sci. Paris 246 (1958), 1655-1656; Math. Rev. 20 (1959), rev. 4324b. [58f] My connection with cybernetics. Its origins and its future, Cyber- IV netica (Belgium) 1 (1958), 1-14. [58g] Time and the science of organization, Part II, Scientia 93 (1958), IV 225-230. [58h] Science: The megabuck era, New Republic, 138 (1958), 10-11. IV 554 BIBLIOGRAPHY OF NORBERT WIENER [58i] Nonlinear Problems in Random Theory, The MIT Press, Cam• bridge, Mass., and Wiley, New York, 1958; paperback edition, The MIT Press, 1966. [59a] A factorization of positive Hermitian matrices (with E.J. Akuto- HI wicz), J. Math. Mech. 8 (1959), 111-120. [59b] Nonlinear prediction (with P. Masani), in Probability and Statis- III tics, The Harald Cramer Volume, ed. U. Grenander, Stockholm, 1959, 190-212. [59c] On bivariate stationary processes and the factorization of matrix- III valued functions (with P. Masani), Teor, Verojatnost, i Primenen. 4 (1959), 322-331. (English transl. Theor. Probability App. 4 (1959), 300-308). [59d] Man and the machine (Interview with N. Wiener). Challenge (The IV Magazine of Economic Affairs) 7 (1959), 36—41. [59e] The Tempter (novel), Random House, New York, 1959. [60a] The application of physics to medicine, in Medicine and Other IV Disciplines, New York Academy of Medicine, ed. I. Galderston, International Universities Press, 1960, pp. 41-57. [60b] The brain and the machine (Summary of an address), in Dimensions IV of Mind, ed. S. Hook, Collier Books, 1960, (Proceedings of Third Annual New York Univ. Institute of Philosophy held on May 15-16, 1959), pp. 113-117. [60c] Kybernetik, Contribution to Worterbuch der Soziologie, F. Enke IV Verlag, Stuttgart, 1960, pp. 620-622. [60d] Some moral and technical consequences of automation, Science 131 IV (1960), 1355-1358. [60e] The duty of the intellectual, Tech. Rev. 62 (1960), 26-27; reprinted IV almost in entirety in The grand privilege, Sat. Rev. 43 (1960), 51-52; also in Technion 18 (1961), 86-87—"A professor tells what a professor is." [60f] Preface to Cybernetics of Natural Systems, by D. Stanley-Jones, IV Pergamon Press, London, 1960, pp. v-viii. [60g] Possibilities of the use of the interferometer in investigating IV macromolecular interactions, in Fast Fundamental Transfer Pro• cesses in Aqueous Biomolecular Systems, ed. F.O. Schmitt, Department of Biology, MIT, Cambridge, Mass., June 1960, pp. 52-53. [61a] Uber Informationstheoreie, Naturwissenschaften 48 (1961), IV 174-176. [61b] Science and society, Voprosy Filosofii (1961), No. 7, 117-122; IV reprinted in Estratto Rivista Methodos 13 (1961), 1-8, and in Tech. Rev. 63 (1961), 49-52. Excerpts in Science 138 (1962), 651. [61c] Cybernetics, Second edition of [481] (revisions and two additional chapters). The MIT Press and Wiley, New York, 1961; paperback edition, The MIT Press, 1965. [62a] A verbal Contribution to Proc. of the International Symposium IV on the Application of Automatic Control in Prosthetics Design, August 27-31, 1962, Opatija, Yugoslavia, pp. 132-133. [62b] The mathematics of self organizing systems, in Recent Develop- IV ments in Information and Decision Processes, Macmillan, New York, 1962* pp. 1-21. BIBLIOGRAPHY OF NORBERT WIENER 555 [62c] Short-time and long-time planning, originally presented at 1954 IV ASPO National Planning Conference. Jersey Plans, An ASPO Anthology (1962), 29-36. [63a] Random theory in classical phase space and quantum mechanics III (with Giacomo Delia Riccia), Proc. Internat. Conference on Functional Analysis, Massachusetts Institute of Technology, Cambridge, Mass., June 9-13, 1963; Analysis in Function Space, The MIT Press, Cambridge, Mass., 1964, pp. 3-14. [63b] Introduction to neurocybernetics (with J. P. Schade) and Epilogue, IV in Progress in Brain Research, vol. 2 of Nerve, Brain and Memory Models, Elsevier Publishing Co., Amsterdam, 1963, pp. 1-7, 264-268. [63c] The lonely nationalism of Rudyard Kipling (with K. Deutsch), Yale IV Rev. 52(1963), 499-517. [64a] On the oscillations of nonlinear systems, Proc. Symposium on Sto- IV chastic Models in Medicine and Biology, Mathematics Research Center, U.S. Army, June 12-14, 1963, ed. John Gurland, Uni• versity of Wisconsin Press, Madison, Wisconsin, 1964, pp. 167-177. [64b] Dynamical systems in physics and biology, Contribution to series IV "Fundamental Science in 1984", The New Scientist (London) 21 (1964), 211-212. [64c] Machines smarter than men? (Interview with N. Wiener), U.S. IV News and World Rept. 56 (1964), 84-86; abbreviated in Reader's Digest 84 (1964), 121-124. [64d] Intellectual honesty and the contemporary scientist (Transcript of IV talk given to Hillel Group at Massachusetts Institute of Technol• ogy), Tech. Rev. 66 (1964), 17-18, 44-45, 47. [64e] God, Golem, Inc.—A Comment on Certain Points Where Cyber• netics Impinges on Religion, The MIT Press, Cambridge, Mass., 1964; paperback edition, The MIT Press 1966. [64f] Selected Papers of Norbert Wiener with expository papers by Y. W. Lee, Norman Levinson, and W.T. Martin, The MIT Press, Cambridge, Mass., 1964. [65a] L'homme et la machine, Proc. Colloques Philosophiques Inter- nationaux de Royaumont, July, 1962; Le concept d'information dans la science contemporaine, Gauthier-Villars, Paris, 1965, pp. 99-132. [65b] Perspectives in cybernetics, Progress in Brain Research 17 (1965), IV 399-408. [65c] Cybernetics in Collier's Encyclopedia, U.S.A., ed. William D. Hal- IV sey, The Cormwell-Collier Publishing Co., New York, 1965, pp.598-599. [66a] Wave mechanics in classical phase space, Brownian motion, and IV quantum theory (with G. Delia Riccia), J. Math. Phys. 7 (1966), 1372-1383. [66b] Differential Space, Quantum Systems and Prediction (with A. Siegel, B. Rankin, W.T. Martin), The MIT Press, Cambridge, Mass., 1966. [66c] Generalized Harmonic Analysis and Tauberian Theorems (paper• back edition of [30a] and [32a]), The MIT Press, Cambridge, Mass., 1966. 556 BIBLIOGRAPHY OF NORBERT WIENER [75a] Cybernetics (with F. Landis), in Funk and Wagnall's New Ency- IV clopedia, Funk & Wagnall, New York, 1975, p. 228. [85a] Letter covering the memorandum on the scope, etc., of a suggested IV computing machine (September 21, 1940), Coll. Works, IV, pp. 122-124, cf.