# Mathematics

Mathematics is the science of numbers and spatial relationships.

### Mathematics

Mathematics is the science of numbers and spatial relationships. It is customary to distinguish pure mathematics and applied mathematics. Pure mathematics can be divided roughly into 3 major fields: geometry and topology, algebra and arithmetic, and analysis. It is complemented by logic, which deals with sets, the basic mathematical objects, their axioms and rules of inference.

### Geometry

Geometry studies figures, especially with regard to their rigid properties, and as a deductive science it began, like mathematics itself, with the Greeks. Straight lines, triangles, circles, spheres and cubes are among the first figures studied. A typical theorem, or general statement arrived at deductively, is that a right-angled triangle is characterized by the Pythagorean property (eg, the square of the hypotenuse is equal to the sum of the squares of the other 2 sides).

In the early period (600 BC - 100 AD), the theory of conic sections (ellipses, hyperbolas, parabolas) was developed, as was trigonometry, important for astronomy. In addition, formulas for the area or volume of special figures were discovered and proved. The principal later developments were the introduction of co-ordinate geometry (Descartes) and of curvature (Gauss), the central notion of differential geometry. The discovery of non-Euclidean geometry is, in comparison, minor.

In co-ordinate geometry the points of a plane are labelled by 2 co-ordinates, its distances from the 2 axes, and familiar figures become the loci of points satisfying an algebraic equation. Thus, geometry is incorporated into algebra and the way is opened for the introduction of spaces of arbitrarily large dimension, important in mechanics. Curvature, in its simplest form, is a number attached to a surface at a point. The flatter the surface, the smaller the number (positive for a sphere, negative for a saddle-shaped surface).

### Topology

Topology studies properties of figures or spaces that are invariant under deformation. It has had its greatest successes in higher dimensions and studies spaces by attaching algebraic and numerical invariants to them. For example, closed oriented surfaces are characterized by their genus. For the surface of a ball, g = 0; of a doughnut, g = 1; and of a pretzel, g = 2. The classification of 3-dimensional spaces, of much current interest, is still incomplete. The Gauss-Bonnet theorem connects topology and geometry in a typical way: the integral of the curvature over a closed surface equals 4 Pi (1-g).

### Algebra

Algebra studies general properties of the solutions of one or several equations; arithmetic or number theory is the study of solutions in specific domains or fields of numbers: eg, solutions of xn + yn= zn in whole numbers as in Fermat's theorem. Linear algebra, the highly developed theory of equations of degree one, is of importance thoughout mathematics. The notion of eigenvalues and eigenvectors, which appear geometrically as the principal axes of an ellipsoid, is critical. All of algebra is pervaded by the concept of a group, a collection of elements which can be multiplied, such as operations on a Rubik's cube. The theory of finite groups has made great strides in recent years.

### Analysis

Analysis begins with calculus, the calculation of velocities and tangents (derivatives), lengths, areas and distances traversed (integrals), and maxima and minima. The central notion is that of a function, which expresses the way one variable depends upon another. The integrals of very few functions can be calculated explicitly; the attendant problems for algebraic functions have greatly influenced the development of geometry and arithmetic.

A differential equation is one connecting a function and its derivatives. Differential equations arise in all sciences. The equation is ordinary if there is only one independent variable and 2 of the basic problems are stability and the existence of periodic solutions. Much is known but the empirical data, found by computer, outstrip our theoretical understanding.

The equation is called partial if there is more than one independent variable. Partial differential equations appear in the study of the propagation of waves and matter. In spite of an enormous theory, there are still basic phenomena that we do not understand, such as turbulence, and to which ideas from probability, a discipline in its own right, are often applied. Derivatives and integrals require the use of limits, as do expansions of functions in infinite series, such as a power series or Fourier series. The use of Fourier series, which express a function as an infinite sum of sines and cosines, is indispensable in the study of partial differential equations.

ROBERT P. LANGLANDS

### Applied Mathematics

Applied mathematics is any valid mathematics that arises in the evolution and dispatch of real-world problems. It may use all of the techniques of "pure" mathematics, but the 2 differ in the sources of problems addressed and the uses to which solutions will be put. Therefore, applied mathematics can involve engineering, physics and other fields, in addition to pure mathematics. Canadian mathematician John L. Synge once noted that applying mathematics to a real-world problem involves three stages. The first stage is to dive from the world of reality into the world of mathematics. The second stage is a swim in the ocean of mathematics. The third stage is a climb back from the world of mathematics into the world of reality and, importantly, with a prediction in your teeth.

