It is a great privilege and honour for me to have the opportunity to address this Congregation. On behalf of my fellow graduates I would like to thank His Excellency the Chancellor Sir David Wilson, Vice-Chancellor Dr Wang, and the Senate of the University of Hong Kong for this real honour.
If I am in any way qualified for this unusual honour, I must be very near the lower threshold of such a qualification. Since among the graduates today I am the most junior, it is wit considerable trepidation that I stand here and speak. Coming from my own alma mater, this honorary degree carries for me a very special meaning. Standing here, I cannot help but feel a sense of nostalgia. This hall brings back to me memories of the green gown which every undergraduate in my days had to wear for official occasions. I remember thirty years ago it was in this very hall that I had my green gown on for the first time attending the matriculation ceremony. Three years later I sat here again for my final examinations and I wore my green gown for the last time.
As one of a small number of students in the Faculty of Arts doing nine papers in mathematics, I found the faculty accessible and the climate of the Department of Mathematics conducive to the development of close relationships between faculty and students. When I was a student here, Professor Y C Wong gave me a copy of the book An Introduction to the Theory of Numbers by G H Hardy and E M Wright and Dr K T Leung gave me a copy of Herman Weyl's book Die Idee der Riemannschen Flaechen. Those gifts of classical books have exerted considerable influence on my mathematical career.
In this university I was taught the fundamentals of mathematics. I still remember the intellectual excitement I felt when I was first exposed to the structural beauty of rigorous arguments in the foundation of mathematics. I was fascinated by the paradox of Russell, a popular form of which is the following story about a barber. Once there was a barber whose policy was to cut the hair of those and only those who do not cut their own hair. Should the barber cut his own hair? No matter what your answer is, you get a contradiction. After I graduated from the University of Hong Kong I have been devoting myself to mathematical research. It is an old and exciting subject, constantly renewed by rejuvenating discoveries.
I would like to talk about the nature of mathematics and its place in the modern world. As the popularity of the public lecture goes, any discourse on mathematics would certainly receive the lowest rating from the audience. At parties, when you mention that you are a mathematician, people would usually look awe-struck (thought it is hard to tell whether the respect is genuine) and mumble, "I was never good at it" or "I cannot even balance my checkbook" and then try to change the subject or walk away on the pretext of getting the glass filled.
Among the intelligentsia, especially in places like Hong Kong where specialisation takes priority over general liberal education, there seems to be a clear dichotomy between those quantitatively inclined and those verbally and qualitatively inclined. Actually mathematics is more an art than a science. It is with good reason that the Department of Mathematics here is in both the Faculty of Arts and the Faculty of Science. Though the development of mathematics has been motivated to a certain extent by the need of other natural or experimental sciences, mathematics is driven by a desire to understand the structural beauty of quantity and space and for this reason is akin to poetry, music and the fine arts. Personally I find the division of the intelligentsia into quantitative and qualitative groups highly unnatural and undesirable.
Mathematics is universal both in terms of space and time. It transcends national boundaries and language barriers. Like music the language of mathematics is a universal language. Whereas theories in experimental sciences may have to be modified or discarded in the light of new discoveries, mathematics which is valid now is going to be valid in the indefinite future. As I delved deeper into mathematics, I was quite pleasantly surprised to find out that one of the mathematical achievements of ancient China, the Chinese Remainder Theorem, is still an important tool in the sophisticated theory of numbers today. According to legend, the Chinese Remainder Theorem was used by General Han Xin to count his soldiers. As you may recall, Han Xin was a general of Emperor Liu Bang of the Han Dynasty and as a young man, facing unfavourable circumstances, he was flexible enough to swallow his pride and suffered the indignity of crawling under the crotch of a bully. To count his soldiers, he asked them to form groups of equal size and see how many were left over. After several such groupings using a different group size each time, he could quickly find out the exact number of soldiers in his command. In case you wonder how Han Xin could find the time to play mathematical games when he was reviewing his troops to prepare for battle, there was a Chinese poem giving the algorithm for the calculation of the answer. The ancient Chinese, as you can see, were a very poetic people. Just imagine writing poems today as a mnemonic device for algorithms of computer programs!
Mathematics is the study of quantity and space, corresponding very roughly to algebra and geometry. There is a division of mathematical methods into the discrete and the continuous. The former uses finite processes and the latter infinite ones. Infinite processes can be illustrated by the following story of the turtle and the hare. One day a turtle and a hare decided to have a race. Since the hare can run much faster than the turtle, they agreed on a handicap race. The turtle is allowed to go some distance first before the hare begins to run. Before the start of the race, the turtle says to the hare that there is no need to go through with it, because by logical reasoning the hare can never catch up with the turtle no matter how much faster the hare can run. Here is the turtle's reasoning. As soon as the hare gets to the turtle's starting position, the turtle would have already moved some distance. Then as soon as the hare gets to this new position of the turtle, the turtle would again be some distance away. So the turtle argues that the hare can never catch up with it. The fallacy of the argument, the time for each step gets shorter and shorter and the sum of all such times is still infinite. Such infinite processes that lead to finite answers at the core of continuous mathematics such as calculus.
Practically nobody would raise any question about the usefulness of mathematics. Some parents in Hong Kong may object to their children devoting too much time to what they consider "trivial" pursuits such as sports or art, but never would they object to their children using their time for mathematics. Everybody realises the importance of knowing enough mathematics. However, the kind of mathematics the man in the street finds useful is not the kind that present day professional mathematicians do. Some fifty years ago the English mathematician G H Hardy wrote a short, interesting book A Mathematician's Apology. His thesis was that the mathematics which real mathematicians do, both applied and pure, is almost wholly useless and pure mathematicians glory in the uselessness of their work and make it a boast that it has no practical applications. He quoted the statement usually attributed to the great German mathematician Gauss that "if mathematics is the queen of the sciences, then the theory of number is, because of its supreme uselessness, the queen of mathematics".