{M 10}. [85b] Memorandum on mechanical solution of partial differential equa- IV tions, Coll. Works, IV, pp. 125-134, cf. {M10}. [85c] Muscular Clonus: Cybernetics and Physiology (with A. Rosen- IV blueth and J. Garcia Ramos), Coll. Works, IV, pp. 466-510, cf. {M10}. Addenda [43c] N. Wiener, Elements of Calculus, pp. 459-499, in Practical Mathemat• ics, Editor Reginald Stevens Kimball, vol. 2, Issue 8, June 1943. [93a] N. Wiener, Invention: The Care and Feeding of Ideas (Introduction by Steve Joshua Heims), MIT Press, Cambridge, MA, 1993. Proceedings of Symposia in Applied Mathematics Volume 52, 1997 Defense Department Documents1 la. S.H. Caldwell, Proposal to Section D2, NDRC (3 p.), November 22, 1940. lb. N. Wiener, Principles governing the construction of prediction and compensating apparatus (8 p.) accompaniment to la, November 22, 1940. II. K.T. Compton, Letter to Dr. Warren Weaver, NDRC, May 13, 1941. III. J.H. Bigelow, Minutes of Conference held at Bell Laboratories on June 4, 1941. IV. N. Wiener, Letter to Dr. Warren Weaver, NDRC, December 1, 1941. V. G.R. Stibitz, Note on prediction networks a la Wiener (14 p.), February 22, 1942. VI. Warren Weaver, Letter to Dr. J.C. Boyce, MIT, March 24, 1942. VII. N. Wiener, A. A. Directors, Summary Report of Demonstration (17 p.), June 10, 1942. VIII. Demonstration by Wiener and Bigelow at MIT, July 1, 1942, Diary of G.R. Stibitz, Chairman, Division D2, July 23, 1942. IX. N. Wiener: Final report on Section D2, Project No. 6 (8 p.) submitted to Dr. Warren Weaver, NDRC, December I, 1942.2 X. N. Wiener, Letter to Dr. Warren Weaver, NDRC, January 15, 1943. XI. R.S. Phillips and P.R. Weiss, Theoretical calculation on best smoothing of posi• tion data for gunnery prediction, MIT Radiation Laboratory Report 532, February 16, 1944. XII. N. Wiener, Automatic Control Techniques in Industry, Industrial College of the Armed Forces, Washington, D.C., 1952-1953. 1 NDRC stands for National Defense Research Committee. 2 This report is accompanied by a Report to the Services, No. 59, of March 27, 1945, entitled "Statistical Method of Prediction in Fire Control". Reprinted from Defense Documents of Norbert Wiener, Norbert Wiener: 1894-1964, Vita Mathematica Series 5, Birkhauser Basel, 1990. 557 Norbert Wiener Centenary Congress Participants List David R. Adams, Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, Phone: 606-257-6818, Fax: 606-257-4078, E-mail: dave@ ms.uky.edu Sergio Albeverio, Fakultat fur Mathematik, Ruhr-Universitut Bochum, Gebaude Na 3/33, D-44780 Bochum, GERMANY, Phone: 011 49 234 700 5599, Fax: 011 49 234 709 4242 Madjid Amir, Institute for Mathematical Economics, University of Mannheim, A5,6,68165 Mannheim, GERMANY, Phone: 011 49 621 292 3154, Fax: 011 49 621 292 5007, E-mail: [email protected] S. Balaji, Stat-Math Unit, Indian Statistical Institute, R.V. College Post, Bang- alore-560 059, INDIA, E-mail: [email protected] Harm Bart, Economic Institute, Erasmus University Rotterdam, Postbus 1763, 3000 DR ROTTERDAM, THE NETHERLANDS, Phone: 31 10 4081253, Fax: 31 10 4527746, E-mail: [email protected] John J. Benedetto, Department of Mathematics, University of Maryland, College Park, MD 20742, Phone: 301-405-5161, Fax: 301-314-0827, E-mail: [email protected] Nirmal K. Bose, Elect. & Computer Engin. Dept, The Pennsylvania State Uni• versity, 121 Electrical Engineering East, University Park, PA 16802-2705, Phone: 814-865-3912, Fax: 814-865-7065, E-mail: [email protected] James E. Brassert, C/O Frau Smieja, Kalvarienbergstr.33, D-87500 Immenstadt, GERMANY, Phone: 011 49 8323 8363 Donald L. Burkholder, Department of Mathematics, University of Illinois, Urbana, IL 61801-2917, Phone: 217-333-9674, Fax: 217-333-9576, E-mail: donburk@ symcom.math.uiuc.edu Eric Carlen, Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, Phone: 