The Power of Numerics

September 10, 2001

Book Review
Philip J. Davis

Mathematics Unlimited---2001 and Beyond. Edited by Björn Engquist and Wilfried Schmid, Springer-Verlag, Berlin, 2001, 1236 pages, $44.95.

A few initial facts and figures: This book contains 63 articles written by 91 contributors (whose photographs and short biographies appear at the back). There are 179 figures, 95 in color. The articles are arranged in alphabetic order, according to the name of the first author. In a number of the articles, particularly those based on interviews, "pulls" have been placed in the margins to emphasize major points. In the margins of Henri Cohen's article on number theory is a list of 22 "Gems." Physically, the volume is of top-notch quality as regards layout, printing, illustrations, and binding. There is no Web version.


Fifty years ago, Milton Abramowitz, the principal architect of the famous collection of tables and formulas known as Abramowitz and Stegun, a.k.a. Handbook of Mathematical Functions, or AMS 55, used to joke that all a practicing numerical analyst needed to know was linear interpolation and Simpson's rule. In retrospect, I think he was parodying what the average mathematician knew or cared to know about a subject that was often regarded as having low prestige, a subject that one picked up "behind the garage" in the unlikely event that one needed it.

All this, together with many aspects of our daily lives, has been changed by the digital computer. Mathematics and its applications are increasingly built around numerical methods. Numerical analysis has spawned deep theories, theorems, applications, and experiences, and its applications have produced material beyond what can be formally derived by classical methods. The best algorithms are now often to be found within commercial numerical software, and we even have Web software houses today.

As a (far from adequate) measure of the distance numerical analysis has come in half a century, allow me to borrow from two of the articles (by Risto M. Nieminen, and by Björn Engquist and Gene Golub) that, quoting in turn the magazine Computing in Science and Engineering, list the "top ten" algorithms of the 20th century:

(1) Metropolis (Monte Carlo), (2) Dantzig's simplex method for linear programming, (3) Krylov subspace methods, (4) decomposition approach to matrix computations (e.g., Cholesky, Schur), (5) Fortran optimizing compiler, (6) QR algorithm for eigenvalue computation, (7) Quicksort, divide and conquer, and randomized divide and conquer, (8) FFT, (9) Ferguson and Forcade's integer relation detection algorithm, (10) fast multipole algorithm (Pincus and Scheraga).

The interested reader can draw up his/her own list and submit it to Price, Waterhouse for the centennial algo-Oscar awards. Engquist and Golub do suggest three more of their own. In any case, it's a far cry from linear interpolation and Simpson's rule.

The new millennium has given rise to a number of books and articles that describe the accomplishments of the 20th century, survey the current scene, and project forward. Some emphasize pure mathematics, e.g., Mathematics: Frontiers and Perspectives, put out by the International Mathematical Union (which I reviewed in SIAM News, July/August 2000; ). The present book stresses---but not to the exclusion of all else---the applied and computational aspects of mathematics and the use of mathematics in other sciences. The authors were instructed that they need not produce survey articles and could, if they wished, simply explore particular topics. Several authors did take the survey route.

A good half dozen of the articles require no specialist knowledge, and I found that only parts of many articles made for good general reading and should be accessible to those who have no deep understanding of the particular topics. Nonetheless, this is a book that all graduate students and professionals in industrial and applied mathematics ought to examine.

To give a whiff of what the authors think about the future of mathematics, I append a selection of quotes. Clipped from their context, they may occasionally sound banal or give a false impression, but they may also induce readers of this review to consult the individual articles.

Throughout the book there are, as one might expect, expressions of enthusiastic optimism:

"We expect the amount of mathematics produced in the 21st century to dwarf that of all the centuries that came before."---John C. Baez and James Dolan.

"The 'Blue Gene' will deliver petaflop operations (1015 /sec). . . . Computer simulations will be used as a guide in many cases where intuition fails. . . . The payoff of the new computational mechanics could be enormous: new materials, new drugs, chemicals, predictive surgical procedures reviewed for each subject on the basis of detailed computational models, reliable predictions of weather, environmental flows, galactic phenomena, new sub-atomic devices for thousands of applications."---Ivo Babuška and J. Tinsley Oden.

"As visual effect artists get bolder and attempt even more realistic and fantastic effects, mathematics will be behind every image generated."---Doug Roble and Tony Chan.

A few expressions of caution are voiced, tamping down total hubris:

"At present, nobody can guarantee that we will even be able to develop a consistent set of mathematical laws describing nervous systems."---Erik De Schutter.

