Speaker: Professor K Ramasubramanian is in the department of Humanities and Social Sciences at IIT Bombay, India. He is one of the authors who prepared the Explanatory Notes of the celebrated work, Ganit-yuktibhaasha (Ratiional in Mathematical Astronomy) of Jyeshthadeva (c. 1530) which brings out the seminal contributions of the Kerala School of astronomers and Mathematicians in the field of mathematical Analysis. The book Ganit Yuktibhaasha has already been published by Hindustan Book Agency, New Delhi in two volumes and has been reprinted by Springer. Besides this Prof. Ramasubramanian has edited the (500 years old) work on Tantrasangraha with Prof. Sriram and Prof. Srinivas, published by the Indian Institute of Advanced Study, Simla (India). He believes that one of the main reasons why not much headway has been made in arriving at a comprehensive understanding of development of science in India is that we are yet to develop a community of scholars who are well-versed in Sanskrit to understand classical Indian scriptures and at the same time have necessary expertise in science. Professor Ramasubramanian is a traditionally trained Sanskrit scholar and also holds a doctorate in Theoretical Physics.
ABSTRACT: In this talk, I propose to examine the ways in which different mathematical traditions of the past have been characterised by historians of mathematics. A litmus test of a valid mathematical practice today is "proof" and a number of criticisms have been levied against Indian mathematics because of the perceived absence or lack of rigour in their proof procedures (seen today as the litmus test of whether we are "doing" real mathematics or doing it well). By taking specific examples, I propose to examine the validity of these criticisms and indicate the relevance of alternative proof traditions today.
ABSTRACT: Now a flourishing branch of our discipline, with its own journal since 1992, "experimental" mathematics has been characterized as "the utilization of modern computer technology as an active tool in mathematical research". Taken literally, this definition obviously confines the history of the subject to the past few decades. But in fact the psychology and methodology implied by the definition have much earlier incarnations, some of which I shall here try to point out. In particular one can ask, in his tercentenary year, to what extent Euler may be said to have anticipated modern developments, and the answer is that he could serve as "poster boy". I shall take all my examples from number theory, including the Riemann hypothesis.
ABSTRACT What does mathematics, which often provokes interesting ontological and epistemological questions, have to say about philosophy's other long-standing topic, that of aesthetics? While the Ancient Greek mathematicians certainly saw a strong affinity between mathematics and aesthetics, modern views of mathematics--be they those of philosophers, mathematicians themselves, and, especially, schoolchildren!--often see aesthetic considerations as belonging almost exclusively to the domain of art. In this talk, I will draw on some of the constructs used in philosophy, and art criticism in particular, to argue that the aesthetics is deeply enmeshed in the development and understanding of mathematical knowledge.
ABSTRACT: Although there was great interest in the classical method of analysis during the early modern period, after about 1800 explicit foundational discussions of this method appeared much less frequently in the writings of mathematicians. Analysis came to mean simply all calculus-related parts of mathematics and basic questions centered on the definition of the function concept, the properties of infinite series, the nature of the numerical continuum, existence of integrals and the representation of functions. When foundational investigations picked up at the end of the nineteenth century, the concern was with the place within mathematics of logic and set theory and the role of the axiomatic method. As Michael Mahoney has observed, during this later period the subject of Greek analysis had become the object of historical inquiry and evaluation. The paper reflects on the place of analysis in the eighteenth century, a time that was intermediary between the early modern period, when notions of analysis rooted in Greek antiquity occupied a vital place, and the nineteenth century, when these notions were seemingly only of historical interest.
ABSTRACT: "The significant problems we face cannot be solved by the same level of thinking that created them."--Albert Einstein
"The existence of analogies between the central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to these central features".--E.H. Moore
Since ancient times, mathematicians have struggled with the precise formulation of continuity. An abstract, yet a simple formulation, was given by F. Riesz in 1908 [Ri], using the concept of nearness. Nearness is one of the rare concepts in the whole of mathematics that is at once intuitive and which can be made rigorous with little effort. Since we use the words "near" and "far" in day-to-day life, the concept of "nearness" is--even for non-mathematicians--easy to understand. It can be explained to "the first person one meets in the street", as the great mathematician Joseph Louis Lagrange said. And it can be made rigorous by formulating precise axioms. Its simplicity and depth provide a powerful tool in research in Topology and Analysis.
In this talk I'll attempt to give a brief history of the concept in its various incarnations: TOPOLOGY, PROXIMITY, NEARNESS, UNIFORMITY and their use in teaching CALCULUS, ADVANCED CALCULUS, BEGINNING TOPOLOGY etc. Topics include Compactifications, Extensions of functions from dense subspaces, Reflective Functors, Hyperspaces, Function Spaces, Metrization, Hahn-Banach Theorem, Open Mapping and closed graph theorems, Duality etc.
