The work of Indalecio Liévano on the foundations of real numbers

We discuss the work of the nineteenth century Colombian mathematician, Indalecio Liévano, on the foundations of real numbers. His work (from 1856) pre-dates that of Cantor, Dedekind and Méray.     

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.     

The invariance of dimension: Problems in the early development of set theory and topology [1]

In 1878 Georg Cantor proved that unique, one-to-one mappings could be constructed between spaces of arbitrary yet different dimension. This paper is devoted to a detailed analysis of the earliest attempts to deal with the implications of that proof. Dedekind was the first to suggest that continuity was a key to the problem of dimensional invariance. Lüroth, Thomae, Jürgens and Netto offered solutions, Netto's being the most interesting in terms of the specifically topological character of his paper. Cantor finally offered a faulty proof in 1879 that domains of different dimension could not be mapped continuously onto each other by means of a one-to-one correspondence. Finally, consideration is given to the reasons why Netto's and Cantor's faulty proofs went unchallenged for twenty years, until Jürgens criticized them both in 1899.     

Documents on the mathematical education of Edmund Külp (1800–1862), the mathematics teacher of Georg Cantor

Despite some studies by the historian Wilhelm Lorey, Edmund Külp is rather unknown today. His role in the development of mathematics and mathematics teaching in the nineteenth century, however, deserves closer attention. Having been the director of the höhere Gewerbe—und Realschule in Darmstadt, he can be counted among the founders of the Technische Hochschule Darmstadt. Moreover, he had been, still at the Realschule, the mathematics teacher of Georg Cantor. Recently detected documents concerning Külp’s mathematical formation in Brussels by A. Quetelet permit revealing insights into the evolution of Külp’s mathematical ideas and of his views on the context of mathematics in Germany. The contribution presents extracts from these documents (in French) and analyses them. Furthermore, the paper discusses possible influences exerted by Külp on Cantor.     

论高斯

高斯(C.F.Gauss,1777—1855)生于德国Brunswick,1795—98求学于Gttinsen大学。1807—55任天文观测台台长及Gttingen大学教授。 大家都知道高斯的名言:“数学是科学的女王,算术是数学的女王”这里“算术”是在古希腊人的意义下理解的,指的初等数论,而区别于近代的解析数论。     

Pure mathematics applied in early twentieth-century America: The case of T.H. Gronwall,consulting mathematician

Thomas Hakon Gronwall (1877–1932) was a Swedish-American mathematician with a broad range of interests in mathematical analysis, physics, and engineering. Though he was primarly known for his results in pure mathematics, his career as a “consulting mathematician” in America from 1912 to his death in 1932 provides a backdrop against which one can discuss contemporary issues involved in the increasing application of mathematics to engineering, industrial, and scientific problems. This paper attempts a summary of his major mathematical contributions to industrial, governmental, and academic institutions while relating his often difficult life during these years.     

Robert Richard Anstice (1813–1853): a Hertfordshire bicentenary

R R Anstice was a little known English clergyman who made significant contributions to Combinatorics, anticipating results in modern design theory. The year 2013 is the 200th anniversary of his birth. Anstice took the first steps in treating Combinatorial Design Theory as part of Algebra, in retrospect anticipating many later developments. In particular he touched on an important structure now known as a ‘Room square’, named after the twentieth century Anglo-Australian mathematician T G Room. Here we review Anstice's short life and give an account of his mathematics.     

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.     

Spaces to discuss mathematics: communities of practice on an online discussion board

In theories of learning that adopt a situated stance to knowledge the notion of identity is vital; how learners position themselves in relation to, and are mutually positioned by, the situation within which they are learning will have a strong bearing on the learning outcomes. One of the challenges for learning mathematics in school is that learners position themselves, and are positioned, as pupils rather than as mathematicians. This paper focuses on discussion boards designed for secondary school mathematics students, and we use Wenger's (1998) model of communities of practice, building on earlier work by the authors (Back and Pratt 2007; Pratt and Kelly 2007) in which ‘idealised communities’ are constructed and used, to consider a case study of one participant who engages in developing his identity as a mathematician doing mathematics, as well his identity as a learner and a teacher of mathematics.     

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.     

Colin Maclaurin (1698–1746): a Newtonian between theory and practice

The Scottish scientist Colin Maclaurin (1698–1746) is mainly known as a mathematician who focused on pure mathematics. But during his life he was interested in the application of mathematics in all branches of knowledge. This article considers the relationships between theory and practice in Maclaurin's works.     

Beyond mathematical content knowledge: A mathematician's knowledge needed for teaching an inquiry-oriented differential equations course

In this research report we examine knowledge other than content knowledge needed by a mathematician in his first use of an inquiry-oriented curriculum for teaching an undergraduate course in differential equations. Collaboratively, the mathematician and two mathematics education researchers identified the challenges faced by the mathematician as he began to adopt reform-minded teaching practices. Our analysis reveals that responding to those challenges entailed formulating and addressing particular instructional goals, previously unfamiliar to the instructor. From a cognitive analytical perspective, we argue that the instructor's knowledge — or lack of knowledge — influenced his ability to set and accomplish his instructional goals as he planned for, reflected on, and enacted instruction. By studying the teaching practices of a professional mathematician, we identify forms of knowledge apart from mathematical content knowledge that are essential to reform-oriented teaching, and we highlight how knowledge acquired through more traditional instructional practices may fail to support research-based forms of student-centered teaching.     

