Dual review of
Mathematics by Experiment, by Jonathan Borwein and David Bailey
Experimentation in Mathematics, by Jonathan Borwein, David Bailey, and Roland Girgensohn.
How large a role will computer computations play in the mathematics of tomorrow? The books under review are about many things, but it is clear that the authors are focused on this question. Their answer: very large. Their attitude: we should embrace this change.
I agree on both points. Do you? In this review, I'll try to convince you. Naturally I'll quote from both the first book [BB] and the second book [BBG]. These two books are clearly influenced by two papers written by the founding editors of the journal Experimental Mathematics, [ELL] in 1991 and [EL] in 1995; so I'll quote from them too. Also I'll quote from a 2002 report [Rep] to the NSF written by 22 computationally-inclined research mathematicians.
[BB], [BBG], [ELL], and [EL] all have some form of "Experimental Mathematics" in their titles, and this may appear to some as an oxymoron. Let's agree that an "experiment" in mathematics means a calculation designed to find or test a conjecture. So it's something all of us do to at least some extent.
Experiments from long ago. Since we are making predictions about the future, we should start with a look at the past. [BB] and [Rep] each refer to the same well-known example, so let's look at this "canonical example."
Gauss writes that as a teenager in the early 1790s he experimentally found "the density of primes around t is 1/log t, so that the number of primes up to a given bound x is approximately the integral from 2 to x of dt/log t." Surely a startling conjecture, but the young Gauss kept it to himself. The corresponding prime number theorem wasn't proved until 1896.
In the late 1850s, Riemann investigated the error of the integral approximation and found out it was tied up with the location of the non-real zeros of the zeta function. He conjectured that all real parts of these zeta zeroes are 1/2. How did he arrive at this conjecture? One ingredient was that he computed the first several zeros to several decimal places, a fact completely buried from public view until Siegel unearthed it in 1929. Of course, this Riemann hypothesis is still not proved.
What can we learn from this two-part canonical example? We see in each part that experimental work played an important role. We are reminded that conjecture can sometimes outrun proof even by centuries. Also we see very clear illustrations of experimental work being hidden, an aspect of our community's culture which is elegantly decried as pervasive and dysfunctional in [ELL] and [EL].
Recent experiments. Since we are making predictions about the future, we should take a real close look at the present. [EL] says "It is probably the case that most significant advances in mathematics have arisen from experimentation with examples." Section 1.1 of [Rep] lists a number of such experiments. Another example, unusual because the experimental work was prominently published, concerns "Monstrous Moonshine." This was a title of a 1979 paper by John Conway and Simon Norton reporting on numerical agreement between coefficients of modular functions and characters of the Monster group, itself with origin the humble observation that the modular function coefficient 196884 almost agrees with the Monster character degree 196883. The agreement has since been explained by the Fields medal work of Richard Borcherds.
[EL] also says "theory and experiment feed on each other." Again, one does not have to look far for instances. The entire history of fractals provides a popularly known instance. Theoretical work of Fatou and Julia in the 1920s seemed to have little consequence for decades. But as soon as Julia sets became visible via computers in the 1980s, their work was revived. Many further theoretical results were obtained, most inspired by careful inspection of computer generated examples.
A more detailed view of experimental mathematics. So far we have taken "Experimental Mathematics" to mean the conjectural component of the traditional research process. The two books [BB] and [BBG] communicate a more refined understanding. They do this without comprehensively surveying the field. In fact, as they state in each preface, they mostly stay away from "highly abstract" and "esoteric mathematics." Rather they focus on aspects of analysis and number theory which are familiar to most mathematicians.
A major focus of both books is a specific experimental technique, inverse symbolic computation. Suppose one has a real number c given for example by an integral or an infinite series. One would like to express c if possible in terms of known constants, such as algebraic numbers, π, e, special values of zeta functions, etc. The technique is to compute c numerically to high precision, say 100 significant figures. Then, one inputs this approximation into an inverse symbolic calculator to guess a formula. Then, using the structure of the guessed formula as a critical clue, one tries to prove the formula. [BB] and [BBG] are replete with successful instances of this procedure.