Other mathematicians have amplified the preceding 3 stages to 6 stages: recognition, data collection, formulation, solution, computation and communication. Each stage requires different skills. While there are no sharp boundaries between the stages, they do catch the applied mathematician in different attitudes.

### Data Collection

After a problem has been recognized, some specific data will be required to define it. Such data may be experimental, statistical or both. Therefore, experimental design and statistical analysis are important tools of the applied mathematician.

### History in Canada

Every Canadian university has a mathematics department and offers one or more programs in this field. The same is probably true of almost every university throughout the world, a reflection of the importance of mathematics in contemporary society. Mathematics came to prominence as a scientific discipline after the Renaissance, during the period historians call the Scientific Revolution (1450-1700), when brilliant astronomer-mathematicians such as Copernicus and Newton discovered the true nature of the solar system, with the sun at the centre and the planets revolving around it. The role that mathematics played in these major discoveries gave the discipline the stature it has maintained to this day.

### Formulation

When enough data has been collected to define the problem, it should be formulated in a way that is precise enough to work on, ie, a mathematical model must be made of the situation. The model must be simple enough to permit a complete analysis, but also sufficiently close to reality to be relevant to the real problem being considered. In the process, all irrelevant details, and all details of only minor importance, must be suppressed. This narrowing focus permits concentration on major effects. To decide what is of major and what is of minor importance demands considerable savoir-faire and makes model building probably the most valuable and difficult task of the applied mathematician.

### Solution

After recognition, data collection and formulation comes the solution, which may not be easy to achieve. Different formulations of a problem are usually possible and, as one formulation may be easier to handle than another, the solutions may vary in complexity. Often, general mathematical methods that are applicable in principle are not actually useful. This situation is especially true when a numerical answer, correct to a specified degree of accuracy, and at a reasonable cost of time and labour, is required. The neatness and simplicity of most textbook problems cannot be guaranteed in the real world; however, neatness or elegance in a solution frequently comes with real understanding of the problem.

### Computation

Most problems require not only understanding but also an actual numerical solution. Computation of the relevant numbers frequently may be done more quickly and economically without using expensive computers; however, if extensive computer time is required, efficient programming is important. Good mathematicians can effect considerable savings through the way they prepare the problem for computation. Such problems arise in high-dimensional combinatorial problems (travelling salesman-type problems), the computation of high-dimensional multiple integrals and 3-dimensional partial differential equations involved in calculations of elasticity, weather predictions, etc. At the solution and computation stages, there should be feedback to the first three stages to be sure that the problem actually being solved is indeed the problem that was supposed to be solved and not some other, albeit interesting, one related to it. This may require several passes through the feedback loop.

### Communication

Applied mathematicians must make their findings accessible to the people they work for, and must communicate their work in a style less compact and easier for a nonspecialist to read than that common in most mathematical journals.

MURRAY KLAMKIN

### Recognition

In industry, the realization that a problem exists will usually come from an engineer, scientist or manager who is involved in the practical applications of a technology and is in a position to recognize that something needs improvement or that something is going wrong.

### Early Québec History

The evolution of mathematics in New France followed closely on the heels of this newly acquired stature. Although there were no new discoveries, the quality of teaching was virtually equal to that found in colleges in France. The Jesuits founded Collège de Québec in 1635 and started to teach intermediate mathematics there in 1651. Until 1760 students were taught arithmetic, the rudiments of second-degree or quadratic equations, trigonometry, geometry and a little differential and integral calculus - all in one of the 2 final years of the 8-year course of studies. The first full professor was Martin Boutet de St-Martin. In 1678 Louis XIV appointed him to the new royal chair of mathematics and hydrography in Québec City, his wish being that pilots for the St Lawrence River and surveyors and cartographers be trained in the colony. The chair was not abolished until the end of the French regime.

The most celebrated appointee to this chair was Louis Jolliet, the discoverer of the Mississippi River. Soon after Jolliet's death, the chair officially passed to the Jesuits.

After the Conquest of 1759-60, the Collège de Québec had to close, but the Séminaire de Québec, which took over its operations, retained the same classical course structure. Encouraged by Abbé Jérôme Demers, the teaching of science and mathematics flourished, particularly around 1840. Soon, however, for sociological and religious reasons, it fell into disfavour. Even at the École polytechnique de Montréal, an engineering school founded in 1873, only intermediate mathematics was taught until 1910. It was only until 1920 that the sciences were recognized as valuable among Québec francophones. In that year, Université Laval (Québec City) organized its École supérieure de chimie (which became its faculty of science in 1937) and Université de Montréal established its Faculty of Science.

