Peter M Neumann OBE (1940–2020)

Introduction

The history of mathematics has lost one of its major champions and a great scholar and personality in the death on 18 December 2020 of Peter M Neumann, Emeritus Fellow of The Queen's College, Oxford, and past President of the British Society for the History of Mathematics.

Peter (which he always signed with a ‘Π’) was the son of mathematicians Bernhard and Hanna Neumann, whom he followed into a career of research in abstract algebra (principally group theory). He was born in Oxford on 28 December 1940, but was raised and educated in Hull, and entered The Queen's College, Oxford, as an undergraduate in 1959. He graduated BA in 1963, by which time he had already begun his career in mathematical research with a paper on wreath products and varieties of groups, co-authored with his parents (subsequently known as the ‘3N’ paper: Neumann et al. 1962). Peter spent a ‘year abroad’ as a Senior Scholar at Merton College, Oxford in 1963–1964, before returning to Queen's as a Junior Research Fellow. He went on to complete a DPhil dissertation entitled A study of some finite permutation groups under the supervision of Graham Higman FRS in 1966. Peter was elected a Tutorial Fellow of Queen's in the same year, and remained a much-loved member of the college for the rest of his life. In 1967, he was appointed to a University Lectureship in Oxford, and continued to teach pure mathematics to undergraduates, both in college and in the Mathematical Institute, well into retirement.¹

During his long career, Peter served in numerous capacities within the wider mathematical community: as well as being Chair of the UK Mathematics Trust (1996–2004), he was heavily involved in the publications of the London Mathematical Society, serving also as that society's Vice President (1990–1992); in addition, he was an IMU representative on the Executive Committee of the International Commission on the History of Mathematics (2007–2010), and was President of both the Mathematical Association (2015–2016) and the British Society for the History of Mathematics (2000–2002). Peter received a DSc degree from the University of Oxford in 1976, and in 1987 he was presented with the Lester R Ford Award of the Mathematical Association of America for expository excellence for a review of Harold M Edwards' Galois theory (1984), published in The American Mathematical Monthly, and incorporating historical comments on the development of Galois theory. In 2003, he received a Senior Whitehead Prize from the London Mathematical Society, and in the 2008 New Year Honours List was appointed Officer of the Order of the British Empire (OBE) for services to education. In that same year, he received a Lifetime Teaching Award from the University of Oxford. This was followed in 2012 by the IMA–LMS David Crighton Medal for services to mathematics and to the mathematical community. In 2016, Peter was awarded an honorary DSc degree by the University of Hull.

Mathematics

Peter was internationally known and greatly respected for his contributions to mathematics, to its history, and to education. He achieved eminence in all three fields with works of lasting influence. Peter's mathematical publications, comprising around 100 papers and books (he placed the emphasis on the former: Peter was of the opinion that there are already too many books in the world), spanned permutation groups, combinatorics, and computational group theory. As Cheryl Praeger and Martin Liebeck observe:

Each of his publications is beautifully crafted, and its place in mathematics carefully thought out and explained, together with insightful comments on where further work might lead. (Praeger and Liebeck 2021, 65)

Particularly noteworthy is his joint work with Cheryl Praeger in computational group theory on a recognition algorithm for special linear groups, as well as work co-authored with S A Adeleke on a general theory of tree-like relational structures, which led to new lines of enquiry concerning infinite permutation groups, model theory, and graph theory. However, I leave a fuller description of his mathematical works for other, more competent commentators,² and focus here on his work in the history of mathematics.