404-894-8380, Fax: 404-853-9112, E-mail: carlen@ math.gatech.edu Henri Carnal, Universitat Bern, Institut fur Mathematische Statistik, CH-3012 Bern, SWITZERLAND, Phone: 011 41 31 631 88 11, Fax: 011 41 31 631 38 70, E-mail: [email protected] 559 560 PARTICIPANTS LIST Maria Carvalho, Dep. Matematica, Faculdade Ciencas de Lisboa, R Ernesto Vasconcelos, bloco CI, 1700 Lisboa, PORTUGAL, E-mail: sao.ptifm.bitnet Barry Cipra, 305 Oxford St., Northfield, MN 55057, Phone: 507-663-1353, E-mail: [email protected] Jiu Ding, Department of Mathematics, Sou St a Box 5045, Hattiesburg, MS 39406, Phone: 601-266-5505, E-mail: [email protected] Jamshid Farshidi, Department of Mathematics, Hampton University, Hampton, VA 23668, Phone: 804-727-5549 Hans G. Feichtinger, Department of Mathematics, University Vienna, Vienna, AUSTRIA, Phone: 011 43-1-40480-696, Fax: 011-43-1-40480-697, E-mail: fei@ ty che. mat. univie. ac. at George D. Gale, Department of Philosophy, University of Missouri, Kansas City, MO 64110, Phone: 816-235-2816, Fax: 816-235-1717, E-mail: ggale@ umkc.edu Leonard Gross, Department of Mathematics, Cornell University, Ithaca, NY 14853, Phone: 607-255-4110, E-mail: [email protected] Francis Henderson, 7420 Westlake Terrace-1312, Bethesda, MD 20817-6580, Phone: 301-469-7721 A. Alexandrou Himonas, Mathematics Department, Mail Distribution Center, University of Notre Dame, Notre Dame, IN 46556-5683, Phone: 219-631-7245, Fax: 219-631-6579 Leonid Hurwicz, Department of Economics, University of Minnesota, Minneapolis, MN 55455, Phone: 612-625-6353, Fax: 612-927-8317 Gerald W. Johnson, Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0001, Phone: 402-472-7229 Gopi Kallianpur, Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, Phone: 919-962-2187, Fax: 919-962-1279, E-mail: [email protected] Ron Kerman, 4 Elmbridge Dr., St. Catharines, Ont L2T 2A6, CANADA, Phone: 905-688-5793 Vladimir V. Kisil, Vakgroep Wiskundige Analyse, Universiteit Gent, Galglaan 2, B-9000, Gent, Belgium, E-mail: [email protected] John R. Klauder, Department of Mathematics, University of Florida, Gainesville, FL 32611, Phone: 904-392-8667/8747, Fax: 904-392-8357, E-mail: klauder@ neptune.phys.ufl.edu Lawrence Klein, Department of Economics, University of Pennsylvania, Philadel• phia, PA 19104, Phone: 215-898-7001, Fax: 215-898-4477 NORBERT WIENER CENTENARY CONGRESS 561 Christopher J. Lennard, Department of Mathematics, University of Pittsburgh, PA 15260, Phone: 412-624-8345 Raoul D. LePage, Department of Statistics & Probability, A428 Wells Hall, Michi gan State University, East Lansing, MI 48824, Phone: 517-353-3984, Fax: 517-432-1405, E-mail: [email protected] Shlomo Levental, Department of Statistics & Probability, Michigan State Univer• sity, East Lansing, MI 48824 Paul Malliavin, Department of Mathematics, Pierre et Marie Curie, 4 Place Jussieu, 75230 Pris Cedex, France, E-mail: [email protected] V. Mandrekar, Department of Statistics & Probability, A413 Wells Hall, Michigan State University, East Lansing, MI 48824, Phone: 517-353-7172, Fax: 517-432- 1405, E-mail: [email protected] Robert W. Mann, Newmann Laboratory for Biomechanics & Human Rehabilitation, Massachusetts Institute of Tech., Cambridge, MA 02139, Phone: 617-253- 2220, E-mail: [email protected] Anders Martin-Lof, Department of Mathematics, Stockholm