Many mathematicians nowadays take exception to Hardy's opinion and feel that a true and deep understanding of the structure concerning quantity and space will eventually lead to useful applications that promote the material well-being and comfort of mankind. Many examples bear out the eventual usefulness of at least some of the mathematics real mathematicians do. Even the theory of numbers which Hardy considered supremely useless now finds applications in the real world, for example coding theory.
Coding theory tries to devise ways to transmit messages which minimize transmission errors and at the same time have built-in error-correcting capabilities. It also tries to find schemes to hide messages from unauthorised parties for security reasons. Deep results in algebraic geometry have been used to produce good error-correcting codes. Advanced number theory has been used in cryptography which is the art of hiding messages and deciphering hidden ones. One method of cryptology makes use of the fact that there is no quick way of breaking down a large number into factors. It is easy to multiply numbers together to get a large number, but the other direction is very time consuming. The secret message is built up by some simple process of multiplying and dividing numbers, but cracking it involves the difficult task in number theory of factoring a large number.
I would like to cite another example in which a branch of algebra, called the representation theory of finite groups, is used to construct efficient computer networks. Suppose we have a large number of computers we want to connect together in a network so that each one is directly connected to, let us say, precisely four other computers. The connection is efficient if the number of computers liked up in any loop is not too small compared to the total number of computers. In technical terms the goal is to have the former dominate the logarithmic order of the latter. The Russian mathematician Margulis succeeded in using the representation theory of finite groups to explicitly construct such efficient networks.
Research in pure mathematics is guided by intellectual curiosity and aesthetic reasons, usually with no specific application in mind. In many cases people in other areas find applications for the results. One beautiful example is the use of differential geometry by Einstein in formulating his theory of general relativity. One could start the story with Euclid. Euclidean geometry derives complicated consequences from self-evident axioms by using only simple logic. One axiom, the fifth, is equivalent to the statement that the sum of the three angles of a triangle equals two right angles. Mathematicians considered the axiom too complicated to be self-evident and tried to derive it from other axioms. After numerous futile attempts they discovered that the fifth axiom is actually independent from the others. If one interprets a line as a great circle on a sphere, then the other axioms are satisfied by the sum of the angles of a triangle formed by great circles can be three right angles. The explanation for this geometrical phenomenon is that the sphere is curved. Riemann, a German mathematician, proposed the study of geometry for spaces which are curved in a variable way, instead of being uniformly curved like the sphere. This branch of mathematics is known as differential geometry. Einstein used this formulation for his theory of general relativity which asserts that the space and time we are in is curved or warped in a variable way and how much it is curved depends on the amount of matter present there. A consequence of the theory is that light when near a massive star may not propagate in a straight line. When differential geometry was introduced, nobody could have envisioned such an important application.
Ever since the discovery of calculus mathematics and physics have been very closely linked. Many methods of mathematics were developed to help solve problems posed by physicists. After the advent of quantum theory in the twenties, mathematics and theoretical physics began to drift apart. Some recent developments point to a reunification of the two subjects. For example recently both mathematicians and theoretical physicists are interested in string theory, in which particles which are zero dimensional objects in a space are replaced by strings which are one dimensional objects in space. There is also a trend for the convergence of different branches of mathematics. A great deal of research in mathematics now straddles several fields.
The widespread availability of computers has considerable impact on mathematics. As a result discrete mathematics is assuming a more important role. Mathematical theory is distilled from concrete examples and in the past such examples had to be computed by hand. Now with the computer the calculation of a large number of more complicated examples becomes feasible, especially in discrete mathematics. However, pure mathematics is by no means in danger of becoming obsolete. The computer can only help with the computation of examples or the verification of a finite number of cases but cannot help with the development of the mathematical theories themselves. For this reason even with computers checking as many cases as we like, famous problems involving integer solutions of polynomial equations such as Fermat's Last Theorem still remain unsolved. A German mathematician G Faltings recently made a very important contribution in this area by proving a conjecture made by the English mathematician Mordell.
The use of computers gave rise to a new branch of mathematical investigation called fractal geometry. By iteration of some simple rules the computer can generate some very irregular sets of points called fractals, also known as sets of fractional dimension. These fractals, though irregular, carry interesting geometric structures. Some simple rules of iteration can produce fractals which look like landscapes, foliage and other naturally occurring images of the real world. These fractals for example can be used in image compression. Instead of transmitting complicated images one can transmit the simple rules of iteration which produce them.
The availability of powerful computer programmes that directly manipulate formulas instead of computing numerical values relieves us of tedious routine calculations. The widespread use of such computer programmes in the future will profoundly affect mathematical education and emphasis will be shifted away from drilling routine calculations to better calculations to better conceptual mathematical understanding. Such a shift, I believe, would be most welcome by educators and students alike.
The research of mathematics requires practically no special equipment. What is most important for research mathematicians is a library well stocked with good books and journal subscriptions and the opportunity to keep in touch with researchers at the major centres of research through academic conferences and professional visits. In terms of return for the investment, financially supporting mathematical research is without doubt a very good bargain. With the human and financial resources in Hong Kong there is no reason why Hong Kong could not develop into a major centre of mathematical research.
