


History of Mathematics / Histoire des mathématiques Org: Thomas Archibald (Acadia, Dibner Institute MA), Rich O'Lander (St. John's), Ron Sklar (St. John's) and/et Alexei Volkov (UQAM)
 THOMAS ARCHIBALD, Acadia University and Dibner Institute, MIT
The Reception of Fredholm's work on Integral Equations

In 1900 Ivar Fredholm presented a method for the solution of integral
equations. The method was extremely useful in showing the existence
of solutions for certain types of boundaryvalue problems in partial
differential equations, and attracted a good deal of international
attention. There were two main lines of reception of the work. The
more conservative consisted of detailed investigations of its meaning
in the field of differential equations, with important contributions
by Poincaré, Picard, T. Boggio and G. Lauricella, among others. The
more radical was initiated by Hilbert, and lies at the origin of the
theory of linear operators on what we now term Hilbert spaces. In
this paper we examine these two threads in the early reception of
Fredholm's work, which persisted as separate though interacting
traditions.
 RYAN BEATON, McGill

 JON BORWEIN, Dalhousie Faculty of Computer Science, 6050 University Drive
Philosophical Implications of Experimental Mathematics

I will discuss the philosophical implications of my work in
experimental mathematics. Philosophers have frequently distinguished
mathematics from the physical sciences. While the sciences were
constrained to fit themselves via experimentation to the `real' world,
mathematicians were allowed more or less free reign within the
abstract world of the mind. This picture has served mathematicians
well for the past few millennia but the computer has begun to change
this. The computer has given us the ability to look at new and
unimaginably vast worlds. It has created mathematical worlds that
would have remained inaccessible to the unaided human mind, but this
access has come at a price. Many of these worlds, at present and
perhaps for ever, can only be known experimentally. Thus, work in
experimental mathematics challenges the standard view of mathematics
as a subject in which proof is the sole pathway to knowledge.
David Bailey and I make this case at length in our newly published
books on Experimental Mathematics
(www.expmath.info).
We start
by observing that: One of the greatest ironies of the information
technology revolution is that while the computer was conceived and
born in the field of pure mathematics, through the genius of giants
such as John von Neumann and Alan Turing, until recently this
marvelous technology had only a minor impact within the field that
gave it birth.
The future of mathematical computing is expected to rely on a mixture
of symbolic and numeric (hybrid) computation that will increasingly
call for significant computing power. This is obvious at the level of
"grand challenge problems" in fluid dynamics, meteorology and
elsewhere, but is equally true in the finance and banking sectors, as
for ecommerce and, growingly, for pure mathematics. Indeed, it is my
personal conviction that the success of the computer as an inference
assistant and insight generating engine demands massively parallel
computation. Only when many small things are done by the computer on a
real time scale (as the mathematical equivalent of a
"spellchecker") can insight follow easily and freely.
 LOUIS CHARBONNEAU, UQAM, C.P. 8888, Succ. Centreville, Montréal, QC,
H3C 3P8
L'algèbre de François Viète et sa réception en
France au XVIIe siècle

En donnant à l'algèbre son indépendance face à
l'arithmétique et à la géométrie et en la situant dans le
cadre d'un programme analytique, au sens des Anciens, Viète ouvre de
nouvelles perspectives en mathématiques. Mais l'oeuvre
du
pionnier est reçue de diverses façons par les mathématiciens
qui lui succèdent. Nous verrons comment progressivement la pensée
viètienne est appliquée dans les travaux de ses premiers
disciples, Anderson et Ghetaldi, de reconstructions
d'oeuvres
grecques perdues, mais aussi de disciples plus tardifs, aujourd'hui
oubliés, comme James Hume, Pierre Herigone et Nicolas Durret. Leurs
oeuvres illustrent le passage graduel de l'algèbre dite vulgaire
à une algèbre qui nous est plus familière, comme celle de
Descartes et des autres grands mathématiciens de la fin du
siècle.
 DAVID COYLE, Université de Montréal and Dawson College
Henri Cartan's filters in the Bourbaki archives

