Ivor Grattan-Guinness (23 June 1941 – 12 December 2014)

As disciplines go, the history of mathematics is relatively small. But in common with most academic pursuits, it is a wide and varied field of research. Nevertheless, the focus of most individual research in the subject is often necessarily narrow and specialised. There is nothing particularly unusual or objectionable in this, but a consequence is that it is entirely possible for one historian of mathematics to be completely unacquainted with the work of another scholar working outside his or her immediate area of interest. Indeed, small though our community may be, it is hard to think of a contemporary historian of mathematics whose name is familiar to every other scholar in our field. However, if such a distinction were ever possible, it is arguable that it was held by Ivor Grattan-Guinness.

The sheer enormity of Ivor's scholarly output is overwhelming. In a career spanning over four decades, he published twelve books and more than two hundred and fifty research papers, as well as a staggering array of shorter articles, book chapters, book reviews, dictionary and encyclopedia entries, and other works. It was not just the volume of his productivity that was immense: the breadth and range of his contributions to the subject were also remarkable. Ivor wrote on everything from the use of number and magnitude in Euclid's Elements to the influence of Max Newman on Alan Turing. His writings encompassed not only the history of mathematics, but related disciplines such as the history of science, the history and philosophy of logic, the history of numerology, the philosophy of mathematics, and the uses of history in mathematics education. His death of heart failure on 12 December 2014 brings a close to both a remarkable career and a formative chapter in the development of the modern discipline of the history of mathematics.

Ivor Owen Grattan-Guinness was born on 23 June 1941 in Bakewell, Derbyshire, the son of Henry Grattan-Guinness, a mathematics teacher in Bakewell and later Deputy Director of Education in Huddersfield, and his wife Helena (née Brown). Schooled at Huddersfield New College, Ivor entered Wadham College, Oxford as a mathematics scholar in 1959, graduating with a BA in 1962. After a brief career in industry, working as a research mathematician for EMI, he re-entered higher education in 1964, joining Enfield College of Technology in north London as a lecturer on the mathematics for business degree course. In 1963 he had met Enid Neville while a member of the BBC Choral Society, an amateur choir in which they both sang. Their shared love of music soon blossomed into romance and they were married in January 1965. Enid would remain his best friend and constant companion for half a century.

It was Ivor's experience as a mathematics undergraduate at Oxford that initiated the curiosity which would ultimately lead to his entrée into the history of mathematics. While there, he had been frustrated by the lack of interest (and in some cases knowledge) on the part of the lecturers with regard to questions concerning both the theoretical motivation and historical development of the subject matter. In an attempt to begin to answer these questions, and simultaneous with the start of his teaching career, Ivor entered the London School of Economics (LSE) as a part-time student in the department of philosophy, logic and scientific method, founded and then run by Karl Popper. Although the departmental atmosphere was not quite as collegial as he might have wished, the intellectual effect on Ivor was both stimulating and influential, and included his first exposure to mathematical logic, sparking a fascination which was to reverberate throughout his work for the rest of his life. He graduated with an MSc in the philosophy of science in 1966.

Equipped with a philosophical background in which to frame his unanswered questions, Ivor now sought to focus on a particular area of mathematics in which his historical and educational concerns were prominent. Having already learnt of the connection between Fourier series and fundamental concepts in analysis and set theory, his attention was drawn to the foundational work of Fourier and Cauchy. Being new to the field, he sought guidance from a scholar who, at this time, was one of the only professional historians of mathematics in the country, then fully immersed in the preparation of the first volume of Newton's mathematical papers. From Tom Whiteside he received much valuable advice on matters concerning archival research, which would very soon become Ivor's forte; Whiteside also helped to establish a working relationship with another scholar who had done recent work on Fourier, Jerome Ravetz at the University of Leeds.

By 1967, Ivor was working on two complementary, but parallel, projects as a doctoral student in the LSE's department of mathematics. Under the supervision of the analyst and head of department, Cyril Offord, with Ravetz serving as his external supervisor, the study of Cauchy became his eventual thesis, for which he received a PhD from the University of London in 1969. With minor changes, this work was published as his first book, The development of the foundations of mathematical analysis from Euler to Riemann, in 1970. Meanwhile his collaboration with Ravetz on Fourier became a full-scale study, not only of the then unpublished 1807 memoir on heat diffusion, but also of his life, with prodigious use of a large number of previously unpublished manuscript sources—a feature that would become a defining characteristic of much of Ivor's work.