The history of mathematics

With regard to historical work, Peter defined himself as a ‘mathematician historian’: a scholar who is trained and operates primarily as a mathematician but who has developed a real interest in the history of their subject. More precisely, he considered himself to be 80% mathematician, 20% historian (Neumann 2008, 169). Peter's interest in the history of mathematics was largely connected with the development of group theory. An awareness of the historical background to the mathematics that he was studying was evident at the very start of his career: in one of the final sections of his DPhil dissertation, headed ‘An unfortunate theorem of Cauchy …’ (Neumann 1966, §IV.13), Peter provided counterexamples to a ‘theorem’ that Cauchy had published without proof in a paper of 1845 (see also Neumann et al. 1968). Indeed, a perusal of Peter's list of mathematical publications suggests a continued interplay between mathematics and history throughout his career: as well as being at the cutting edge of modern group theory, Peter's work was also on occasion inspired by historical problems. One piece of work, for example, was drawn from an 1873 paper of Émile Mathieu (described by Peter as ‘very enjoyable’: Neumann 1976, 52), whilst another solved a problem first posed by Gottlob Frege in 1903 (Adeleke et al. 1987). In each case, the original papers are cited fully, and as the comment about Mathieu shows, these were no mere token references. Peter's papers (mathematical and historical alike) were written with the rigour of a mathematician but also with the precision of a historical scholar, leading one contributor to Mathematical Reviews to observe in connection with one of Peter's papers:

The reviewer would also like to think that the author's example could infect other research mathematicians to be a little more scholarly in their attributions. (MR0562002)

In considering mathematical work inspired by historical problems, I cannot omit Peter's treatment of Alhazen's Problem: given a spherical mirror and points A, B in space, how can a point P on the mirror be found, where a ray of light is reflected from A to B via P? Peter proved that the problem, considered by Ibn al-Haytham (Alhazen) in the 11th century but first formulated by Ptolemy in 150 CE, cannot be solved by ruler and compass constructions (Neumann 1998).³

The Gaussian maxim ‘few but ripe’ might readily be applied to Peter's publications in the history of mathematics. Much of Peter's work on the history of group theory centred around particular individuals in its development, the first of whom was William Burnside. Although Peter's first fully historical paper (‘A lemma that is not Burnside's’) was one that traced the history of Burnside's Lemma in group theory and ended up concluding that it ought not to be attributed to Burnside but rather to Cauchy and Frobenius, his interest in Burnside's contributions to group theory remained over many years. Along with Tony Mann and Julia Tompson, Peter co-edited the two volumes of Burnside's collected papers (Burnside 2004).

Other figures in the development of group theory who received attention from Peter were Helmut Wielandt, Issai Schur, and Thomas Penyngton Kirkman,⁴ but it was the two early nineteenth-century French pioneers of the mathematical notion of a group who were the subjects of his most detailed historical work: Augustin-Louis Cauchy and Évariste Galois. Peter's writings on the two are characterized by the enormous precision and meticulous attention to detail that he brought to all of his work. A full understanding of the place of Cauchy's ideas within the overall story, for example, required the construction of an exhaustive chronology of his writings, which Peter duly supplied.

But nowhere is the precision of Peter's scholarship more clearly in evidence than in his comprehensive edition of Galois's mathematical writings, the first full English edition, carefully compiled from Parisian archive materials (in part during the ‘gap year’ that Peter took upon retirement), and published in 2011 to coincide with Galois's 200th birthday. Transcriptions of the original French appear alongside English translations, with multiple levels of mathematical and historical commentary, all set within the context of critiques of prior studies of Galois's works. This is a truly great piece of historical scholarship, which has opened up Galois's writings to a wider mathematical readership.⁵

Elsewhere, Peter dealt with broader themes within the history of group theory: for example, the emergence of the now-accepted definition of a group, and the twentieth-century classification of finite simple groups.⁶ In a lecture delivered at the BSHM's Research in Progress meeting in 2008, he used the development of the notion of a group as a lens through which to view the difficulties facing the historian who would treat mathematical topics. In other writings, Peter's long connection with mathematics in Oxford made him ‘the obvious person’ to supply a new chapter for the second edition of the volume Oxford figures (Fauvel et al. 2000), bringing the history of Oxford mathematics up to date –‘and of course he did a superb job on it’.⁷

That a mathematician of Peter's stature should engage in historical work so fully and rigorously has surely encouraged other mathematicians to take the discipline more seriously. It seems appropriate to end this very brief survey of Peter's historical works with some comments of his own on the nature of research in the history of mathematics. At the end of the Research in Progress lecture mentioned above, he expressed two hopes in particular: ‘that mathematicians can appreciate that history is hard and exciting’, but also ‘that historians of mathematics will always appreciate the need to engage with the mathematics’ (Neumann 2008, 177).