Applications to General Relativity, Mathematical Music, Digital Images, and Mathematical Economics will be noted.
References:
[CHN] P. Cameron, J. G. Hocking and S. Naimpally, Nearness- a better
approach to continuity and limits, Amer. Math. Monthly 81 (1974), 739-745.
[ DL ] R. Z. Domiaty and O. Laback, Semi-metric spaces in General
Relativity (On Hawking-King-McCarthy's path topology), Russian
Math.Surveys 35; 3 (1980), 57-69.
[Na 1] S. A. Naimpally, Reflective functors via nearness, Fund.
Math.85 ( 1974 ), 245-255.
[Na 2] S. A. Naimpally and B. D. Warrack, PROXIMITY SPACES, Cambridge
Tract # 59 ( 1970 ).
[ Na 3 ] S. A. Naimpally, NEARNESS APPROACH TO TOPOLOGICAL PROBLEMS (
Submitted to Cambridge University Press ).
[ Ri ] F. Riesz, Stetigkeitsbegriff und abstrakte Mengenlehre, Atti
IV Congress International Mathematics Roma ( 1908 ), 18-24.
ABSTRACT: I will discuss Dedekind's mathematical work, focusing on his path-breaking contributions, including the founding of algebraic number theory, the definition of the real numbers in terms of "Dedekind cuts", and the definition of the natural numbers in terms of sets.
ABSTRACT: This talk deals with the difficulties of the acceptance of positional numeration, of negative numbers, of complex numbers, and of infinite numbers.
ABSTRACT: During Han (-206 to +220) times there was a shift in the models of the cosmos used by Chinese astronomers. In this talk these models will be described and the variety of reasons for the change discussed.
ABSTRACT: The application of mathematics to the physical world was rather more problematic for the Greeks than it is for us. I shall here be concerned not with the achievements of the Greeks in this line, which are generally well known, but with the underlying conditions, the philosophical and cultural assumptions which tended to encourage or to inhibit the endeavour. Relevant issues include the compartmentalization of knowledge, debate over the desirability and possibility of abstraction, and the widespread doubt that we can have true knowledge of the changeable. Internal features of Greek mathematics are also germane.
ABSTRACT: Over the past three decades mathematics education researchers have made important advances in understanding how children learn mathematics; however, there are many important issues still to be investigated. In a 2003 report, the RAND mathematics Study Panel chose three goals for immediate research attention:
. Developing teachers' mathematical knowledge in ways that are directly useful for teaching;
. Understanding the teaching and learning skills used in mathematical thinking and problem solving;
. Studying the teaching and learning of algebra from kindergarten through the 12th grade.
To effect change in mathematics teaching and learning in Ontario we must develop a research culture and research community capable of work on a large scale to address major issues such as these. As a step in this process, researchers associated with the Fields Mathematics Education Forum are asking:
. What research is being done in Ontario universities and colleges?
. How can we work together?
. What are the possibilities for graduate students in this research?
. Where are the gaps?
In this session I will provide an overview of some of the projects currently underway and discuss the implications for mathematics education in Ontario.
ABSTRACT: During the Song and Yuan dynasties in China, there were several Chinese mathematicians who were doing mathematics of a uniquely high quality and originality. In this talk I shall have a look at some of their accomplishments in algebra.
ABSTRACT: Although our evidence will not allow us to write a real history of Greek trigonometry, by examining a number of disparate texts, we may form a rough sketch of its development. Trigonometry appears to have been developed as a problem-solving tool by mathematical astronomers and never to have entered the theoretical tradition during the Greek period. Nevertheless, the development of trigonometry exhibits important features of Greek mathematical thought that are not currently discussed in the secondary literature.
ABSTRACT: The Principle of Continuity says, broadly speaking, that what holds in a given case also holds in what appear to be like cases. It has played a significant role in the evolution of mathematics in the seventeenth, eighteenth, and nineteenth centuries-in algebra, analysis, geometry, and number theory. At various times it implied one or another of the following assertions: What is true up to the limit is true at the limit; what is true for finite quantities is true for infinitely small and infinitely large quantities; what is true for polynomials is true for power series; what is true for circles is true for other conics; what is true for positive numbers is true for negative numbers; what is true for real numbers is true for complex numbers; and what is true for ordinary integers is true for (say) Gaussian integers. Such analogies, even when they failed to materialize, were often starting points of fruitful theories.
In the main part of the talk I will focus on examples of the Principle of Continuity in its historical context. I will conclude with several observations with implications for teaching.
