Date: 
Wednesday, October 8, 2003 
Speaker: 
Prof. Ivor GrattanGuiness
Middlesex University at Enfield
ivor2@mdx.ac.uk

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
hardygrant@yahoo.com

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
curtin@nku.edu

Title: 
Scenes from the Prehistory 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
JDauben@gc.cuny.edu

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 AntiDühring [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 (196676), 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 allChina 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
bburdick@rwu.edu

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
ssvitak@qcc.cuny.edu

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 FermatPascal 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
andrea@breard.com

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?

