




History of Mathematics / Histoire des mathématiques (Richard O'Lander and Ronald Sklar, Organizers)
 MICHAEL BARR, Department of Mathematics and Statistics, McGill University
Reminiscences of category theory and Homological algebra

I will talk about some of the early history of these two subjects based
mainly on my own experiences as well on personal encounters with
Eilenberg, Mac Lane and others.
 ISMAIL BISKRI, Université du à QuébecTrois Rivières, Québec
Combinatory logic, categorial grammar and natural language
processing in the framework
of applicative and cognitive grammar

Applicative and Combinatory Categorial Grammar is an extension of
Steedman's Combinatory Categorial Grammar by a canonical association
between rules and Curry's combinators on the one hand and metarules
which control typeraising operations on the other hand. This model is
included in the general framework of Applicative and Cognitive
Grammar. Applicative and Cognitive Grammar (Desclés 90) is an
extension of the Universal Applicative Grammar (Shaumyan 87). It
postulates three levels of representations of languages: (i) Phenotype
level (or phenotype) where the particulary characteristics of natural
languages are described (for example order of words, morphological
cases, etc...). The linguistic expressions of this level are
concatened linguistic units, the concatenation is noted by:
`u1u2¼un'; (ii) Genotype level (or genotype) where
grammatical invariants and structures that are underlying to sentences
of phenotype level are expressed. The genotype level is structured like
a formal language called ``genotype language''; it is described by a
grammar called ``applicative grammar''; (iii) The cognitive level
where the meanings of lexical predicates are represented by semantic
cognitive schemes. Representations of levels two and three are
expressions of typed combinatory logic (Curry & Feys 58) (Shaumyan
87). The aim of the talk is: (i) an automatic parsing of phenotype
expressions that are underlying to sentences; (ii) the constructing of
applicative expressions.
 JONATHAN BORWEIN, Simon Fraser University, Burnaby, BC V5A 1S6
Aesthetics for the working mathematican

If my teachers had begun by telling me that mathematics was pure
play with presuppositions, and wholly in the air, I might have become a
good mathematician..., (George Santayana, 1945)
Most research mathematicians neither think deeply about nor are
terribly concerned about either pedagogy or the philosophy of
mathematics. Nonetheless, as I hope to indicate, aesthetic notions have
always permeated (pure and applied) mathematics.
Through a suite of examples, I aim to explore what that means at the
research mine face. I also will argue that the opportunities to tie
research and teaching to aesthetics are almost boundlessat all
levels of the curriculum. This is in part due to the increasing power
and sophistication of visualization, geometry, algebra and other
mathematical software.
The transparencies, and other resources, for this presentation are
available at www.cecm.sfu.ca/personal/jborwein/talks.html and
www.cecm.sfu.ca/personal/jborwein/mathcamp00.html
 STAN BURRIS, University of Waterloo, Waterloo, ON N2L 3G1
Boole's equational treatment of particular statements

Boole has been universally condemned for trying to use equations to
give an algebraic treatment of the particular categorial statements.
The goal of this talk is to show that his ideas are actually quite
sound.
 JOAN DEBELLO AND RONALD SKLAR, St. John's University, Jamaica, New York 11439, USA
Woody Bledsoe's peopleoriented approach to automated theorem proving

Woody Bledsoe was originally a proponent of the resolution approach to
Automated Theorem Proving. However, he soon realized that resolution
would not allow one to prove ``hard'' theorems in mathematics. He
therefore turned to a nonresolution technique called natural
deduction, which tries to incorporate the way mathematicians reason
when they attempt tp prove a theorem. We will discuss the
contributions and successes of Bledsoe's approach to Automated Theorem
Proving.
 ERIK DEMAINE, MIT Laboratory for Computer Science, Cambridge,
Massachusetts 02139, USA
History of geometric constructions by paper folding

Paper folding (origami) has been used as a tool for illustrating
geometric constructions since at least 1840. In particular, T. Sundara
Row's influential 1893 book illustrates how various Euclidean
constructions can be executed accurately by only folding paper, without
the usual straight edge and compass. Various researchers have
independently attempted to axiomatize the set of basic paperfolding
constructions, and to characterize the resulting constructible
numbers. As early as 1936, it was known that paper folding can
construct nonEuclidean numbers, and indeed solve all polynomial
equations of degree at most four. The various axiomatizations over
time have sometimes omitted an axiom or two that prevent such
constructions, but the majority of the axiomatizations are equivalent.
The first to characterize the origami constructible numbers were
Humiaki Huzita and Benedetto Scimemi in 1989. These results have since
been rediscovered several times, and are still not generally
wellknown, probably because the relevant literature is difficult to
access. We summarize the history of these and related results, and
present a connection between paper folding and a geometric construction
device called the Mira which provides an alternate proof using a result
from 1994. We also present new axiomatizations that make paper folding
much more powerful.
 LISA FAN AND BRIEN MCGUIRE, Department of Computer Science, University of Regina, Regina,
Saskatchewan S4S 0A2
An overview of an undergraduate student mentoring program