ABSTRACT: The study of human nature had traditionally been the realm of novelists, philosophers, and theologians, but has recently been studied by cognitive science, neuroscience, research on babies and on animals, anthropology, and evolutionary psychology. In this talk I will show--by surveying relevant research and by analyzing some mathematical "case studies"--how different parts of mathematical thinking can be either enabled or hindered by aspects of human nature. This new theoretical framework can add an evolutionary and ecological level of interpretation to empirical findings of math education research, as well as illuminate fundamental classroom issues. Consider, for example, the well-documented phenomenon that students tend to confuse between a theorem and its converse. This phenomenon can now be understood as resulting from a clash between Mathematical Logic and the Logic of Social Exchange--a fundamental part of human nature.
This example points to the general thesis of this talk: people fail in some mathematical tasks not because of a weakness in their mental apparatus, but because of its strength! This seeming paradox stems from the evolutionary origins of our brain and mind, and the selection pressures that influenced their "design" over millions of years. The point is, what may have been adaptive in the ancient ecology in which our stone-age ancestors lived (and is still adaptive today under similar conditions), may often clash with the requirements of modern civilization.
ABSTRACT: The science of Geometry originated in India in connection with the construction of fire altars (Yajna vedis) needed for the performance of Vedic Yajnas. According to the strict injunctions laid down in the Vedic texts, each Yajna must be performed in an altar of prescribed size and shape. The slightest departure from the prescribed injunction or the slightest irregularity was supposed to nullify the very object of the Yajna. Thus the greatest care was required to strictly ensure the right shape and size of the altar. This Yajna, in fact, was a central theme around which the entire Vedic culture and civilization developed in the earliest times. It was this Yajna only that led to the origin of Vedas and the ensuing Vedic literature. So in order to understand the origin of geometry in the Vedic age, it is essential to understand the nature and types of Yajna as well as the nature and types of fire-altars required for them. It will also be interesting to know about the Vedic texts dealing with the various aspect of Yajna, particularly the construction of altars and various geometric operations carried out in the concerned Vedic texts in connection with the construction of altars of various geometric shapes and with relative dimensions.
In view of the above, the focus of the talk will be to discuss the origin of the concept of Yajna, the origin of Vedic literature centered around Yajna, fire altars with various geometric shapes and dimensions prescribed for the Yajna, geometric operations laid down for the construction of altars in the concerned Vedic texts, and geometric technical terms used therein.
ABSTRACT: We all know that in the three centuries before Euclid mainstream Greek culture (with, of course, some dissenters) assigned to mathematics a unique significance and prestige of several kinds -- ontological, epistemological, methodological. How and why did this happen? My attempted overview will set our subject's internal progress against a social and intellectual background that includes the philosophical watershed defined by Parmenides, the birth of democracy, the development of rhetoric, the moral and epistemological relativism of the sophists, and more. If time allows I shall venture a few remarks on the subsequent legacy of this rise of mathematics to special status.
ABSTRACT: Numerical notation systems are graphic systems for representing numbers, such as the Roman and Hindu-Arabic numerals. Historians of mathematics have traditionally classified these systems on the basis of only one criterion: whether or not they use place-value (positional notation) to express numbers. Such typologies inadequately represent the various structural features of systems, and place excessive weight on positionality as the only relevant or interesting issue. In their place, I propose a two-dimensional typology that better reflects the variety of systems encountered historically. This new typology emphasizes that all numerical notation systems have advantages and disadvantages, and allows interesting historical questions to be posed regarding the evolution of numerical notation from antiquity to the present.
ABSTRACT: The first Fields medals were awarded in 1936 at the International Congress of Mathematicians in Oslo to Professors Lars Ahlfors (Harvard University) and Jesse Douglas (Massachusetts Institute of Technology). Thus began the tradition of awarding the Fields medals to mathematicians of the highest calibre at the International Congress of Mathematicians, held every four years (except during the years of World War II) .
This talk will be divided into three parts, roughly as follows:
(i) Who was Professor Fields, after whom the medal was named?
(ii) Why, when, and how was the medal established?
(iii) Who were the Fields medalists, and what was the work of some of
them
about?
ABSTRACT: Standard interpretations of Frege's logicism take at face value a drastically oversimplified picture of nineteenth century mathematics. Against this background, Frege can easily seem to be outside the mathematical mainstream, and many commentators have recently concluded exactly this. This paper digs into the historical background to show that Frege (and nineteenth century foundations more generally) was more profoundly engaged with ongoing mathematics than has been realized. Among other things that are relevant to assessing the mathematical thrust of Frege's work are a contrast between the Riemann - inspired "conceptual" style of Gottingen and the arithmetical style of Weierstrass and Berlin, differences between Riemann and Weierstrass on definitional practices, and the early applications in number theory of (what is now called in English) the theory of cyclotomic extensions. This historical background is not just interesting in its own right, but it also prompts a revised assessment of what Frege was trying to do in Grundlagen, and in turn suggests a reevaluation of the proper relation between the philosophy of mathematics and mathematical practice.