### Early English Canada History

Nothing significant in the field of mathematics occurred in English Canada until 1855. Of the few English-language universities in Canada, only the University of Toronto offered programs with specializations, one being in mathematics and natural philosophy (the latter term signifying the physical sciences). However, each university had a mathematics and natural philosophy professor. Trained in Great Britain, these few professors brought with them the idea that science and technology were central to the Industrial Revolution.

A Canadian scientific community thus began to take shape and the need for communication among its members was felt almost immediately. In 1856 the *Canadian Journal of Science, Literature and History*, published under the aegis of the Royal Canadian Institute, accepted articles on mathematics and continued to do so until 1912. Professor J. Bradford Cherriman (U of T) was in charge of the section on mathematics and natural philosophy.

In the 1870s the idea arose of more specialized university studies. In 1877 U of T launched its mathematics and physics programs, which became models for the rest of Canada during the first half of the 20th century. Other universities, eg, Queen's, McGill and Dalhousie, gradually moved in the same direction. During this time, science departments were being subdivided, and by 1890 almost all universities (with the exception of McGill) had at least one mathematics (no longer a "mathematics and natural philosophy") professor. At the same time, bursaries were offered for studies in mathematics, 2 each at University of Toronto and Dalhousie.

In addition to Cherriman, 3 mathematicians merit special mention for the impetus they gave to the establishment and development of mathematics programs at their respective universities: James Loudon of U of T, Alexander Johnson of McGill and Nathan Fellowes Dupuis of Queen's. In 1890 all were members of the Royal Society of Canada. Founded in 1882, this society was originally divided into 4 sections, one of which was mathematics and natural philosophy. The society reserved a place for the mathematics publications of its members in its *Proceedings and Transactions*, thereby offering a new means of communication for the mathematics community. In 1878 the *American Journal of Mathematics* was founded at Johns Hopkins University, Baltimore, followed in 1886 by the American Mathematical Society which, from 1891 on, published its* Bulletin* and from 1900 on, its *Transactions*. Canadian mathematicians contributed regularly to these journals.

### Early 20th Century

The number of mathematics departments increased during the first 2 decades of this century, as a result of the growing importance of mathematics in professional fields (eg, engineering). The University of Toronto was the first North American university to move into the field of actuarial science (ie, the calculation of insurance and annuity premiums and dividends). The Canadian Institute of Actuaries, a professional society, was founded in 1907.

In 1915 U of T awarded the first Canadian doctorate in mathematics to Samuel Beatty, who later became the head of that institution's mathematics department. The U of T increasingly took the lead in Canada and held it until the end of the 1950s. One of the most notable figures in the department was undoubtedly J.C. Fields, renowned for his work in algebraic functions, and one of those who managed to revive the International Congress of Mathematics, meetings of which had been suspended after WWI. The first meeting after the war was held in Toronto in 1924

Fields, reacting to the lack of a Nobel Prize for the field of mathematics, began working to establish an equivalent prize. These efforts were successful in 1932, a few months after his sudden death. The Fields Medal, named in his honour, is now universally recognized as the greatest honour that can be conferred on a mathematician. In 1936 the algebraist and geometer Harold S.M. Coxeter joined the department.

An example of the high calibre of teaching then being provided in Toronto can be seen in the first years of the American William Lowell Putnam Mathematics Competition. Only undergraduates could participate in this competition and each university entered a team of 3 students. In the first year, 1938, the U of T team won first place over all the North American universities. The competition rules prevented U of T from entering a team the following year, but in 1940, it won again, as it did in 1942 and 1946. In the 1986 competition, teams from 2 Canadian universities (UBC and Waterloo) ranked among the top 10 (the others being Harvard, Washington U St Louis, U of California Berkeley, Yale, MIT, California Institute of Technology, Princeton and Rice. Since then, Canada has continued to do well in the competition, which has been dominated by Harvard and Duke since 1985. In 1998, University of Waterloo placed 5th, and had 2 students in the top15; Simon Fraser University had 1 student in the top 15; Dalhousie had 1 in the top 26; and University of Toronto had a 3-member team that received an honourable mention.

The end of WWII was a turning point for mathematics in Canada. During the war Canadian mathematicians became aware of their isolation, even within Canada. If they wished to meet, they had to participate in meetings of the American Mathematical Society. They therefore organized the first Canadian mathematical congress, held in Montréal in June 1945. This very successful gathering led to the creation of the Canadian Mathematical Congress, which in 1978 changed its name to the Canadian Mathematical Society.