The BSHM

Peter's interest in the history of mathematics went hand-in-hand with his long-standing support of the BSHM. I have already mentioned his presidency of the society (2000–2002). Indeed, if one browses old copies of the BSHM Newsletter, Peter's name is clearly in evidence throughout, often as an organizer of meetings, and as a frequent speaker. Most notably, he was for many years the organizer (often in collaboration with his great friend Jackie Stedall or with others) of ‘Research in Progress’, the society's annual meeting for research students in the history of mathematics. Instituted in 1992, these meetings have been held in Oxford since 1998, usually at The Queen's College, where Peter served as a genial host and encourager of young scholars.⁸

Peter gave several talks at BSHM meetings. The earliest, with a title typical of Peter's wry humour, was ‘Looking at nineteenth-century group theory with the disadvantage of a twentieth-century education’, delivered at a meeting at the University of Nottingham in July 1978. Other lectures over the following decades covered the work of Burnside, Galois, Arthur Cayley, and the use of history in the teaching of mathematics. Two of the most recent were delivered at Gresham College in London: ‘The Memoirs and Legacy of Évariste Galois’ (2011) and ‘Hanna Neumann: A Mathematician in Difficult Times’ (2015).⁹

In 2008, Peter made a generous donation to the BSHM to found a prize for books dealing with the history of mathematics and aimed at a broad audience; the society subsequently named the prize in Peter's honour, as ‘a longstanding supporter of and contributor to the BSHM’. The Neumann Prize was awarded for the first time in 2009.

Teaching and outreach

For Peter , teaching was at least as important as research, and therefore something to be taken seriously and prepared for thoroughly. As a result, he became legendary as a teacher of mathematics, and is fondly remembered by his former undergraduate students, many of whom have gone on to mathematical careers, along with the 39 research students whose doctorates he supervised (just one full supervision in the history of mathematics: Nicholson 1993; and one co-supervision: Cretney 2015). Peter was also involved in the training of new generations of mathematics educators, and in the promotion of good practice in mathematics teaching: he was at one time in charge of university teacher training in Oxford (1995–1999), and served as Faculty Teaching Advisor within the Mathematical Institute (2009–2013).

Peter's lifelong commitment to teaching also extended to the history of mathematics: he was a co-founder, along with Raymond Flood, Jackie Stedall, and Robin Wilson, of the Oxford Mathematical Institute's undergraduate module in the history of mathematics. Owing much to the Open University's celebrated history of mathematics course MA290, the Oxford module emphasizes the use of primary sources, and here Peter was in his element. He was at home with the writings of the mathematicians of several centuries, and, in stressing the use of primary materials, was once heard to recommend Euclid's Elements to a student as a ‘rollicking good read’.

Peter was a passionate advocate of mathematics schools outreach, and was actively involved with the UK Mathematics Trust, serving as the first chair of its council, as well as enthusiastically setting problems for competitions and running events. Indeed, Peter's focus shifted towards this work during retirement. It is interesting to note that in his mathematics master classes for Year 10 students, he drew inspiration from the intersection of his mathematical and historical interests by having the students investigate a version of the Burnside Problem (Neumann 2014). For Peter, mathematics, history, and teaching were all but inseparable.

Personal note

I first met Peter at a mathematical conference in Porto, a few months before I came to Oxford as Jackie Stedall's postdoc, and it was here that I soon saw how kind and open he was towards others, particularly younger scholars, in whom he took a great interest. And this was also the first time that I had a taste of Peter the raconteur. As a postdoc, and indeed subsequently, I received much feedback from Peter on my work in the form of critiques that were always fair but firm, always constructive and encouraging. Over several years, and until a few months ago, my work has been much improved by Peter's input and his critical eye – a critical eye with a twinkle in it, of course.