University, S-106 91 Stockholm, SWEDEN, Phone: Oil 46 8 162000, Fax: Oil 46 8 626717, E-mail: [email protected] Pesi R. Masani, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, Phone: 412-624-8378, Fax: 412-624-8397, E-mail: prmas@ vms.cis.pitt.edu Brockway McMillan, P. O. Box 27, Sedwick, ME 04676-0027, Phone: 207-359-8930 Sanjoy Mitter, Laboratory for Information & Decision Systems, MIT-Room 35- 308, Cambridge, MA 02139-4307, Phone: 617-253-2160, Fax: 617-258-8553, E-mail: [email protected] Stanislav A. Molchanov, Department of Mathematics, University of North Carolina, Charlotte, NC 28223, Phone: 704-547-4561, Fax: 704-547-3218 Dennis Murphy, Concordia University-Montreal, CANADA, Phone: 514-848-2559, Fax: 514-848-4512 Boris Paneah, Technion, Department of Mathematics, Israel Institute of Technol• ogy, 32000 Haifa, ISRAEL, Fax: 011 972 4 324654 Oliver Penrose, Herriot-Watt University, Edinburgh, UNITED KINGDOM, Phone: 011 44 31 449 5111, Ext. 3225, Fax: 011 44 31 451 3249, E-mail: oliver@ cara.ma.hw.ac.uk D. H. Phong, Department of Mathematics, Columbia University, New York, NY 10027, Phone: 212-854-4308, Fax: 212-854-8962, E-mail: dp@ math.columbia.edu John W. Poduska, Sr., 155 Somerset St., Belmont, MA 02178, Phone: 617-484-7763 562 PARTICIPANTS LIST Karl H. Pribram, Center for Brain Research and Information Sciences, Radford University-Box 6977, Radford, VA 24142, Phone: 540-831-6107, Fax: 540-831-6113, E-mail: [email protected] Jean Ramaekers, Department of Computer Science, University of Namur, B-5000 Namur, BELGIUM, Phone: Oil 32 81 72 4992, Fax: Oil 32 81 72 4967, E-mail: [email protected], President: International Association for Cyber• netics, Palais des Expositions, B-5000 Namur, BELGIUM, Phone: Oil 32 81 73 52 09 Philip Richard, 5963 York Way, East Lansing, MI 48823, Phone: 517-337-9628 Jorma Rissanen, IBM Research ARC-K52-802, 650 Harry Road, San Jose, CA 95120-6099, Phone: 408-927-1813, IBM San Jose Station: 408-927-1080, Fax: 408-927-2100, E-mail: [email protected] Dennis Roseman, Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, Phone: 319-335-0779, Fax: 319-335-0627, E-mail: roseman@ dimension4.math.uiowa.edu Habib Salehi, Chairperson, Department of Statistics & Probability, A413 Wells Hall, Michigan State University, East Lansing, MI 48824, Phone: 517-355- 9589, Fax: 517-432-1405, E-mail: [email protected] Joerg Schaefer, Fakultaet fuer Mathematik, Rhur-Universitaet Bochum, NA 3/32, 44780 Bochum, GERMANY, Phone: 011 49 234 700 5677, Fax: 011 49 234 709 4242, E-mail: [email protected] Irving E. Segal, Department of Mathematics, MIT, Cambridge, MA 02139-4307, Phone: 617-253-4985, Fax: 617-253-4358, E-mail: [email protected] David Skoug, 399 A Canyon Dr., Pocatello, ID 83204, Phone: 208-232-8734 Henry Stapp, Lawrence Berkeley Laboratory, University of California, 1 Cyclotron Road, Berkeley, CA 94720, Phone: 510-486-4488, Fax: 510-486-6808, E-mail: [email protected] Kenneth R. Stunkel, Dean, School of Arts and Sciences, Monmouth College, West Long Branch, New Jersey 07764, Phone: 908-571-3419, E-mail: kstunkel@ mondec.monmouth.edu Erik G. F. Thomas, University of Groningen, Department of Mathematics, P. O. Box 800, 9700 AV Groningen, THE NETHERLANDS, Phone: 011 31 50 633987, E-mail: [email protected] Lev N. Volkov, 664033 Irkutsk, 120 Lermontov