At Ravetz's suggestion, the external examiner on Ivor's PhD committee was Sir Edward Collingwood, a senior figure in British analysis at the time and shortly to become President of the London Mathematical Society. Collingwood had a deep interest in the history of mathematics in general, but was particularly fascinated by the work of Georg Cantor. The developmental connections between Cantorian set theory and the Weierstrassian mathematical analysis so closely related to Ivor's recent research now led him into a new, but related, area of research: the history of set theory. Collingwood had somehow learned that a key to finding more archival information lay in the former home of Weierstrass's Swedish disciple Gösta Mittag-Leffler, just outside Stockholm, and in 1968, Ivor was dispatched at Collingwood's own expense on a research trip to Sweden. What occurred there on that first visit and again in 1970 was to have a profound impact on Ivor's research: in the course of searching through the largely unsorted documents, he stumbled across literally hundreds of manuscripts not widely known to the history of mathematics community at large. These documents concerned not only Cantor and Weierstrass, but also many of their late-nineteenth-century contemporaries including Sofia Kovalevskaya, Henri Poincaré, Philip Jourdain and Bertrand Russell. The publication of his findings in 1971 helped establish for Ivor a reputation, both for his ability to locate archival sources and for his eagerness to share that knowledge with the wider scholarly community.

His association with Collingwood now led him to meet a woman who would influence his career for many years. Cecily Tanner was born in Göttingen in 1900 as Rosalind Cecilia Hildegard Young, the daughter of William and Grace Chisolm Young, the British mathematicians who had championed the new set and measure theory in the opening decades of the twentieth century. Like her parents, she had studied mathematics at Cambridge and like them she was an analyst, receiving a PhD on the theory of Stieltjes integration in 1929. She had collaborated with Collingwood when they were both working at the University College of Wales in Aberystwyth in the 1920s, and for much of her career she had been an active researcher in the history of mathematics, retiring from a lectureship in mathematics at Imperial College London in 1967. The Youngs had been good friends with Cantor, and Tanner was still in touch with his grandchildren; it was via this link that Ivor was granted access to Cantor's Nachlass, which involved taking a trip behind the Iron Curtain to Halle University in East Germany in 1969. Ivor's friendship with Tanner also led to his detailed study of the mathematical collaboration of her parents, the first of its kind in the history of mathematics.

The network of connections continued to grow wider, and more international. For many years, via her German connections, Tanner had been one of the few non-Germans invited to attend J E Hofmann's annual meetings on the history of mathematics at the Mathematisches Forschungsinstitut in Oberwolfach, which had been running since 1954. At her request, Ivor received an invitation to the meeting of 1969 and his participation at that meeting marked the beginning of a new phase in his career: not only was he introduced to a whole new community of historically-minded mathematicians, able to impart advice on research and point him in new directions, but that community was also to be the source of a number of lifelong friendships, perhaps most notably with Christoph Scriba. The Oberwolfach meetings also marked the start of a truly international career as Ivor soon became a familiar figure at conferences first in Europe, then America, then around the world. For the best part of the next forty years, Ivor circled the globe, visiting archives, attending conferences and giving over five-hundred invited lectures, or ‘gigs’ as he often used to call them. As he later put it: ‘My research life has always centred largely on two kinds of journey: to the centre of London, to use the libraries; and to a station or airport, to travel abroad and tell foreigners about the findings and give lecture courses to their students’. But throughout his career, it was the intellectual immersion, relaxed collegiality and breathtaking scenery of Oberwolfach that he always viewed with a special fondness, and he and Enid would be regular fixtures at the history of mathematics meetings there for over three decades.

Back home however, no such community of historians of mathematics existed. The British history of science community was largely apathetic—and indeed the very subject was still viewed by most practicing mathematicians in Britain merely as something to do in one's retirement. So when, at the beginning of the 1970s, John Dubbey and Arthur Morley proposed the creation of a learned society specifically devoted to the history of mathematics, Ivor considered the idea to be ‘hopelessly optimistic’. Nevertheless, on 2 July 1971, he was one of five speakers at a special meeting held at Thames Polytechnic (now the University of Greenwich) in London. This turned out to be the first meeting of the British Society for the History of Mathematics, now the oldest such society in the world. Elected its sixth president in 1986, he pushed through a formal constitution and forged stronger relationships with the international history of mathematics community, via invitations to speak at the society's meetings and active recruitment of members from overseas. As a result, by the end of his presidency in 1988, one quarter of the society's growing membership came from outside the United Kingdom.

