Joseph W. Dauben

Sophie Germain : an essay in the history of the theory of elasticity

1. Introduction.- 2. Sophie Germain.- 3. Respectfully Yours, Gauss.- 4. Setting the Prize.- 5. The One Entry.- 6. The Molecular Mentality.- 7. An Award with Reservations.- 8. Publication.- 9. Emergence of a Theory.- 10. Final Years.- Notes.

De Minimis Risk: A Proposal for a New Category of Research Risk

De Minimis Risk: A Proposal for a New Category of Research Risk Rosamond Rhodes a , Jody Azzouni b , Stefan Bernard Baumrin c , Keith Benkov a , Martin J. Blaser d , Barbara Brenner a , Joseph W. Dauben c , William J. Earle c , Lily Frank c , Nada Gligorov a , Joseph Goldfarb a , Kurt Hirschhorn a , Rochelle Hirschhorn d , Ian Holzman a , Debbie Indyk a , Ethylin Wang Jabs a , Douglas P. Lackey c , Daniel A. Moros a , Sean Philpott e , Matthew E. Rhodes f , Lynne D. Richardson a , Henry S. Sacks a , Abraham Schwab g , Rhoda Sperling a , Brett Trusko a & Arnulf Zweig h a Mount Sinai School of Medicine b Tufts University c The Graduate Center, CUNY d New York University Medical School, CUNY e Union Graduate College f Pennsylvania State University g Indiana University, Purdue h University of Oregon (Emeritus)

Ancient Chinese mathematics: the (Jiu Zhang Suan Shu) vs Euclid’s Elements. Aspects of proof and the linguistic limits of knowledge

Abstract The following is a preliminary and relatively brief, exploratory discussion of the nature of early Chinese mathematics, particularly geometry, considered largely in terms of one specific example: the Download : Download full-size image (Gou-Gu) Theorem. In addition to drawing some fundamental comparisons with Western traditions, particularly with Greek mathematics, some general observations are also made concerning the character and development of early Chinese mathematical thought. Above all, why did Chinese mathematics develop as it did, as far as it did, but never in the abstract, axiomatic way that it did in Greece? Many scholars have suggested that answers to these kinds of questions are to be found in social and cultural factors in China. Some favor the sociological approach, emphasizing for example that Chinese mathematicians were by nature primarily concerned with practical problems and their solutions, and, therefore, had no interest in developing a highly theoretical mathematics. Others have stressed philosophical factors, taking another widely-held view that Confucianism placed no value on theoretical knowledge, which, in turn, worked against the development of abstract mathematics of the Greek sort. While both of these views contain elements of truth, and certainly play a role in understanding why the Chinese did not develop a more abstract, deductive sort of mathematics along Greek lines, a different approach is offered here. To the extent that knowledge is transmitted and recorded in language, oral and written, logical and linguistic factors cannot help but have played a part in accounting for how the Chinese were able to conceptualize—and think about—mathematics.

The intersection of history and mathematics

Mathematics: An Historians Perspective.- Une Methode de Restitution - quelques examples dans le cas de Pascal.- The Birth of Maxwells Electro-Magnetic Field Equations.- Complex Curves - Origins and Intrinsic Geometry.- From Gauss to Weierstrass: Determinant Theory and Its Historical Evaluation.- The Reciprocity Law from Euler to Eisenstein.- Three Aspects of the Theory of Complex Multiplication.- The Establishment of the Takagi-Artin Class Field Theory.- Where Did Twentieth-Century Mathematics Go Wrong?.- Indian Mathematics in Arabic.- The Tetsujutsu Sankei (1722), an 18th Century Treatise on the Methods of Investigation in Mathematics.- The Adoption of Western Mathematics in Meiji Japan, 1853-1903.- The Philosophical Views of Klein and Hilbert.- Hermann Weyls Contribution to Geometry in the Years 1918 to 1923.- The Origins of Infinite Dimensional Unitary Representations of Lie Groups.- Dispelling a Myth: Questions and Answers about Bourbakis Early Works, 1934-1944.- Questions in the Historiography of Modern Mathematics: Documentation and the Use of Primary Sources [Abstract].- List of Invited Speakers at the Tokyo History of Mathematics Symposium 1990.- List of Speakers and Titles of Their Lectures at Session B (Short Communications).

Mathematicians and World War I: The international diplomacy of G. H. Hardy and Gösta Mittag-Leffler as reflected in their personal correspondence

Abstract Gosta Mittag-Leffler was the founding editor of the journal Acta Mathematica. In the early 1870s it was meant, in part, to bring the mathematicians of Germany and France together in the aftermath of the Franco-Prussian War, and the political neutrality of Sweden made it possible for Mittag-Leffler to realize this goal by publishing articles in German and French, side by side. Even before the end of the First World War, Mittag-Leffler again saw his role as mediator, and began to work for a reconciliation between German and Allied mathematicians through the auspices of his journal. Similarly, G. H. Hardy was particularly concerned about the reluctance of many scientists in England to attempt any sort of rapprochement with the Central European countries and he sought to do all he could to bring English and German mathematicians together after the War. His correspondence with Mittag-Leffler survives in the Archives of the Institut Mittag-Leffler, Djursholm, Sweden, and serves as the basis for this article, which focuses upon the attempts of Mittag-Leffler to reconcile mathematicians after the War, and to renew international cooperation.

