This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwrizm, Cardano, and Descartes to Euler, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form of the symbolic language of algebra. Thus the (...) paper develops further the method of reconstruction which the author introduced for the analysis of the development of geometry. (shrink)

There is a common tradition in European education going back to the Middle Ages which long played a part in providing the curriculum of schools which catered both for the wealthy and for able sons of less well-to-do families. Originally published in 1974, this volume examines the relationship between education and society in the different countries of Europe from which differences in tradition and practice emerge. The countries discussed include: France, Germany, the former Soviet Union, Poland and Sweden.

Originally published in 1972, this book is concerned with education as part of a larger social history. Chapters include: The roots of Anglican supremacy in English education The Board schools of London The use of ecclesiastical records for the history of education Topographical resources: private and secondary education from the sixteenth to the twentieth century.

In the early 1930s W. O. Kermack and W. H. McCrea published three papers in which they attempted to prove a result of E. T. Whittaker on the solution of differential equations. In modern parlance, their key idea consisted in using quantized contact transformations over an algebra of differential operators. Although their papers do not seem to have had any impact, either then or at any later time, the same ideas were independently developed in the 1960–1980s in the framework (...) of the theory of modules over rings of microdifferential operators. In this paper we describe the results of Kermack and McCrea and discuss possible reasons why such promising papers had no impact on the mathematics of the twentieth century. (shrink)

Originally published in 1971, this volume examines the relationship between the history and sociology of education. History does not stand in isolation, but has much to draw from and contribute to, other disciplines. The methods and concepts of sociology, in particular, are exerting increasing influence on historical studies, especially the history of education. Since education is considered to be part of the social system, historians and sociologists have come to survey similar fields; yet each discipline appears to (...) have its own aims and methodology. (shrink)

Part of the educational system in England has been geared towards the preparation of particular professions, while the identity and status of members of some professions have depended significantly on the general education they have received. Originally published in 1973, this volume explores the interaction between education and the professions. It also looks at the education of the main professions in sixteenth century England and at how twentieth century university teaching is a key profession for the training of new recruits (...) to other professions. (shrink)

In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would (...) suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

Politics and Modernity provides a critical review of the key interface of contemporary political theory and social theory about the questions of modernity and postmodernity. Review essays offer a broad-ranging assessment of the issues at stake in current debates. Among the works reviewed are those of William Connolly, Anthony Giddens, J[um]urgen Habermas, Alasdair MacIntyre, Richard Rorty, Charles Taylor and Roy Bhaskar. As well as reviewing the contemporary literature, the contributors assess the historical roots of current problems in the works of (...) Nietzsche and Max Weber. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability (...) to recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

'The Greate Invention of Algebra' casts new light on the work of Thomas Harriot, an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, optics, geometry, and algebra. Although on his death Harriot left behind over four thousand manuscript sheets, much of his work remains unpublished. This book focuses on one hundred and forty of Harriot's manuscript pages, those concerned with the structure and solution of equations. The original material has been carefully ordered, (...) translated, and annotated to provide the first complete edition of his work on this subject, and an extended introduction provides the reader with a lucid background to the work and explains its contents. Illustrations from the manuscripts provide fascinating reference material. The appendix discusses correlations between Harriot's manuscripts and the texts of his contemporaries Viète, Warner, and Torporley. The clear and concise exposition makes this an excellent reference volume for historians of mathematics and those interested in the history of science. This is an important new resource for understanding the development of algebra in seventeenth-century England. (shrink)

We present the history of the conference Applications of Algebra in Logic and Computer Science, whose twenty-third edition will be held in March, 2019. At the end we outline some plans for the future.

