- Email: [email protected]
The lure of modern science: Fractal thinking
802
BOOK REVIEWS
devotes his attention to the Lorenz model, which has little to do with geophysics where the motion is chaotic. This is a serious flaw in exposition. The proofreading of the volume was perfunctory, at best. On page 30, the presentations of three different decompositions of some equations are all the same! The English language in this volume appears not to have been edited by the publisher; on page 43, we have, “In some cases to fulfill relation (4.15) strong perturbation, which could be difficult to realize in practice is necessary.” Finally, the discussion of secure communication starting on page 47 is weak indeed. The methods he describes, and stronger ones, have been broken easily by Short (1994,1996) and others. The author does injustice to these authors by not knowing their work. In the days of the World Wide Web, one can find this information without any trouble, and one should find it. Overall, this is a weak book. One need not have it on one’s shelf. Perhaps one can thank the author for collecting the papers contained in the second half of the volume, but then one can get them elsewhere as well. REFERENCES Isidori, A. 1995. Nonlinear Control Systems. New York: Springer-Verlag. Kaplan, D. and L. Glass. 1995. UnderstandingNonlinear Dylurmics. New York: SpringerVerlag. Short, K. M. 1996. Unmasking a modulated chaotic communications scheme. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 6,367-315. Short, K. M. 1994. Steps towards unmasking secure communications. InternationalJournal of Bifurcation and Chaos in Applied Sciences and Engineering 4, 959-977. Solari, H. G., M. A. Natiello and G. B. Mindlin. 1996. Nonlinear Dynamics: A Two Way Trip from Physics to Math. Bristol: Institute of Physics Publishing.
HENRY D. I. ABARBANEL Department of Physics and Scripps Institution of Oceanography, University of California at San Diego, La Jolla, CA 92093-0402, U.S.A. (E mail: [email protected])
PII: soo92-s24cN97Noo19-o
The Lure of Modem Science: Fractal Thinking, by Bruce J. West and Bill Deering, World Scientific, Singapore, 1995. $58.00 (cloth), viii + 421 pp. The Lure ofModem Science is another book about fractals and chaos. An overwhelming number of such books already compete for our time and money. Even the authors point out that “there have probably been more
BOOK REVIEWS
803
books published on fractaZsand chaos in the past decade than on any other topic in the physical/mathematical sciences in any comparable period of time.” What makes this book different from the others is that, as West and Deering put it, “it is a montage rather than a well argued scholarly work.. . .“l Unfortunately, it is exactly this hodgepodge character which limits the book’s audience. For most people and purposes, another book will be a better choice. The book is broken into six chapters: an introduction, “Lure of Modern Science”; “Linear Spaces and Geometry in Natural Philosophy”; “Noise in Natural Philosophy”; “Self-Similarity, Fractals and Measurements”; “Maps and Dynamics”; and “Dynamics in Fractals Dimensions.” Continuing in self-similar fashion, chapters are divided into sections, sections into subsections, and subsections into paragraphs, each of which is labeled in the margin with a three- or four-word phrase. (These phrases are not very informative, but they are great for remembering where you last stopped reading.) Every chapter except the first is followed by an appendix containing the mathematical nitty-gritty. Topics covered include both the analysis of models (e.g., linear and nonlinear oscillators, Brownian motion, nonlinear l-d maps and maps of the plane, linearization, and bifurcations) and techniques for the analysis of data (e.g., correlation functions, spectral analysis of time series, l/f-spectra, inverse power laws, normal vs lognorma1 distributions, attractor reconstruction an dimension estimation). Dr. West has written about some of this material before. Much of his earlier book (West, 1985) is covered again here along with new material. So why are you looking for a book on chaos and fractals? Do you need an introductory textbook? The Lure of Modem Science does have a number of interesting applications that would be good for engaging students in the classroom (l/f spectra in art and music, patterns of bronchial branching in lung tissue, income distributions, etc.) But it is not a textbook: too much mathematical experience is assumed, no topic is covered in enough detail, and there are no exercises. I would suggest Drazin (1992) (which has many good exercises) or Ott (1993) for good graduate-level mathematical introductions. A good advanced undergraduate text is Strogatz (1994), which has the right blend of theory and applications. Do you want a more thorough and technical mathematical treatment? Then this book is probably not for you either. More advanced audiences will already be familiar with most of the techniques. (Among the technical
‘The authors claim that, “It is a montage rather than a well argued scholarly work because this modem science is still in its formative stages, and we do not want to be guilty of killing it off by unnecessarily restricting its process of growth.”
