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Mathematical and other Memories

John Knopfmacher, as both my father and principal mathematical collaborator, influenced my life in major ways, not only on a personal level but also as a mentor and role model in my own mathematical career. To John, mathematics was not a just a career but a whole way of life. A truly creative person, his abilities and interests ranged far beyond his own speciality in Number Theory to science and the arts in general.

As a young man he composed a little music and poetry as a hobby and painted abstract art. His interest in painting returned to him later in life and he continued to produce further works when in his forties. In studying them, one somehow can sense the mathematical talent that underlies their creation. Fortunately most of these still survive with his family and I hope soon to produce photographic size versions of these, to be displayed on his web page, so that others may perhaps glean further insight to his creative visions.

As a university student he trained in both pure and classical Applied Mathematics and he maintained his interests in the broader areas of mathematically related sciences through extensive reading throughout his life. An avid collector of books, particularly in mathematics, since his early career, he compiled ultimately a very extensive and valuable personal mathematics library. It was his wish to leave this as a legacy to the Mathematics Department at Wits University. It is hoped that a library in his honour will shortly be established at the university. For many years he also ordered the books for the Mathematics library at Wits and the breadth of material available there owes much to his careful acquisitions despite ongoing restrictions and budget limitations.

John s Career

As a high school student John s marks were not exceptional, a fact that perhaps can be linked to an unhappy home life. John s father was himself a holder of a professorship and Phd (in Engineering) but John received little encouragement from his parents. He related for example, how while occupied with a painting he was working on, it was suggested that he would be better employed by painting the house. He did however become reconciled with his mother towards the end of her life, which allowed us children to know a grandparent for the first time in our lives.

After studying unhappily in accountancy for a year he switched to science and his mathematical genius immediately began to flourish. He related to me that as only a first year student, he became inspired by the famous problem of odd perfect numbers and derived what he believed to be a proof that none existed. One of his lecturers realised that while not a proof of this, it was in fact a new proof of the formula that describes all even perfect numbers, and this result became his first publication in a mathematics journal. After completing a double honours at Wits in Pure and Applied Mathematics, he left for the University of Manchester with his wife Rose and 5 month old child (myself) to work towards a Phd.. He spoke little of this period, but I gather that he largely chose his own area of study and completed the work with little supervision. My impression is that the academic culture and traditions of England struck a chord with him and that he returned to South Africa, now with the addition of a baby daughter Nadine, mainly to please Rose who greatly missed her family here.

At the time he joined Wits Mathematics Department as a lecturer, only he and the head of Department, held PhD degrees. From this time in 1965 until the early eighties he found himself as regards research, in a position of both geographical and mathematical isolation. Despite this, his inner drive never wavered and perhaps his most significant works, in abstract analytic number theory, a subject he invented and developed himself, appeared during this time.

In the early years, he found the South African Mathematics Society to be largely a clique with interests mainly unrelated to the advancement of research. He allowed his membership to lapse and rejoined only in the eighties when South African Mathematics appeared to him to be going through a renaissance in research. Thereafter, he served for many years as an Associate Editor of the South African journal Quastiones Mathematicae and in 1995 was awarded the Society s medal for life time achievement.

During the earlier part of his career he felt that his own research and research in general were unappreciated and he turned away from outward awards, achieving his satisfaction from his own knowledge of his achievements and that of his peers overseas. He declined to apply for promotion, but finally in 1978 was asked to accept a Chair in the Department. Initially he even felt some reluctance about this, feeling that he might be called upon to spend too much of his time on administrative tasks. Ultimately the lack of a suitable alternative head made him accept this position, which to his initial surprise he found he managed comfortably while still maintaining his research output. His initial 3 year appointment was subsequently renewed twice so that he served as head for nine years.

One of his major adjustments was to turn the department much more strongly in a research direction. Under his leadership, research in Mathematics at Wits began to blossom, and he made sure that no deserving talent failed to get support and recognition.

In 1992 he founded the Centre for Applicable Analysis and Number Theory which has grown to be one of the finest research units at Wits. After his retirement in June 1997 the reigns were passed to Doron Lubinsky, who has continued the fine traditions that John fostered. John recognised Doron s talents when Doron was just 32 and was responsible for Doron s appointment as full Professor at age 33, jumping him through the ranks from initial position as a part-time lecturer. It was important for him that his own struggle for local advancement should not happen to the next generation of South African mathematicians. Finally, in his fifties, John began to receive personal recognition within South Africa, although at no stage did he seek this or feel the need to promote himself.