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. A leading example, touched upon in both books, is the recent progress in multizeta functions and their applications to quantum field theory.
The experimental approach is capable of breathing new life into unfashionable traditional areas. For example, computing quantities to extremely high precision has had for some of us a rather pointless feel. But now, seeing as high precision is needed for the input to inverse symbolic calculators, things are different!
A characteristic feature of experimental mathematics is that practitioners are often awash in a sea of data, out of which they have extracted various conjectures. Often evidence for a conjecture seems quite substantial. The traditionalist leanings of the authors are seen here too, as they offer several cautionary examples of "high precision fraud."
In short, computers can be harnessed to make research into mathematics more efficient. The main prediction for the future is that this will indeed happen, and accordingly our conception of mathematics will shift somewhat.
An enlarged vision of mathematics. The last chapter of [BB] begins with a stirring paragraph leading up to "Like it or not, the world of the mathematician is becoming experimentalized." Why should we like it? Part of my personal answer is that the computational approach gives one an enlarged vision of mathematics. In particular, examples and well-supported conjectures are valued more.
A narrow vision of mathematics is that it is a list of its theorems, together with a proof or perhaps several for each. Moreover, for a mathematical statement to be respected as a theorem, it needs to be not only established as true, but also to satisfy certain unspoken criteria. Any statement just about the primes less than 1015 say, would be dismissed in this vision as "just an observation." A statement about all primes would get closer inspection as a candidate for theoremhood.
The enlarged vision of mathematics that I like says that mathematical reality is complicated and our job is to expose it. We cannot expect to fully capture it in short theorems. The colored ring on the cover of [BB], drawn larger and more clearly as Plate VII of [BBG], provides a convenient example. It indicates where certain polynomials have their roots, with color indicating a root density. This picture to me is the reality. Theorems about this situation have a certain humble aspect: they capture only some of the computationally evident reality.
There are other, non-visual, parts of mathematical reality that stand out enough to be worthy of careful study. Consider again the Conway-Norton data tables. The information presented cannot be condensed into a pithy paragraph to be italicized. Computationalists view this information as valuable in its own right, not only as a step towards a traditional theorem. After all, the information involved has the same immutable quality we admire of theorems. In this vein, [Rep] talks about the value of online databases as worthy output of mathematical labor.
A more accessible mathematics. A second and closely related reason we should embrace the computerization of pure mathematics is that it will render pure mathematics much more accessible to non-mathematicians. At present we are something of an alien culture to our students and our scientific colleagues, not to mention the general public. Students enter our freshman courses with a twelve-year running start in mathematics, and still often run into difficulties. Tellingly, these very same students may sometimes do fine in statistics or computer science courses.
[BB] writes in its preface that the experimental approach helps "to make mathematics tangible, lively, and fun for both the professional researcher and the novice." While researchers have individual styles, in my experience all students profit from computer explorations.
Conclusion. The back cover to the second book says that the two books are "testaments to a paradigm shift in the making." My feeling is that the incorporation of computers into mathematical research constitutes an enormous and very positive change, although not quite a paradigm shift. I encourage you to look through both books and form your own opinion.
[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.
David Bailey maintains an extensive web site related to the two books. Included are many pages of excerpts from the two books and many links.
[ELL] Experimental Mathematics: Statement of Philosophy and Publishing Criteria , by David Epstein, Sylvio Levy, and Rafael de la Llave. At the homepage of the journal Experimental Mathematics.
[EL] Experimentation and Proof in Mathematics. David Epstein and Sylvio Levy. Notices of the American Mathematical Society , June/July 1995, 670-674.
[Rep] Computational Opportunities in Algebra, Number Theory, and Combinatorics, 22 authors, 2002.
David Roberts is associate professor of mathematics at the University of Minnesota, Morris.