This talk gives an overview of an undergraduate student mentoring
program at the Computer Science Department, University of Regina. The
undergraduate mentoring program enrolls good senior undergraduate
students in senior courses as mentors for junior students enrolled in a
lower level course. The mentoring program is the only program formally
involving undergraduate students in undergraduate teaching at the
faculties of science and engineering in the University. The program has
been offered for two years. Both the senior students as mentors and
the junior students taking the lower level course have responded to the
program very positively. This talk describes the design and
implementation of the program. The authors also try to explore some
philosophical implications of the mentoring program to undergraduate
teaching. Further improvements for the program are also proposed.
 MICHAEL FRIENDLY, Department of Psychology York University, York,
Ontario M3J 1P3
The graphical works of Charles Joseph Minard

Charles Joseph Minard is most widely known for a single work, sometimes
said to be the ``best statistical graphic ever produced,'' and a work
which ``defies the pen of the historian.'' This poignant flowmap
depicts the terrible fate of Napoleon's Grand Army in the disasterous
1812 Russian campaign.
Even more than this, Minard was a true pioneer in thematic cartography
and in statistical graphics. He developed many novel graphic forms to
depict databut always with the goal to let the data ``speak to the
eyes.'' This paper reviews Minard's contributions to statistical
graphics, and some background behind the famous March on Moscow
graphic.
As time permits, we also look at some modern revisions of this graph
from an information visualization perspecitive, and examine some
lessons this graphic provides as a test case for the power and
expressiveness of computer systems or languages for graphic information
display and visualization.
 CHARLES GALLANT, St. Francis Xavier University, Antigonish, Nova Scotia B2G 2X1
Mathematics B.C.when dinosaurs roamed the earth

Technological advance has radically changed the mathematics classroom.
The handheld calculator and the computer have made penandpaper
calculations obsolete. Indeed we are unlikely to ever demand these
algorithms of students todayor are we???
The curriculum of secondary schools and universities demand much more
than mere computational facility. Perhaps, then, dusting off some of
these old algorithms would be appropriate, and might help achieve these
other objectives e.g. of experiencing algorithmic thinking, of
understanding the constituent concepts, and of exploring problem
solving situations.
Since the old penandpaper algorithms were always so streamlined and
therefore devoid of meaning for most people, let's review some of the
surprisingly accurate and ingenious techniques and see (for the first
time??) why they worked.
 TOM HURD, McMaster University, Hamilton, Ontario L8S 4K1
Martingales and their role in mathematical finance

Since the pioneering works of Black, Scholes, Merton and others in the
1970s and earlier, finance has been hit a never decreasing wave of
mathematization which makes particularly heavy use of the theory of
continuous time stochastic processes. By crystalizing the idea of a
fair game, the somewhat specialized notion of martingale has proved
remarkably well adapted for describing the fundamental concepts of
financial modelling. This talk will review the main developments of
modern mathematical finance and have a look at some of the reasons for
the current status of the martingale as the central unifying concept.
 VELIMIR JURDJEVIC, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
S. Kowalewski and the mathematical legacy of the top

The prize winning paper of S. Kowalewski of 1889 concerning the motion
of a rigid body around its fixed point in the presence of gravity is,
without doubt, the most dramatic contribution to the theory of the
top. Bold, original and eccentric, the paper produces a new integral
of motion for the top under exceptional relations among the principal
moments of inertia (I_{1} = I_{2} = 2 I_{3}) and under the condition that
the center of mass of the body is in the equatorial plane relative to
the coordinates of the body measured by an orthonormal frame affixed to
the body. The existence of this integral of motion is deduced through
a search for the meromorphic solutions of complex time, and then the
equations in this exceptional case are integrated in terms of
hyperelliptic functions based on an unexplained change of coordinates.
Compelling and bizzare, the findings of Kowalewski have fascinated and
challenged the mathematical community all the way to present, and it
was only recently that the mystery behind these results has been
essentially lifted.
My lecture, a journey through mathematical history starting with
L. Euler in 1775 concerning the arclength of the lemniscate to the
paper of A. Weil on the addition formulas for elliptic curves and to
the contemporary studies of Hamiltonian system on Lie groups, provides
a mathematical context for understanding Kowalewski's contributions to
mathematics through her work on the top.
 STANLEY KOCHMAN, Department of Mathematics and Statistics, York University,
Toronto, ON M3J 1P3
Maple labs for calculus

In Winter 2001, students in my section of the mainstream calculus
course at York University were required to complete six Maple labs.
Each student learned the basic Maple calculus commands independently
through an interactive lab manual. After each lab the student chose
one of several nontrivial problems to solve using Maple. Then the
student used Maple to create an exposition of the solution which was
submitted for grading. This talk will detail the methodology of this
learning experience as well as its pedagogical benefits.
 JIM LAMBEK, McGill University, Montreal, Quebec
The number systems in Greek philosophy