After an initial reluctance to be subjected to rating, I persuaded him to apply to the Foundation for Research and Development for research support. Pleasantly surprised by the outcome, he pursued mathematical research with renewed vigour. This resulted in him achieving A rated status, reserved for world leaders in their scientific fields, a status he maintained until his retirement. He was also elected a Fellow of the Royal Society of South Africa and played a role in getting more mathematicians recognised by this body thereafter.

Research Collaboration

Up until the mid eighties almost all of John s research was done on his own, a further tribute to his drive and strength of character. A notable exception to this were his two papers with Wolfgang Schwarz. In 1986 I had just completed my Phd in Approximation Theory under the supervision of Doron Lubinsky and was suffering unhappily through two years of compulsory national service which was largely instrumental in my losing a sense of mathematical direction. John suggested to me that a recent paper by Rieger, on construction of the real numbers using continued fractions, could perhaps be modified so that unit fraction representations could be used instead. This little beginning lead us to turn further towards the study of unit fraction representations themselves and ultimately to pursue analogues of these ideas for complex,p-adic numbers and formal power series. Initially I was still living at home when we began our work together, so our conversations on mathematics continued from the office to the dinner table at night.

Undoubtedly I gained from this collaboration with an older, established and more talented mathematician but I think that he gained too, at last having someone around to speak to about his mathematical interests on a daily basis. Initially it was very much the case of him as teacher and myself as learner but as I mastered the ideas I began to suggest further extensions and generalizations and using his initial ideas I began to produce further works myself. Computers and programming were among the few areas of science in which John seemed to show little interest and indeed he adopted even email only after several years of persuasion. I, by contrast, embraced computer algebra and enjoyed producing conjectures based on experimental evidence. With our research collaboration being of mutual benefit to us, it became clear that a balance was being maintained, not necessarily in each individual joint paper, but rather as a whole in terms of the originator of the idea, the solution of various parts of the problem and the time consuming but crucial phase of writing up and polishing our rough results.

This fruitful collaboration was at its peak from 1986 until 1992 and indeed this period resulted in some twenty eight or so joint papers. The happy coincidence that the recently established Foundation for Research and Development gave both recognition and research grants to support our work was undoubtedly a further stimulus. However, research itself was the motivation for our work, and the recognition that followed a year or two afterwards was welcomed, but neither expected or envisaged when we began.

The founding of the Centre by John in 1992 was a tremendous boost to the department as a whole. However the regular influx of visitors and new collaborators unfortunately meant less time to continue our personal collaboration. Nevertheless a further 6 or so joint papers followed in 1993 and thereafter. Despite being in his mid fifties, a time by which the creative impulses of many mathematicians has passed, John was always ready to learn new subjects. After our study of basic properties of expansions was completed, one area of further study was to investigate ergodic and metric properties of these, an area of mathematics foreign to both of us. John undeterred, proceeded to read up and acquire the necessary techniques and then translate them to the context we needed. This led deservedly to some publications under his name only. With his work as a gentle introduction for me to follow, I subsequently saw how to improve and extend some of his results and some further joint papers followed on this new topic.

In the most recent years, having to some extent exhausted our interest in expansions we began to look again at some topics relating to his earlier and most important work. With my own tastes having moved more to Combinatorics, I noticed that various families of trees and polyhedra could in fact be studied by defining a new axiom in the spirit of Johns first book. John pursued this idea with great enthusiasm and our most recent paper, on the abstract prime number theorem under this axiom, has just appeared. Just prior to his departure for Australia in July 1997 John also began to work on other consequences on this axiom following developments from his first book. Here he was way ahead of me and I could make little immediate contribution, thus leaving this work on hold to be continued at a later stage. Of course life never follows the expected path but the fact remains that the results are largely done and I hope I will still be able to complete what will possibly be his final publication, in a manner befitting his high standards.

Although I dont know enough to write much on this topic, in recent years John has also had other collaborators, notably Richard Warlimont, Stefan Porubsky and most importantly his work with W.B. Zhang on a greatly enlarged second edition of his second book, "Analytic Arithmetic of Algebraic Function Fields". This is the work that has largely occupied him since his move to Australia and despite his untimely death, the book is in its final stages and should appear in the next year or two. This new book with Zhang, together with his first book now republished by Dover will be his most enduring contributions to the subject that he loved so dearly.

