Research papers

'Oxford mathematics at a low ebb? An 1855 dispute over examination results', to appear in Annals of Science [Open Access]

Between December 1855 and March 1856, a public dispute raged, in British national newspapers and locally published pamphlets, between two teachers at the University of Oxford: the mathematical lecturer Francis Ashpitel and Bartholomew Price, the professor of natural philosophy.  The starting point for these exchanges was the particularly poor results that had come out of the final mathematics examinations in Oxford that December.  Ashpitel, as one of the examiners, stood accused of setting questions that were too difficult for the ordinary student, with the consequence that, in Price’s view, further mathematical study in Oxford – never as robust as in Cambridge – would be discouraged.  We examine this short-lived affair, and use it not only to gain insight into the status of mathematical study in Oxford in the mid-nineteenth century, but also to point towards the increasing importance of competitive examinations in British public life at that time.

'T. E. Peet, a mathematician among Egyptologists?' (with R. B. Parkinson), in Research in History and Philosophy of Mathematics: The CSHPM 2021 Volume (Maria Zack and David Waszek, eds.), Annals of the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques, Cham: Birkhäuser, 2023, pp. 183–198

During the 1920s, the study of ancient Egyptian mathematics was particularly vigorous, with the emergence of new sources, and new editions of old ones.  A central figure, and virtually the only professional Egyptologist in this activity, was Thomas Eric Peet (1882–1934).  Before embarking on an archaeological career, Peet had studied mathematics at university, which probably accounts for the major interest that he subsequently took in Egyptian mathematics.  Although he never pursued mathematics professionally, he sometimes referred to himself as a 'mathematician' presumably as a means of distinguishing himself among Egyptologists.  Elsewhere, however, he used the word in a much less complimentary way to refer to those scholars who took a mathematically-focused and uncontextualised approach to the study of ancient mathematics.  This article examines three different ways in which Peet employed the word `mathematician', thereby illuminating both his own career trajectory and self-presentation, and also the way in which different disciplines interacted within the study of ancient mathematical texts.

'Contrasting aims and approaches in the study of ancient Egyptian mathematics in the 1920s' ('Objectifs et approches contrastés dans l’étude des mathématiques de l’Égypte ancienne dans les années 1920') (with R. B. Parkinson), Revue d'histoire des mathématiques 28(2) (2022), 183–286

The modern academic study of ancient Egyptian mathematics emerged in the mid-nineteenth century as the decipherment of ancient texts revealed the arithmetical and geometrical notions and processes employed by the ancient Egyptians; most of what is now known stemmed from the discovery and study of the Rhind Mathematical Papyrus in the 1860s and 1870s.  However, despite the unearthing of a small number of additional sources, the study of ancient Egyptian mathematics remained quite closely focused on the Rhind Papyrus, with many texts simply restating what had already been written about it.  In this paper, we discuss how the topic re-emerged in the 1920s in a more fully contextualised form.  Particular attention is paid to the contributions of the Egyptologist Thomas Eric Peet (1882–1934) and the historian of mathematics Otto Neugebauer (1899–1990).  We argue that by the end of the 1920s, a topic that had hitherto largely been the preserve of Egyptologists had passed into the hands of mathematicians.

L’étude des mathématiques de l’Egypte ancienne s’est constituée en champ académique au milieu du XIXe siècle lorsque le déchiffrement des textes anciens a révélé les notions et les processus arithmétiques et géométriques employés par les anciens Égyptiens; la majorité de nos connaissances actuelles découle de la découverte et de l’étude du papyrus mathématique Rhind dans les années 1860 et 1870.  Cependant, malgré la découverte d’un petit nombre de sources supplémentaires, l’étude des mathématiques de l’Égypte ancienne est restée assez étroitement centrée sur le papyrus Rhind, de nombreux textes ne faisant que reprendre ce qui avait déjà été écrit à son sujet.  Dans cet article, nous discutons de la façon dont le sujet a réémergé dans les années 1920 sous une forme plus pleinement contextualisée.  Une attention particulière est accordée aux contributions de l’égyptologue Thomas Eric Peet (1882–1934) et de l’historien des mathématiques Otto Neugebauer (1899–1990). Nous soutenons qu’à la fin des années 1920, ce sont les mathématiciens qui se sont saisi de ce sujet, qui avait jusque-là été largement l’apanage des égyptologues.

