Cultivating a research imperative: Mentoring mathematics at Stockholms Högskola, 1882–1887

Though the central role of Gösta Mittag-Leffler in the promotion of specialized, research-oriented mathematics at Stockholms Högskola is widely acknowledged, the specific social and technical means by which he sought to cultivate a fledgling research community there during the early- to mid-1880s have received little attention. In particular, a detailed study of the relationship of his own research activity to that of his first Swedish students is absent from the existing literature.Through the juxtaposition of their research activities and unpublished correspondence, this paper explores Mittag-Leffler's active and deliberate efforts to engage his students Ivar Bendixson and Edvard Phragmén in open problems within his own research agenda, support them through his institutional connections, and instill within them norms concerning research ideologies, practices of communication and criticism, and frameworks for shared knowledge. It also illuminates the extent to which his teachings took root in at least one student to emerge from his program, who would perpetuate the mathematical practices set in place by his teacher and set forth on the international stage to promote his newly-acquired system of values.     

The rejected parts of Brouwer's dissertation on the foundations of mathematics

Brouwer launched his intuitionist attack on the formalistic trends in mathematics in his now famous dissertation “On the Foundations of Mathematics” in 1904. In the autumn of 1976, the author found what turned out to be the first version of Brouwer's dissertation. He also discovered that his supervisor disapproved and rejected what Brouwer considered to be the most important part of his dissertation. The extreme solipsistic views held by Brouwer in his later years are well known. The first version of the dissertation shows that these views were held before Brouwer began his intuitionist campaign and that they determined his philosophy of mathematics. The deleted parts not found in the final version of the dissertation are presented here in an English translation. In a brief introduction the author gives some of the historical background, based partly on the correspondence between Brouwer and his supervisor, Korteweg.     

Boscovich's geometrical principle of continuity,and the “mysteries of the infinity”

In this paper we give a detailed account of Boscovich's geometrical principle of continuity. We also compare his ideas with those of his forerunners and successors, in order to cast some light on his possible sources of inspiration and to underline the elements of novelty in his approach to the subject.     

Peano's concept of number

Giuseppe Peano's development of the real number system from his postulates for the natural numbers and some of his views on definitions in mathematics are presented in order to clarify his concept of number. They show that his use of the axiomatic method was intended to make mathematical theory clearer, more precise, and easier to learn. They further reveal some of his reasons for not accepting the contemporary “philosophies” of logicism and formalism, thus showing that he never tried to found mathematics on anything beyond our experience of the material world.     

Shiing-Shen Chern��s Centenary

Shiing-Shen Chern was an editor of our journal Results in Mathematics from 1984 to 2004, the year he passed away at Tianjin. This article honors one of the greatest mathematicians of the twentieth century, in particular remembering his studies at Hamburg University during the years 1934?C1936. This period strongly influenced his mathematical work and was decisive for his later career. We survey the situation of the Department of Mathematics there, Chern??s studies, his visits to Germany in later years, his honours and awards from German institutions, and finally exemplarily his influence on the next generations of Chinese mathematicians studying in Germany.     

Interview with Ludwig Reich

On the occasion of his 80th birthday, Ludwig Reich, editor-in-chief of Aequationes Mathematicae from 1996 to 2008, gave an interview conducted via Zoom. This paper presents his answers to several questions regarding his education and academic life.

     

According to Davis: Connecting principles and practices

In this article, the author allows Robert B. Davis to state for himself his own Principles concerning how children learn, and how teachers can best teach them. These principles are put forward in Davis’ own words along with detailed documentation. The author goes on compare Davis’ words with his practices. A single Davis video (Towers of Hanoi) is analyzed to determine if, and to what extent, his principles are evident in his teaching of this lesson.     

The difficult birth of stochastics: Jacob Bernoulli's Ars Conjectandi (1713)

Jacob Bernoulli (1654–1705) did most of his research on the mathematics of uncertainty – or stochastics, as he came to call it – between 1684 and 1690. However, the Ars Conjectandi, in which he presented his insights (including the fundamental “Law of Large Numbers”), was printed only in 1713, eight years after his death. The paper studies the sources and the development of Bernoulli's ideas on probability, the reasons behind the delay in publishing and the circumstances under which his masterpiece eventually reached the public.     