The archives for the Bourbaki group have been recently enlarged and
expanded. They may be broadly divided into two categories:
(1) Successive drafts for different sections of the final
publications.
(2) Informal communications in the form of personal
correspondence and internal newsletters.
After a short description of the early Bourbaki group to provide a
context for what follows, I present sample documents from the
archives, focusing on integration theory and, in particular, the
filters of Henri Cartan. While the more formal drafts are of central
importance for the historian in tracing the development of the
Bourbaki and their mathematics, it is the entertaining informal
communications which provide the clearest clues to their motivations,
which illustrate most clearly what it means to do mathematics, and
which are in consequence of greatest interest to the student. It is,
moreover, the informal items which explain the Bourbaki style of
presentation, and expose most starkly the many internal contradictions
of their project.
 JOSEPH DAUBEN, Lehman College, City University of New York
Suan Shu Shu (A Book on Numbers and Computation): Problems in
Collating, Interpreting and Translating the Most Ancient
YetKnown Chinese Mathematical Text

In December and January of 19831984, archaeologists excavating the
tomb of an ancient Chinese nobleman at a Western Han Dynasty site near
Zhangjiashan, in Jiangling county, Hubei Province, discovered a number
of books on bamboo strips, including works on legal statutes, military
practice, and medicine. Among these was a previously unknown
mathematical work on some 200 bamboo strips, the Suan Shu Shu,
or Book of Numbers and Computation. As the earliest yet
discovered work devoted specifically to mathematics from ancient China
(no later than 186 BCE), it has stirred considerable interest among
historians of Chinese mathematics.
While some sections of this work are straightforward and have been
understood with little disagreement, otherswhether because of
missing, misplaced, or incomplete parts of the texts on the bamboo
strips comprising this earliest of the yetknown mathematical works
from ancient Chinahave been open to diverse and often divergent
interpretations. In some cases, related methods serve as clues to
help interpret the meaning of given problems, as is the case for the
three problems devoted to Fu Tan, Lu Tang, and Yu
Shi. But for another pair of seemingly related problems, Yi
Yuan Cai Fang and Yi Fang Cai Yuan, there has been little
agreement about whether these are inverse or quite different problems,
and virtually everyone who has approached these two problems has
understood them differently in trying to account for the statements,
answers, and methods given for these two problems. This presentation
will be devoted to discussion of the various collations and
explanations offered for these especially challenging parts of the
Suan Shu Shu, and what they may tell us about early Chinese
mathematics in general.
 ROBERT DAWSON, St Mary's University, Halifax, NS B3H 3C3
The Slide Rule in the 21st Century Classroom: Cultural Icon
and Manipulative

When many of the parents of today's high school students were
themselves in high school, the slide rule was ubiquitous: mysterious
to many, but as iconic a badge of the scientist as the doctor's
stethoscope or the farmer's pitchfork. Today, three decades later,
not a single one of the famous brandsKeufel and Esser, Hemmi, Faber
and Castell, etc.are still manufactured. For the
inflationadjusted cost of a basic slide rule in 1970, you can buy a
calculator which can do everything that the slide rule could, and much
more. Moreover, the calculator is easy enough to use that it has
achieved a level of penetration into the schoolroom that the slide
rule never did.
Nonetheless, it can be argued that the slide rule had pedagogical
potentials that the calculatorno matter how sophisticateddoes
not. It makes many of the properties of exponents and logarithms
clear in an intuitive, tactile way. It encourages orderofmagnitude
calculation and provides a good motivation for scientific notation.
And, finally, it is today a link with living mathematical history; for
some decades yet, many school classes will be able to find a parent or
grandparent who used a slide rule professionally.
This talk suggests some ways in which the slide rule might be included
in the modern high school classroom. In particular, I will look
briefly at ways of getting around the current (and, it is to be
feared, continuing) slide rule shortage, and some ways in which the
design of the slide rule might be modified for an environment in which
it is primarily a manipulative rather than a calculating device.
 PAUL DEGUIRE, Université de Moncton
Teaching history of maths to non scientists