'Two letters from Otto Neugebauer to Thomas Eric Peet on ancient Egyptian mathematics' (with R. B. Parkinson), Historia Mathematica 52 (2020), 66–98

We describe two letters of 1926 from the historian of mathematics Otto Neugebauer to the Egyptologist Thomas Eric Peet (Griffith Institute Archive, Oxford: Peet MSS 4.9).  The letters concern Neugebauer's study of certain parts of the Rhind Mathematical Papyrus, as found in the doctoral dissertation that he had completed in Göttingen that year, for which Peet's 1923 edition of the papyrus was a major source.  In the letters, we see the young Neugebauer establishing himself within the wider Egyptological community.  The letters open up a discussion of Peet's own work on ancient Egyptian mathematics.

Wir beschreiben zwei Briefe des Mathematikhistorikers Otto Neugebauer an den Ägyptologen Thomas Eric Peet aus dem Jahr 1926 (Griffith Institute Archive, Oxford: Peet MSS 4.9). Diese Briefe handeln von Neugebauers Studien über bestimmte Passagen des mathematisches Papyrus Rhind, die er in seiner Doktorarbeit aus demselben Jahr vornahm. Neugebauer hatte nämlich als Hauptquelle auf Peets Edition des Papyrus von 1923 zurückgegriffen. In diesen Briefen können wir beobachten, wie sich der junge Neugebauer im erweiterten Kreis der Ägyptologen etabliert. Die Briefe eröffnen zugleich eine Diskussion über Peets eigene Arbeiten zur altägyptischen Mathematik.

'The early mathematical education of Ada Lovelace' (with Ursula Martin and Adrian Rice), BSHM Bulletin: Journal of the British Society for the History of Mathematics 32(3) (2017), 221–234  [Open Access]

Ada, Countess of Lovelace, is remembered for a paper published in 1843, which translated and considerably extended an article about the unbuilt Analytical Engine, a general-purpose computer designed by the mathematician and inventor Charles Babbage. Her substantial appendices, nearly twice the length of the original work, contain an account of the principles of the machine, along with a table often described as “the first computer program”. In this paper we look at Lovelace’s education before 1840, which encompassed older traditions of practical geometry; newer textbooks influenced by continental approaches; wide reading; and a fascination with machinery. We also challenge judgements by Dorothy Stein and by Doron Swade of Lovelace’s mathematical knowledge and skills before 1840, which have impacted later scholarly and popular discourse. 

'"Nobody could possibly misunderstand what a group is": a study in early twentieth-century group axiomatics', Archive for History of Exact Sciences 71(5) (2017), 409–481  [Open Access]

In the early years of the twentieth century, so-called 'postulate analysis' — the study of systems of axioms for mathematical objects for their own sake — was regarded by some as a vital part of the efforts to understand those objects.  I consider the place of postulate analysis within early twentieth-century mathematics by focusing on the example of a group: I outline the axiomatic studies to which groups were subjected at this time, and consider the changing attitudes towards such investigations.

'The Lovelace–De Morgan Mathematical Correspondence: A Critical Re-Appraisal' (with Ursula Martin and Adrian Rice), Historia Mathematica 44(3) (2017), 202–231  [Open Access]

Ada Lovelace is widely regarded as an early pioneer of computer science, due to an 1843 paper about Charles Babbage's Analytical Engine, which, had it been built, would have been a general-purpose computer. However, there has been considerable disagreement among scholars as to her mathematical proficiency. This paper presents the first account by historians of mathematics of the correspondence between Lovelace and the mathematician Augustus De Morgan from 1840–41. Detailed contextual analysis allows us to present a corrected ordering of the archive material, countering previous claims of Lovelace's mathematical inadequacies, and presenting a more nuanced assessment of her abilities.