Street, Siberian Institute of Ener• getics, RUSSIA, E-mail: [email protected] Quoc Phong Vu, Department of Mathematics, Ohio University, Athens, OH 45701, Phone: 614-593-1278, Fax: 614-593-0406, E-mail: [email protected] Robert Warnock, Stanford Linear Accelerator Center, Stanford University, P. O. Box 4349, MS 26, Stanford, CA 94309, Phone: 415-926-2870, Fax: 415-926- 4999, E-mail: [email protected] NORBERT WIENER CENTENARY CONGRESS 563 Shinzo Watanabe, Department of Mathematics, University of Kyoto, Kyoto, JAPAN, Fax: Oil 75 753 3711, E-mail: [email protected] Georg Zimmermann, Department of Mathematics, University of Maryland, College Park, College Park, MD 20742, Phone: 301-405-5098, E-mail: georg® math.umd.edu Acknowledgements The American Mathematical Society gratefully acknowledges the kindness of these institutions and individuals in granting the following permissions. Birkhauser Verlag AG Academic Vita of Norbert Wiener, Norbert Wiener: 1894-1964, Vita Mathe- matica Series 5, Birkhauser Basel, 1990. Doctoral Students of Norbert Wiener, Norbert Wiener: 1894-1964, Vita Mathematica Series 5, Birkhauser Basel, 1990. Defense Documents of Norbert Wiener, Norbert Wiener: 1894-1964, Vita Mathematica Series 5, Birkhauser Basel, 1990. Donna M. Coveney/MIT [Figure 10, p. 436] Tadoma method of speech communication Robert Crawford [Photograph, p. 523] President Jean Ramaekers of the International Associa• tion of Cybernetics presenting the citation to Mrs. Elizabeth Shannon. David J. Edell [Figure 3, p. 429] Neuroelectrode inserted into a transected peripheral nerve in a rabbit, in A peripheral nerve information transducer for amputees: Long-term multichannel recordings from rabbit peripheral nerves, by D. J. Edell,IEEE Trans, on Biomedical Eng. 33(2), 1986, pp. 203-213. [Figure 13, p. 439] Schematic of retina-stimulating system [Figure 7, p. 433] Concept for free floating cortical-signal system Institute of Electrical and Electronic Engineers, Inc. [Figure 3, p. 429] Neuroelectrode inserted into a transected peripheral nerve in a rabbit, in A peripheral nerve information transducer for amputees: Long-term multichannel recordings from rabbit peripheral nerves, by D. J. Edell, IEEE Trans, on Biomedical Eng. 33(2), 1986, pp. 203-213. [Figure 4, p. 430] Ten-electrode active cortical microprobe, Solid-state microsensors for cortical nerve recordings, Najafi, K., IEEE Eng. in Med. and Biol, June 1994, pp. 375-387. 565 566 ACKNOWLEDGEMENTS [Figure 5, p. 431] Single implantable microstimulator, Solid-state microsensors for cortical nerve recordings, Najafi, K., IEEE Eng. in Med. and Biol, June 1994, pp. 375-387. [Figure 6, p. 432] Spinal-cord-injured subject walking with hybrid brace-FES system, in Design of a controlled-brake orthosis for FES-aided gait, M. Goldfarb and W. Durfee, IEEE Tran. Rehab. Eng. 4(1), 1996. [Figure 11, p. 437] Schematic of visual prosthesis system, Visual neuro- prosthetics—Functional vision for the blind, R. A. Normann, IEEE Eng. in Med. and Bio., Jan./Feb. 1995, pp. 77-83. [Figure 12, p. 438] 3D, 100 electrode silicon array, Visual neuroprosthe- tics—Functional vision for the blind, R. A. Normann, I IEEE Eng. in Med. and Bio., Jan./Feb. 1995, pp. 77-83. Liberty Mutual Insurance Company [Figure 1, p. 427] Unilateral above-elbow amputee manipulating boxes wearing a Liberty