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Report on the ICHM Co-Sponsored Session on the History of Mathematics

Sunday, 7 January, 2007, 1:00-6:00 pm

by Karen V.H. Parshall

This session of ten talks was co-organized by Joseph W. Dauben (City University of New York), Patti W. Hunter (Westmont College), and Karen V. H. Parshall (University of Virginia) and took place at the Joint Meetings of the American Mathematical Society held in New Orleans, Louisiana, USA. The talks (titles, speakers, and abstracts below), each of which lasted a half-hour, drew audiences of between 50 and 150.

"The Ellipse Seen from China"

Andrea Bréard, Université des Sciences et Technologies de Lille; Laboratoire Paul Painlevé (France)

The transmission of “Western” mathematics and astronomy into China during the seventeenth century consisted not only of the introduction of certain methodological tools but also of the integration of new geometric objects, the ellipse being one of them.  This talk examined how the calculation of the circumference of the ellipse, for example, was dealt with in early modern China within the traditional mathematical framework, and how ancient procedures were rewritten to help in the solution of the rectification problem

"The Other Book Nobody Read: Georg Rheticus and the Opus Palatinum"

Glen R Van Brummelen, Quest University (Canada)

Georg Rheticus is known to most of us as the man who encouraged his mentor, Nicolas Copernicus, to publish his heliocentric planetary theory.  A strong mathematician himself, Rheticus led a tumultuous life that nevertheless allowed him to make important contributions mathematics, and especially to trigonometry.  His Opus Palatinum, published in 1596 long after his death, is a massive 1300-page tome, half of which comprises an unprecedentedly massive set of tables of all six trigonometric functions.  His description of how he assembled the tables sheds light on his meeting with Gerolamo Cardano, a meeting he had hoped would help his mathematical research.

"Communicating Mathematics in the Journal des savants (1675-1737)"

Jeanne Peiffer, CNRS Paris (France)

The Journal des savants, created in 1665 in Paris, addressed a general readership in the vernacular French (contrary to Leipzig’s Acta eruditorum which was published in Latin).  The presence of mathematics, categorized between 1675 and 1737 under the general heading “mathematics,” was rather limited (a bit more than 3% of all the published “papers”).  After briefly describing the mathematics on which the journalists wrote, the style in which they wrote, and the readership they sought to address, this talk examined the role of controversies in creating new audiences for mathematics.  In particular, the practice of public challenge problems delineated small elite circles, the members of which operated inside narrow disciplinary boundaries, but managed through controversial public debate greatly to publicize their activities.

"Problems of Infinitesimals: Descartes, Leibniz, and Peirce"

Maria Sol de Mora, University of Basque Country (Spain)

The connection between very small quantities (or infinitesimals) and the infinite has always been more troubled than the idea of the infinitely large (or transfinite).  Almost all civilizations have been able to think about the infinite, even if they eventually rejected it on account of the difficulties that is occasions, including the many paradoxes associated with the concept, as was the case of classical Greece.  Only some Pythagoreans dared to support an infinite universe without bounds.  Among the mathematicians, the infinitely small, which was so useful for the infinitesimal calculus in the seventeenth century, was rejected afterwards with the assertion that there is only one infinite, the “great” one.  But this idea was reconsidered by Robinson and others in the twentieth century.  In philosophy and science, authors like Descartes, Leibniz, and C. S. Peirce have approached this difficult issue with various results.  This paper analyzed their results.

"Motivation and Context for B. Peirce's Linear Associative Algebra"

Deborah A. Kent, Simon Fraser University (Canada)

Throughout the nineteenth century, the mathematical work of Harvard professor, Benjamin Peirce, primarily involved applied or analytic areas as well as astronomy.  Nonetheless, Peirce’s most well-known and widely read work today is Linear Associative Algebra, which contains results foundational to the structure theory of algebras.  He presented this work to a mystified audience at the National Academy of Sciences in 1870, and it finally appeared in the American Journal of Mathematics in 1881 after Peirce’s death.  Peirce’s paper initially seems anomalous given his earlier research interests, yet—considered in the context of his motivation and understanding of the discipline—Linear Associative Algebra comes into focus as the culmination of a life’s mathematical work.