Ada Lovelace wird generell als frühe Pionierin der Informatik angesehen. Dies vor allem wegen des 1843 erschienenen Artikels über Charles Babbages ‘Analytical Engine’, die, wäre sie damals gebaut worden, einen Allzweckcomputer dargestellt hätte. Allerdings gibt es beträchtliche Meinungsverschiedenheiten unter Historikern hinsichtlich Lovelaces mathematischer Kenntnisse. Dieser Artikel präsentiert den ersten Bericht von Mathematikhistorikern über die Korrespondenz der Jahre 1840–41 zwischen Lovelace und dem Mathematiker Augustus De Morgan. Detaillierte Kontextanalyse erlaubt es uns, eine korrigierte Anordnung des Archivmaterials vorzulegen, die bisherigen Meinungen über die mathematischen Unzulänglichkeiten von Lovelace entgegenwirkt und die eine nuanciertere Bewertung ihrer Fähigkeiten erlaubt.

'A tale of mathematical myth-making: E. T. Bell and the 'arithmetization of algebra'', BSHM Bulletin: Journal of the British Society for the History of Mathematics 31(1) (2016), 69–80

Throughout E. T. Bell's writings about mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the 'arithmetisation of algebra') would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell's own mathematical work. I investigate this apparent contradiction.

'Three approaches to inverse semigroups', European Journal of Pure and Applied Mathematics 8(3) (2015), 294–323

I give a historical survey of the three main approaches to the study of the structure of inverse semigroups. The first is that via inductive groupoids, as studied by Charles Ehresmann. The second concerns the notion of a fundamental inverse semigroup and its Munn representation. Finally, the third centres upon the concept of an E-unitary or proper inverse semigroup and its representation (due to McAlister) by a so-called P-semigroup.

'The acceptance of abstract algebra in the USSR, as viewed through periodic surveys of the progress of Soviet mathematical science', Historia Mathematica 42(2) (2015), 193–222

During the several decades of the USSR's existence, Soviet mathematicians produced, at intervals, a number of volumes of survey articles which provide us with a series of 'snapshots' of Soviet mathematics down the years. In this paper, I introduce these volumes as a resource for historians of Soviet mathematics, and consider the picture they paint of the development of abstract algebra in the USSR, paying particular attention to the aspects in which these surveys differ from later, retrospective accounts of Soviet algebra.

В течение нескольких десятилетий существования СССР советские математики с определенными интервалами выпустили несколько томов обзоров, которые дали целую серию “мгновенных снимков” советской математики за прошедшие годы. Настоящая статья имеет целью введение этих изданий в научный оборот в качестве источников по истории советской математики. В ней также исследуется создаваемая ими картина развития абстрактной алгебры в СССР. Особое внимание обращается на отличия этих обзоров от последующих ретроспективных отчетов о советской алгебре.

'Embedding semigroups in groups: not as simple as it might seem', Archive for History of Exact Sciences 68(5) (2014), 641–692

We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra. Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry. We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Øystein Ore. After disposing of A. K. Sushkevich's erroneous contribution in this area, we present A. I. Maltsev's example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general. We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Pták and Joachim Lambek. We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally.

'Investigating a claim for Russian priority in the abstract definition of a ring', BSHM Bulletin: Journal of the British Society for the History of Mathematics 29(2) (2014), 111–119

In this article, I scutinize an assertion that the Russian-Ukrainian mathematician S. O. Shatunovskii (1859–1929) should be credited with the first modern definition of a ring. Shatunovskii's claim is compared with that of Abraham Fraenkel, who defined a notion very close to the current concept of a ring in a paper of 1914.

'The struggle against idealism: Soviet ideology and mathematics', Notices of the American Mathematical Society 60(11) (2013), 1448–1458

This article provides a brief account of Soviet ideology of mathematics, beginning with a short introduction to the underlying philosophy of dialectical materialism, and then examining in turn the three distinct ideological phases identified by Alexander Vucinich: before, during and after Stalin's period in power.