ABSTRACT: The famous 15th-century cardinal appears on at least one list of "great" mathematicians; on the other hand his contemporary Regiomontanus dismissed his efforts in mathematics as "ridiculous". But whatever his technical competence, it is certain that Nicholas's perception of mathematics shaped and illumined his views on such issues as the limits of human knowledge and the relation of man to God. I shall try to sketch from both perspectives -- the technical and the philosophical -- the place of mathematics in the world-view of this fascinating figure.
ABSTRACT: Algebraic number theory is the study of number theory using the tools of abstract algebra. I will discuss the rise of the subject, focusing on the contributions of Euler, Gauss, Kummer, and Dedekind. I will also touch on more recent developments in algebraic number theory.
ABSTRACT: What is the goal of education? Surely it is to connect students into the larger community - a community that extends over space and time--as well as prepare them for the demands of career and citizenship. In mathematics, this means more than covering the basic techniques of arithmetic, algebra and geometry. We need to take account of the fact that mathematics has a past and a cultural imprint and see our role as providing one more way in which a person reaching maturity can direct his interests and enthusiasms. I will argue that in doing this, we actually touch more strongly on important mathematical issues that will help even the students who will need a strong mathematical background for their careers and for higher education.
ABSTRACT: Euclid's systematization of mathematics was the crowning achievement of ancient science. His ordering of virtually all of the known propositions of mathematics into a coherent logically entailed system built upon reasonable axioms was seen as the path to certain knowledge. When Isaac Newton chose the same format for physics in the Principia Mathematica, the axiomatic system was established as the template for a scientific theory, regardless of the subject matter. The 18th and 19th centuries saw the creation of dozens of systematic formulations that purported to be scientific analyses of varieties of human endeavors that would have the same reliability and certainty as did Euclid's Elements. The only problem was getting true axioms. This talk will review some of those efforts to make social studies "Euclidean".
ABSTRACT: Mathematical techniques of great importance, involving elements of the calculus, were developed between the 14th and 16th centuries in Kerala, India. Kerala during that time was a region which had been in continuous contact with the outside world, with China amongst others to the East and with Arabia to the West. And after the pioneering voyage of Vasco da Gama in 1499, there was a direct conduit to Europe. Despite these communication routes Kerala mathematics, according to current knowledge, lay localised in Kerala until an Englishman, Charles Whish, ‘re-discovered' it in the 19th century. The talk is based on some of the findings of an ongoing research project which examines the epistemology of the calculus of the Kerala school, its transmission to Europe and the consequential educational implications.
ABSTRACT: Cuneiform tablets recovered from the sites of Babylon and other cities in what is now Iraq provide extensive material for studying Babylonian astronomy. These texts mainly date to the first millennium BC and attest to widespread astronomical activity by Babylonian scribes. This included both a regular programme of astronomical observation that apparently continued uninterupted from circa 750 BC to the first century AD, and the development of arithmetical techniques for calculating planetary and lunar phenomena such as first visibilities and eclipses of the sun and moon. In this talk I will provide an overview of Babylonian astronomy and the theoretical developments which led to the world's first astronomical models. I will then present some examples of recent research on the techniques of Babylonian mathematical astronomy, and what it may have been used for in Babylonian society.
ABSTRACT: I will present the contents of a paper under the above title by the eminent Russian mathematician V.I. Arnol'd.
ABSTRACT: I will discuss aspects of the number-theoretic work of Lagrange, Gauss, Dirichlet, and Riemann.
ABSTRACT: Dynamic geometry is the exploration of geometric relationships by observing geometric configurations in motion. Although, in most classrooms, these configurations are constructed by the students themselves, sketches pre-constructed by the teacher or downloaded from a website can also be used. There is a continuing discussion in the educational technology community about whether it is better to give students powerful general-purpose programming and construction tools or to have them interact with pre-constructed, interactive models. Some strongly support student constructions because they believe that students develop a deeper understanding of the object by explicitly connecting the parts. Others believe that pre-constructed models are valuable as learning tools because ability to recognise connections between geometric objects is a necessary stage before students can effectively carry out many constructions.
In an attempt to inform this debate my research investigated the benefits and limitations of using JavaSketches--web-based, interactive, dynamic geometry sketches--with senior high school students in geometric activities related to proof. In this seminar I will present some of my findings, and demonstrate the features of pre-constructed sketches that helped students focus attention on mathematically meaningful details. By reflecting on my results in relation to the extensive research on Geometer's Sketchpad and Cabri, I will attempt to characterise situations in which pre-constructed sketches can play a purposeful role in the geometry program.
ABSTRACT: 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 of topology 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. We conclude with excerpts from a video, due to W. Thurston, which illustrates Smale's theorem on "turning a sphere inside out".