Ivor's published output was so vast and so wide-ranging that it would take a monograph-length paper to provide a detailed survey; and though what follows is solely a summary of his contributions to the history of mathematics, it will be seen that even this is necessarily incomplete.

Ivor's research began in an area that was to permeate much of his work, particularly in the first half of his career: the history of the foundations of mathematics. Initially, as seen above, this was motivated by a dissatisfaction with the presentation of mathematics in his own education, and this led to an early focus on the development of analysis with the work of Fourier and Cauchy. His first book, The development of the foundations of mathematical analysis from Euler to Riemann (1970), centred on an investigation of the mathematics of Cauchy and his contemporaries, including Abel and Dirichlet. It also contained a valuable appendix on the history of convergence tests, which still remains a useful resource. His joint work with Ravetz on Joseph Fourier appeared in 1972. Up to this point, Fourier's key 1807 memoir ‘Sur la propagation de la chaleur’ only existed in manuscript form in Paris and all subsequent historical studies of Fourier had been forced to rely on the greatly expanded and amended published version, Théorie analytique de la chaleur (1822). Working in the Paris archives, Ivor studied over six-thousand pages of Fourier's manuscripts, including many in which the handwriting was barely legible, to come up with his annotated version. The previously unpublished 1807 memoir appeared in print for the first time in the collaborative book Joseph Fourier, 1768-1830, in which Ivor provided a detailed commentary on the paper and on related aspects of Fourier's life and work.

Ivor's consequent expertise, not only in the history of nineteenth-century analysis, but also in the history of post-revolutionary French mathematics, led to invitations to contribute several biographical studies to the Dictionary of scientific biography (DSB), then in the process of compilation by Charles Gillispie. In addition to the entries on Fourier, Laurent, Mathieu, Riesz and Stäckel, Ivor also wrote a section on the history of the Laplace transform for Gillispie's book-length biography of Laplace, first published in volume 15 of the DSB in 1978, and later re-published in book form by Princeton University Press.

Work on French mathematics continued through the 1980s, much of which was occupied by the preparation of what was arguably his magnum opus: the three-volume Convolutions in French mathematics, 1800–1840: from the calculus and mechanics to mathematical analysis and mathematical physics, published in 1990. This one thousand six hundred-page work remains the most comprehensive study to date of early nineteenth-century French mathematics and is a veritable treasure trove of previously unpublished sources, biographical and institutional information and reference. It treated the development of the mathematical sciences in France not as a sequence of great leaps, or ‘revolutions’, nor as a story of gradual progress, ‘evolution’, but rather as a series of twists and turns; hence the ‘convolutions’ of the title. Rejecting the one-dimensional internalist approach to the history of mathematics, Ivor situated the mathematical developments within the Parisian mathematical and broader French scientific milieu, so that publication venues, institutional histories, and developments in higher education were all integral to the main narrative. Consequently, since people like Cauchy, Fourier, and Laplace did not operate in isolation it was not just the famous names that featured, but a plethora of minor (but significant) figures were also interwoven into the account to provide a fuller picture of the intellectual environment. Finally, Ivor avoided the common error of viewing mathematics as ‘pure mathematics’ by treating developments in analysis, calculus, mechanics, heat diffusion, elasticity, electricity and magnetism, and optics as interrelated components of a cohesive field of study. This inclusive attitude to mathematics, or rather, ‘the mathematical sciences’, was yet another characteristic feature of Ivor's work.

The path which had led from the history of analysis to that of set theory via Georg Cantor, soon led to an interest in the life and work of Bertrand Russell, to whom in later life he developed a strong physical resemblance. Ivor was one of the very few people who actually attempted to read Russell and Whitehead's Principia mathematica and was familiar not only with its content, but had also researched the tortuous process of its creation. Again, substantial archival research produced significant contributions to our understanding of Russell's mathematical work, including a reconstruction of the reasoning which led to Russell's 1901 discovery of his paradox, and an edition of his correspondence with Philip Jourdain, Dear Russell—Dear Jourdain, published in 1977. Ivor's The search for mathematical roots, 1870–1940: logics, set theories and the foundation of mathematics from Cantor through Russell to Gödel, which appeared in 2000, was the culmination of decades of immersion in the collections of the Russell archive. This book traced the development of a variety of logical systems in a wide range of national settings, with the work of Russell forming the centrepiece of the study. Key features included the de-emphasis on Frege as an influence on Russell, with more focus on the stimulus of the Peano school. Ivor also made an important distinction between practitioners of what he called ‘algebraic logic’—namely those who used mathematical notation to facilitate the study of logic (for example, Boole, De Morgan, Schröder)—and those of ‘mathematical logic’ (for example, Frege, Russell, Whitehead)—who sought to use logic to base the study of mathematics—and remarked upon the distinct lack of communication between the two sides.