In 1949 the society began publishing the* Canadian Journal of Mathematics*, an internationally recognized publication, to which was added the *Canadian Mathematical Bulletin* in 1958 and the *Canadian Mathematical Congress Notes* in 1969. In 1950, still under the aegis of the congress, Professor R.L. Jeffery, from Queen's, organized a Summer Research Institute in Kingston, which brought together 10 mathematicians to conduct joint research. This type of meeting proved so fruitful that summer research institutes have been held annually in some universities.

### Late 20th Century

In the 1950s mathematics departments developed very quickly. Toronto lost its leadership, not because its quality of teaching had declined, but because departments in other Canadian universities had improved - a case of students surpassing their teachers.

Graduate studies burgeoned everywhere. Because of the development of statistics, operations research and computer science, industry showed increasing interest in mathematics graduates. However, forecasts of major increases in the student population in the 1960s indicated that there would be a high demand for university professors. With the launching of the first Sputnik by the USSR in 1957, mathematics once again found itself in the public eye, and the "New Math" movement began. In the same year, the National Research Council of Canada began to give grants to mathematician-researchers and 2 new societies were founded: the Canadian Information Processing Society and the Canadian Operational Research Society.

The prodigious development of computer science and diversified interests among mathematicians and computer scientists later led to the subdivision of mathematics departments in almost all universities into mathematics and computer science departments, the most striking example being the creation in 1966 of the Faculty of Mathematics at the University of Waterloo, with its 5 constituent departments: pure mathematics, applied mathematics, combinatorics and optimization, applied analysis and computer science, and statistics.

In Québec francophone universities returned to the Canadian mainstream around 1945 after successfully overcoming the difficulties generated by the prolonged lack of a scientific tradition in that province. By 1970 the department of mathematics at Université de Montréal had acquired an international reputation under the direction of Maurice L'Abbé, and had established a Centre recherches mathématiques, which is highly regarded internationally.

The early 1970s were truly a golden age for mathematicians in Canada. In 1961 Canadian universities had awarded 11 PhDs in mathematics; this number increased to 94 in 1973. The NRC in 1960-61 gave $87 500 for mathematics research; the figure for 1972-73 was $2 461 500. The figure in 1986-87 was $8 419 000.

The Statistical Sciences Association of Canada, later known as the Statistical Society of Canada, was founded in 1971; it publishes the *Canadian Journal of Statistics*. In 1973, the Canadian Society for the History and Philosophy of Mathematics began its activities. Another association, the Canadian Applied Mathematics Society, was created in 1980. It publishes *Applied Mathematics Notes* and sponsors *Canadian Applied Mathematics Quarterly* founded in 1992.

In 1991, a group of Ontario universities sponsored a second research center, the Fields Institute for Research in Mathematical Sciences. Initially in Waterloo, it is now located on the U of T campus.

Today the number of members of the Canadian mathematical community has grown to around 2400. The Canadian Mathematical Society, which celebrated its 50th birthday in 1995, had a membership counting 1209 individuals and 39 institutional members in 1994. According to a survey based on papers reviewed in the international *Mathematical Reviews*, Canada's impact on international mathematics is now much greater that the population alone would predict (73 mathematical research papers per year per million of the population in 1990. US and Holland followed with 47 mathematical research papers per year per million of the population).

In 1961 there were about 250 university mathematics professors (assistant rank and higher) in Canadian universities; in 1973, about 1300.

### Canadian Mathematicians

The Canadian mathematics community, like other scientific communities, is affected by the movement of promising graduates to the US, a fact that further testifies to the level of excellence of instruction in Canadian departments. Mathematicians with international reputations who received their early training in Canada include Cathleen Morawetz (U of T), Courant Institute of Mathematical Sciences, New York; Robert Langlands (UBC), Institute for Advanced Study, Princeton; Israel Herstein (U Man), U of Chicago; Irving Kaplansky (U of T), U of Chicago; Louis Nirenberg (McGill), New York U; G.F. Duff (U of T), emeritus professor, U of T; Leo Moser (d 1970) (U Man), U of A (d 1970); W.O. Moser (U Man), McGill; Raoul Bott (McGill), Harvard.

The Canadian mathematics community faces a number of challenges. Connections seem to have been made among university departments, industry and government in such fields as statistics and computer science, where the future appears to hold promise. However, other areas of mathematics appear more sensitive to difficult economic conditions.

LOUIS CHARBONNEAU

Authors contributing to this article:

## Suggested Reading

Canadian Mathematical Society/Société mathématique du Canada,

*Canadian Mathematical Society/Société mathématique du Canada 1945-1995, Vol 1: Les mathématiques au Canada*(1995).

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