Gauss as a geometer

In an attempt to reveal the breadth of Gauss's interest in geometry, this account is divided into six chapters. The first mentions the fundamental theorem of algebra, which can be proved only with the aid of geometric ideas, and in return, an application of algebra to geometry: the connection between the Fermat primes and the construction of regular polygons. Chapter 2 shows his essentially ‘modern’ approach to quaternions. Chapter 3 is a sample of his work in trigonometry. Chapter 4 deals with his approach to the geometry of numbers. Chapter 5 sketches his differential geometry of surfaces: his use of two parameters, the elements of distance and area, his theorema egregium, and the total curvature of a geodesic polygon. Finally, Chapter 6 shows that he continually returned to the subject of non-Euclidean geometry, which was so precious and personal that he would not publish anything of it during his lifetime, and yet did not wish to let it perish with him.     

The Leibniz catenary and approximation of e — an analysis of his unpublished calculations

Leibniz published his Euclidean construction of a catenary in Acta Eruditorum of June 1691, but he was silent about the methods used to discover it. He explained how he used his differential calculus only in a private letter to Rudolph Christian von Bodenhausen and specified a number that was key to his construction, 2.7182818, with no clue about how he calculated it. Apparently, the calculations were never divulged to anyone but were discovered later among his personal papers. They may be the earliest record of an accurate approximation of the number we label e and a demonstration of its role as the base of the natural logarithm and exponential function.This, at that time, was a remarkably precise estimate for e, accomplished more than 22 years before Roger Cotes published e to 12 significant digits, and some 57 years before Euler's treatment of the logarithm in his Introductio in Analysin Infinitorum. The Leibniz construction reveals a hyperbolic cosine built on an exponential curve based on his estimated value, which implies that he understood the number as the base of his logarithmic curve. The sheets of arithmetic used by Leibniz preserved at the Gottfried Wilhelm Leibniz Bibliothek (GWLB) in Hannover, confirm this.Those sheets show how Leibniz calculated e and applied it to his catenary construction. The data actually yield e to 12 significant figures: 2.71828182845, missed by Leibniz because of a misplaced decimal point. We summarize the construction and examine the worksheets. The unpublished methods seem entirely modern to us and could serve as enrichening examples in modern calculus texts.     

Charles Peirce's place in philosophy

Peirce's publications on the method of scientific investigation (as distinct from his work in formal logic and mathematics) are his most important and valuable contributions to philosophy. His views on this subject are superior in clarity and cogency to his voluminous writings on metaphysics and cosmology. He subscribed to a fallibilistic conception of knowledge that is poles apart from a wholesale skepticism; his formulations of the conditions for meaningful discourse and of the pragmatic maxim, though not free from difficulties, have been fruitful sources of much subsequent philosophical and scientific analyses; and his classification of and discussions of types of argument or reasoning employed in scientific inquiry continue to be valuable and insightful clarifications of this important subject. In contrast to his account of scientific method, Peirce's evolutionary theory of ultimate reality, though marked by originality and ingenious speculation, has little merit as a contribution to genuine knowledge.     

Peirce e dedekind: La definizione di insieme finito

Starting from Peirce's repeated claims of priority with respect to Dedekind's definition of finite set [R. Dedekin, Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888), Definizione 64], this paper traces the history of Peirce's definition and its role in his research on the foundations of arithmetic. This brings to light some remarkable and neglected achievements of Peirce in this field. It also shows that his priority claims are unjustified, although understandable in terms of his desire for acknowledgment of his pioneering work on the foundations of arithmetic.     

From anomaly to fundament: Louis Poinsot's theories of the couple in mechanics

In 1803 Louis Poinsot published a textbook on statics, in which he made clear that the subject dealt not only with forces but also with ‘couples’ (his word), pairs of coplanar non-collinear forces equal in magnitude and direction but opposite in sense. His innovation was not understood or even welcomed by some contemporary mathematicians. Later he adapted his theory to put forward a new relationship between rectilinear and rotational motion in dynamics; its reception was more positive, although not always appreciative of the generality. After summarising the creation of these two theories and noting their respective receptions, this paper considers his advocacy of spatial and geometrical thinking in mechanics and the fact that, despite its importance, historians of statics who cover his period usually ignore his theory of couples.     

