CMS/CSHPM Summer Meeting 2009
Memorial University of Newfoundland, St. John's, Newfoundland, June 6 - 8, 2009

History and Philosophy of Mathematics (CSHPM)
Org: Tom Drucker (Wisconsin-Whitewater)

BRUCE BURDICK, Roger Williams University, Bristol, RI 02809, USA
Mathematics and Astrology in the Printed Works of the Americas before 1700

The talk will explore how the popularity of astrology before 1700 led mathematically trained authors in the New World to publish locally printed books in that field to supplement what was available from Europe. Such authors often were able to include mathematical exposition at a level that would not be associated with popular astrology today. Examples will include an astrologically motivated history from Mexico, an astrology text from Peru, and almanacs from British North America.

GREGG DE YOUNG, The American University in Cairo, Cairo, Egypt
Ziy¯ad¯at Literature: An Early Form of Commentary in the Arabic Euclidean Tradition

The use of ziy¯ad¯at (additions of propositions) is one of several forms that commentary on the medieval Arabic Euclidean corpus could assume. Like many genres of Euclidean commentary, its boundaries are not sharply and clearly delineated. Sometimes these additions occur individually, at other times they occur in blocks, almost as a separate sub-unit in the Euclidean tradition. These propositions are usually introduced either to fill a perceived logical lacuna in the Euclidean text or to complete a topic that Euclid did not, apparently, judge to be essential to his mathematical argument. For purposes of this paper, I shall confine myself to blocks of propositions which were specifically identified as additions. The best-known ziy¯ad¯at are those ascribed to al-Jawhar¯i (fl. 3rd century AH / AD 9th century) and to Ab¯u Sahl al-Q¯uh¯i (fl. early 4th century AH / AD 10th century). This paper uses these early ziy¯ad¯at, along with a block of newly analyzed propositions added to Book VI by al-Ant¯ak¯i (died 376 AH / AD 987) in his now incomplete Arabic commentary on the Elements, to introduce the general characteristics of ziy¯ad¯at literature. I argue that these ziy¯ad¯at, although one of the less studied forms of Euclidean commentary in Arabic tradition, offer insight into the concerns of early Islamic mathematicians as they encounter the Euclidean corpus.

ROGER GODARD, RMC, 92 Florence Street, Kingston, ON, K7M 1Y6
Some examples of eigenvalue problems in mathematical physics

In his 1834 "Essay on the Philosophy of Sciences", André-Marie Ampère made the distinction between the elementary general physics and the mathematical physics. The first branch of physics was related to observations and experiments. The second branch considered physical laws, correlations with experiments, the explanation of phenomena. Among these problems of mathematical physics, the eigenvalue problems and the Helmholtz equation are fundamental. In this work, we comment on a important and thick memoir written by Siméon Denis Poisson in 1829, "Sur l'équilibre et le mouvement des corps élastiques", Lord Rayleigh's influence on Irwin Schrödinger, and finally the first attempts of the numerical solutions of the Helmholtz equation with Runge, Liebmann, and R. G. D. Richardson.

HARDY GRANT, York University, Toronto
Eighteenth-century mathematics: the naysayers

Some skepticism attended the dramatic Enlightenment progress made in mathematics and in its applications. The most interesting dissent came from certain observers who could claim to be well informed and even sympathetic. Diderot and Buffon urged that mathematics is too tautologous and abstract to have more than a limited role in natural philosophy, and that its study was therefore destined to become increasingly marginal. I shall try to set these assertions in context, and I shall cite another famous figure of that age who countered them-cogently, as history would prove.

W. JIM JORDAN, University of Waterloo, 200 University Avenue W., Waterloo, ON, N2L 3G1
Structuralist Mirages on the Road from Antiquity: Why the Ancient Greek Mathematicians Were Not Mathematical Structuralists

H. G. Zeuthen and B. L. van der Waerden understand ancient Greek mathematics to be purely algebraic, even the vast body of geometrical work within it. Ian Mueller, through an examination of Euclid's Elements, shows that Euclid, at least, did not have anything resembling modern algebra and algebraic structure in view when he set down his principles and demonstrations. I give some examples of how a structuralist interpretation of Euclid could possibly be discerned. I then show that Euclid, though he had a structure to his work, likely did not do his work within a modern mathematical structuralist framework. I continue with an examination of three later ancient mathematicians (Nicomachus, Diophantus, and Pappus), and show that, like Euclid, they did not reveal any structuralist understanding in their works. I conclude that the attribution of mathematical structuralism as an intentional aspect of ancient Greek mathematics is mistaken.

GREGORY LAVERS, Concordia University
Frege the conventionalist and Carnap the Fregean

I begin by identifying passages in Frege's work that count against the standard interpretation of Frege as being a platonist of the most extreme sort. The goal is not to argue that Frege was not a platonist, but that there are at least some conventionalist tendencies in Frege's work. I then outline Carnap's position on matters in the foundations of logic and mathematics. I argue that the differences between the positions of these two philosophers can be traced to a disagreement about just a few theses. I point out that given technical developments between Frege and Carnap's time, Frege's position on these points becomes untenable. In this sense we can see Carnap as holding a maximally Fregean position on the nature of logical and mathematical knowledge.