The second half of his career was dominated by work on edited volumes and works of reference. One of the first of these was his remarkable Companion encyclopedia of the history and philosophy of the mathematical sciences (1994). Featuring one hundred and seventy-five contributions from no fewer than one hundred and thirty-four authors, this two-volume work is a masterpiece of editorial organization and a valuable source of articles on many topics completely ignored by other compendia. Ivor's general survey of the history of the mathematical sciences, The rainbow of mathematics, appeared as the sixth volume of the Fontana History of Science series in 1997. Like the Encyclopedia, this substantial work distinguished itself from many of its predecessors by its concentration on post-1800 mathematical developments, its inclusion of important but lesser-known figures, its consideration of national characteristics in the practice of mathematics, and its emphasis on the importance of mathematical applications as a stimulus of mathematical progress. A final tour de force was delivered in 2005 with the publication of his Landmark writings in western mathematics 1640-1940, a thousand-page collection of seventy-seven chapters by sixty-five authors surveying eighty-nine important mathematical ‘writings’, from Descartes’ Géométrie to Hilbert and Bernays’ Grundlagen der Mathematik. All these works codified vast amounts of information in accessible formats, combining first-rate scholarship with deft editorial skill to produce valuable and high quality works of reference.

This editorial skill had been honed over many years as a journal editor, for in addition to serving on the editorial board of Historia Mathematica and a number of other journals for much of his career, Ivor rescued the ailing Annals of Science from pending extinction in 1974, serving as its editor-in-chief until 1981. He also founded a new journal, History and Philosophy of Logic in 1979, which he edited until 1992. When the project to update the century-old Dictionary of national biography (DNB) was announced in 1994, Ivor was the obvious choice for the associate editor in charge of mathematics and statistics, and when the Oxford DNB finally appeared ten years later, the scores of articles on mathematicians and statisticians had all been commissioned, edited and in some cases, written, by him.

Ivor's work found many admirers, but it was not without controversy. His assertion that Cauchy had plagiarised key ideas in analysis from Bolzano's 1817 paper on the intermediate value theorem was hotly disputed, while a later claim that the substitutional theory of classes and relations formed the missing link between Russell's theory of denoting and the Principia mathematica also proved contentious. But in truth, Ivor never intended his work to be the last word on any subject, merely to provide the basis for further work; and if he stirred up debate along the way, that would serve as the impetus for future research on the subject. Elements of his style may not have been to everyone's taste, but in his opinion it was far better for the discipline if his work was criticised than ignored. Thus, whether they agreed with him or not, subsequent historians of mathematics were forced to engage with his work and respond to his conclusions.

His published output was characterised by the use of vast amounts of archival research, the meticulous study of primary sources, often in several languages, and the fusion of a wide-range of mathematical ideas. Reoccurring themes included an intense belief in the use of the history of mathematics in mathematics education—the influence of which has begun to appear in recent years; a conviction that the history of mathematics without reference to its applications is incomplete; and an insistence on the distinction between the notions of ‘history’ and ‘heritage’ when discussing mathematical developments. By the latter term he referred to a view of history held by many mathematicians, which centres on questions like ‘how did we get here?’ as opposed to the concern of most historians, which tends to focus more on asking ‘what happened in the past?’ This distinction, as well as its many ramifications, was a subject of immense importance to Ivor and, after many years of gestation, formed the subject of one of his last major papers.

