Schedule of Talks for the 2003-2004 Academic Year
Date: Wednesday, October 8, 2003
Speaker: Prof. Ivor Grattan-Guiness
Middlesex University at Enfield
Title: History or Heritage? An Important Distinction in Mathematics and for Mathematics Education
Abstract: During recent decades there has been a remarkable increase in work in the history of mathematics, including its relevance to mathematics education. But at times considerable differences of opinion arise, not only about its significance but even concerning legitimacy - that is, whether or not an historical interpretation counts as history at all. In this lecture I consider the latter issue, and note some consequences for education.
Date: Wednesday, November 5, 2003
Speaker: Prof. Hardy Grant
Department of Mathematics
York University
Title: Greek Mathematics in Cultural Context
Abstract: I shall attempt an overview of the "career" of Greek mathematics in its cultural setting. Features of the background include Eleatic philosophy, the birth of the "liberal arts" tradition, the rivalry of professions, the rise of rhetoric as a technique of demonstration, and Aristotle's formalization of logic. A unifying thread is the emergence of the familiar view -- not, of course, uncontested -- of mathematics as attaining a unique exactness, certainty, and insight into the nature of things.
Date: Wednesday, December 3, 2003
Speaker: Prof. Dan Curtin
Department of Mathematics and Computer Science
Northern Kentucky University
Title: Scenes from the Pre-history of Calculus
Abstract: I will discuss the approaches to tangents and maxima of Fermat, Descartes, and Hudde. Descartes and Fermat proposed quite different procedures, and carried on a rather acrimonious discussion of the issues in the pages of their letters to Mersenne. Although some of Descartes' criticisms were valid, ultimately Fermat's approach won out and Descartes' fell by the wayside. Still, I will show how Descartes' approach can be used as the basis for student projects in calculus courses. Finally, Hudde's approach to finding maxima is a precursor of our modern rule for the derivative of polynomials, and his paper on maxima and minima has a number of curious twists.
Date: Wednesday, February 4, 2004
Speaker: Prof. Joseph Dauben
Graduate Center
City University of New York
Title: The Politics of Infinitesimals: Marx, Mao and Mathematics
Nonstandard Analysis and the Cultural Revolution.
Abstract: The Mathematical Manuscripts of Karl Marx were first published (in part) in Russian in 1933, along with an analysis by S.A. Yanovskaya. Friedrich Engels was the first to call attention to the existence of these manuscripts in the preface to his Anti-Dühring [1885]. A more definitive edition of the Manuscripts was eventually published, under the direction of Yanovskaya, in 1968, and subsequently numerous translations have also appeared. Marx was interested in mathematics primarily because of its relation to his ideas on political economy, but he also saw the idea of variable magnitude as directly related to dialectical processes in nature. He regarded questions about the foundations of the differential calculus as a touchstone of the application of the method of materialist dialectics to mathematics.

Nearly a century later, Chinese mathematicians explicitly linked Marxist ideology and the foundations of mathematics through a new program interpreting calculus in terms of nonstandard analysis. During the Cultural Revolution (1966-76), mathematics was suspect for being too abstract, aloof from the concerns of the common man and the struggle to meet the basic needs of daily life in a still largely agrarian society. But during the Cultural Revolution, when Chinese mathematicians discovered the mathematical manuscripts of Karl Marx, these seemed to offer fresh grounds for justifying abstract mathematics, especially concern for foundations and critical evaluation of the calculus. In 1975, a study group in the Department of Mathematics at Zhejiang Teachers College issued its own account of "The Brilliant Victory of Dialectics: Notes on Studying Marx's Mathematical Manuscripts."

Inspired by nonstandard analysis, introduced by Abraham Robinson only a few years previously, some Chinese mathematicians adapted the model Marx had laid down a century earlier in analyzing the calculus, and especially the nature of infinitesimals in mathematics, from a Marxist perspective. But they did so with new technical tools available thanks to Robinson but unknown to Marx when he began to study the calculus in the 1860s. As a result, considerable interest in nonstandard analysis has developed subsequently in China, and almost immediately after the Cultural Revolution was officially over in 1976, the first all-China conference on nonstandard analysis was held in Xinxiang, in Henan Province, in 1978.

