The Digital Intuitionist: A Profile of Jonathan Borwein (1951–2016)
Jonathan Michael Borwein was a mathematical polymath whose work bridged the gap between the abstract rigors of pure mathematics and the burgeoning power of high-performance computing. A pioneer of "Experimental Mathematics," Borwein transformed the computer from a mere calculator into a laboratory for mathematical discovery. Over a career spanning four countries and four decades, he reshaped how we understand numbers, optimization, and the very process of mathematical inquiry.
1. Biography: A Life of Perpetual Motion
Jonathan Borwein was born on May 20, 1951, in St Andrews, Scotland, into a family of formidable intellectual pedigree. His father, David Borwein, was a distinguished mathematician and former head of the department at the University of Western Ontario, which set the stage for Jonathan’s lifelong immersion in the field.
Education and Early Career:
Borwein’s academic journey was marked by rapid ascent. He earned his Bachelor of Arts from the University of Western Ontario in 1971 before moving to the University of Oxford as a Rhodes Scholar. At Oxford, he completed his D.Phil. in 1974 under the supervision of the renowned functional analyst Michael Dummett and Marcus Wright.
Academic Trajectory:
His career was characterized by a series of prestigious appointments across the globe:
- Dalhousie University (1974–1991): Where he rose to the rank of Professor.
- University of Waterloo (1991–1993): Serving as the Shrum Professor of Science.
- Simon Fraser University (1993–2004): Here, he founded the Centre for Experimental and Constructive Mathematics (CECM), a landmark institution in his career.
- University of Newcastle, Australia (2008–2016): He finished his career as a Laureate Professor and Director of the Centre for Computer-Assisted Research Mathematics and its Applications (CARMA).
Borwein’s sudden passing in August 2016 while in London, Ontario, left the mathematical community mourning one of its most energetic and prolific voices.
2. Major Contributions: The Experimental Frontier
Borwein’s most significant contribution was the formalization and promotion of Experimental Mathematics. While mathematicians had always used examples to find patterns, Borwein argued that modern computing allowed for a more systematic "experimental" approach to conjecture and proof.
Key Theories and Discoveries:
- High-Precision Calculation of π: Alongside his brother Peter Borwein and David H. Bailey, Jonathan developed algorithms capable of calculating π to billions of decimal places. Their work on the "Borwein Algorithms" provided some of the fastest-converging series for π ever discovered.
- Nonsmooth Analysis and Optimization: He made foundational contributions to convex analysis and optimization theory. He developed techniques to handle functions that are not differentiable (smooth), which is critical for modern logistics, economics, and engineering.
- The Borwein Integrals: He identified a fascinating class of integrals (involving the sinc function) that appear to follow a predictable pattern for several iterations before suddenly breaking it. This discovery served as a cautionary tale against assuming a pattern holds based only on initial data.
- Functional Analysis: He contributed to the understanding of Banach spaces and the geometry of operators, often finding surprising links between abstract space theory and practical optimization.
3. Notable Publications
Borwein was staggeringly prolific, authoring or co-authoring over 500 refereed articles and more than 25 books.
- Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987): Co-authored with Peter Borwein, this is considered a masterpiece of modern number theory, linking the Arithmetic-Geometric Mean (AGM) to the calculation of transcendental constants.
- Convex Analysis and Nonlinear Optimization: Theory and Examples (2000): Written with Adrian Lewis, this remains a standard graduate-level textbook in the field.
- Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004): Co-authored with David Bailey and Roland Girgensohn, this book served as a manifesto for the use of computers in mathematical research.
- Experimental Mathematics in Action (2007): A practical guide that demonstrated how computational tools could be used to solve "unsolvable" problems.
4. Awards & Recognition
Borwein’s brilliance was recognized by the highest echelons of the scientific community:
- Fellow of the Royal Society of Canada (1994): Elected at a relatively young age for his contributions to Canadian science.
- Chauvenet Prize (1993): Awarded by the Mathematical Association of America for outstanding expository writing in mathematics.
- Fellow of the American Mathematical Society (2012): Recognized for his contributions to optimization and experimental mathematics.
- Honorary Doctorates: He received honorary degrees from several institutions, including the University of Limoges.
- The David Borwein Distinguished Career Award (2016): Posthumously honored by the Canadian Mathematical Society.
5. Impact & Legacy: A New Way of Thinking
Jonathan Borwein’s legacy is not just found in his theorems, but in the methodology of modern mathematics. Before Borwein, many pure mathematicians viewed computer-aided research with skepticism, seeing it as "cheating" or "lesser" than pen-and-paper proof.
Borwein changed this culture. By founding the CECM at Simon Fraser University, he created a space where mathematicians, programmers, and physicists collaborated. He proved that computers could discover identities that humans might never stumble upon, such as the Bailey–Borwein–Plouffe (BBP) formula, which allows one to calculate the n-th digit of π without calculating all the preceding digits—a feat previously thought impossible.
Today, the "Experimental Mathematics" journal and the widespread use of software like Maple and Mathematica in pure research owe a direct debt to his advocacy.
6. Collaborations: The "Borwein Clan" and Beyond
Borwein was a quintessential collaborator, rarely working in isolation.
- The Borwein Family: His most frequent collaborator was his brother, Peter Borwein. Together, they were the "Wright Brothers" of number theory. His father, David, also collaborated with him on several papers, making the Borwein name synonymous with Canadian mathematics.
- David H. Bailey: A Chief Technologist at NASA, Bailey and Borwein formed a decades-long partnership that defined the field of computational number theory.
- Adrian Lewis: Their work on optimization theory bridged the gap between pure functional analysis and applied mathematics.
- Mentorship: Borwein supervised dozens of PhD students and postdoctoral fellows, many of whom now hold prestigious chairs in optimization and analysis globally.
7. Lesser-Known Facts
- The "Pattern-Breaking" Integrals: Borwein was delighted by a specific set of integrals that equal π/2 for the first seven cases, leading a casual observer to conclude they always equal π/2. However, the eighth integral suddenly drops to a value slightly less than π/2. He used this as a pedagogical tool to teach the necessity of rigorous proof.
- Polymathic Interests: Beyond math, Borwein was deeply interested in the philosophy of science and cognitive psychology. He frequently wrote about how the human brain visualizes mathematical objects.
- Digital Preservation: He was an early adopter of the web for academic dissemination. Long before "Open Access" was a buzzword, Borwein maintained massive digital repositories of mathematical constants and experimental data, ensuring his research was accessible to anyone with an internet connection.
- The "Pi" Obsession: While many saw his work on π as a hobby, he viewed it as a "stress test" for hardware. His algorithms were often used to check the integrity of new supercomputers—if a computer could calculate π to a billion places without a single bit-error, it was deemed reliable.