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.     

Alpha-theory: An elementary axiomatics for nonstandard analysis

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number . The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.     

Continuity between Cauchy and Bolzano: issues of antecedents and priority

In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his 1821 Cours d'Analyse, may have plagiarized Bolzano's Rein analytischer Beweis (RB), first published in 1817. That paper was subsequently discredited in several works, but some of its assumptions still prevail today. In particular, it is usually considered that Cauchy did not develop his notion of the continuity of a function before Bolzano developed his in RB and that both notions are essentially the same. We argue that both assumptions are incorrect, and that it is implausible that Cauchy's initial insight into that notion, which eventually evolved to an approach using infinitesimals, could have been borrowed from Bolzano's work. Furthermore, we account for Bolzano's interest in that notion and focus on his discussion of a definition by Kästner (in Section 183 of his 1766 book), which the former seems to have misrepresented at least partially.     

Reform of the construction of the number system with reference to Gottlob Frege

Due to missing ontological commitments Frege rejected Hilbert’s Fundamentals of Geometry as well as the construction of the system of real numbers by Dedekind and Cantor. Almost all of school mathematics is ontologically committed. Therefore, H.-G. Steiner considered Frege’s viewpoint of mathematics fundamentals, refined by Tarski’s semantics, as suitable for math education. Frege committed numbers ontologically by using measurement to define numbers. He invented the concept of quantitative domain (Größengebiet), which – it is now known by reconstruction of that concept by the New-Fregean Movement – agrees with the concept of quantity domain (Größenbereich) as established in the German reform of the application-oriented construction of the system of real numbers. Concepts of quantity (ratio-scale) and interval-scale in comparative measurement theory – going beyond Frege – show the way how the negative numbers can be ontologically committed and the operations of addition and multiplication can be included. In this work it is shown how Frege’s viewpoint of mathematics fundamentals, as propagated by H.-G. Steiner, can be better implemented in the current construction of the system of real numbers in school.     

The emergence of open sets,closed sets,and limit points in analysis and topology

General topology has its roots in real and complex analysis, which made important uses of the interrelated concepts of open set, of closed set, and of a limit point of a set. This article examines how those three concepts emerged and evolved during the late 19th and early 20th centuries, thanks especially to Weierstrass, Cantor, and Lebesgue. Particular attention is paid to the different forms of the Bolzano–Weierstrass Theorem found in the latter's unpublished lectures. An abortive early, unpublished introduction of open sets by Dedekind is examined, as well as how Peano and Jordan almost introduced that concept. At the same time we study the interplay of those three concepts (together with those of the closure of a set and of the derived set of a set) in the struggle to determine the ultimate foundations on which general topology was built, during the first half of the 20th century.     

Breaking social barriers: Florentia Fountoukli (1869–1915)

During the nineteenth century the foundations were laid for the advance of female education. At the same time it was still not easy for women to enter the fields of mathematics and science. In Greece the situation was worse than in some other countries: women were not allowed to matriculate at Athens University until the last decades of the century. Florentia Fountoukli was the first Greek woman to attend lectures at the mathematics department. Her dream of becoming a mathematician was partly fulfilled after a long struggle to persuade university professors to accept her in their lectures as a regular student. Florentia was a talented person who in the end had little choice but to teach philosophy and pedagogy in an Arsakeion school. This paper sheds light on her professional life in the school in Corfu. Finally after four years of service she went to Athens where she set up and managed her own school.

(For another article on the mathematical education of women in the nineteenth century see Marit Hartveit's ‘How Flora got her cap’ in BSHM Bulletin, 24 (2009), 147–158.)     

John von Neumann, the Mathematician

Imagine a poll to choose the best-known mathematician of the twentieth century. No doubt the winner would be John von Neumann. Reasons are seen, for instance, in the title of the excellent biography [M] by Macrae: John von Neumann. The Scientific Genius who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Indeed, he was a fundamental figure not only in designing modern computers but also in defining their place in society and envisioning their potential. His minimax theorem, the first theorem of game theory, and later his equilibrium model of economy, essentially inaugurated the new science of mathematical economics. He played an important role in the development of the atomic bomb. However, behind all these, he was a brilliant mathematician. My goal here is to concentrate on his development and achievements as a mathematician and the evolution of his mathematical interests.     

Contributions of Lo Yang to Value Distribution Theory

Professor Lo Yang is a world famous mathematician of our country. He made a lot of outstanding achievements in the value distribution theory of function theory, which are highly rated and widely quoted by domestic and foreign scholars. He also did a lot of work to develop Chinese mathematics. It can be said that Professor Yang is one of the mathematicians who made main influences on the mathematical development in modern China. This paper briefly introduces Professor Yang’s life, mainly discusses his academic achievement and influence, and briefly describes his contributions to the Chinese mathematics community.