The Greek mathematician Diophantos of Alexandria lived during the third century CE. Apart from his age, very little else is known about his life. Even the exact form of his name is uncertain, and only a few incomplete manuscripts of his greatest work, Arithmetica, have survived. In this impressive scholarly investigation, first published in 1885, Thomas Little Heath meticulously presents what can be gleaned from Greek, Latin and Arabic sources, and guides the reader through the algebraist's idiosyncratic style of mathematics, (...) discussing his notation and originality. This was the first thorough survey of Diophantos' work to appear in English. Also reissued in this series are Heath's two-volume History of Greek Mathematics, his treatment of Greek astronomy through the work of Aristarchus of Samos, and his edition in modern notation of the Treatise on Conic Sections by Apollonius of Perga. (shrink)

In this paper I introduce a broader context, and sketch an integrated account with the purpose of examining the significance of Neurath’s attention to logic in early works and subsequent positions. The specific attention to algebraic logic is important in integrating his own interest in mathematics and combining, since Leibniz, the ideals of a universal language and of a calculus of reasoning. The interest in universal languages constitutes a much broader, so-called tradition of pasigraphy that extended beyond philosophical projects. I (...) argue that Neurath’s works can be embedded in a richer intellectual landscape that includes developments in logic and their local reception in Vienna, and that his attention to logic developed a sustained symbolic standpoint – with semiotic and typographic expressions –; that specific aspects of the work in algebraic logic became a standard and a resource in subsequent work often thought independent, while its value was steadily challenged by the separate goal of empirical theorizing and practical application in social domains – including in the areas of economics, history and visual communication –; that, in particular, the presentation of systems of algebraic logic by Neurath’s sources such as Stanley Jevons and Schröder was not isolated from discussions of political economy; and finally that some of his positions in matters of language, unity and epistemology in the articulation of logical empiricism and its debates are better understood in terms of shared but diversified acquaintance with pasigraphy, formal standards and logical projects. (shrink)

More than any other aspect of the Second Scientific Revolution, the remarkable revitalization or British mathematics and mathematical physics during the first half of the nineteenth century is perhaps the most deserving of the name. While the newly constituted sciences of biology and geology were undergoing their first revolution, as it were, the reform of British mathematics was truly and self-consciously the story of a second coming of age. ‘Discovered by Fermat, cocinnated and rendered analytical by Newton, and enriched by (...) Leibniz with a powerful and comprehensive notation’, wrote the young John Herschel and Charles Babbage of the calculus in 1813, ‘as if the soil of this country [was] unfavourable to its cultivation, it soon drooped and almost faded into neglect; and we now have to re-import the exotic, with nearly a century of foreign improvement, and to render it once more indigenous among us’. (shrink)

Algebraic quantum field theory (AQFT) puts forward three ``causal axioms'' that aim to characterize the theory as one that implements relativistic causation: the spectrum condition, microcausality, and primitive causality. In this paper, I aim to show, in a minimally technical way, that none of them fully explains the notion of causation appropriate for AQFT because they only capture some of the desiderata for relativistic causation I state or because it is often unclear how each axiom implements its respective desideratum. After (...) this diagnostic, I will show that a fourth condition, local primitive causality (LPC), fully characterizes relativistic causation in the sense of fulfilling all the relevant desiderata. However, it only encompasses the virtues of the other axioms because it is implied by them, as I will show from a construction by Haag and Schroer (1962). Since the conjunction of the three causal axioms implies LPC and other important results in QFT that LPC does not imply, and since LPC helps clarify some of the shortcomings of the three axioms, I advocate for a holistic interpretation of how the axioms characterize the causal structure of AQFT against the strategy in the literature to rivalize the axioms and privilege one among them. (shrink)

This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated to (...) the ideas and methods at the heart of the usual mathematical curriculum – algebraic equations, functions, derivatives, analytic geometry – if it becomes an essential subject for mathematics, then the attempt to find a place for history of mathematics requires refining our understanding of the nature of mathematics education itself. Thus, the paper asks not only how can history of mathematics be incorporated into mathematics education but also how the idea of mathematics education may need to be adjusted to accommodate history. (shrink)

This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.