In the last sixty years, computers have increased the speed at which we can do mathematical computations by a factor of roughly a billion. Further increases are sure to come. How should the presence of computers affect the conduct of research into pure mathematics?

The extreme traditionalist answer is "not at all." In support of this position, there continue to be many spectacular non-computer-based advances. For example, we've seen proofs of the Poincaré and Serre conjectures in just the past few years.

The extreme revolutionary answer is that computers will change the face of mathematics. Computers assist greatly at finding and supporting conjectures, but not so much at proving them. Moreover, proofs are growing longer and more esoteric and their value is consequently decreasing. The future will see strongly supported conjectures take over the role of theorems.

Bailey and Borwein addressed this question in [BB] and then, joined by Girgensohn, in [BBG]. Several years later, the fellowship has expanded to now include Calkin, Luke, and Moll. The authors return to the question yet again in the book under review, [BBCGLM].

So where on the traditionalist-revolutionary spectrum are our authors? As I indicated in my MAA online review of [BB] and [BBG], they are in the reasonable middle. I wrote then that "One way [BB] and [BBG] may prove to be influential is that in many ways they have a traditionalist tone. Experimental techniques are just one more ingredient in the old mix, sometimes just enough extra to remove roadblocks to progress." The moderation continues in [BBCGLM].

To the authors, the traditionalist extreme amounts to a dysfunctional rejection of a very powerful research tool. The revolutionary extreme amounts to an unnecessary and unappealing retreat from traditional values. As [BBCGLM] says on page 25, it is instead a question of "rebalancing...our community's valuing of deductive proof over inductive knowledge."

The three books [BB], [BBG], and [BBCGLM] all indicate the authors' best guesses of what the new equilibrium will look like. When a key parameter is changed by a factor of a billion, one should expect some response in a system. Certainly, as sketched by all three books, there will be a substantial response. However it is a testament to the vitality of traditional mathematics that the new equilibrium will be an enrichment of traditional mathematics, not at all its overthrow.

**The organization of the book.** [BBCGLM] began as a two-day course given by the six authors at the joint AMS-MAA meetings in San Antonio in January 2006. The eight lectures there became the eight main chapters of the book, each consisting of about twenty to thirty pages. The final chapter consists of exercises, roughly five to ten pages worth for each of the previous eight chapters, and then twenty pages of additional exercises.

The books [BB] and [BBG] were criticized somewhat by reviewers for their scrapbook feel. The organization of [BBCGLM] is somewhat tighter. Chapters 1 and 8, by Borwein, provide a general frame for the course. Chapters 2 and 3, by Bailey, describe many of the algorithms which underlie modern computer-based mathematics. Chapters 4, 5, 6, and 7, by Luke, Girgensohn, Calkin, and Moll respectively, pursue more specialized topics. Chapters 1, 2, 3, 5, and 8 have considerable overlap with [BB] and [BBG].

The authors of the various chapters are not indicated in the book itself, but are revealed on the Bailey-Borwein Experimental Mathematics website. I have highlighted the principal authors below, since I feel knowing authorship aids in understanding the material.

**The Borwein chapters: Secure mathematical knowledge.** Chapter 1 is a "Philosophical introduction." It begins by acknowledging a general distaste for philosophy among scientists, but says "we are of the opinion that mathematical philosophy matters more now than it has in nearly a century." The moderation, as defined above, is captured in the sentence "while I appreciate fine proofs and aim to produce them when possible, I no longer view proof as the royal road to secure mathematical knowledge."

An example of unproved but secure mathematical knowledge is provided by a relation between multizeta functions, 8^{N} ζ({-2,1}_{N}) = ζ({1,2}_{N}), with definitions given in the text. A small part of the security comes from the fact that the relation is proved for N=1. Most of the security comes from the fact that the relation has been numerically checked to 1000 decimal places for many N. As a rule, we should view relations such as this unproven zeta identity as likely deeper than similar relations which we can prove. Should we keep relations like this one hidden away on hard disks until the unknown and perhaps never to come day that they are proved? Of course not. The authors ask of the community that experimentally supported statements like this zeta identity be given considerable respect. In return, it is always acknowledged that these statements are not proved. Proofs are to be highly valued too, especially when they add insight.

