Introduction

Teaching mathematics and training mathematicians are two fundamental responsibilities of a Mathematics Department. The problem of determining what to teach, how to teach and how much mathematics to teach to students is not a mathematical problem with a unique solution. It is a controversial issue, in which many mathematicians, scientists and engineers do not agree with one another. Different people hold different views and different expectations of mathematics. Although the aim of teaching mathematics in general may be different from that of training mathematicians, the two activities have an overlapping set of objectives. I will:

• Present some of the problems we encounter in our attempts to train mathematics graduates through rigorous mathematics education;
  
• Illustrate the difficulties for pure mathematics to reach out to science and engineering; and
  
• Discuss the steps taken by the Department to address these problems.

The Controversy in Teaching Mathematics

Many mathematicians who are consciously or unconsciously influenced by the Bourbaki¹ philosophy and tradition share the view that all mathematics students must see the proof of every theorem at least once in their life. Although the “axiom-theorem-proof” style in teaching mathematics has been a popular approach in the training of mathematicians, the Russians hold a contrasting view. The following excerpt from the article by Arnold V.I.², On Teaching of Mathematics, delivered in Palais de la Découverte, Paris, March 1997, sums up the Russian school of thought:

“When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old classical calculus course of French type... . These facts capture the imagination so much that (even given without any proofs) they give a better and more correct idea of modern mathematics than whole volumes of the Bourbaki treatise.”

The teaching of mathematics to physics and engineering students is even more controversial. Some engineers believe that engineering students should be taught mathematics the way mathematics students are taught. This actually happens in top US universities, like MIT and Cornell, all French Grand Écoles, top French universities and most Russian institutes of higher learning. Partly a result of practical considerations, many physicists and engineers however, do not agree with this philosophy. They are of the view that their students need to learn advanced mathematical methods and concepts such as Fourier transforms, partial differential equations, operators and Hilbert spaces as early as possible. Mathematicians often describe such mathematics syllabuses as unrealistic, too ambitious and impossible to teach effectively. This problem is encountered in many universities and very much so in NUS.

Many scientists, engineers and mathematicians believe that a better understanding of physical laws can be achieved through a proper understanding of the underlying mathematical principles. Likewise, many mathematicians also believe that mathematics students should know enough science, engineering or economics to which they can relate their mathematical knowledge. However, mathematical knowledge is so diverse that not all can be related to the physical world³.

Pure Mathematics, Applied Mathematics or Mathematics

Is there a characteristic distinction between pure and applied mathematics? Many scientists and mathematicians are of the opinion that there is none. Some even maintain that there is no such thing as pure mathematics or applied mathematics. There are only mathematics and applications of mathematics. My personal view is different. For all intents and purposes, the distinction between pure and applied mathematics has become a reality, although there is no boundary separating them. There are differences in philosophy, in culture and values that distinguish a pure mathematician from an applied mathematician. Let me illustrate this view with anecdotes and examples.

Pure mathematics is a perfect game in mountaineering according to Arnold V.I. (1995):

“At the beginning of this century…the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and separate from all other sciences.”

Arnold appears to be very unforgiving about this trend and blames mainly Hilbert for what he calls the self-destructive advancement of mathematics. However, I believe that if democracy survives, (which I hope it will), then this trend will continue and intellectual competition will go on, and this is only half of the aim of an enterprise that I call pure mathematics. I personally believe also that the ability to divorce from the physical world into a world of imagination is one of the strengths of mathematics. We should let the mind wander where it can reach. The downside is that, whereas in real mountaineering all players know exactly where are the challenging peaks to climb, in mathematical mountaineering the peaks are often created by leading groups of mathematicians.

Because they are judged purely on their intellectual strength, pure mathematicians strive for perfection. Pure mathematics becomes a perfect game. Results without rigorous proofs are not acceptable, gaps and holes, no matter how tiny they are, are not tolerated. Results without conditions are considered to the best. Anything short of necessary and sufficient condition is usually not appreciated.

The other half of pure mathematics enterprise is to increase the body of knowledge by developing theories without necessarily considering its practical consequences. However, sometimes it can spring surprises, when mathematicians, scientists or engineers twig results in pure mathematics and shape them to fit the imperfect world. For instance, prime power factorisation of integers, which is an activity in number theory, is the principle behind cryptography and computer security. This is an example of an activity in applied mathematics.
It is impossible to define mathematics, pure mathematics or applied mathematics. There are no clear boundaries separating applied mathematics from science, engineering and economics as there is no clear boundary separating applied mathematics from pure mathematics. Like pure mathematics, applied mathematics is also an enterprise that strives to increase the body of knowledge, except that the knowledge is perceived to be relevant to science, technology or economics.

Mathematics that can impact and create breakthroughs in science and technology are very often not sophisticated mathematics. An example is the Rivest-Shamir-Adleman (RSA) Cryptosystem, which has enormous impact on the secure exchange of information. However, the underlying mathematics was not deep, not difficult and not new. To understand why it was a breakthrough, we need to know the problem it solved and the state-of-art then (Tay, Y.C. 2001).

I would envisage an applied mathematician as one with multidisciplinary background and wide knowledge in mathematics as well as in science, engineering or economics. Here are some examples to illustrate my point.

Nobert Wiener was the father of Cybernetics, the mathematical study of control and communication in the animal and machine. Though his works that are permanently displayed along the Infinite Corridor at the MIT exhibit ingenuity and richness of ideas, they are not highly regarded by pure mathematicians.

