[This is a condensed version of an article written by Michel Théra, to appear in the *Journal of Optimization Theory and Applications*.]

Jonathan Borwein left us suddenly on August 2, 2016 at age 65. He leaves his wife Judith, three children Naomi, Rachel and Tova; five grand children Jakob Joseph, Noah Erasmus, Skye, Zoe and Taj. Jonathan Borwein was born in 1951 in St. Andrews, Scotland, to a family of intellectuals. In fact, both of his parents belonged to the world of research.His mother Bessie was an anatomist while his father, David, was a mathematician, holding a position in the Department of Pure Mathematics at the University of Western Ontario from 1967 through 1989, and served as President of the Canadian Mathematical Society in the period 1984-1986.

Jon began his studies with the firm intention of becoming an historian. He often told me that he was never subjected to the least pressure from his parents with respect to his future career. The only exception occurred in 1957, when his father had wagered with his colleagues at St. Andrews (the prize being a large quantity of cheese) that he could teach his son who was six at that time to solve two equations in two unknowns through a game. It was with this method that in post-war Great Britain his father taught him the solution of such a system without he really understanding the reasoning: “I love playing the mysterious game and I taught it myself to my best friend” he said to me. He also added,

From David and Bessie, I learned to appreciate the calm rewards of an intellectual life, but without all the time finding myself being confined to an ivory tower. I acquired a penchant for work well-executed, and the particular pleasure and satisfaction associated with putting the finishing touches on an article, following many other earlier adjustments. But finally, my most powerful memory is of my father, with respect to evenings frequently organized by my mother whose is more the extroverted of the two. My father played the role of a welcoming host, but not particularly exuberant. In the middle of the evening, his eyes opened slightly, and he pulled together a pile of papers filled with his tight, slanted handwriting. I felt extraordinarily privileged to be included in familial mathematical projects. I worked with great energy and on equal footing with my brother Peter as well as with my father. Furthermore, I felt my father to be an intellectual companion.

While a student at Oxford (at Jesus College) he claimed to be profoundly influenced by the philosopher (also mathematician and linguist) Michael Dummet who taught him Frege and Heting. At Jon’s oral examination, “Dummet expected me to be able to prove the independence of the continuum hypothesis.” He wrote his doctoral thesis at Oxford under the direction of Michael Dempster. He began his brilliant academic career in 1975 at Dalhousie University (Halifax, Nova Scotia, Canada), where Michael Edelstein, a strong functional analyst, took him under his wing, and provided an example for Jon of what a mathematician should be. After several years, he briefly held a position in the United States, at Carnegie Mellon (Pittsburgh, Pennsylvania) from 1980-1982 where he had the opportunity to work with a number of captivating researchers in the areas of operations research and mathematical economics such as Dick Duffin, an emeritus mathematician and engineer, who had earlier guided John Nash. Dick Duffin insisted to Jon that “a passion for the sciences is not well served by garish or pretentious packaging.”

He returned from time to time to Dalhousie, before accepting a position in 1991 at the Department of Combinatorics and Optimization at the University of Waterloo (Ontario, Canada) where he stayed through 1993. After these two years at Waterloo, he moved to Simon Fraser University (Burnaby, British Columbia, Canada), where he founded the Centre for Experimental and Constructive Mathematics (CECM). Jon’s purpose in this endeavor was for CECM to be become a centre of excellence that would attract world-class scholars interested in the interaction between mathematics and computer applications. He left CECM in 2003 to return briefly to Dalhousie, holding the title *Canada Research Chair in Distributed and Collaborative Research*. Finally, he became Laureate Professor in the School of Mathematical and Physical Sciences at the University of Newcastle in Australia. There, he created a new centre for Computer-Assisted Research Mathematics and its Applications (CARMA), in order to promote in Australia collaborative research in computational analysis, the theory of numbers, discrete mathematics, optimization and simulation, with an eye to addressing real-world problems facing scientists, engineers and quantitatively-inclined managers.

Jon’s vision and initiative were crucial for the creation and development of CECM and after that of CARMA. Much earlier, he held the view that an enormous change was taking place in the overall field of mathematics. He viewed mathematics as the language of high technology. He viewed researchers as being involved in wide thematic endeavors, ranging over computational number theory, optimization, pure and applied functional analysis, and experimental mathematics. The researchers in these centers were always profoundly involved with one or more of the following: web-based calculation, the administration of high-performance networks, distance learning, a humane conception of user interfaces, construction of interactive dictionaries and “living books,” medical imaging, financial mathematics, and crisis interventions. There has been involvement with missions to Mars among other things.