For the past 6 years I have presented conferences to groups of liberal
art students whose formation includes no maths and no science beyond
high school. They are students in a specific 1 year program called
"Odyssée Humaine" in which all the different classes are
interrelated, that is the teacher in a philosophy class for instance
knows exactly where the students are in their geography or history
class and builds his own course upon this knowledge.
We are in the process of creating a specific course on History of
Maths and Science that will be integrated into a modified version of
l'Odyssée Humaine that is supposed to begin in September 2005. Of
course, it will probably take a few years to create a satisfactory
description of this course. Nevertheless, it starts next September
and in this talk, I'll present the starting point of my thoughts
concerning this presentation of Math History to nonscientists.
One of the important difference between this course and a similar
course that is intended for math students is the objective. Knowledge
of specific results is not an important part of this new course, but
the integration of mathematical knowledge and scientific knowledge in
the broader scope of the evolution of the western culture is.
Two examples: First, Mathematics in Ancient Greece probably played an
important rôle in the apparition of Greek rationalism and philosophy.
Second, new successes in maths, applied as well as pure, played an
important part of the fast evolution of Europe after the Middle Ages.
Just knowing that they were able to do things that the Ancient Greeks
and Romans never did was enough to create a climate of confidence that
lead to the scientific revolution of the 17th century.
 HAROLD HASTINGS, MARYSIA WEISS, YIHREN WU, Department of Physics, 151 Hofstra University, Hempstead, NY
115491510
A Century of Topological Dynamicsreflections on its impact
upon education and research

Many of us were drawn to topological dynamics in the 1970s and 1980s
by the apparently surprising behavior and broad applicability of maps
of the unit interval: a sequence of bifurcations leading to the onset
of chaos showing apparently random behavior. At the same time, many
key results are readily accessible to undergraduate students, and
topological dynamics has found its way into the undergraduate
curriculum. This talk will survey some high points of historical
development of topological dynamics.
Partially supported by the NSF.
 MICHEL HELFGOTT, East Tennessee State University, Department of Mathematics,
P. O. Box 70663, Johnson City, TN 37614, USA
Thomas Simpson and Problems of Maxima and Minima

Thomas Simpson (17101761) was a selftaught English mathematician
who wrote several remarkable books, among them "The Doctrine and
Application of Fluxions" (1750). The fact that in 1823 it was still
being published attests to its wide and deserved popularity. Changing
the word "fluxion" by "derivative", the problems on maxima and
minima that can be found in the abovementioned work stand out as
particularly interesting and can be read with profit nowadays.
Keeping a historical framework we will discuss some of these problems.
 ROBERT IRWIN, SUNY Oswego, Oswego, NY 13126
Early Computability Theory

A survey of early computability theory will be presented. Here,
"early" means the 1920s and 1930s. The main different, but
equivalent, approaches to formalizing (or defining) the notion of
effective calculability in this period will be discussed, as will be
the motivations of the prime movers of the theory, including Gödel,
Kleene, Church, Turing and Post. Modern continuations of some
research threads originating in the early period will be briefly
discussed.
 ALEXANDER JONES, Classics, University of Toronto, 97 St. George Street
Patterns of deduction in Ptolemy's Almagest

Ptolemy's treatise on celestial mechanics, the Almagest, is
not composed as an assemblage of reasonings but as a single reasoning,
the biggest deductive argument in all ancient applied mathematics.
This is exceptional in the history of astronomical writing, even (so
far as we can tell) in antiquity. Other writers recognize that the
solutions of some problems depend on others, but they generally assume
that some degree of isolation between problems is possible. But for
Ptolemy, mathematical astronomy is like a puzzle that can only be
taken apart piece by piece in a particular order starting with a
particular key piece.
This paper will consider three aspects of the logical structure of the
Almagest:
(1) the three levels of argument (nontechnical deduction,
technical quantitative deduction, and model confirmation);
(2) the largescale ordering of the topics; and
(3) the use of recursive deductions.
 NIKY KAMRAN, McGill University, Montreal, Quebec, Canada
Einstein, Hilbert and the field equations of gravitation

It is a remarkable fact that Einstein and Hilbert discovered the field
equations of the relativistic theory of gravitation almost
simultaneously, following totally different paths. I shall review
some of the highlights of this fascinating episode, including the
important contribution that Emmy Noether made to the development of
Hilbert's approach.
 JIM LAMBEK, Mathematics Department, McGill University
Remarks on the History of Categorial Grammar