During the time that I knew him, Peter was always someone to whom I could turn for advice, which would be given generously. And I was lucky to be able to interact with him in a range of contexts: research and conference organization, for example, as well as college life, and also teaching, whose importance was always stressed. I am conscious that I learnt an enormous amount from Peter in all of these different settings. I could not have had a better role model, as an academic and as a person – a view that will surely be shared by the generations of school children, university students, and other scholars whom he influenced. My abiding memory of Peter will be of someone who was a generous teacher in all things. Peter was a meticulous and insightful scholar and a kind supporter of others. He will be greatly missed by all who had the honour and delight of knowing him.

Peter's historical publications

I provide here what I hope to be a reasonably complete list of Peter's publications relating to the history of mathematics (including biographies, but excluding reviews¹⁰ and conference reports). I would be grateful for information about any omissions.

‘A lemma that is not Burnside's’, The Mathematical Scientist 4/2 (1979), 133–141.

‘Review: Galois Theory by Harold M. Edwards’, The American Mathematical Monthly 93/5 (1986), 407–411.

‘On the date of Cauchy's contributions to the founding of the theory of groups’, Bulletin of the Australian Mathematical Society 40/2 (1989), 293–302.

‘W. L. Ferrar [obituary]’, The Queen's College Record 6/6 (December 1990), 5–7.

‘Dom George Temple [obituary]’, The Queen's College Record 6/8 (December 1992), 12–16.

‘Obituary: William Leonard Ferrar’, Bulletin of the London Mathematical Society 26/4 (1994), 395–401 (with ME Rayner).

‘Helmut Wielandt on permutation groups’, in Helmut Wielandt, Mathematische Werke/Mathematical works. Vol. 1. Group theory (Bertram Huppert and Hans Schneider, eds), Berlin: Walter de Gruyter & Co, 1994, pp 3–20.

‘Brooke Benjamin [obituary]’, The Queen's College Record 7/1 (December 1995), 14–16.

‘A hundred years of finite group theory’, The Mathematical Gazette 80/487 (1996), 106–118.

‘What groups were: a study of the development of the axiomatics of group theory’, Bulletin of the Australian Mathematical Society 60/2 (1999), 285–301.

‘The life of Issai Schur through letters and other documents’, in Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progress in Mathematics 210, Boston, MA: Birkhäuser Boston, 2003, pp xlv–xc (with Walter Ledermann).

‘Frank Smithies, 1912–2002’, BSHM Newsletter 48 (Spring 2003), 1–2.

‘The context of Burnside's contributions to group theory’, in The collected papers of William Burnside. Vol. 1. Commentary on Burnside's life and work; papers 1883–1899 (Peter M Neumann, AJS Mann, Julia C Tompson, eds), Oxford: Oxford University Press, 2004, pp 15–37.

‘Introduction to Walter Ledermann's paper on Issai Schur's posthumous notes on elementary number theory’, BSHM Newsletter 50 = BSHM Bulletin 1 (Spring 2004), 16–17.

‘The concept of primitivity in group theory and the second memoir of Galois’, Archive for History of Exact Sciences 60/4 (2006), 379–429.

‘The history of symmetry and the asymmetry of history’, BSHM Bulletin 23/3 (2008), 169–177.

‘VI.33 Niels Henrik Abel’, in The Princeton Companion to Mathematics (Timothy Gowers, June Barrow-Green, Imre Leader, eds), Princeton: Princeton University Press, 2010, pp 760–762.

‘VI.41 Évariste Galois’, in The Princeton Companion to Mathematics (Timothy Gowers, June Barrow-Green, Imre Leader, eds), Princeton: Princeton University Press, 2010, pp 767–768.

‘VI.58 Ferdinand Georg Frobenius’, in The Princeton Companion to Mathematics (Timothy Gowers, June Barrow-Green, Imre Leader, eds), Princeton: Princeton University Press, 2010, pp 783–784.

‘VI.60 William Burnside’, in The Princeton Companion to Mathematics (Timothy Gowers, June Barrow-Green, Imre Leader, eds), Princeton: Princeton University Press, 2010, pp 785–785.

The mathematical writings of Évariste Galois, Heritage of European Mathematics, Zürich: European Mathematical Society, 2011.

‘Galois and his groups’, European Mathematical Society Newsletter 82 (2011), 29–37.