Robert Aboolian, University of Toronto, will give a talk entitled "Spatial Interaction Location Models with Exponential Expenditure Function" from 10:30a.m. to 11:30a.m. in N638 Ross.
ABSTRACT: In this paper, we introduce a new model for locating competitive facilities, which captures key aspects of competitive facility location models such as market expansion and market cannibalization, while retaining computational solvability that allows for realistic-size applications. The model presents a direct generalization for the class of traditional spatial interaction models .
The new model is formulated as an integer program with a ''nasty'' non-linear objective function. We develop a novel approach to solve problem by using the TLP linear approximation that yields an efficient piece-wise linear approximation within a specified relative error bound. We develop a tight worse case error bound for the greedy heuristic, which is somewhat unexpected in view of the non-linearity of the underlying model.
We also develop efficient computational approaches - both exact (Branch and Bound) and approximate ( optimal with controllable error bound ), allowing for solution of fairly large-scale models.
Refreshments will be served in N620 Ross at 10:00a.m.
ABSTRACT: I will discuss a number of classical properties of the Schur function basis revealing the importance of combinatorics in symmetric function theory. I will then consider a filtration of the symmetric function space, and introduce new symmetric functions appearing, from their combinatorial properties, to be the Schur functions of the subspaces associated to this filtration. I will finish by discussing the connection between these new symmetric functions and Macdonald polynomials.
ABSTRACT: I will discuss a number of classical properties of the Schur function basis revealing the importance of combinatorics in symmetric function theory. I will then consider a filtration of the symmetric function space, and introduce new symmetric functions appearing, from their combinatorial properties, to be the Schur functions of the subspaces associated to this filtration. I will finish by discussing the connection between these new symmetric functions and Macdonald polynomials.
ABSTRACT: The "class number" of a field or of a set of quadratic forms is an ostensibly unremarkable concept which in fact has surprising connections to such perennial fascinations as Fermat's Last Theorem and the Riemann Hypothesis. I shall try to sketch something of the relevant history, with particular attention to the origin and resolution of Gauss's famous conjecture that for quadratic fields of negative discriminant d the class number tends to infinity with -d. Another focus will be the interplay, in the work of Dirichlet, between the development of a "class number formula" and the beautiful proof that certain arithmetic progressions contain infinitely many primes.
ABSTRACT: I will follow up the first general talk on mathematics in visual form with some specific connections between the visual (and eye-hand) cognitive processes and the learning of modern geometry. I will draw on current understandings of geometry via Klein's Hierarchy, and of geometry education via modified forms of the van Hiele levels, as well as my own experiences in discrete applied geometry. Technology is another factor which supports visual learning and explorations in geometry. My goal is support of current models for the learning geometry, with technology and open-ended investigations that connect with how geometry is applied today, and with current research in mathematics education.
This is the second of two talk.
ABSTRACT: There is a lot of evidence, both anecdotal and from cognitive science, that much of what mathematicians do is carried out with the visual (and eye-hand coordination) parts of the brain. What a student 'sees' has a lot to do with how the student will do mathematics. We can change what we see, and such change is an important part of the cognitive apprenticeship we can offer a student of mathematics. I will describe some of this evidence and give examples of visual thinking in mathematics to illustrate the ways we can help students learn to 'see like a mathematician'. I will draw some tentative conclusions for mathematics education, including issues of diversity and support of students with certain disabilities.
A follow-up talk will be given on November 9.
ABSTRACT: To many math historians outside India, the history of Indian mathematics may well be described as 'a mystery wrapped up in an enigma'. There is a common perception, especially in India, that standard histories of mathematics generally tend to ignore, distort or even devalue the Indian contribution to the subject. There is also a view, subscribed to even in India, that innovative mathematical work ceased more or less in the twelfth century AD. In this talk I propose to examine some of the difficulties in studying the history of Indian mathematics and also provide a broad survey of the development of Indian mathematics from around 2500 BC to the era just before the introduction of modern mathematics in India. In undertaking this survey, I hope to construct a chronology for Indian mathematics which will take account of some the more recent research findings on the subject. The talk will end with a discussion of possible transmissions from India to the rest of the world.
ABSTRACT: That Aristotle adumbrated noneuclidean geometry has been urged for many years by Imre Toth, and has been denied just as repeatedly by G.E.R. Lloyd, our leading modern authority on Greek science. The question resurfaced last summer on one of the e-mail discussion groups on the history of mathematics. I shall try to sketch the competing positions and to set the debate amid wider issues of axiomatics and mathematical necessity in Aristotle and beyond.
ABSTRACT: I will discuss aspects of the nearly 4000-year history of number theory, from the Babylonians through Fermat, Euler, Gauss and others, to 1994. (This will require at least two talks, the second to be given in March 2002.)
