Bulletin Autumn‧Winter 1980

Needless to say, the solving of these pro blems require strong back-up from very advanced mathematical theories. Therefore it can be seen that sometimes it is very difficult to draw ahard and fast line between pure mathematicians and applied mathematicians. Q. You have mentioned more than once the ‘beauty o f mathematics '. Could you enlighten us on the exact meaning o f this? A. The first thing that is important for amathemati cian to acquire is an understanding of simplicity and beauty in mathematics. Mathematics has the qualities of both a science and an art. It is as d ifficult to define beauty in mathematics as in art. Normally, we would say a theorem is beauti ful if it is simple aswell as significant. Let me illustrate this with an example in physics. Newton invented the Inverse Square Law, which states that two spheres separated by distance R w ill attract each other with a force proportional to R-2. This is avery simple law but subsequently it was found to be applicable to the calculation of orbits in astronomy. We have found in this law enormous beauty because of its applicability in other sophisticated and signifi cant fields. Q. That means simplicity is an important com ponent o f beauty, and applicability as well A. Simplicity is extremely important. As for appli cability, let me put it this way. An old unsolved problem is like a blockage in the river of mathe matical development, and its solution, just like removal of the blockage, would enable us to find answers to other old unsolved problems and thus contribute to advancement in the whole field. We would then consider the solution of this problem very beautiful. There is another thing which you may be interested to know. Although we seldom embark on our research from a practical standpoint or think of the role it would play in the practical world, we have found that the laws discovered by mathematicians by way of logical deduction often correspond to the laws of the physical world, and they are beautiful in our eyes for this reason. Most probably what we consider beauti ful in our minds corresponds closely to what is true in the physical world. I must admit that this is something I am unable to explain. To illustrate my point further. As we all know, one dimensional space is a line, two dimensional space is a plane and three dimensional space is the ordinary space. It was com monly believed that dimension beyond the third was non-existent in the physical world, but mathematicians s till ventured to study higher dimensions. Soon it was discovered that not only does there exist a close connection between higher dimensions and the physical world but they have even a direct bearing on physics and engineering, etc. So, you can see that beauty and applicability are closely related. Q. Could you please tell us what ‘simplicity' in mathematics is? A. Our idea of ‘simplicity’ is something that changes with time and circumstances. As our knowledge of the world grows, our idea of simplicity be comes more and more sophisticated. This is all the more so in the twentieth century when modern technology and the mass media have made us better informed and more knowledge able. As I have just said, judgement of what simpli city is forms part of the basic training for mathematicians, and such training usually starts at the graduate school level. Q. Is it essential fo r mathematical proofs to satisfy the requirement o f rigour? A. In the late nineteenth century, mathematicians found that up to that time mathematical proofs were not rigorous enough. Many famous mathe maticians therefore called for a greater effort at rigour in the solution of mathematical problems so as to ensure their correctness. The require ment for rigour, initiated by David Hilbert (1862-1943), received general support from other mathematicians. The requirement for rigour in mathematics has its merit. Making rigour a requirement enables mathematicians to have complete con fidence in the correctness of their work —once a theorem is proved it is proved, leaving no room for doubt. However, this requirement has also some adverse effect — mathematics has, as a result, become more abstract. Q. But it seems that applied mathematicians as well as scientists in other fields have often been criti cized by pure mathematicians fo r using less rigorous methods in solving problems. Has this really happened? A. Yes, even very famous applied mathematicians have been criticized. Some pure mathematicians think that applied mathematicians have 'never 13