Jon viewed traditional mathematics as a particularly conservative discipline to the extent that most mathematicians embraced an idealistic vision of their subject and a distrust of experimental mathematics as he conceived of it. He said that mathematics in the age of computers could be entertaining and that homo sapiens could also be homo ludens. In fact, the problems he met while teaching classes to students in management science and in business (at Carnegie-Mellon and upon his return to Dalhousie in 1982) gave him the impetus for his first foray into computer-assisted mathematics. He began to write very simple software that could help his students solve linear programming problems whose solution on the blackboard would have been very unpleasant and tedious.

His work on interactive dictionaries began in 1985 as part of a sabbatical leave, in a company on the outskirts of Limoges, and lead to the creation of a society of educational software. “We are leaving our particular research areas in order to establish liaisons with other scientists, educators, the business community, and the media.”

Jon was a prolific and innovative mathematician. He was internationally known for his work in analysis, be it pure, numerical, or computational; in optimization; and in high-speed computation. His stature resulted in his recognition at the highest level and in various directions. I first mention his Chauvenet Prize for expository writing awarded by the Mathematical Association of America (1993) for his contribution “Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi.” He was a recipient of a fellowship from the Royal Society of Canada (1994), from the American Association for the Advancement of Science (2002), as well as from the Australian Academy of Sciences (2010), the American Mathematical Society, and the Royal Society of New South Wales (2015). He was also admitted as a foreign member of the Bulgarian Academy of Sciences (2003). I had the immense pleasure and honor to arrange his being awarded a Doctorat Honoris Causa from the University of Limoges (1999).

Jon became involved over his career in many varied administrative activities, among which are the following: Governor-at-large for the Mathematical Association of America (2004-2007); President of the Canadian Mathematical Society (2000-2002), and past chair of (the National Science Library) NRC-CISTI’s Advisory Board (2001-2003), reporting to the International Math Union’s Committee on Electronic Information and Communications (2002-2008). Since his arrival at Newcastle, he was in charge of the Scientific Advisory Board of the Australian Mathematical Sciences Institute (AMSI) and was a member of the Council of the Australian Mathematical Society (2009-2016).

Jon had numerous editorial responsibilities over his career. He co-edited the Canadian Mathematical Society’s book series, and was Associate Publisher of the Canadian Mathematical Society, in charge of Books and Rich Media. He belonged to the editorial boards of many journals; among the most prestigious were: *Notices of the American Mathematical Society* (2010-2015), *Proceedings of the American Mathematical Society* (1998-2006), *American Mathematical Monthly* (2012-2016), *Journal of Optimization Theory and Applications* (2011-2016), and the *Society of Industrial and Applied Mathematics Journal of Optimization*. At the time of his death, he was co-editor-in-chief (with George Willis) of the *Journal of the Australian Mathematical Society*.

Finally, in what may not be common knowledge to many mathematicians, he created at least one successful start-up devoted to the technology of computation. He was a scientific consultant to Apple Computers. A good portion of his ideas came from his ability to quickly draw subtle images on his computer and to develop dependable ideas of his own. He liked to cite David Berlinski’s review of T. W. Korner’s book *The Pleasures of Counting*: “In its time, the computer has changed the very nature of the mathematical experience, suggesting for the first time that mathematics, like physics, could become an empirical discipline, an area where one could discover things because one sees them.” He thus devoted a large portion of his mathematical life in the hope of refuting Picasso, who had said “Computers are useless, only capable of rendering answers.”

Let me give an example of what he did.

In 1982, with his brother Peter, who had ended up living in the same apartment house, they decided to study Pi together. At age thirty, “I thought it would be nice to write a little paper together on Pi, the area of a unit circle and how to compute it and other elementary constants quickly”. Over the years, that little paper has generated many joint publications, including an article in Scientific American, several shared prizes and four joint books. While preparing his talk for his Doctorat Honoris Causa, Jon told me that “Pi introduced Peter and him to wildly diverse people: some cranks and hobbyists but also including the great Indian astro-physicist Chandrasekhar, Freeman Dyson of the Institute for Advanced Study, John Todd and Olga Tausky from Cal Tech and the extraordinary Russian Chudnovsky brothers. Moreover, it gave me a topic about which I could and frequently do lecture fruitfully to non-mathematicians, and talk about to the press.”

In 1982, only two million decimal digits of Pi had been computed. Many of the computations to that date had employed variations of the same classical formulas that had been used in the 19th century, although a few had employed the newly discovered Brent-Salamin algorithm, which is quadratically convergent–every iteration approximately doubles the number of correct digits. What Jon and Peter Borwein did was to discover and prove algorithms that converge to even higher order — third, fourth and even p-th order for any prime p. Their fourth order algorithm is particularly simple and efficient.