ABSTRACT: While it is true that during the 17th century algebraic modes of thought and algebraic techniques acquired a growing relevance for mathematics, the relationships between algebra and geometry during the formative years of the modern language of mathematics are far from obvious. Until recently (until now, indeed, in layman accounts of the history of mathematics), it has often been suggested that algebra and geometry embodied at that time two distinct and incongruent, if not antagonistic, perspectives on mathematical objects, problems, and proofs. I would like to argue here that however opposed the two approaches may appear today, they were then complementary rather than incompatible. Actually, they grew ever more intertwined, and for most of the century geometrical and physical models were a precondition for the existence of mathematical objects--even if they were defined or handled algebraically. Geometry and algebra were two layers of mathematical language rather than two conflicting views about its purpose and nature.
Professor Pat Rogers, York University will speak on "Equity Issues in Mathematics Education" from 12:00p.m. to 2:00p.m. in The Harry Crowe Room, Atkinson.
ABSTRACT: Much is now known about the major inequities in mathematics outcomes for women, certain ethnic groups, speakers of English as a second language, and lower socioeconomic groups in North America. Although the problems are easy to identify, understanding their causes is more difficult for they are complex and influenced by many factors that differ by group. Recent efforts to achieve equity have focussed on the mathematics curriculum itself, for example by introducing practical mathematics, mathematics for all, or multicultural mathematics.
My talk will survey the shifts that have occurred over the past three decades in approaches to gender equity in mathematics education and identify some of the problems with various current curricular reforms. I will argue instead for an approach to mathematics education that neither pretends mathematics is easy, nor attempts to de-skill teachers, but trusts students and teachers to explore difficult ideas together in a community that values difference.
ABSTRACT: A detailed understanding of the design methods and technology of the Roman and medieval land surveyors and stonemasons requires an approach that attempts to work both backwards and forwards through the original sequence of the design and construction process. Moving backward through the construction and creative process from the extant town or building remains to the original design involves the application of current surveying and computer technology, and modern statistical methods and mathematics. In contrast, moving forward in the creative and construction process requires a) making working models of Roman and medieval instruments, such as the Roman groma, measuring rods, and large masonic compasses, to re-enact the design, surveying and building stages, and b) applying only mathematics known to the crafts traditions. These 'backward' and 'forward' approaches help inform and correct each other. Furthermore, the source of the design and the ultimate conformation of the building process were deemed the same, a divine archetype, particularly the Roman templum of the sky, and the Christian Heavenly Jerusalem.
Hardy Grant, Department of Mathematics and Statistics, York University, will speak on "Greek Mathematics in Cultural Context" 12:00p.m. to 2:00p.m. in The Harry Crowe Room, Atkinson College.
ABSTRACT: The often substantial prestige and influence enjoyed by mathematics through the long history of western culture can be traced to beginnings in ancient Greece. Here arose, for example, the vision of mathematical knowledge as both uniquely certain--because attained by rigorous proof from incontestable axioms--and as potentially allowing unique insight into the cosmic order. I shall try to sketch both (i) the development, in cultural context, of the mathematical tradition that culminated in Euclid's exemplary Elements, and (ii) the concomitant influence of contemporary mathematics in the shaping of such characteristically Greek achievements as Plato's theory of Forms, Aristotle's theory of scientific method, and the "liberal arts" tradition in education.
ABSTRACT: Several years ago, while I was musing on the fact that Euclidean geometry can be obtained from projective geometry by deleting one line, it occurred to me that there must be a dual geometry obtainable by deleting one point instead. I pondered this idea for a while, and gradually and gropingly worked out some of the most basic facts about what I dubbed "Euclidual geometry". Incidentally, not least of the several challenges I had to face was that of figuring out what to call the new objects that populate the Euclidual plane. This led to some rather amusing terminology. After some more pondering, I realized that by making further types of deletions, I could produce a sizable family of closely related geometries. In my talk I will describe these intuitive explorations, concentrating mostly on the fairly disorienting world of Euclidual geometry, but also suggesting at least a little of the flavor of the others.
ABSTRACT: Abstract field theory emerged from three "concrete" theories--what came to be known as Galois theory, algebraic number theory, and algebraic geometry. I will describe this process, which began in the early decades of the 19th century and concluded with Steinitz's great work of 1910.
ABSTRACT: Pure mathematics is often taught ahistorically. We believe in teaching the pure essence of a subject, and we adjust our style within the limits of "what comes first: concept or application?". I will illustrate with several examples that we do so at the peril of robbing our students of deeper meaning. My central example occurs in the subject of computability.
Arguably one of the major accomplishments of 20th century pure mathematics is Godel's incompleteness theorem and the definition of commutability. Godel used only a simple form of computable functions (now called primitive recursive functions) in his proof; Turing's definition appeared five years later. Most textbooks on computability treat primitive recursive functions as an aside: a simple idea that leads nowhere. An historical curiosity. But actually these functions are precisely on the edge of most students' understanding of computability; discussing this topic deeply enriches learning in this whole area. Are there topics like this in other courses in mathematics that deserve our attention?