Categorial grammar is an attempt to place most of the grammar of a
natural language into the dictionary. This is accomplished by
assigning to each word in the dictionary one or more types, which are
taken to be elements of an algebraic system or terms of a
substructural logic. There are, at present, two streams of
investigation. One, going back to Ajdukiewicz, relies on residuated
monoids, i.e., monoids equipped with two binary operations of
division that mimic logical implication. The other makes use of two
unary operations instead, which mimic the logical operation of
negation. It turns out that the second approach, which I now favour,
has historical roots in ideas of C. S. Peirce and Z. Harris.
 DUNCAN MELLVILLE, St. Lawrence University
Multiplication in Mesopotamia

We trace the development of the ideas of multiplication in Mesopotamia
from the first archaic evidence to the Old Babylonian period
(20001600 BC). In particular, we will show how the abstract
sexagesimal place value system that was developed at the end of the
third millennium allowed the unification of ideas of area and repeated
addition by freeing these concepts from the metrological systems that
had earlier been in use.
 ROUND TABLE DISCUSSION

 CHRISTIANE ROUSSEAU, Université de Montréal
From Rolles theorem to Khovanskii theory of fewnomials

The modelling of an important number of problems in mathematics leads
to real systems of equations with real unknowns. Depending on the
context we may only be interested in the number of solutions, or the
subproblems of, either finding a bound for the number of solutions, or
showing that the number of solutions are finite. In this lecture I
will discuss the power of Rolles theorem and its generalization by
Khovanskii to the Rolles theorem for dynamical systems. I will
introduce the theory of fewnomials and discuss some applications.
 NATHAN SIDOLI, University of Toronto
Geometrical Analyses in Heron's Measurements

The bulk of Heron's Measurements can be described as applied,
perhaps even practical, mathematics. In the third book, however, he
gives a series of analytic propositions in the pure geometric idiom.
He tells us that these will be useful for solving problems which
cannot be solved by computation (dia ton arithmon). An
investigation of this material is interesting on two counts. In the
first place, it adds to our limited evidence for the actual practice
of ancient geometrical analysis. More interestingly, however, it
makes clear that Heron considered geometrical analysis capable of
solving certain problems that were not susceptible to computation. My
talk will try to elucidate Heron's thinking in these matters.
 GLEN VAN BRUMMELEN, Bennington College
AlSamaw'al's Curious Approach to Trigonometry

The 12thcentury mathematician Ibn Yahya alMaghribi alSamaw'al, now
better known for his algebra, also wrote the extensive treatise
Exposure of the Errors of the Astronomers. This fascinating
understudied work, containing criticisms of a number of astronomers,
provides an interesting study of debates over the proper practice of
medieval astronomy. In particular, alSamaw'al eschews any form of
geometrical approximation, no matter how trivial. One of his
objections is to the methods that had been used to determine the
geometrically unattainable sin (1 degree), in Ptolemy's
Almagest as well as in later Muslim works. To avoid this
apparently unavoidable problem, alSamaw'al presents an alternate
trigonometric table that breaks the circle into 480 rather than
360 parts. We shall present the table as well as one of its uses in
alSamaw'al's work.
 ALEXEI VOLKOV, Université du Québec à Montréal
History of traditional Vietnamese mathematics: the state of
the field

The history of traditional Vietnamese mathematics can be reconstructed
on the basis of the extant mathematical treatises only partially.
Some 20 mathematical treatises are to be found in Vietnamese
libraries, of which the latest ones were compiled in the early 20th
century; as for the earliest texts, the date of their actual
compilation remains uncertain, but most likely they do not antedate
the early 18th century. All these texts are compiled on the basis of
Chinese mathematical treatises of the Ming (13681644) and Qing
(16441911) dynasties.
The paper will focus on several issues pertaining to the study:
(1) the legends of the life and scientific activities of the
state functionary Luong The Vinh (14411496?) conventionally viewed
as the most outstanding Vietnamese mathematician;
(2) the scientific interaction between Vietnamese literati and
Jesuits in the early 17th century;
(3) the revival of mathematics education in Vietnam in the
early 18th and 19th century; and
(4) the decline of traditional mathematics in the context of
introduction of Europeanstyle mathematics education under French
colonial rule.