Boston Elbow Marina von Neumann Whitmann Letter from von Neumann to Wiener, Norbert Wiener 1894-1964, P. R. Masani, ed., Vita Mathematica Series 5, Birkhauser Basel, 1990. MIT Press Bibliography of Norbert Wiener, N. Wiener, Collected Works (P. Masani, ed.), I, II, III, IV, MIT Press, Cambridge, MA 1976, 1979, 1981, 1985. National Academy Press [Figure 9, p. 435] Wearable tactile display for voice fundamental frequency, Speech processing for physical and s ensory disabilities, Levitt, H., Voice communication between humans and machines, Roe, D. B. and Wilpon, J. G., Editors, National Academy Press, Washington, D. C. 1994. Taylor & Francis [Figure 2, p. 428] Schematic of surgically implanted neuroelectric signal detection and transmission system, emphControl of upper-limb prostheses: A case for neuroelectric control, J. of Med. Eng. and Tech., 2 1978, pp. 57-61. Time Inc. [Figure 8, p. 434] Dr. Norbert Wiener and the Felix tactile speech display, by Alfred Eisenstaedt, Life Magazine. Selected Titles in This Series (Continued from the front of this publication) 23 R. V. Hogg, editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980) 22 G. H. Golub and J. Oliger, editors, Numerical analysis (Atlanta, Georgia, January 1978) 21 P. D. Lax, editor, Mathematical aspects of production and distribution of energy (San Antonio, Texas, January 1976) 20 J. P. LaSalle, editor, The influence of computing on mathematical research and education (University of Montana, August 1973) 19 J. T. Schwartz, editor, Mathematical aspects of computer science (New York City, April 1966) 18 H. Grad, editor, Magneto-fluid and plasma dynamics (New York City, April 1965) 17 R. Finn, editor, Applications of nonlinear partial differential equations in mathematical physics (New York City, April 1964) 16 R. Bellman, editor, Stochastic processes in mathematical physics and engineering (New York City, April 1963) 15 N. C. Metropolis, A. H. Taub, J. Todd, and C. B. Tompkins, editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962) 14 R. Bellman, editor, Mathematical problems in the biological sciences (New York City, April 1961) 13 R. Bellman, G. Birkhoff, and C. C. Lin, editors, Hydrodynamic instability (New York City, April 1960) 12 R. Jakobson, editor, Structure of language and its mathematical aspects (New York City, April 1960) 11 G. Birkhoff and E. P. Wigner, editors, Nuclear reactor theory (New York City, April 1959) 10 R. Bellman and M. Hall, Jr., editors, Combinatorial analysis (New York University, April 1957) 9 G. Birkhoff and R. E. Langer, editors, Orbit theory (Columbia University, April 1958) 8 L. M. Graves, editor, Calculus of variations and its applications (University of Chicago, April 1956) 7 L. A. MacColl, editor, Applied probability (Polytechnic Institute of Brooklyn, April 1955) 6 J. H. Curtiss, editor, Numerical analysis (Santa Monica City College, August 1953) 5 A. E. Heins, editor, Wave motion and vibration theory (Carnegie Institute of Technology, June 1952) 4 M. H. Martin, editor, Fluid dynamics (University of Maryland, June 1951) 3 R. V. Churchill, editor, Elasticity (University of Michigan, June 1949) 2 A. H. Taub, editor, Electromagnetic theory (Massachusetts Institute of Technology, July 1948) 1 E. Reissner, editor, Non-linear problems in mechanics of continua (Brown University, August 1947)