ABSTRACT: The Mathematics Learning Centre where I work is a small department in the College of Sciences and Technology at The University of Sydney. Students at the university attend the Centre voluntarily to get help with their elementary mathematics and statistics courses. My teaching covers a wide range of topics and is usually on a small group or individual basis. I am particularly involved with second year Psychology students who seek assistance with statistics. Many of the students I teach lack the prerequisite background in mathematics. Some may be termed reluctant learners as they do not have an intrinsic interest in mathematics or statistics.
My research looks at learning basic mathematics and statistics at university from the perspectives of the students--their diverse points of view. In order to try and teach students in ways that are appropriate for them, I draw on activity theory based on the powerful ideas of Vygotsky and Leont'ev (for example, Leont'ev, 1981; Vygotsky, 1978). This theory explains that people actively develop knowledge on the basis of life experiences that are rooted in the ongoing practical and communal life by which societies organise and reorganise themselves. My perspective emphasises that the ways students learn mathematics or statistics are inseparable from a complex web of personal, social and cultural factors. This is important because students' activities determine the quality of their mathematical learning and emerging knowledge. As one student explained: " Mathematics is what you make of it. It can be artistic, practical, creative or routine". Some questions that arise from my teaching and research are these: What could it mean to educate students in mathematics and how can we effect this?
REFERENCES: Leont'ev, A. N. (1981). The problem of activity in psychology. In J. V. Wertsch (Ed.), The
Concept of Activity in Soviet Psychology, (pp. 37-71). New York: M. E. Sharpe.
Vygotsky, L. S. (1978). Mind in Society. Cambridge, MA: Harvard University Press.
ABSTRACT: In the 13th and 14th centuries in China algebraic techniques reached a very advanced stage vis-a-vis the rest of the world. In this talk I shall discuss some of the details of these techniques and consider questions of influence on and from other cultural areas.
ABSTRACT: In this presentation, I will reflect on over ten years of integrating the use of Maple in a large enrolment Applied Calculus course. Some of the issues to be discussed are: Maple in the exploration and development of ideas in Calculus; the laboratory as a learning environment; the evolution of my teaching using computer technology; and the effects of technological developments on faculty and students in the Mathematics Department at Brock.
Professor Emeritus Hardy Grant, York University, will speak on "Factoring Since the Dark Ages" from 4:00p.m. to 5:00p.m. in N638 Ross.
ABSTRACT: In this context the "dark ages" end not around 1000 A.D. but around 1970. I shall offer a (necessarily sketchy) overview of the main methods developed since then, showing in particular their evolution from what went before (what is arguably the "mainstream" goes back continuously to Fermat). If time allows, I shall mention one or two sociological sidelights.
ABSTRACT: Western scholars have, so far, thought that Aryabhata's (b. 476 A.D.) Indian methods for calculating square and cube roots were the same as the ones Theon of Alexandria (c. 390 A.D.) and Heron of Alexandria (c. 200 A.D.) were using. I propose that this is a false claim, coming from a poor understanding of Aryabhata's methods. During the lecture I will demonstrate how these methods work, after which the audience can judge for itself if they are identical or not.
Professor Abe Shenitzer, Department of Mathematics and Statistics, York University, will speak on "Riemann's Dissertation and its Influence on Mathematics" from 4:00p.m. to 5:00p.m. in N638 Ross.
ABSTRACT: Content of the Dissertation: (a) The notion of a Riemann surface; (b) The notion of a function as a mapping; (c) The idea of defining a function by the positions and nature of its singularities rather than by a formula; and, (d) The Riemann Mapping Theorem.
Influence of the Dissertation: (a) introduced analysis on manifolds; (b) is the way we most frequently think of a function; (a) and (c) brought topology into analysis; and, (d) inspired the search for conformal mappings between regions and simplified, at least in principle, much of mathematics.
Israel Kleiner, Department of Mathematics and Statistics, York University will give a talk entitled "From Fermat to Wiles: Fermat's Last Theorem becomes a Theorem" from 12:00 until 2:00p.m. in The Harry Crowe Room, Atkinson College.
ABSTRACT: In the 1630s Fermat claimed to have shown that the equation x^n + y^n = z^n has no nonzero _integer_ solutions, but he did not publish a proof. This "result" (in effect, a conjecture) came to be known as Fermat's Last Theorem (FLT). For the next 350 years some of the greatest mathematicians tried to give a proof of FLT, without success, although considerable progress was made.
The proof was given by Princeton mathematician Andrew Wiles in 1994. The event was likened to the splitting of the atom and the discovery of DNA. The story appeared on the front page of the New York Times and on the nightly news programmes of the US networks.