Presocratic Greek philosophy was largely concerned with the question:
What is more basic, measuring (hence real numbers) or counting (hence
natural numbers)? These views were advocated by Thales and Pythagoras
respectively. When the existence of irrationals was discovered, to the
disappointment of the Pythagorean school, the question arose: how to
define the reals? The modern answers by Dedekind and Cauchy were
anticipated in Plato's Academy by Eudoxus and Theaetetus respectively.
The latter used what we now call continued fractions. Stelios
Negrepontis recently discovered that these enter all of Plato's
dialectical dialogues, either literally or metaphorically.
 MIROSLAV LOVRIC, Department of Mathematics and Statistics, McMaster University,
Hamilton, ON L8S 4K1
Old math textbooksWhy did we abandon them?

In this presentation, I plan to compare old and new textbooks in
mathematicsnot just the way they look or the way they are written,
but also in terms of messages they send to their users. I will try to
argue that, by abandoning old textbooks, we might be in danger of
abandoning some important methods and ideas on teaching and learning
mathematics.
 ANGELO MINGARELLI, School of Mathematics and Statistics, Carleton University, Ottawa,
Ontario
The golden section in early renaissance Italian art

The aim of this talk is mainly to lead the audience to a greater
appreciation of the role and the use of the golden section in early
renaissance Italian art. We present material some of which is dubbed as
folklore or speculative in art history circles as it is often difficult
to determine the intentions of a painter of the early era. On the other
hand, as we will show, in some cases the intent is clearly visible and
perhaps the composition had its roots in the judicious use of golden
rectangles.
 DON ROBINSON, St. Thomas University, Fredericton, New Brunswick E3B 5G3
Anomaly cancellation in the standard model of particle physics

An anomaly occurs when a classical symmetry fails to survive
quantization and regularization (a step in renormalization). In the
late 1960's, anomalies were discovered in the WeinbergSalam model of
leptons and electroweak interactions. Their presence spoiled the
renormalizability of the theory. However, it was subsequently
discovered that when quarks from QCD are inserted into the model, they
too produce anomalies but opposite in sign to those produced by the
leptons. This cancelling of anomalies was subsequently taken to be a
constraint on the construction of further theories. This paper traces
the history of the discovery of anomaly cancellation and discusses this
phenomenon in the context of models the theory selection.
 LUIS SECO, University of Toronto, Toronto Ontario, M5S 3G3
Mathematical finance: past, present and future

One of the remarkable advances in industry that can be directly linked
to sophisticated mathematics is the development of the option pricing
theories of Black and Noble Laureates Merton and Scholes. That topic
has also revolutionized the way in which mathematics and mathematicians
relate to the financial sector. This talk will survey some historical
developments, describe the current state of affairs, and speculate
about its incidence in the near future.
 DAVID SPRING, Glendon College, 2275 Bayview Avenue, Toronto, ON M4N 3M6
The golden age of immersion theory in topology: 19591973

We briefly review the modern history of immersiontheoretic topology,
beginning with the seminal work of S. Smale in the U.S.A. and finishing
with the work of the Leningrad School in Russia, especially the work of
M. Gromov. We discuss also the interesting role that jet spaces of
maps played in the formulation of results for the Leningrad school.
The presentation is nontechnical with emphasis on historical
developments.
 W.A. VAN WIJNGAARDEN, Department of Physics, York University, Toronto, ON M3J 1P3
History of lasers

Lasers are revolutionizing everyday life. A brief history of laser
development and progress over the last four decades will be given,
along with exciting examples illustrating the use of lasers in fiber
optics, biomedicine, environmental monitoring and scientific research.
 WOJCIECH ZIARKO, University of Regina, Regina, Saskatchewan S4V 0L3
Rough sets: the origins and current status

Rough Set theory was originated by Zdzislaw Pawlak in early eighties.
The theory is concerned with the classificatory analysis of imprecise,
uncertain or incomplete knowledge, often expressed in terms of data
acquired from experience. It extends the standard notion of a set by
incorporating the knowledge about the domain (the universe) into the
model. This knowledge is subsequently used to construct, approximate
in general, discriminatory description of any subset of the universe.
The primary notions of the theory of rough sets are the approximation
space and lower and upper set approximations. The approximation space
is a classification of the domain of interest into disjoint
categories. The classification formally represents our knowledge about
the domain and is used to derive the lower and upper set
approximations. The theory of rough sets was used as a basis of
numerous application algorithms and applications for data mining,
machine learning, pattern recognition and data analysis. It also
provided fertile ground for theoretical research in logic of
approximate reasoning and fundamentals of mathematics. The
presentation will introduce the basic ideas of rough set theory and
will review the past and recent developments in the area of rough sets
with particular emphasis on existing and potential applications.