"Robert Leslie Ellis on the Misuse of the Principle of Insufficient Reason"

Byron E. Wall, York University, Toronto (Canada)

The so-called principle of insufficient reason assigns equal probability to events with unknown or indeterminate causes, where there is no known reason to favor one outcome over another.  The classic case is the coin toss.  This principle is easily buried and not made manifest as an axiom in probability calculations, leading to circular “proofs.”  In 1842, the Cambridge mathematician, Robert Leslie Ellis, read a paper to the Cambridge Philosophical Society which explored the ramifications of this.  Ellis’s life was cut short by illness, hence he was not able to pursue or expand upon this topic.  Though some of his key points were taken up by others, some remain unresolved.

"The Mittag-Leffler Theorem: Interpretation and Reception of a Mathematical Result, 1876-1884"

Laura E Turner and Thomas Archibald, Simon Fraser University (Canada)

Gösta Mittag-Leffler studied as a “post-doctoral” student in Paris with Charles Hermite and in Berlin with Karl Weierstrass from 1873 to 1876.  In 1876, Mittag-Leffler extended Weierstrass’s work on the representation of entire functions and proved the theorem associated with his name that asserts the existence of a meromorphic function with the prescribed poles and multiplicities.  Alternative proofs were provided by Hermite and Weierstrass, and the early response to this work was highly enthusiastic.  Between 1876 and 1884, Mittag-Leffler sought to generalize his early results, and his desire to deal with infinite sets o singular points attracted him to Cantor’s work.  His enthusiasm for Cantor’s results was unusual and brought with it disapproval.  Based on Mittag-Leffler’s correspondence with Cantor, Weierstrass, Kronecker, Poincaré, and Hermite, this talk examined his results and their interpretation and reception.  It provides a concrete case study of the role of major figures in the transition from a view of mathematics sometimes referred to as “formula-based” to a more modern view in which abstract entities became widely accepted as valid, interesting objects of mathematical study.

"A Political and Mathematical Unification: The Case of Nineteenth-Century Italy"

Laura Martini, Siena, (Italy)

The Unification of Italy was proclaimed in 1861.  This event marked a key turning point not only in the country’s political life, but also in the development of mathematical studies and research.  This talk discussed the revival and renewal of Italian mathematics in the years following the Unification and showed that, parallel to a political unification, a mathematical unification also took place in Italy in the second half of the nineteenth century.  In particular, in an institutional, historical, political, and mathematical context, this talk focused on algebraic developments: by tracing the contributions of Italian mathematicians throughout the peninsula, it showed that, contrary to conventional wisdom, Italy supported a wide range of algebraic research.

"Mathematics in Brazil during the First Half of the 20th Century: Institutionalization and Professionalization"

Sergio Nobre, UNESP, Rio Claro (Brazil)

This talk discussed the creation in Brazil of the first engineering (polytechnic) school at the end of the nineteenth century and the foundation of the Brazilian Academy of Sciences in the early twentieth century as well as the country’s first university.  From these beginnings, it examined the issue of the professionalization of mathematics in Brazil.

"Mathematics and Pacifism in Cambridge 1915-1916: A Student Perspective"

June Barrow-Green, The Open University, Milton Keynes (United Kingdom)

From January 1915 to July 1916, the Cambridge mathematics student, F. P. White, kept a diary in which he freely recorded the details of his daily life.  During this period, White came top in the Mathematical Tripos, embarked on postgraduate research in applied mathematics, and appeared in front of several Appeal Tribunals as a conscientious objector.  White’s diary not only provides a vivid description of Cambridge mathematical life during the First World War but also gives an insight into the pacifist movement and the mathematicians who supported it.