I will describe the history of attempts to prove FLT, focusing on the ideas involved, and will include an outline of the main steps leading to Wiles' proof.
The talk will be accessible (I hope) to nonmathematicians.
ABSTRACT: This talk will describe mathematics 280 ("Mathematics for the contemporary classroom"), a course that thas been offered by Trent for the past three years. The course is aimed at prospective elementary school teachers, many of whom have little or no mathematical background. Other mathematics courses (calculus, linear algebra, statistics, etc.) are neither particularly relevant nor accessible to this audience; math 280 attempts to fill the gap.
The course is a blend of lectures and activities, and incorporates the use of various audio-visual materials, peer learning, journal writing and project presentations. Student teaching assistants have also played an important role in this course. In addition, research is being conducted to determine the success of such a course in helping these students to overcome or change their attitudes about mathematics and their perceived ability to teach it.
ABSTRACT: In the period between the two world wars, Stanislaw Lesniewski (1886--1939), one of the most influential and prominent Polish logicians of the twentieth century, created his system of the foundations of mathematics, a system comprising three deductive theories: Protothetic, Ontology, and Mereology. Although Lesniewski's system is among the most original and philosophically significant attempts to provide a logically secure foundation for mathematics, it is largely unknown. In this talk I shall briefly present Lesniewski's system: its philosophical and mathematical roots, its development, and its fate.
ABSTRACT: The title refers to mathematicians who deny that mathematically defined, abstract objects exist independent of our thought and outside of the physically existing universe. There are many mathematical agnostics, then, including Kronecker, Brouwer, Weyl, and most applied mathematicians. I will show how many faces agnosticism has, and I will argue that some of its variants must be the mathematical philosophy of the future.
ABSTRACT: In 1870, infinitesimals were revived in Germany by du Bois-Reymond and Thomae, then developed further by Stolz. In the 1890s the Italians Bettazzi, Veronese, and Levi-Civita developed infinitesimals further in the context of "magnitudes", projective geometry, and abstract fields respectively. Cantor was vehemently opposed, arguing that infinitesimals are contradictory, and was supported in this by Peano and Russell. Hilbert (1899) and Hahn (1907) showed the usefulness of non-Archimedean fields. By the time A. Robinson (1961) invented non-standard analysis, infinitesimals were well accepted in algebra but not in analysis, since it was not clear how to use them to prove theorems in calculus in a rigorous way.
ABSTRACT: Hamilton and Graves discovered the quaternions and octonions in 1843 after trying for years to construct a 3-dimensional division algebra. They proceeded in complete ignorance of results in number theory whihc have a strong bearing on this question. Was this good luck or good management? In this talk I shall discuss the history of their discoveries, and the related discoveries they didn't know about.
ABSTRACT: In the years B.C.--meaning, of course, Before Computers; called the "Dark Ages" by some in the field--two main strategies were pursued in attempts to decide the primality of large positive integers. The first was simple trial division, with refinements made possible by progress in number theory; the second, dating from the 1870s, involved in part a delightful connection with an ostensibly remote corner of mathematics. I shall sketch the history of these matters, up to and (I hope) including the first applications of computers to the problem. As time allows, I shall also try to suggest how the "classical" techniques underlie more recent advances--a fact which raises the alarming spectre of, some day, a follow-up talk on the same topic.
ABSTRACT: Through three case studies the paper examines how questions of mathematical existence were handled in different periods of history. The first involves the discussion about Greek geometry initiated by H.G. Zeuthen's 1896 article "Die geometrische Konstruction als 'Existenz beweis; in der antiken Geometrie", including the modern responses of Wilbur Knorr and Ian Mueller. The second is taken from Euler's work on analysis in the eighteenth century, and concerns the concept of the integral as it was depolyed by him in his treatises on analysis. The last examines some of Weierstrass's work in the 1870s in the calculus of variations, and looks at how he employed implicit function theorems in order to establish the existence of mathematical objects. A comparison of the three studies reveals that quite distinct philosophical conceptions of existence prevailed at different periods in the history of mathematics.
Professor Abe Shenitzer, York University, will speak on "Intellectual Vignettes of some of the Leading Mathematicians of the 19th and 20th Centuries based on D. Laugwitz's Book 'Bernhard Riemann'" from 3:00p.m. to 4:00p.m. in N638 Ross.
ABSTRACT: In addition to being a scientific biography of Riemann, Laugwitz's book is also a panorama of the evolution of mathematical ideas in analysis and geometry from Euler to Hilbert. In particular, it provides intellectual vignettes of Cauchy, Riemann, Dirichlet, Dedekind, Cantor, Kronecker, Weierstrass, and Hilbert. Some of these vignettes are due to the author and some are self-definitions.