Peirce's place in mathematics

Abstract This article compares treatments of the infinite, of continuity and definitions of real numbers produced by the German mathematician Georg Cantor and Richard Dedekind in the late 19th century with similar interests developed at virtually the same time by the American mathematician/philosopher C. S. Peirce. Peirce was led, not by the internal concerns of mathematics which had motivated Cantor and Dedekind, but by research he undertook in logic, to investigate orders of infinite sets (multitudes, in his terminology), and to introduce the related concept of infinitesimals. His arguments in support of the mathematical and logical validity of infinitesimals (which were rejected by such eminent mathematicians as Cantor, Peano, and Russell at the turn of the century) are considered. Attention is also given to the connections between Peirces mathematics, his philosophy, and especially his interest in continuity as it was related to his Pragmatism.

Mathematics, ideology, and the politics of infinitesimals: mathematical logic and nonstandard analysis in modern China

I first met Ivor Grattan-Guinness and his wife Enid in the late summer of 1970. I was in England following an intensive course in German at the Goethe Institute in Prien am Chiemsee, and had arranged to spend the month of August in London. Ivor and I had been corresponding about the subject of my PhD thesis, Georg Cantor, for several years, and it was a pleasure to be invited for Sunday tea in High Barnet, where the Grattan-Guinnesses had a very pleasant garden in which we enjoyed tea and scones with strawberry jam. It was the beginning of a long and lasting friendship, one which has included periodic visits with Ivor and Enid in England, most recently when I was at Cambridge for a term at the Needham Institute and Clare Hall. However Ivor and I have also enjoyed numerous international conferences together, many at the periodic meetings for history of mathematics at the Mathematisches Forschungsinstitut in Oberwolfach, and others at International Congresses as well, which have included Moscow, Edinburgh, Berlin/Munich and Zaragoza. Our paths have also crossed in China, first in Hong Kong when we both happened to be there in 1997 and Ivor was giving a lecture at the University of HongKong, at the invitation of Siu Mankeung, on ‘History of mechanics in the eighteenth century’. Two years later we were both again in mainland China, for a meeting atWuhan University, where Ivor spoke on ‘National differences in mathematics in the nineteenth century’. This same trip also gave Ivor the opportunity to lecture at North-West University in Xi’an, where he spoke on general histories of mathematics, and in Beijing at the Institute of Mathematics (at the invitation of Li Wenlin) of the Chinese Academy of Sciences, where Ivor spoke on ‘Mathematical physics in the nineteenth century’. Since my own research is now focused largely on aspects of the history of Chinese mathematics, ancient and modern, it seems fitting to contribute this study to a special issue of History and Philosophy of Logic in his honor. What follows recounts briefly the story of the introduction of modern mathematics to China, as well as the fate of mathematicians there during the Cultural Revolution and the perhaps unexpected role that Karl Marx played in expediting the introduction of nonstandard analysis to China. The Mathematical Manuscripts of Karl Marx were first published (in part) in Russian in 1933, along with an analysis by Sofia Alexandrovna Yanovskaya. Friedrich Engels was the first to call attention to the existence of these manuscripts in the preface to his Anti-Dühring of 1885 (see Engels 1969). A more definitive edition of the Manuscripts was eventually published, under the direction of Yanovskaya, in 1968, and subsequently numerous translations have also appeared. HISTORY AND PHILOSOPHY OF LOGIC, 24 (2003), 327–363

Mathematics: an Historian’s Perspective

This article is based upon remarks originally prepared for the Tokyo History of Mathematics Symposium held at the University of Tokyo, August 31 — September 1, 1990, in conjunction with the International Congress of Mathematicians held in Kyoto the week before. Considering the interest the International Mathematical Union has shown in history of mathematics by virtue of its recent unanimous vote to recognize the International Commission on History of Mathematics as a joint IMU commission with the International Union of the History and Philosophy of Science, it seemed appropriate to consider a question that is by no means new, but one which has often stimulated considerable controversy among mathematicians and historians of mathematics alike — namely, on the subject of history of mathematics, what should the discipline include, and who should be included in defining the discipline?

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

Abstract 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. Luroth, Thomae, Jurgens and Netto offered solutions, Nettos 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 Nettos and Cantors faulty proofs went unchallenged for twenty years, until Jurgens criticized them both in 1899.

The Battle for Cantorian Set Theory

The substance of Georg Cantor’s revolutionary mathematics of the infinite is well-known: in developing what he called the arithmetic of transfinite numbers, he gave mathematical content to the idea of actual infinity. In so doing he laid the groundwork for abstract set theory and made significant contributions to the foundations of the calculus and to the analysis of the continuum of real numbers. Cantor’s most remarkable achievement was to show, in a mathematically rigorous way, that the concept of infinity is not an undifferentiated one. Not all infinite sets are the same size, and consequently, infinite sets can be compared with one another. But so shocking and counter-intuitive were Cantor’s ideas at first that the eminent French mathematician, Henri Poincare, condemned Cantor’s theory of transfinite numbers as a “disease” from which he was certain mathematics would one day be cured.1 Leopold Kronecker, one of Cantor’s teachers and among the most prominent members of the German mathematics establishment, even attacked Cantor personally, calling him a “scientific charlatan,” a “renegade,” and a “corrupter of youth.”2