|Keith Devlin and Jonathan Borwein talk to host Robert Krulwick on stage at the World Science Festival in 2011.|
At the end of this week I fly to Australia to speak and participate in the Jonathan Borwein Commemorative Conference in Newcastle, NSW, Borwein’s home from 2009 onwards, when he moved to the Southern hemisphere after spending most of his career at various Canadian universities. Born in Scotland in 1951, Jonathan passed away in August last year, leaving behind an extensive collection of mathematical results and writings, as well as a long list of service activities to the mathematical community. [For a quick overview, read the brief obituary written by his long-time research collaborator David Bailey in their joint blog Math Drudge. For more details, check out his Wikipedia entry.]
Jonathan’s (I cannot call him by anything but the name I always used for him) career path and mine crossed on a number of occasions, with both of us being highly active in mathematical outreach activities and both of us taking an early interest in the use of computers in mathematics. Over the years we became good friends, though we worked together on a project only once, co-authoring an expository book on experimental mathematics, titled The Computer as Crucible, published in 2008.
Most mathematicians, myself included, would credit Jonathan as the father of experimental mathematics as a recognized discipline. In the first chapter of our joint book, we defined experimental mathematics as “the use of a computer to run computations—sometimes no more than trial-and- error tests—to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search.”
The goal of such work was to gather information and gain insight that would eventually give rise to the formulation and rigorous proof of a theorem. Or rather, I should say, that was Jonathan’s goal. He saw the computer, and computer-based technologies, as providing new tools to formulate and prove mathematical results. And since he gets to define what “experimental mathematics” is, that is definitive. But that is where are two interests diverged significantly.
In my case, the rapidly growing ubiquity of ever more powerful and faster computers led to an interest in what I initially called “soft mathematics” (see my 1998 book Goodbye Descartes) and subsequently referred to as “mathematical thinking,” which I explored in a number of articles and books. The idea of mathematical thinking is to use a mathematical approach, and often mathematical notations, to gather information and gain insight about a task in a domain that enables improved performance. [A seminal, and to my mind validating, example of that way of working was thrust my way shortly after September 11, 2001, when I was asked to join a team tasked with improving defense intelligence analysis.]
Note that the same phrase “gather information and gain insight” occurs in both the definition of experimental mathematics and that of mathematical thinking. In both cases, the process is designed to lead to a specific outcome. What differs is the nature of that outcome. (See my 2001 book InfoSense, to get the general idea of how mathematical thinking works, though I wrote that book before my Department of Defense work, and before I adopted the term “mathematical thinking.”)
It was our two very different perspectives on the deliberative blending of mathematics and computers that made our book The Computer as Crucible such a fascinating project for the two of us.
But that book was not the first time our research interests brought us together. In 1998, the American Mathematical Society introduced a new section of its ten-issues- a-year Notices, sent out to all members, called “Computers and Mathematics,” the purpose of which was both informational and advocacy.
Though computers were originally invented by mathematicians to perform various numerical calculations, professional mathematicians were, by and large, much slower at making use of computers in their work and their teaching than scientists and engineers. The one exception was the development of a number of software systems for the preparation of mathematical manuscripts, which mathematicians took to like ducks to water.
In the case of research, mathematicians’ lack of interest in computers was perfectly understandable—computers offered little, if any, benefit. (Jonathan was one of a very small number of exceptions, and his approach was initially highly controversial, and occasionally derided.) But the writing was on the wall—or rather on the computer screen—when it came to university teaching. Computers were clearly going to have a major impact in mathematics education.
The “Computers and Mathematics” section of the AMS Notices was intended to be a change agent. It was originally edited by the Stanford mathematician Jon Barwise, who took care of it from the first issue in the May/June 1988 Notices, to February 1991, and then by me until we retired the section in December 1994. It is significant that 1988 was the year Stephen Wolfram released his mathematical software package Mathematica. And in 1992, the first issue of the new research journal Experimental Mathematics was published.
Over its six-and- a-half years run, the column published 59 feature articles, 19 editorial essays, and 115 reviews of mathematical software packages — 31 features 11 editorials, and 41 reviews under Barwise, 28 features, 8 editorials, and 74 reviews under me. [The Notices website has a complete index.] One of the feature articles published under my watch was “Some Observations of Computer Aided Analysis,” by Jonathan Borwein and his brother Peter, which appeared in October 1992. Editing that article was my first real introduction to something called “experimental mathematics.” For the majority of mathematicians, reading it was their introduction.
From then on, it was clear to both of us that our view of “doing mathematics” had one feature in common: we both believed that for some problems it could be productive to engage in mathematical work that involved significant interaction with a computer. Neither of us was by any means the first to recognize that. We may, however, have been among the first to conceive of such activity as constituting a discipline in its own right, and each to erect a shingle to advertise what we were doing. In Jonathan’s case, he was advancing mathematical knowledge; for me it was about utilizing mathematical thinking to improve how we handle messy, real-world problems. In both cases, we were engaging in mental work that could not have been done before powerful, networked computers became available.
It’s hard to adjust to Jonathan no longer being among us. But his legacy will long outlast us all. I am looking forward to re-living much of that legacy in Australia in a few days time.