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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 Québec-Trois 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 type-raising 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: `u1-u2-¼-un'; (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 boundless-at 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 and

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 people-oriented 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 non-resolution 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 paper-folding constructions, and to characterize the resulting constructible numbers. As early as 1936, it was known that paper folding can construct non-Euclidean 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 well-known, 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 flow-map 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 data-but 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 re-visions 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 pen-and-paper calculations obsolete. Indeed we are unlikely to ever demand these algorithms of students today-or 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 pen-and-paper 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 (I1 = I2 = 2 I3) 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 arc-length 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 textbooks-Why did we abandon them?

In this presentation, I plan to compare old and new textbooks in mathematics-not 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 Weinberg-Salam 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: 1959-1973

We briefly review the modern history of immersion-theoretic 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 non-technical 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.

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