Instructional School for Lecturers in Numerical Analysis
(Sponsored by National Board for Higher Mathematics)
Mathematics Department, Panjab University, Chandigarh
9 June 2014
Honorary Professor, Mathematics Department, Panjab University, Chandigarh
Let me take you back in time, 10000 years, 50000 years, or even more. Human beings would have noticed sunrise after sunrise, and orbits of the seven geocentric planets. From this, they would have drawn two conclusions: (i) There is order in the Universe; (ii) This order is amenable to mathematics. The order that the ancients saw was a manifestation of rotation within the solar system, but the conclusions they drew are profound indeed; they form the basis of all modern science.
Mathematics is a product of pure thought. That it was a vehicle for comprehending divine scheme of things invested it with mystical powers. Spiritual and practical needs of human beings gave rise to mathematics and ensured its advancement. The abstraction came later. Ritual was an important part of ancient life. It was used to propitiate the gods, ask for forgiveness, express thanks, and seek divine approval and support for future actions. To be effective, ritual must be elaborate and well timed, with clear distinction made between auspicious and inauspicious times.
Half a century ago, Abraham Seidenberg made an interesting point. The Western World was subject to Greek rationalist ideology and therefore apt to scorn priestly works. There is a greater appreciation now that geometry’s origin lies in the ritual. Let me draw your attention to the ancient Vedic text Sulvasutras, although I am sure one would arrive at similar results by an examination of ancient Iraqi, Egyptian and Chinese sources.
Strict rules were laid down for the shapes of fire altars. Circle and square were considered sacred figures. Accordingly, fire altars were designed as circles, semicircles and squares, although other more complex geometries were also used. If you take a square with side of unit length, its area will be one unit, and diagonal √2. If you now construct a square with this as a side, its area will be 2, that is double the first square. Doubling the square and circle was a common practice in the Sulvasutras. Both required evaluating √2. This was done to a fair approximation by series expansion.
I have mentioned that ritual had not only to be elaborate but also well-timed. Since natural timekeepers come from astronomy, study of the skies became a greatly valued intellectual exercise. Beginning with Aryabhata in 499 CE, for more than a millennium, a succession of Indian mathematician-astronomers set up equations and solved them to be able to calculate planetary orbits and predict eclipses. Finding approximate solution to equations arising in the context of astronomy and space science has occupied human mind for a long time now, even though the driving force has varied: desire to appease celestial divinities, need for navigational safety, human curiosity and spirit of adventure.
The historical role of almanac making can be seen from the etymology of the term algorithm. There is a small historical place called Khiva to the south of Aral Sea. It is now part of Uzbekistan. Its ancient name is Khwarizm. A noted 8th-9th century mathematician was associated with this place. Even though he worked in Baghdad, he was known by his short name al-Khwarizmi. Title of a book of his introduced the term algebra in Europe. He also wrote a book on Indian numerals which was translated into Latin. The Latin version of his name was Algoritmi which in turn gave rise to algorithm. Internet search for algorithm lists 38 million entries.
It is customary to glamorize mathematical theorems and exact solutions. It should however be kept in mind that giving out results in numbers has been a very important part of mathematical pursuits through the ages. The goal has remained the same while the tools have become more and more sophisticated. Many celebrated mathematicians of the time have lent their name to approximation methods: Newton-Raphson, Lagrange, Euler, and Gauss. Mechanical calculators were a great advance till they were supplanted by electronic calculators. And now we have powerful computers at our command with the help of which we can take up problems that could not be tackled earlier.
It has now become possible to apply mathematics towards understanding biological systems, earth’s atmosphere and hydrosphere, and even the working of stock and financial markets. Earth is a far more complex system than the Universe as a whole is.( Contrast the simplicity of molecules present in space with the complexity of those making up terrestrial life.) And mathematical modeling of even complex natural phenomena is far easier than addressing problems involving human actions. As far as numerical mathematics is concerned, we are living in exciting times.
The fact that so many participants and instructors have assembled here unmindful of the weather shows that the degree of their commitment is as high as that of the thermometer. I have no doubt that this three-week instructional Course will help teachers of numerical analysis brush up their knowledge. Clearly the most important part of mathematics teaching is conveying its excitement to the students. I am sure this Course will enthuse the participants who in turn will pass on the infection to their students. Lastly, I would like to make a small suggestion for consideration by the Board. When such instructional courses are organized, it may be good idea to have one or two lectures on the history of the subject. If that is sometime not possible, some already published source material can be made available to the participants for reading during the course and for reference later. Thanks