'The Ehresmann–Schein–Nambooripad Theorem and its successors', European Journal of Pure and Applied Mathematics 5(4) (2012), 414–450

The Ehremann--Schein--Nambooripad Theorem expresses the fundamental connection between the notions of inverse semigroups and inductive groupoids, which exists because these concepts provide two distinct approaches to the study of one-one partial transformations. In the case of arbitrary partial transformations, the analogous two approaches are provided by restriction semigroups and inductive categories, the former being generalisations of inverse semigroups, and the latter of inductive groupoids. There is indeed also a generalisation of the Ehremann-Schein-Nambooripad Theorem which encapsulates the connection between these two more general objects. In this article, we will explore the origins of these theorems, and survey the basic theory surrounding them.

'Tropical matrix duality and Green's D relation' (with Mark Kambites), Journal of the London Mathematical Society 86(2) (2012), 520–538

We give a complete description of Green's D relation for the multiplicative semigroup of all n × n tropical matrices. Our main tool is a new variant on the duality between the row and column space of a tropical matrix (studied by Cohen, Gaubert and Quadrat and separately by Develin and Sturmfels). Unlike the existing duality theorems, our version admits a converse, and hence gives a necessary and sufficient condition for two tropical convex sets to be the row and column space of a matrix. We also show that the matrix duality map induces an isometry (with respect to the Hilbert projective metric) between the projective row space and projective column space of any tropical matrix, and establish some foundational results about Green's other relations.

'On conditions for constellations', International Electronic Journal of Algebra 10 (2011), 1–24

A constellation is a set with a partially-defined binary operation and a unary operation satisfying certain conditions, which, loosely speaking, provides a 'one-sided' analogue of a category, where we have a notion of 'domain' but not of 'range'. Upon the introduction of an ordering, we may define so-called inductive constellations. These prove to be a significant tool in the study of an important class of semigroups, termed left restriction semigroups, which arise from the study of systems of partial transformations. In this paper, we study the defining conditions for (inductive) constellations and determine that certain of the original conditions from previous papers are redundant. Having weeded out this redundancy, we show, by the construction of suitable counterexamples, that the remaining conditions are independent.

'Simplified conditions for type-I and type-II monoids', Acta Scientiarum Mathematicarum (Szeged) 77(1-2) (2011), 63–72

In a previous paper, we obtained conditions on a monoid M for its prefix expansion to be either a left restriction monoid (in which case M must be either 'type-I' or 'type-II') or a left ample monoid (M is 'type-Ia' or 'type-IIa'). In the present paper, we demonstrate that there is some redundancy in these conditions. We therefore trim down the sets of conditions and show, by construction of suitable counterexamples, that the reduced sets of conditions are independent.

'Actions and partial actions of inductive constellations' (with Victoria Gould), Semigroup Forum 82(1) (2011), 35–60

Inductive constellations are one-sided analogues of inductive categories which correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids. In this paper, we define the notions of the action and partial action of an inductive constellation on a set, before introducing the Szendrei expansion of an inductive constellation, which is modelled closely on that defined by Gilbert (2005) for inductive groupoids. The main result of the paper is a theorem which uses this Szendrei expansion to link the actions and partial actions of inductive constellations, and is analogous to results previously proved by various authors for groups, monoids, and other objects.

'Extending the Ehresmann-Schein-Nambooripad Theorem', Semigroup Forum 80(3) (2010), 453–476

We extend the '∨-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the 'morphisms' part. However, it is so-called '∧-premorphisms' which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for (ordered) ∧-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.

'Restriction semigroups and inductive constellations' (with Victoria Gould), Communications in Algebra 38(1) (2010), 261–287

The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample and weakly E-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals.

Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial action and expansion for inductive constellations, along the lines of that of Gilbert for inductive groupoids.