MICHAEL MOLINSKY, University of Maine at Farmington
The Mathematical Education of a Founding Father: John Adams

Although John Adams made no significant contributions to the field of mathematics, his life provides an opportunity to investigate the study of mathematics at the time of the American Revolution. This talk will examine Adams' education, explore some of the mathematical works in his personal library, and present several recreational mathematics problems found in his journals.

ANDREW PERRY, Springfield College
Isaac Greenwood And The Earliest American Textbooks

Isaac Greenwood's "Arithmetick Vulgar and Decimal: with the Application thereof to a Variety of Cases in Trade and Commerce" (1729) is now thought to be the oldest mathematics textbook written by an American. Apparently, Greenwood's book was quickly forgotten by Americans, since Nicolas Pike's "A New and Complete System of Arithmetic, Composed for the Use of Citizens of the United States" (1788) was also thought to be the first American textbook at it time of publication. We will consider Greenwood's text in the context of similar volumes of its era.

BRUCE PETRIE, Institute for the History and Philosophy of Science and Technology at the University of Toronto, Toronto, Ontario
Why Euler Didn't Prove the Irrationality of PI: A Chapter in the History of Transcendence

This paper is part of an ongoing study by the author to determine whether Leonhard Euler considered the irrationality of p adequately demonstrated or if he believed that the problem required new tools or methods. Using Euler's criteria of rigor evident in his proof that e is irrational, as set out in his Introductio (1748) and documented by Ed Sandifer (2006), the paper shows that previous proofs of p's irrationality by William Brouncker, John Wallis, and Euler himself failed to satisfy these rigorous standards. A fully satisfactory proof of the irrationality of p was obtained by Johann Lambert in 1768 using new methods devised for this purpose. In his investigation Lambert relied heavily on Euler's exposition of continued fractions. The paper examines the state of Euler's work on the problem in relation to these contemporary researches of Lambert.

V. FREDERICK RICKEY, West Point, West Point, NY 10996, USA
Choosing Department Heads at West Point

If you were looking to hire a mathematician to teach at your institution around 1800, who would you hire and why? Even though this was a Presidential appointment, why would Lagrange or Lacroix move from cosmopolitan Paris to West Point, New York, an isolated outpost on the Hudson River ninety kilometers north of New York City? There was an aversion in the United States army against hiring anyone French. Gauss was famous for discovering Ceres, but not yet for his mathematics. An English speaker was needed, but it was not considered fair to steal a faculty member from another school even by the President. This problem has arisen and has been solved 21 times since the Military Academy was founded in 1802. Were the methods of solution all the same or has the hiring process changed over time? Our purpose here is to illustrate the dramas involved: sometimes the mathematician had a connection with the U.S. President, sometimes there was family connection, and on occasion the person selected was the most qualified army officer for the job. The most interesting case was when a national search was conducted, a search that included some prominent civilian mathematicians.

JOEL SILVERBERG, Roger Williams University, Bristol, Rhode Island, USA
Nathaniel Torporley and his Diclides Coelometricae (1602)-A Preliminary Investigation

Torporley is perhaps one of the more interesting and enigmatic mathematical figures of 15th and 16th century England. Attracting the patronage of Henry Percy, ninth Earl of Northumberland, Torporley served as personal secretary to François Viète, and was a mathematical colleague and trusted friend of Thomas Harriot. He was chosen by Harriot to prepare his manuscripts for posthumous publication.

Yet he was also a figure of controversy. Delambre in his Astronomie Moderne (1821) refers to the tables presented in Diclides as "the most obscure and incommodious that ever were made". Two decades later Augustus De Morgan in The Penny Cyclopedia (1838) and again in a note to the Philosophical Magazine and Journal (1843) both praises and condemns Diclides, giving the work credit for discovering the essence of Napier's Rules twelve years before Napier, while at the same time describing it as "the greatest burlesque on mnemonics we ever saw".

The language, forms of expression and the Latin usage are indeed close to impenetrable. Even the title of the work is obscure and the mathematics filled with everything from rebuses to verse. De Morgan abandoned his attempt to explain this work with the comment that "those who like such questions may find out the meanings of the other parts of the tables". I will describe the nature of this enigmatic work and share such progress in deciphering and decoding the Diclides as I have made at this time.

CHARLOTTE SIMMONS, University of Central Oklahoma, 100 N. University, Box 129, Edmond, OK 73012
Benjamin Gompertz: Pioneer of Actuarial Science

In this talk, we investigate the contributions of actuarial pioneer Benjamin Gompertz, known for his capacity to sustain the complex computation required to generate "tables of lives and tables of stars", to the field of actuarial science. Gompertz is best known today for his Law of Mortality, an extremely powerful tool in the study of mortality and the creation of life tables for actuaries. The significance of this law will be examined. Additionally, we will discuss the contributions of friend and staunch supporter, Augustus De Morgan, to the field, both direct (via his own work) and indirect (via his defense of Gompertz during the Edmonds-Gompertz controversy). Because of De Morgan's efforts, Edmonds is "now remembered only for the disparagement of the work of a man of genius", while Gompertz is remembered "because his outstanding brilliance as a mathematician was equalled by his modesty and generosity".