Jon discovered this algorithm on an 8K portable Radio-Shack computer in 1985 in the back of a car. “In Richard Dawkin’s terms it is a successful “meme,” and it gives me great pleasure to watch it replicate around the world wide web and elsewhere.”

At John Todd’s suggestion he begun to explore Ramanujan’s work on Pi, working largely intuitively and without proof. Peter and Jon began to heavily use an early version of Maple so that they could check his assertions before they tried to prove them or digest them. In the process “we became early exponents of a growing field called ‘Experimental Mathematics’.”

They had some striking early successes. One I call “Pentium farming for binary digits.” This was done when David Bailey, Peter Borwein and Simon Plouffe (1996) discovered an infinite series for Pi that allows one to compute hexadecimal digits of Pi *without* computing prior digits. Prior to this discovery, mathematicians had not believed that this was possible. The key formula was found by a computer. In August 1998 Colin Percival (a 17 year old student at Simon Fraser) used this formula in what is now called an “embarassingly parallel” computation of the *five trillionth bit* (using 25 machines over the world wide web).

I have already spoken a great deal about Jon’s involvement with number theory and experimental mathematics. I would like to emphasize that he had a great interest in mathematics without computation/algorithmics, although from his personal experience, he thought that the computer could be a useful tool for pure as well as applied mathematics.

He also made numerous contributions in other arenas, including variational analysis, optimization, monotone operator theory, functional analysis, convexity, linear algebra and Banach spaces theory. One contribution that particularly fascinated me was his 1987 paper with David Preiss, which published the first *smooth variational principle*.

Given an “almost minimal” point of a lower-bounded and lower semicontinuous function, Ivar Ekeland proved the existence of another point and a suitably perturbed function for which this point is (strictly) minimal. He also provided estimates of the distance between the points and also the size of the perturbation. Although this principle is of an enormous importance in optimization and in various areas of applied analysis, the perturbation guaranteed by Ekeland’s variational principle is non-smooth, even if the underlying space is a smooth Banach space and the function is everywhere Fréchet differentiable. The genius idea of Borwein and Preiss was to use a special class of perturbations determined by the norm; when the space is smooth (i.e., the norm is Fréchet differentiable away from the origin), the perturbations are smooth too. For this reason, this principle turns out to have such powerful applicability in variational analysis, in optimization and in Banach space geometry.

Another very deep contribution related to my domain of expertise is the famous Barzilai-Borwein (BB) algorithm for large-scale unconstrained optimization. According to Google Scholar, this paper has been cited over 1300 times. This is a gradient method that includes modified step sizes motivated by Newton’s method, but does not require computation of the Hessian of the function. Thus the scheme typically converges nearly as fast as the Newton scheme, but at virtually no additional cost compared with standard gradient method. It performs very well on minimizing quadratic and many other functions.

The Barzilai-Borwein algorithm has been extended and utilized in many applications, including sparse optimization, compressive sensing and image/signal processing, where the Barzilai-Borwein method is often used and compared to other state of art algorithms in these areas.

Jon’s most cited paper (according to Mathscinet) was his paper co-authored with his doctoral student Heinz Bauschke, which appeared in *SIAM Review* in 1996. This contribution offers an overview of several types of projection algorithms for which the convergence behavior is investigated into details. Projection methods have a broad utility and applicability in many areas including, as before, sparse optimization, compressive sensing and image/signal processing.

I will conclude by mentioning what Jon said in his address given on the occasion of his Doctorat Honoris Causa at the University of Limoges:

- Mathematics is a human endeavor. It takes part in and adapts itself to culture (it is not a question of abstract reality, immutable, eternal, unearthly constructs as conceived of by Frege).
- Mathematical knowledge is not infallible. In concert with empirical science, mathematics can move forward while errors are made, then corrected, and then perhaps corrected again. (This flawed nature of the subject is brilliantly described in
*Proofs and Refutations*by Lakatos.) - There exist several conceptions of proof and of mathematical rigor, as a function of time, place and other considerations. The use of computers to construct proofs constitutes a nontraditional version of rigor.
- Empirical evidence, numerical techniques, and probabilistic evidence, help all of us to decide what ought to be believed as true in mathematics. Aristotelean logic isn’t always the best means to come to a decision.

His intelligence and quickness of mind, his knowledge, his ideas, his willingness to make himself available, and his sense of humor are going to be missed among those who knew and respected Jon. I of course include myself in this group, and his leaving us took away from me the possibility to honor him once more in person. I will always be personally grateful for what he taught me and for his friendship. His acknowledgement of me last June, on the occasion of my 70th birthday celebrated in Alicante, will remain forever in my memory.

The author wishes to thank Jerry Beer and David H. Bailey for their assistance in the preparation of this article.