'Anton Kazimirovich Suschkewitsch (1889–1961)', BSHM Bulletin: Journal of the British Society for the History of Mathematics 24(3) (2009), 172–179

Anton Kazimirovich Suschkewitsch was a Russian mathematician who spent most of his working life at Kharkov State University in the Ukraine. In the 1920s, he embarked upon the first systematic study of semigroups, placing him at the very beginning of algebraic semigroup theory and, arguably, earning him the title of the world’s first semigroup theorist. Owing to the political circumstances under which he lived, however, his work failed to find a wide audience during his lifetime. We give a brief account of his life and his researches into semigroup theory.

'The early development of the algebraic theory of semigroups', Archive for History of Exact Sciences 63(5) (2009), 497–536

In the history of mathematics, the algebraic theory of semigroups is a relative new-comer, with the theory proper developing only in the second half of the twentieth century. Before this, however, much groundwork was laid by researchers arriving at the study of semigroups from the directions of both group and ring theory. In this paper, we will trace some major strands in the early development of the algebraic theory of semigroups. We will begin with the aspects of the theory which were directly inspired by, and were analogous to, existing results for both groups and rings, before moving on to consider the first independent theorems on semigroups: theorems with no group or ring analogues.

'From right PP monoids to restriction semigroups: a survey', European Journal of Pure and Applied Mathematics 2(1) (2009), 21–57

Left restriction semigroups are a class of semigroups which generalise inverse semigroups and which emerge very naturally from the study of partial transformations of a set. Consequently, they have arisen in a variety of different contexts, under a range of names. One of the various guises under which left restriction semigroups have appeared is that of weakly left E-ample semigroups, as studied by Fountain, Gomes, Gould and Lawson, amongst others. In the present article, we will survey the historical development of the study of left restriction semigroups, from the 'weakly left E-ample' perspective, and sketch out the basic aspects of their theory.

'Partial actions of inverse and weakly left E-ample semigroups' (with Victoria Gould), Journal of the Australian Mathematical Society 86(3) (2009), 355–377

We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α → α+, where α+ is the identity map on the domain of α. We investigate the construction of 'actions' from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

'An analysis of nonpositional numeral systems', The Mathematical Intelligencer 31(2) (April 2009), 15–23

Motivated by the example of Roman numerals, we initiate the development of a classification for numeral systems which is finer than the traditional description of such systems as being either positional or nonpositional. In this paper, we deal exclusively with nonpositional systems, and consider the examples of Sumerian, Minoan, Attic and Milesian numerals. At the end of the paper, we return to the motivating example of Roman numerals.

'Conditions for the prefix expansion of a monoid to be (weakly) left ample', Acta Scientiarum Mathematicarum (Szeged) 73(3-4) (2007), 519–545

We derive necessary and sufficient conditions for the Birget-Rhodes prefix expansion of a monoid to be (weakly) left ample, thereby proving analogues of the results already obtained for the related Szendrei expansion by Fountain, Gomes and Gould and Fountain and Gomes. As a corollary, we obtain conditions for the prefix expansion to be inverse.

'Partial actions of monoids', Semigroup Forum 75(2) (2007), 293–316

We investigate partial monoid actions, in the sense of Megrelishvili and Schröder. These are equivalent to a class of premorphisms, which we call strong premorphisms. We describe two distinct methods for constructing a monoid action from a partial monoid action: the expansion method provides a generalisation of a result of Kellendonk and Lawson in the group case, whilst the approach via globalisation extends results of both Megrelishvili and Schröder and of Kellendonk and Lawson.

'The history of the 2-, 4- and 8-square identities', BSHM Bulletin: Journal of the British Society for the History of Mathematics 21(2) (2006), 111–118

The identity (a2 + b2)(c2 + d2) = (ac + bd)2 + (ad - bc)2 for the product of two sums of two squares has been known since ancient times. There also exist versions of this identity for 4 and 8 squares, these dating from the eighteenth and nineteenth centuries respectively. In this article I survey the history of such identities and explain why those for 2, 4 and 8 squares are the only ones.