J. J. TATTERSALL, Providence College, Providence, RI 02918, USA
A Late Nineteenth Century Mathematical Subculture

In the second half of the nineteenth century, in an attempt to promote mathematics, a number of recreational mathematicians published mathematical journals and edited mathematical columns. The most prominent among them was William J. C. Miller who edited a mathematics column in the Educational Times and published Mathematical Questions and Their Solutions from the Educational Times in London. In America, Artemas Martin published the Mathematical Visitor and Mathematical Magazine, J. E. Hendricks published The Analyst: A Journal of Pure and Applied Mathematics, and Benjamin Finkel the American Mathematical Monthly. Mathematical columns were edited by William Hoover in the Wittenburger, Samuel Hart Wright in the Yates County Chronicle, W. D. Henkle in the Educational Notes and Queries: A Medium of Intercommunications for Teachers, H. A. Wood in the National Educator, E. T. Quimby in the New England Journal of Education, and Archibald MacMurchy in the Canada Educational Monthly and School Chronicle. These publications consist mainly of articles, brief biographies of the contributors, obituaries, and mathematical problems for solution. We discuss a few of the publications, their editors, their contributors, and problems that appeared in them.

MARINA VULIS, Norwalk Community College
Early Mathematics Russian Mathematics Textbooks

In 1703, Leontiy Magnitskiy published Arifmetika, the first Russian mathematics textbook. Magnitsky taught mathematics at the Mathematics and Navigation School, the first school in Russia in which mathematics was an important subject. Arifmetika was the first mathematics textbook written by a Russian author in Russian. It was a comprehensive mathematics textbook, and contained material on arithmetic, algebra, geometry, trigonometry, and navigation.

Arifmetika was an important book in developing mathematics education in Russia. We will discuss the teaching career of Leontiy Magnitskiy and the contents of his Arifmetika.

BYRON WALL, York University, 4700 Keele St., Toronto, ON, M3J 1P3
Some 19th century arguments for the rational assignment of probabilities for possible events in Nature

Countless decisions are made every day by each of us individually and collectively through our governments and other institutions, about what actions to take in the present in order to optimize a future in which many possible outcomes are more than moderately uncertain. At a personal level, we make these decisions intuitively, based on past experience. At the institutional and government level, we increasingly rely upon quantitative statistical projections and risk assessments. A great deal of interesting and well-worked out mathematics goes into these projections. Most of the mathematics is based upon models in which probabilities can be specified with precision. But the usefulness and reliability of these models depends crucially upon how well the tidy world of the model compares to an incompletely understood nature.

The history of probability and statistics is peppered with arguments, sometimes vociferous, over the assignment of a probability to events in nature, both those that are agreed to be highly probable, such as whether the sun will rise tomorrow, and those that are deemed highly improbable, such as what the chances are of snow in July, or living to the age of 200, or invasion from outer space. In the late 19th century, these arguments were carried on by respectable mathematicians and philosophers who were seeking to find solid ground for inference from incomplete information, the basis of statistics. This talk will explore some of that debate.

MARIA ZACK, Point Loma Nazarene University
Robert Hooke and An Attempt to Prove the Motion of the Earth from Observations

After the Great London Fire of 1666, Robert Hooke was appointed to work in the office of the City Surveyor of London. With that appointment, a scientist best known as the Curator of Experiments for the Royal Society whose research encompassed both the microscopic (Micrographia) and the astronomical, embarked on a second career as an architect and surveyor. For the next several decades the massive effort to reconstruct London was lead by Hooke and his long-time friend, fellow scientist and co-founder of the Royal Society, Christopher Wren.

Hooke was involved extensively in all aspects of the rebuilding of London, both the mundane (widening streets and establishing property boundaries) and the creative (designing churches and civic buildings). One of Hooke's few surviving buildings is the column that is the Monument to the Great Fire. This ingenious building is an excellent example of the intersection between Hooke's architectural and scientific work.

At the time of the Monument to the Great Fire's design, Hooke was conducting experiments on both the motion of the earth which he describes in An Attempt to Prove the Motion of the Earth from Observations. Hooke was particularly interested in using the measurement of the parallax to prove that earth revolved around the sun and the Monument was designed to be a zenith telescope to further that research. This talk will discuss Hooke's paper, the history of attempts to measure the parallax and how this scientific work influenced the design and construction of the Monument.

Event Sponsors

Memorial University of Newfoundland    AARMS: Atlantic Association for Research in the Mathematical Sciences    Centre de recherches mathématiques Fields Institute MITACS Pacific Institute for the Mathematical Sciences Canadian Language and Literacy Research Network

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