Institutionally, Ivor remained for his whole career at the Enfield College of Technology, although it underwent substantial changes in that time. In 1973, the college merged with two other north London schools to form Middlesex Polytechnic which, as a new and growing institution, soon sought to improve its research profile. To complement his own research, Ivor was encouraged to supervise doctoral students. Beginning with Tony Crilly, Ivor went on to successfully supervise nine PhD theses, from Crilly's study of the mathematics of Arthur Cayley to Abhilasha Aggarwal's survey of higher mathematics education in British India in 2007. The content of many of these theses, particularly that of Maria Panteki on the relationships between algebra, differential equations and logic in early nineteenth-century Britain, also influenced the course of his own research and the breadth of his publications spread still further.

As a research supervisor, Ivor was unparalleled. Despite what may have appeared to those who did not know him well as an intense and rather brusque persona, he was in fact a tremendously kind and generous man gifted with a powerful intellect and love of his subject. From the moment work began, the student would be showered with papers, articles, newspaper cuttings and other material relevant to their research. Ivor was a consummate professional and expected his students to be likewise, encouraging active participation in meetings, seminars and conferences, and fostering connections with others in the history of mathematics community. Regular weekly or fortnightly meetings with him, usually over lunch near an archive in central London, were the student's opportunity to discuss their work with Ivor and occasionally reveal any discoveries they had made. Such revelations were frequently accompanied by Ivor's wide-eyed exclamations of ‘Really??’ To the novice researcher, such expressions of excitement and interest, the free acknowledgement of gaps in his own knowledge, and the willingness to learn from his students were the source of tremendous encouragement and inspiration. Beyond the PhD, Ivor continued to maintain close and friendly relationships with all of his former graduate students. Indeed, for Ivor, who had no children of his own, this small group of people came to form what was, for him and Enid, almost a surrogate family.

Ivor was a man of many eccentricities. In later years he was wont to appear to fall asleep in seminars and conference talks, particularly after lunch, only to wake up and ask a penetrating question at the end. An erratic driver, car journeys with Ivor were always memorable, not least for having survived them. Tony Crilly recalls being driven back by Ivor after a meeting in Oxford at high speed so that Ivor could get home in time for two editing sessions before bed! In written communications, Ivor's distinctive scrawl was a challenge even to those well trained in reading barely legible manuscripts, and the advent of e-mail only heightened the need for one's cryptographic skills to decipher his messages. One example (of many) was in reply to an earlier message and read simply:

v god. tanks for rude. no way to get plan to pairs. iv

Translation: ‘Very good. Thanks for the ride. On my way to get a plane to Paris. Ivor.’

During a long and distinguished career, Ivor was accorded many honours. In 1978 the University of London bestowed the rare distinction of awarding him the prestigious DSc, its highest doctoral degree. Elected a membre effectif of the Académie internationale d'histoire des sciences in 1991, he also held visiting positions at the Institute for Advanced Study in Princeton, Monash University, and the University of Western Australia. In 1992, Middlesex Polytechnic was granted university status and became Middlesex University. The following year, Ivor, who by this time had progressed from Senior Lecturer to Reader in Mathematics, was appointed Professor of the History of Mathematics and Logic at the university. He would hold this title until his retirement in 2002. The two honours that perhaps afforded him the most pleasure came late in life: in 2009 he received the Kenneth O May Prize and Medal from the International Commission for the History of Mathematics for ‘lifetime scholarly achievement and commitment to the field’, followed by his election to honorary membership of the Bertrand Russell Society in 2010.

The first few years of retirement saw little change in Ivor's output as his research continued at its usual rapid pace. Only with the diagnosis of Parkinson's disease in 2007 and a heart bypass operation the following year did the number of meetings he could attend begin to decrease and the time taken to prepare his papers begin to increase. But he was as responsive as ever to requests for comments on first drafts, with work returned almost immediately with references and comments. His final research paper for Historia Mathematica appeared in February 2014, and his last paper was accepted for publication the week before he died. It was only in the last few weeks of his life that he began to find working at the computer too much of a strain. But his fascination with the history of mathematics was undiminished to the end.

In Britain he leaves the dual legacy of an established professional discipline and a learned society devoted to it, both non-existent at the start of his career and in both of which he played foundational roles. To the Archives of American Mathematics he leaves his Nachlass. And to scholars around the world, he leaves his vast array of publications—an oeuvre on which future generations of historians of mathematics will now continue to build.

Acknowledgments

The author wishes to thank June Barrow-Green, Tony Crilly, Joseph Dauben, Enid Grattan-Guinness, Niccolò Guicciardini, Albert Lewis, and Karen Parshall for their valuable help and advice during the preparation of this article.