John Tukey was well known for his work with Cooley on the Fast Fourier Transform (FFT), which revolutionised the world of digital image processing. FFT is extremely simple with no sophisticated mathematical ramifications.

C.E. Shannon was an engineer as well as a mathematician whose greatest contribution to knowledge was his mathematical theory of communication, which is a simple mathematical problem (on hindsight) in sampling and interpolation. Slepian (1974) wrote:

“Probably no single work in this century has more profoundly altered man’s understanding of communication than C E Shannon’s article, ‘A mathematical theory of communication’….”

Is Mathematics Unnatural to Ordinary Man?

I believe that if mathematics is divorced from reality and taught in a rigorous axiom-theorem-proof tradition in lectures, there is a danger that most students will find it unnatural. The rebellious ones will reject it while the law-abiding ones will force themselves into a habit of drilling to conform to the way things are usually done without real understanding. Only a few students who have the mathematical aptitude will benefit from such a treatment.
Moreover, the mathematics that we teach from textbooks has been beautifully refined over the years. Very often, a good lecturer would give beautiful and elegant lectures, without referring to any notes. This is of course very impressive to students, but it can also create a wrong perception about studying mathematics. Students who have gone through such a teaching environment in all their courses may not be very different from children who are brought up in rich and sheltered environments.

In learning mathematics, I believe that most students would enjoy learning if they can acquire the general understanding of the ideas and principles in the beginning. Those exceptional ones who yearn for deeper understanding, for details and rigour are the ones who have the traits of a mathematician and they can be given special attention like what the Department has done in its programme to nurture special mathematics talents.

It is obvious that no one approach in teaching will be effective for all students. I prefer a curriculum that has a few paths to cater to students with different habits and abilities in learning.

A Way Forward

The Department of Mathematics is a relatively large department with expertise in pure as well as applied mathematics. Traditionally, it is strong in pure mathematics, and has established its strengths in a number of areas, including representation of Lie groups, topology and geometry, logic and set theory, probability, algebra and number theory. To increase the international visibility of the Department as a whole, we will continue to strengthen the traditional areas of mathematics. At the same time, we are building up new emerging strategic areas that are driven by the development of modern science, technology and economics. These new emerging strategic areas will form an outer core of the Department to engage in multidisciplinary activities in mathematics with science, engineering, economics and finance. This outer core will be a bridge for the Department of Mathematics to reach out to the other departments within the university, as well as research institutes and the industry.

These emerging strategic areas are organised into two overlapping research groups: Coding Theory & Information Security and Computational Biology, and three faculty-based centres:

• Centre for Wavelets, Approximation and Information Processing (CWAIP), which has already established an international reputation in its research in wavelets and their applications to information processing;
  
• Centre for Industrial Mathematics (CIM), which focuses on scientific computing, optimisation and industrial aspects of information security;
  
• Centre for Financial Engineering (CFE), which will be developed into a self-funded centre to lead in multidisciplinary activities combining both theory in financial mathematics and practice in financial engineering.

The CWAIP, CIM, CFE together with the two strategic research groups of Coding Theory & Information Security and Computational Biology will form the outer core of the Department of Mathematics. They will develop capabilities and mathematical resources to engage in multidisciplinary activities in teaching, research and development in areas such as high performance computing, biomedical sciences, information processing, security & communications, and quantitative finance.

These capabilities and resources will also be extremely useful in particular to the teaching of mathematics to non-mathematics majors, an activity that is important to Mathematics. It would be a paradigm shift in the philosophy of service mathematics teaching, which would not only benefit the students, but also help to foster multidisciplinary activities within the University, thus providing opportunities to students and staff to have a wider and better understanding of applications of mathematics and its relations with other subjects.

References

Arnold, V.I. ‘On Teaching of Mathematics’. An extended text of the address at the discussion on teaching of mathematics in Palais de Découverte in Paris on 7 March 1997 is available at http://pauli.uni-muenster.de/~munsteg/arnold.html.

Arnold, V.I. (1995). ‘Will Mathematics Survive?’ The Mathematical Intelligencer. Vol. 17, No. 3, pp. 6–10.

Slepian, D. (ed.). (1974). Key Papers in the Development of Information Theory. New York: IEEE Press.

Tay, Y.C. (2001). ‘Understanding, Assessing and Rewarding Multidisciplinary Research and Industrial Applications.’ Report prepared for the Department of Mathematics.

1. Bourbaki, N. is a group of mostly French mathematicians, which was formed in the 1930s with the aim of writing a thorough unified account of all mathematics from a single source of axioms. The stated purpose of its treatise is to provide a rigorous foundation for the whole body of modern mathematics. Bourbaki had tremendous influence on the way mathematics has been taught, not only in France, but also in Europe and the US. <Back to Article>
   

2. Arnold, V.I. is a prominent Russian mathematician who holds a joint appointment at the Steklov Mathematical Institute in Moscow, Russia and the Université Paris 9 in France. He is the winner of the 2001 Wolf Prize (a mathematical award equivalent to the Nobel Prize), in recognition of his achievements in mathematics. <Back to Article>
   

3. This reminds me of the following remark by a French mathematician in response to a question from the audience during a talk he delivered in the early seventies, “I am a pure mathematician, and I don’t care how the mathematics is used.” Even now I can remember vividly that being charismatic, confident and flamboyant, he gave an excellent and impressive lecture. That was how young mathematical enthusiasts were won. <Back to Article>