‘The editors and editions of the writings of Évariste Galois’, Historia Mathematica 39/2 (2012), 211–221.

‘Recent developments’, in Oxford figures: Eight centuries of the mathematical sciences (John Fauvel, Raymond Flood, Robin Wilson, eds), 2nd ed, Oxford: Oxford University Press, 2013, pp 337–358.

‘Jacqueline Anne Stedall (4 August 1950–27 September 2014)’, Historia Mathematica 42/1 (2015), 5–13.

‘Jacqueline Anne Stedall, 1950–2014’, Oxford Mathematics Newsletter 14 (Spring 2015), 3.

‘Jacqueline Anne Stedall [obituary]’, The Queen's College Record 9/1 (December 2015), 81–83.

‘Inspiring teachers’, The Mathematical Gazette 100/549 (2016), 385–395.

‘A professor at Greenwich: William Burnside and his contributions to mathematics’, in Mathematics at the meridian. The history of mathematics at Greenwich (Raymond Flood, Tony Mann, Mary Croarken, eds), Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series, Boca Raton, FL: CRC Press, 2020, pp 139–153.

Acknowledgments

I am very grateful to Philip Beeley, Howard Emmens, Raymond Flood, Tony Mann, Ursula Martin, Adam McNaney, Richard Parkinson, Karen Parshall, Cheryl Praeger, Adrian Rice, Brigitte Stenhouse, and Robin Wilson for their feedback on an earlier version of this obituary. Thanks are also due to Tessa Shaw of The Queen's College Library for her help in getting hold of some of Peter's publications. This obituary reuses material from Hollings (2021) and from a tribute to Peter on The Queen's College website: https://www.queens.ox.ac.uk/news/peter-m-neumann-obe-1940-2020. It also draws upon other published tributes and obituaries (Bridson 2021; Cullingworth 2021; McBride 2021; Neumann 2021; Praeger and Liebeck 2021), as well as Peter's own biographical notes on his college webpage, which was still live at the time of writing. Further tributes to Peter may also be read on the blogs of Peter Cameron (https://cameroncounts.wordpress.com/2020/12/24/memories-of-peter-neumann/), Tony Mann (https://tonysmaths.blogspot.com/2020/12/memories-of-peter-neumann.html?m=1), and Ken Regan (https://rjlipton.wpcomstaging.com/2021/01/01/peter-m-neumann-1940-2020/) (all URLs here last accessed 29 March 2021). The image of Peter is reproduced by kind permission of the Provost, Fellows, and Scholars of The Queen's College, Oxford.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 Although, as Martin Bridson has noted, ‘“Retirement” was not the most apt of words in Peter's case’ (Bridson 2021, 19).

2 See, in particular, the forthcoming obituary by Cheryl Praeger and Martin Liebeck in the Bulletin of the London Mathematical Society.

3 See also Highfield (1997), though as Peter himself observed, the title of the latter is not quite accurate.

4 Peter's work on the latter was sadly not completed.

5 Having made extensive use of Peter's edition of Galois's writings prior to arriving in Oxford, a visiting student was star-struck upon meeting Peter for the first time, and later confided that it was ‘like meeting a movie star’. When this assessment was relayed to Peter, he dismissed it with a modest harrumph, but was clearly rather chuffed.

6 A mathematical lecture entitled ‘A breakthrough in algebra: Classification of the finite simple groups’ that Peter gave on this topic in 1992 is available online: https://www.youtube.com/watch?v=s88bfJzyA78 (last accessed 24 March 2021).

7 Robin Wilson, private communication, 19 March 2021.

8 Alongside strictly BSHM-related activities, Peter was also the founder of the monthly Oxford History of Mathematics Forum, which was subsequently run by Jackie Stedall, and now by Robin Wilson and the author.

9 These lectures are available online: https://www.gresham.ac.uk/lectures-and-events/the-memoirs-and-legacy-of-evariste-galois and https://www.gresham.ac.uk/lectures-and-events/hanna-neumann-a-mathematician-in-difficult-times (both last accessed 24 March 2021).

10 With one exception – for reasons indicated above.