804
BOOK REVIEWS
treatments of nonlinear dynamics, I personally like Guckenheimer and Holmes, 1983.) It seems that the authors have aimed for a middle-ground audience, hoping to hit both mathematically trained scientists and neophytes. This approach inevitably produces some tradeoffs between precision and readability. For example, on page 32, we read, “An attractor derives its name from the fact that no matter how a system is started, it is attracted to this structure.. . .” Yet, on page 33, we find the apparently contradictory statement: “Another feature of certain nonlinear systems is the simultaneous existence of multiple attractors.” In some cases, this imprecision can seriously mislead untrained readers. On page 12, we are told that the “multiplicative process” Xi+ r = 2Xj is not linear, in contrast to the linear “additive process” Xi+ r = Xi + 1. Both processes are, in fact, linear; the former simply has nonlinear solutions. Perhaps you want a book on chaos and fractals for biologists. Z%eLure of Modem Science is not a bad choice. Probably half of the examples in the book are biologically motivated. But these examples are drawn overwhelmingly from anatomy and physiology, particularly mammalian lung architecture and physiological time series such as electroencephalographs and electrocardiographs. Other fields, particularly ecology, get short shrift. Most of the other examples are taken from physics, sometimes without much in the way of introduction. Try Kaplan and Glass (1995) for a good introductory text specifically aimed at a biological audience (although ecology gets less attention there too).2 Neither a textbook nor a popular treatment, neither introductory nor advanced, not particularly for biologists or physicists or mathematicians, this book will obviously have a small audience. However, if your interests fall between the traditional disciplinary cracks, and you do not find yourself in either extreme of the experience distribution-i.e., if you are like me-then in The Lure of Modem Science you would probably find a readable introduction to a few techniques with which you are not familiar or an engaging analysis of a novel application or two. What prevents me from recommending this book to you unconditionally is something else that you will find. Throughout the book, but especially in the preface and first chapter, West and Deering discuss the implications of nonlinearity for the philosophy of science. While they raise some interesting issues (for example, the implications of chaotic dynamics both for irreversibility and determinism, and the role of reductionism in science), these are dealt with more
*See the review by Pascual in the May 19% volume of the Bulletin.
BOOK REVIEWS
805
thoroughly elsewhere. 3 Here, they are at best distracting and at worst irksome. For example, the authors emphasize that the concepts of chaos and fractals have “led to a new kind of science and a new way of thinking both in and out of science.” I am not sure what this really means-and I am not sure that I care. As the Nobel laureate Steven Weinberg put it, “ . . . recently scientists . . . have claimed a deep philosophical significance for work on nonlinear dynamics, a subject that is interesting enough without the hype” (Weinberg, 1996). Amen to that. The author thanks Hal Caswell for illuminating discussions which improved this review.
REFERENCES Ayala, F. J. and Th. Dobzhansky (Eds). 1974. Studies in the Philosophy of Biology. Berkeley, CA: University of California Press. Bricmont, J. 1995a. Science of chaos or chaos in science? Physicalia Magazine 17, 159-208. Bricmont, J. 1995b. The last word from the rearguard. Physicalia Magazine 17, 219-221. Bricmont, J. 1996. Science of chaos or chaos in science? In The Flight from Science and Reason, P. R. Gross, N. Levitt and M. W. Lewis (Ed& pp. 131-175. Drazin, P. G. 1992. Nonlinear Systems. Cambridge: Cambridge University Press. Guckenheimer, J. and P. Holmes. 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag. Kaplan, D. and L. Glass. 1995. Understanding Nonlinear Dynamics. New York: SpringerVerlag. Mayr, E. 1985. How biology differs from the physical sciences. In Evolution at a Crossroads, D. J. Depew and B. H. Weber (Eds), pp. 43-64. Cambridge, MA: MIT Press. Mayr, E. 1988. The limits of reductionism. Nature 331, 475. Ott, E. 1993. Choas in Dynamical Systems. Cambridge: Cambridge University Press. Strogatz, S. 1994. Nonlinear Dynamics and Chaos. Reading, MA: Addison-Wesley. Weinberg, S. 1987. Newtonianism, reductionism and the art of congressional testimony. Nature 330,433-437. Weinberg, S. 1988. The limits of reductionism (reply). Nature 331, 475-476. Weinberg, S. 1994. Dreams of a Final Theory. New York: Vintage Books. Weinberg, S. Aug. 8, 1996. Sokal’s hoax. New York Review of Books, pp. 11-15. West, B. J. 1985. An Essay on the Importance of Being Nonlinear, Lecture Notes in Biomathematics, Vol. 62. Berlin: Springer-Verlag.
MICHAEL G. NEUBERT Biology Department, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, U.S.A. PII: SOO92-8240(97jOOO20-7
3For a debate on chaos, determinism and irreversibility, see Bricmont (1995a, b, 1996) and Prigogine and Antoniou (1995). For a discussion of reductionism in the sciences, and biology in particular, see Ayala and Dobzhansky (19741, Mayr (1985, 1988) and Weinberg (1987,1988, 1994).
