At the start of each semester I ask my students to write a "math autobiography," so I can get some idea of who I am teaching — their mathematical skills, interests, hopes and fears. While there have always been a fair number of physics majors in calculus, linear algebra and even some upper division courses, in the last ten years I have observed a sharp rise in those majoring in biology and economics. Indeed, these are often my best students, and some double-major in mathematics as well. I believe these numbers are a reflection of a trend in contemporary social and natural sciences-the ubiquity and increasing level of sophistication of mathematical modeling. This phenomenon has spawned many books, workshops, meetings, special editions of journals, and new journals to accommodate the burgeoning interest in mathematical applications in ____________ (fill in the blank).
In late June of the final year that the words "this century" could refer to the 20th century, a conference was held in Italy to bring together an interdisciplinary group of physicists, biologists, mathematicians, and historians of science, to "share information on their work and reflect on the way mathematics and mathematical models are used in the natural sciences today and in the past" (p. vii). The volume under review, which takes its title from the meeting, is the proceedings, and the papers included also discuss mathematical models in the social sciences — in particular, economics.
I will get my gripes out of the way first. There are numerous typographical and spelling errors, and the book would have benefited enormously from a good proofreader — there may be similar kinds of errors in the mathematics, but I did not catch any in a superficial scan. The index is quite minimal (one and a half pages) and really not very useful. An English language editor might have helped smooth out the many instances of awkward English usage — most of the authors are Italian and I believe the American Ted Porter is the only one whose first language is English — but none render the text unintelligible.
The book is divided into three sections: Physics, Biology, and History of Science. The physics section includes chapters with titles like "An Optical Geometric Model of the Betatronic Motion," "Pattern Induced by Parameter Modulation in Spatiotemporal Chaos," and "Mathematical Models in Beam Dynamics." Embodying the interdisciplinary spirit of the conference, I was intrigued by Vieri Benci's paper, "Some Remarks on the Theory of Relativity and the Naïve Realism," which has a decidedly philosophical cast. He starts with two "basic assumptions of modern physics" and from them derives "a model of a relativistic universe which can be very easily interpreted in the spirit of the physics of the nineteenth century" (p. 40). All this to prove his claim that "the most radical consequences [of Relativity and Quantum Mechanics] are not implied by the theory and the experiments, but rather they reflect the cultural environment where they were born."
Reading Accardi and M. Regoli's paper "Quantum Probability and the Interpretation of Quantum Mechanics: A Crucial Experiment," I got the distinct impression that it was but one more salvo in an ongoing skirmish between the authors (whose website documents this and other examples of the "sociology of academic repression" (p. 3)) and editor of the Italian language edition of Scientific American ("Le Scienze"), who apparently has repeatedly refused to publish the articles they submit. Although I am not qualified to judge the merit (scientific or mathematical) of their work, it does seem that these authors (and perhaps a few of the others in this volume) are scholars positioned on the margins of the mainstream scientific and mathematical communities. This does not necessarily detract from the interest and value in reading these papers — indeed their relative obscurity (especially to American readers), and in some cases nonstandard approaches and conclusions, might be value added. Readers should not expect to find the traditional exemplars of mathematical physics here.
The first two papers in the biology section (which include among the authors two of the editors of the volume) "continue the enterprise of elaborating an axiomatic framework suitable for modern Biology" (p. 139), a program originated by E. De Giorgi in Pisa in the 1980s. This goal of this project is "not to provide safe and unquestionable grounds to scientific activity, but rather to develop conceptual environments where this activity can be carried out" (p. 139, italics in original) — that is, it is less ambitious than Hilbert's program and more along the lines of the Bourbakist attempt to unify knowledge. The authors invite criticisms and contributions from scholars in other fields, and seem eager for feedback from mathematicians in particular. Another chapter in this section discusses fractal complexity of membrane structures, and there is a paper on population dynamics that applies mathematical models used in economics to "supply-side ecology."