Philosophy of mathematics

Since to John, mathematics was both an art and a science, the importance of the mathematical concept and its implications and applications was paramount. Long technical lemmas and theorems with obscure or unmotivated statements did not pass the test. In his books, having shown how the ideas of ordinary number theory could be extended to find, for example, mean values of "abstract" arithmetical functions, he left it as a possible item for further research to sharpen these asymptotoc mean values. In his view, having found the leading term and error estimate for such a function, it was the work of a technician (albeit highly skilled) to change say O(x^1/2) to O(x^0.46). Indeed, unless a new such error estimate could be shown to be best possible, he did not consider such results to have lasting value. Here is another debt I owe to John, learning how to write papers well, by imitating his masterful presentation of mathematical ideas.


In July 1997 John moved to Melbourne, where he and his second wife Bev, hoped to find a better quality of life. He loved the beautiful coastal scenery and the tranquillity of his new environment but at the same time greatly missed Wits and his three children. He was an honorary visitor at Melbourne University but had hoped to find a proper mathematical position, even at a junior level, but was unsuccessful in this. This unfortunately left him feeling disappointed and unsure of the value of his mathematical works. During his final weeks of life, as a visiting Professor in Austria, his e-mails showed that he had regained his enthusiasm for both research and teaching, perhaps a very small consolation to us all.

Personal Recollections

If John had to describe a passion outside of mathematics it would probably be for chocolate, especially dark chocolate, a taste he has passed on to his three children. He told me once that as a child he regarded working in a chocolate factory to be his desired career choice. Another liking passed on particularly to his sons Arnold and Kevin, is his love of comedies in both film and radio . He was also a regular subscriber to Punch magazine in its heydays in the sixties and seventies. As a teenager I became a fan of the dry cartoons of Punch and John and I began a collection of these, a few of which I have placed on my home page so far. Perhaps his favourite cartoonist was Handelsman and favourite Punch writer, Alan Coren. John himself, although usually reserved and quietly spoken with outsiders, possessed a zany even silly sense of humour. It was never vicious or directed against others, just a gentle amusement.


Yet another area in which he studied deeply was Philosophy, perhaps his second choice of academic interest after Mathematics. An agnostic since an early age, he found that the existence of a God, however comforting or prevalent such beliefs are, was not consistent with any scientific or serious philosphical evidence. He had no wish to offend and never shared these ideas with those whose upbringing or inclinations would find this upsetting. His personal philosophy, based on a deep and considered study of writers like Bertrand Russel and A. J. Ayer, made him a good person, more so in many ways than some devoutly religious folk whose beliefs are based partly on a fear of divine retribution or of an expectation of spiritual reward in some life hereafter. Despite the easy comfort such beliefs would offer to him and to us now at this difficult time, John accepted that death was an ending, he wished only for some small degree of immortality in his contribution to the advancement of mathematics - in this he has his wish.

A man of liberal belief, John found the apartheid government of South Africa repugnant, but it was not in his nature to fight, whether in politics or even university administration. My impression was that he regarded politicians in general as untrustworthy. He did however, have an admiration for a few of the great leaders in history. As a child I remember reading of Galileo being forced to recant his beliefs by the Catholic Church, this being described as if it were a failure of Galileo as a scientist. John s reaction - if a maniac threatens your life unless you swear that the Earth is flat, by all means tell them this. On another occasion I read of the triumph of the Roman Civilization over the Greeks after they defeated them militarily. Again John gently admonished, the ability to kill is no measure of the accomplishments of a civilisation.

Of course like all men John had his faults; to claim otherwise would be to devalue any words of praise above. As someone far more inclined to cynicism than he was, it is a source of recurrent amusement to me how men and women with the morals and values of Hitler are transformed miraculously upon death into the selfless saviours of disabled children. Perhaps mathematics again best describes this situation. On a "goodness" scale John had a high mean value with small variance, but even such a mathematical distribution posseses a tail of low probability that lies in the realm of jealousy, distrust and deceit. It is to John s great credit, that viewed over his lifetime, little of his character falls within these domains.

Although he never had a chance to meet his first granddaughter Caylee, born just three weeks prior to his untimely death, John gained great pleasure in hearing of her birth and fortunately saw photos of her almost immediately by means of the internet. As her parents, Sharon and I hope to let John live on in yet another way, through the memories of her grandfather that we pass on to her as she grows.

Arnold Knopfmacher