The need for much more mathematical cross-fertilization is a concern of many contributors:

"The past twenty years has seen a continuing diminution in the amount of mathematics learned by physical scientists and engineers and in the amount of physical science learned by mathematicians."---Stuart S. Antman.

"I feel it is important for academic mathematicians to be exposed to mathematicians from industry."---David Eisenbud.

This concern feeds into discussions of mathematical education (at the professional level). Its current weaknesses are underlined:

"Since pure mathematicians, in general, seem to ignore or sometimes fight against the possibilities offered by computers, it is appropriate to ask whether they will be able to keep their monopoly on teaching mathematics."---Hans Petter Langtangen and Aslak Tvieto.

"The greatest challenge is the education of new generations of computational scientists who have a thorough understanding of mathematics, computer science and applications."---Björn Engquist and Gene Golub.

Practically every article contains challenges for the future, some in very specific problems and others along more general lines:

"The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact."---Massimo Bertolini and Henri Darmon.

"Interesting new challenges [in insurance mathematics] are mainly due to the coincidence of two factors: the increase in catastrophic claims in the 1970s and 1980s and the development of the financial market."---Claudia Klüppelberg.

"The challenge [in econometrics] for the 21st century is to narrow the gap between theory and practice. Many feel that this gap has been widening with theoretic research growing more and more abstract and highly mathematical without an application in sight or a motivation for practical use."---Bad H. Baltagi.

"We are exposed to many [polls], and we regularly have the feeling that information drawn from them is definitely improper. . . . Getting to know how to estimate the error margin and how polls can be falsified is therefore essential."---Jean-Pierre Bourguignon.

Some authors foresee major paradigm shifts in the mathematics of the future:

"It will be a significant challenge to us, as mathematicians, in the immediate future to convince applied scientists and engineers to shed their perceived psychological security found in generating explicit solutions. The replacement by the companion approaches of (1) computational solutions, and (2) a (global) qualitative analysis of the behavior of a system supplies the same level of information and this latter approach is more widely applicable."---Christopher K.R.T. Jones.

"Understanding the structure and dynamics of DNA, RNA, and proteins, in general, may very well require the forging of new mathematical tools."---Kishore B. Marathe.

"Will computer mathematics systems eventually achieve such intelligence that they discover deep new mathematical results, largely or entirely without human assistance? Will new computer-based mathematical discovery techniques enable mathematicians to explore the realm, proved to exist by Gödel, Chaitin and others, that is fundamentally beyond the limits of formal reasoning?"---David H. Bailey and Jonathan M. Borwein.

Here and there, complaints are voiced:

"A mathematician's response to a very practical problem is often that it is too complicated."---Achim Bachem.

"It's completely impossible to find referees of high quality who are willing to check in detail all the papers that are being printed. The refereeing is supposed to be like a stamp that the paper is correct, but I think this is an illusion."---Lennart Carleson.

An occasional statement raised my eyebrows:

"Mathematicians, who are part of a rich tradition that has its roots in antiquity, realize that in the grand scheme of things, it does not make much difference whether their big paper appears this year or next."---Neal Koblitz.

Aware of the impatience of contributors to technical journals, I think Koblitz is expressing his vision of a nonexistent ideal mathematical world. Has "self-publishing" on the Web alleviated researchers' anxieties for establishing priorities?

"Mathematics is a possible antidote for authoritarianism."---Jean-Pierre Bourguignon.

But, as the author is aware, mathematics creates its own kind of rigidities and authoritarianisms.

"By learning how to create, store, process, and transmit quantum information faithfully, we come closer to an understanding of the laws and limitations of Nature."---Jozef Gruska.

If, as some philosophers of science believe, the world is infinite in its variety, and if, as the title of this book proclaims, mathematics is unlimited, to what are we getting closer?

There are pleas for modesty of thought:

"I do not claim that the new problems [in algebraic geometry] I identify are particularly important or meaningful or that solving them should deserve millions of dollars."---Marie-Françoise Roy.

"I now realize that what I used to see as so critical in my own scientific area is really quite unimportant."---Achim Bachem.

The remembrance of things past also makes an appearance:

"The golden age of mathematics is now. . . . It is going to be different in the future because these basic questions don't exist any more."---Lennart Carleson.

There is even a plug for SIAM:

One of the most successful activity groups founded by SIAM is in dynamical systems.---Christopher K.R.T Jones.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at [email protected].

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