BOOK REVIEWS
devotes his attention to the Lorenz model, which has little to do with geophysics where the motion is chaotic. This is a serious flaw in exposition. The proofreading of the volume was perfunctory, at best. On page 30, the presentations of three different decompositions of some equations are all the same! The English language in this volume appears not to have been edited by the publisher; on page 43, we have, “In some cases to fulfill relation (4.15) strong perturbation, which could be difficult to realize in practice is necessary.” Finally, the discussion of secure communication starting on page 47 is weak indeed. The methods he describes, and stronger ones, have been broken easily by Short (1994,1996) and others. The author does injustice to these authors by not knowing their work. In the days of the World Wide Web, one can find this information without any trouble, and one should find it. Overall, this is a weak book. One need not have it on one’s shelf. Perhaps one can thank the author for collecting the papers contained in the second half of the volume, but then one can get them elsewhere as well. REFERENCES Isidori, A. 1995. Nonlinear Control Systems. New York: Springer-Verlag. Kaplan, D. and L. Glass. 1995. UnderstandingNonlinear Dylurmics. New York: SpringerVerlag. Short, K. M. 1996. Unmasking a modulated chaotic communications scheme. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 6,367-315. Short, K. M. 1994. Steps towards unmasking secure communications. InternationalJournal of Bifurcation and Chaos in Applied Sciences and Engineering 4, 959-977. Solari, H. G., M. A. Natiello and G. B. Mindlin. 1996. Nonlinear Dynamics: A Two Way Trip from Physics to Math. Bristol: Institute of Physics Publishing.
HENRY D. I. ABARBANEL Department of Physics and Scripps Institution of Oceanography, University of California at San Diego, La Jolla, CA 92093-0402, U.S.A. (E mail: [email protected])
PII: soo92-s24cN97Noo19-o
The Lure of Modem Science: Fractal Thinking, by Bruce J. West and Bill Deering, World Scientific, Singapore, 1995. $58.00 (cloth), viii + 421 pp. The Lure ofModem Science is another book about fractals and chaos. An overwhelming number of such books already compete for our time and money. Even the authors point out that “there have probably been more
BOOK REVIEWS
803
books published on fractaZsand chaos in the past decade than on any other topic in the physical/mathematical sciences in any comparable period of time.” What makes this book different from the others is that, as West and Deering put it, “it is a montage rather than a well argued scholarly work.. . .“l Unfortunately, it is exactly this hodgepodge character which limits the book’s audience. For most people and purposes, another book will be a better choice. The book is broken into six chapters: an introduction, “Lure of Modern Science”; “Linear Spaces and Geometry in Natural Philosophy”; “Noise in Natural Philosophy”; “Self-Similarity, Fractals and Measurements”; “Maps and Dynamics”; and “Dynamics in Fractals Dimensions.” Continuing in self-similar fashion, chapters are divided into sections, sections into subsections, and subsections into paragraphs, each of which is labeled in the margin with a three- or four-word phrase. (These phrases are not very informative, but they are great for remembering where you last stopped reading.) Every chapter except the first is followed by an appendix containing the mathematical nitty-gritty. Topics covered include both the analysis of models (e.g., linear and nonlinear oscillators, Brownian motion, nonlinear l-d maps and maps of the plane, linearization, and bifurcations) and techniques for the analysis of data (e.g., correlation functions, spectral analysis of time series, l/f-spectra, inverse power laws, normal vs lognorma1 distributions, attractor reconstruction an dimension estimation). Dr. West has written about some of this material before. Much of his earlier book (West, 1985) is covered again here along with new material. So why are you looking for a book on chaos and fractals? Do you need an introductory textbook? The Lure of Modem Science does have a number of interesting applications that would be good for engaging students in the classroom (l/f spectra in art and music, patterns of bronchial branching in lung tissue, income distributions, etc.) But it is not a textbook: too much mathematical experience is assumed, no topic is covered in enough detail, and there are no exercises. I would suggest Drazin (1992) (which has many good exercises) or Ott (1993) for good graduate-level mathematical introductions. A good advanced undergraduate text is Strogatz (1994), which has the right blend of theory and applications. Do you want a more thorough and technical mathematical treatment? Then this book is probably not for you either. More advanced audiences will already be familiar with most of the techniques. (Among the technical
‘The authors claim that, “It is a montage rather than a well argued scholarly work because this modem science is still in its formative stages, and we do not want to be guilty of killing it off by unnecessarily restricting its process of growth.”