Date: Wednesday, March 3, 2004
Speaker: Prof. Bruce Burdick
Department of Mathematics
Roger Williams University
Title: A Mathematical Romp through Spanish Colonial Taxation and Currencies
Abstract: The Sumario Compendioso (Mexico, 1556) has been justly celebrated as the earliest work on arithmetic and algebra to be printed in the Western Hemisphere, even though most of it consists of tables to be used for currency exchange, taxation, and transactions with silver and gold. The section on arithmetic and algebra contains, among other things, word problems, quadratic problems which are solved by completing the square, and a treatment of square numbers that appears to be influenced by Fibonacci's Liber Quadratorum. In spite of the attention that this book has received, its author, Juan Diez Freyle, is an enigma. We have gradually come to the position that nearly everything that has been written about him for the last eighty years is in error. In the first part of this talk we will give reasons why we have taken this position, and conclude that there is startlingly little that can be said with confidence about the real Juan Diez Freyle.

One theory, due to Joaquin Garcia Icazbalceta, is that Juan Diez Freyle was actually from Lima, not Mexico, and came to Mexico to print his book because Lima did not yet have a printing press. In the absence of historical evidence for this we look for textual evidence. In the second part of the talk we survey similar works from the New World up to 1700 to see if there is a difference between the accounting texts of Mexico and Lima in this period, and if so, whether the Sumario Compendioso resembles more the one or the other. The evidence, while very consistent with Garcia Icazbalceta's theory, does not completely dispose of the question. However, the survey takes on a life of its own since many items of intrinsic interest reveal themselves. Especially representative is the contrast between Jo'an de Belveder's Libro General (Lima, 1597), the only other book on commerce from this period to include a list of word problems in the back, and Felipe de Echagoyan's Tablas de Reducciones (Mexico, 1603), which not only gives tables for how the royal tax on silver should be collected but also tables for how the tax actually is collected.

Date: Wednesday, April 14, 2004
Speaker: Prof. Sylvia Svitak
Department of Mathematics and Computer Science
Queensborough Community College
City University of New York
Title: The Confluence of Probability and Statistics in the History of Mathematics and Science
Abstract: From ancient musings on chance and early record keeping of floods and commerce, probability and statistics grew and conjoined to be a theoretical base of scientific investigation. The theory of probability advanced from the Fermat-Pascal correspondence in the 1600s to axiomatic status in the 1930s as given by Kolmogorov. Statistical laws, used to estimate errors of astronomical measurements in the 18th century, came to describe the nature of nuclear particles in the 20th, a century also called "the first measured century in world history." In this presentation, I will map the histories of these two rivers of thought and focus on where they met and joined.
Date: Wednesday, May 5, 2003
Speaker: Prof. Andrea Bréard
Graduate Center
City University of New York
Title: The Concept of "Series" in China before and after Zhu Shijie (1303)
Abstract: Zhu Shijie's treatment of finite series in his Jade Mirror of Four Unknowns (Si yuan yu jian, first published in 1303) is one of the most celebrated cases in the history of Chinese Yuan Dynasty mathematics. In historiography it has further become a commonplace that Zhu relied on the work of Yang Hui (Southern Song Dynasty, 13th century) in formulating his procedures. Yet the precise relation between Zhu's approach to "series" and his predecessors' has never been analyzed, nor have the different modes of expression of "series" been properly described in the framework of a textual tradition.

In an attempt to do so, it will prove fruitful to compare first how Zhu Shijie and Yang Hui read the ancient works of their precursors, and then, to observe the different ways in which Zhu Shijie himself was read by later mathematicians and historians of mathematics, in particular during Qing times. Including Zhu's Introduction to Mathematics (Suan xue qi meng, first published in 1299) in our analysis, the following questions will be addressed:

1) The problems concerning "series" were usually grouped together with other mathematical or astronomical problems. What were the changing patterns of their classification, and what information on Zhu's methodology can be implied from the different modes of classification?

2) Zhu's treatment of series was conducted within the context of the former treatments. What exactly were the structures that he identified and adopted from former texts in his mode of description? The same question will be raised for the operations performed by Qing Dynasty mathematicians reading Zhu's text at the end of the 19th century.

3) What is implied by Zhu Shijie's system of names for discrete accumulations, and how do these names affect later interpretations of the mathematical objects in Zhu's text?