Infinitely small quantities in Cauchy's textbooks

The fundamental role of infinitely small quantities for his teaching of the calculus was underlined by Cauchy himself in the introduction to his Cours d'analyse of 1821 and in the announcements of his later textbooks. First steps toward theories of such quantities which are briefly denoted as variables having zero as their limit were made by Cauchy, who represented them by sequences converging to zero (in the Cours) or by functions vanishing at zero (since 1823). It is shown that the famous so-called errors of Cauchy are correct theorems when interpreted with his own concepts. A few gaps in his proofs are explained by the hypothesis that he tacitly assumed continuity. No assumptions on uniformity or on nonstandard numbers are needed. Finally, some possible completions of Cauchy's rudimentary theories of infinitesimals are ventured.     

Die Habilitation von John von Neumann an der Friedrich-Wilhelms-Universität in Berlin: Urteile über einen ungarisch-jüdischen Mathematiker in Deutschland im Jahr 1927

The mathematician John von Neumann was born in Hungary but principally received his scientific education and socialization in the German science system. He received his Habilitation from the Friedrich-Wilhelms–Universität in Berlin in 1927, where he lectured as a Privatdozent until his emigration to the USA. This article aims at making a contribution to this early part of Neumann’s scientific biography by analyzing in detail the procedure that led to his Habilitation as well as the beginnings of Neumann’s research on functional analysis. An analysis of the relevant sources shows that in Berlin in the year 1927 Neumann was not yet regarded as the outstanding mathematical genius of the 20th century. Furthermore it will be seen that Neumann had great difficulties in developing the fundamental concepts for his path breaking work in spectral theory and only managed to do so with the support of the Berlin mathematician Erhard Schmidt.     

Johann Heinrich Lambert,mathematician and scientist, 1728 – 1777

1977 is the two hundredth anniversary of the death of Johann Heinrich Lambert, a little known but nonetheless intriguing figure in 18th century science. In the general histories of science and mathematics Lambert's contributions are often described piecemeal, with each discovery and invention usually divorced both from the method by which he arrived at it and from the totality of his intellectual endeavour. To the student of optics he is remembered for his cosine law in photometry, to the astronomer for his work on comets, to the meteorologist for his design of a gut hygrometer, and to the mathematician for his work on non-Euclidean geometry and his demonstration of the irrationality of π and e. There is no doubt that each of these contributions had a definite importance of its own; but it is not the aim of the present article to enumerate in this way the high points of Lambert's scientific and mathematical work, rather to describe it for once as a unified whole, and to relate it to the contemporary intellectual outlook.     

The notion of variable quantities ω in Bolzano's early works

Amid the debate over infinitesimals, Bolzano introduced the alternative notion of variable quantities ω in his 1816 work on the binomial theorem. It has often been assumed that his 1817 definition of continuity using them is practically the modern one. This paper explores Bolzano's early mathematical works and diaries to gain insight into the subtleties in his definition of ω and some of his mathematical procedures. We show that those quantities are not clearly ‘proto-Weierstrassian’ and argue that Bolzano was in the process of refinement and increasing abstraction of the idea of quantity that eventually led to the development of a theory of real numbers.     

Kenneth O. May, 1915–1977 : His early life to 1946

Kenneth Ownsworth May graduated from the University of California at Berkeley in 1936 with highest honors in mathematics. The following year he received his Masters degree and became a fellow of the Institute of Current World Affairs, and during the next two years he traveled to England, Europe, and Russia. On his return to the United States he became active in the Communist Party, the consequences of which would plague him for years. He joined the United States Army in 1942, serving with distinction, and after the war returned to Berkeley, where he obtained his Ph.D. in 1946. He immediately accepted an assistant professorship at Carleton College in Northfield, Minnesota, later moving to the University of Toronto.This part of May's biography focuses on the events up to his accepting a position at Carleton College. In this early phase his openness, his emphasis on good communications in the process of education, and his interest in practical procedures emerge which later set the background for his successful career as a leading historian of mathematics and the founding editor of Historia Mathematica.     

John Hawkes (1944-2001)

In his curriculum vitae, John Hawkes lists his research interestsas geometric measure theory, probability (Lévy processes),and potential theory (probabilistic). In fact, he made significantcontributions to all three areas, and there are strong relationshipsbetween them. He used both geometric measure theory and potentialtheory as tools for his study of the trajectories of particularLévy processes, but in many cases he needed to developthe tool before it was ready to be used. We will summarise hisresearch later, but first we discuss what is known of his lifehistory.