804
BOOK REVIEWS
treatments of nonlinear dynamics, I personally like Guckenheimer and Holmes, 1983.) It seems that the authors have aimed for a middle-ground audience, hoping to hit both mathematically trained scientists and neophytes. This approach inevitably produces some tradeoffs between precision and readability. For example, on page 32, we read, “An attractor derives its name from the fact that no matter how a system is started, it is attracted to this structure.. . .” Yet, on page 33, we find the apparently contradictory statement: “Another feature of certain nonlinear systems is the simultaneous existence of multiple attractors.” In some cases, this imprecision can seriously mislead untrained readers. On page 12, we are told that the “multiplicative process” Xi+ r = 2Xj is not linear, in contrast to the linear “additive process” Xi+ r = Xi + 1. Both processes are, in fact, linear; the former simply has nonlinear solutions. Perhaps you want a book on chaos and fractals for biologists. Z%eLure of Modem Science is not a bad choice. Probably half of the examples in the book are biologically motivated. But these examples are drawn overwhelmingly from anatomy and physiology, particularly mammalian lung architecture and physiological time series such as electroencephalographs and electrocardiographs. Other fields, particularly ecology, get short shrift. Most of the other examples are taken from physics, sometimes without much in the way of introduction. Try Kaplan and Glass (1995) for a good introductory text specifically aimed at a biological audience (although ecology gets less attention there too).2 Neither a textbook nor a popular treatment, neither introductory nor advanced, not particularly for biologists or physicists or mathematicians, this book will obviously have a small audience. However, if your interests fall between the traditional disciplinary cracks, and you do not find yourself in either extreme of the experience distribution-i.e., if you are like me-then in The Lure of Modem Science you would probably find a readable introduction to a few techniques with which you are not familiar or an engaging analysis of a novel application or two. What prevents me from recommending this book to you unconditionally is something else that you will find. Throughout the book, but especially in the preface and first chapter, West and Deering discuss the implications of nonlinearity for the philosophy of science. While they raise some interesting issues (for example, the implications of chaotic dynamics both for irreversibility and determinism, and the role of reductionism in science), these are dealt with more
*See the review by Pascual in the May 19% volume of the Bulletin.
BOOK REVIEWS
805
thoroughly elsewhere. 3 Here, they are at best distracting and at worst irksome. For example, the authors emphasize that the concepts of chaos and fractals have “led to a new kind of science and a new way of thinking both in and out of science.” I am not sure what this really means-and I am not sure that I care. As the Nobel laureate Steven Weinberg put it, “ . . . recently scientists . . . have claimed a deep philosophical significance for work on nonlinear dynamics, a subject that is interesting enough without the hype” (Weinberg, 1996). Amen to that. The author thanks Hal Caswell for illuminating discussions which improved this review.
REFERENCES Ayala, F. J. and Th. Dobzhansky (Eds). 1974. Studies in the Philosophy of Biology. Berkeley, CA: University of California Press. Bricmont, J. 1995a. Science of chaos or chaos in science? Physicalia Magazine 17, 159-208. Bricmont, J. 1995b. The last word from the rearguard. Physicalia Magazine 17, 219-221. Bricmont, J. 1996. Science of chaos or chaos in science? In The Flight from Science and Reason, P. R. Gross, N. Levitt and M. W. Lewis (Ed& pp. 131-175. Drazin, P. G. 1992. Nonlinear Systems. Cambridge: Cambridge University Press. Guckenheimer, J. and P. Holmes. 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag. Kaplan, D. and L. Glass. 1995. Understanding Nonlinear Dynamics. New York: SpringerVerlag. Mayr, E. 1985. How biology differs from the physical sciences. In Evolution at a Crossroads, D. J. Depew and B. H. Weber (Eds), pp. 43-64. Cambridge, MA: MIT Press. Mayr, E. 1988. The limits of reductionism. Nature 331, 475. Ott, E. 1993. Choas in Dynamical Systems. Cambridge: Cambridge University Press. Strogatz, S. 1994. Nonlinear Dynamics and Chaos. Reading, MA: Addison-Wesley. Weinberg, S. 1987. Newtonianism, reductionism and the art of congressional testimony. Nature 330,433-437. Weinberg, S. 1988. The limits of reductionism (reply). Nature 331, 475-476. Weinberg, S. 1994. Dreams of a Final Theory. New York: Vintage Books. Weinberg, S. Aug. 8, 1996. Sokal’s hoax. New York Review of Books, pp. 11-15. West, B. J. 1985. An Essay on the Importance of Being Nonlinear, Lecture Notes in Biomathematics, Vol. 62. Berlin: Springer-Verlag.
MICHAEL G. NEUBERT Biology Department, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, U.S.A. PII: SOO92-8240(97jOOO20-7
3For a debate on chaos, determinism and irreversibility, see Bricmont (1995a, b, 1996) and Prigogine and Antoniou (1995). For a discussion of reductionism in the sciences, and biology in particular, see Ayala and Dobzhansky (19741, Mayr (1985, 1988) and Weinberg (1987,1988, 1994).