Chapter 8 is a "Computational Conclusion." One of many topics is the answer to Problem 9 of the "Experimental Mathematics Problem Set" of [BB]. We learn that the experimentally discovered identity Σ_{i>j>k>l>0} 1/(i^{3} j k^{3} l) = 2 π^{8}/10! is in fact correct.

**The Bailey chapters: Algorithms as poetry.** Chapters 2 and 3 are about the infrastructure of experimental mathematics. Much of this infrastructure is coded into programs like Mathematica and accordingly perhaps taken too much for granted by casual users.

Chapter 2 focuses on a Bailey-Borwein speciality, recognizing symbolic forms of numbers from high precision decimal forms. This highly successful procedure does not have an antecedent in traditional mathematics.

Chapter 3, in contrast, could well be titled "Classical Mathematics Revisited." Section titles include "Prime Number Computations," "Roots of Polynomials," "Numerical Quadrature," and "Infinite Series Summation." In each case, the feel is very practical, along the lines of "we really need the answer." It is not enough to know that an arbitrary positive integer factors uniquely into primes; we need the best algorithms to actually do it. It is not enough to know that a degree n polynomial has n complex roots; we need to actually find the roots. It is not enough to do Simpson approximation to get definite integrals to four decimal places; we might need ten thousand decimal places. It is not enough to naively approximate infinite sums; we need to be likewise more clever to get better accuracy.

In traditional mathematics, one proves theorems by appealing to earlier theorems. In experimental mathematics, outputs of one algorithm are often inputs of the next. Chapters 2 and 3 make the case that the networking nature of experimental mathematics is on a par with that of traditional mathematics.

**The remaining chapters: a broader spectrum.** [BB] and [BBG] both suffered from a certain lack of balance. On the one hand, the main subject, describing a "paradigm shift" within mathematics, was extremely broad in scope. On the other hand, the supporting examples were heavily skewed towards the personal experience of the authors. [BBCGLM] has a better balance because of the extra authors with new areas of expertise.

Luke's Chapter 4 is on inverse scattering. The chapter serves as a reminder that experimental techniques which are new to pure mathematicians have a lot in common with traditional applied mathematics. Girgensohn's Chapter 5 takes it as well-known that computers are great when it comes to working with smooth functions; it examines the difficulties which arise in the study of say continuous but nowhere differentiable functions. Calkin's Chapter 6 is about the experimental analysis of a factoring algorithm. Here again, the development reminds us that experimental techniques have a lot in common with traditional computer science. Moll's Chapter 7 is about how the computer can be used to provide clues towards proving general formulas for definite integrals.

Different readers are going to find different parts of Chapters 4-7 most appealing. My personal favorite is Section 5.3, an expanded version of Section 2.4.3 of [BBG]. It concerns the probability measure μ_{q} giving the distribution of "random geometric series" x= ± 1 ± q ± q^{2} ± q^{3} ± ..., for q a fixed number in (1/2,1). For most but not all such q, the measure is absolutely continuous and so can be written in terms of a density function, μ_{q} = f_{q}(x)dx. The humble experimental problem is to graph the "strange" functions f_{q}(x), and in particular to understand more intuitively the rather wild dependence on q. The section appeals to me because of the elementary context, the elegant conceptual framework from the 1930s involving Fourier analysis and Pisot numbers, the fundamental issues still unresolved, and the insight brought by the experimental approach.

**Conclusion.** [BBCGLM] includes two quotes also in [BB], from Thomas Kuhn and Max Planck respectively:

The transfer of allegiance from paradigm to paradigm is a conversion experience that cannot be forced.

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents die and a new generation grows up that is familiar with it.

The new book [BBCGLM] differs from its predecessor [BB] by following up these quotes with "This transition is certainly already apparent." I'm not sure who has died, but there certainly is a new generation rising using computers adroitly to expand the boundaries of pure mathematics.

**References:**

[BB] Mathematics by Experiment: Plausible Reasoning in the 21st century, by Jonathan Borwein and David Bailey. A K Peters, 2003.

[BBG] Experimentation in Mathematics: Computational Paths to Discovery*,* by Jonathan Borwein, David Bailey, and Roland Girgensohn. A K Peters, 2004.

[BBCGLM] *Experimental Mathematics in Action* , by Jonathan Borwein, David Bailey, Neil J. Calkin, Roland Girgensohn, D. Russell Luke, and Victor H. Moll. A K Peters, 2007.

David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.