The history section was of greatest interest to me, and the one I read most carefully. I was familiar with three of the authors in this section, all historians of mathematics: Giorgio Israel, Jean Dhombres, and Theodore Porter. Israel's paper is an insightful evaluation of the present state of mathematical modeling, and along the way he presents a concise history of its development into "this century" (i.e., the 20th). In support of his thesis that "the use of mathematical models to describe real phenomena is a recent scientific practice" (p. 233), he contrasts Isaac Newton's image of science with that of John von Neumann. Newton's science — "natural philosophy" — was guided by a search for truth, its goal to discover and explain the causes of phenomena, in order to find the First Cause. Following Galileo's dictum that the book of nature was written in the language of mathematics, for Newton and his contemporaries mathematical models contained the very essence of phenomena. Von Neumann, on the other hand, viewed models as descriptive tools, images to be judged according to how useful or effective they were (for making predictions, allocating resources, preventing the spread of disease, etc.). And this more recent view explains why "contemporary science appears to be increasingly addicted to making models, mathematical models" (p. 234) (which perhaps verifies my conjecture about the reason for the growing interdisciplinary audience for math classes — there is, after all, a time lag for students!).
Jean Dhombres' discussion of the role of functions around 1750 is constructed on a detailed treatment of the Buffon-Clairaut controversy which traces the development of the concept of the function through both mathematics and physics and connects with Israel's paper by claiming that "the establishment of the concept... could be looked at as the mathematical answer, that is a quantitative answer, to the very old metaphysical question about causes and effects" (p. 210). Unfortunately, the passages used as examples and evidence are all in French. Readers who are not able to translate them will completely miss the key points in his arguments.
I imagine one might have seen some sparks fly in a discussion between Israel and Dhombres at this conference, and I wish I could have attended a Q&A session with both of them present. Dhombres takes a swipe at the human and social sciences where "a very superficial exhibition of mathematics has more than once been used as a proof for the validity of other kinds of arguments," and recommends the "intellectual and pleasant reaction by J. Bricmont and A. Sokal" (co-authors of Les impostures intellectuelles, a scathing critique of cultural science studies) (p. 209). Israel, on the other hand, views mathematics and science as inextricably embedded in a socio-cultural context. He believes that "science is governed by metaphysical hypotheses in its conceptual choices." After demonstrating that one of the characteristics of classical science is the assumption that "linearization is the mathematical process used to penetrate the heart of a physical process and to reveal the simple inner structures by which it is governed," he makes the observation that "it [linearization] was accepted not because it had proved to correspond to the facts but because it expressed faith in the idea that nature is what is simplest to imagine from the mathematical point of view" (p. 237).
At a more recent conference that I did attend, the annual meeting of the History of Economics Society conference (Duke University, July, 2003), Ted Porter was the keynote speaker. He spoke on themes similar to those in his paper in this volume, his continuing work on the "culture of quantification." In his 1995 book Trust in Numbers: The Pursuit of Objectivity in Science and Public Life, he explores the connections between the discovery/invention of successful methods of quantification and social and technological power. In his essay in this volume, "Models, Analogies, and Statistical Reason, 1760-1900), Porter shows how "statistics was not merely a set of tools for analyzing data; it provided the rudiments of a theory of society. It came to function also as a model in the sense that its basic conceptual structure could be transported into other fields..." (p. 279). The rest of the papers in this section are also well worth reading, and include such tempting topics as the relevance of music for the history of science and the role of game theory the mathematization of the social sciences.
Hopefully I have convinced you that this volume serves up a rich fare — different readers will prefer different courses, but it's a varied enough buffet that I think professional mathematicians, scientists, philosophers and historians with an interest in application of mathematics to the natural and social sciences will all find something to their taste.
Bonnie Shulman (email@example.com) is associate professor of mathematics at Bates College in Lewiston, Maine. She was trained as a mathematical physicist, and has also become interested in applications of mathematics to biology and economics — due primarily to working with students in these areas. Her current research interests are in the history and philosophy of mathematics, and she enjoys integrating these topics into standard mathematics classes, from Calculus to Real Analysis.