A View Of Mathsland

A view of Mathsland To most people, mathematics is that subject they have always had difficulty understanding. It is a form of communication, a kind of strange language in which complete sentences must have something called an equals sign or some other equally strange symbol. It appears to be a form of the English language but interlaced with rows of austere symbols and incomprehensible formulae (some Martian language!). For different reasons, the majority of the world's `educated' population, by the time they graduate from high school, have already made up their minds that mathematics is difficult and that nothing new ever happens in mathematics. My suspicion is that this uninformed majority spreading these unfounded rumors have no personal experience with mathematics. They probably heard this story from a friend who in turn had heard rumors from elder brothers and sisters that mathematics is a difficult subject. Believing this lie and obviously lacking self-confidence and motivation, most decide to give up before they even give it a try. The world of mathematics is an ever-changing one. It is a world highly endowed with provocative ideas, very rich in poetry and full of vivid images. Mathematics has been likened to a land-mass -- a continent (like Africa!) comprising a mainland and a couple of off-shore islands. The mainland consists of countries with such exotic names as Algebra, Analysis and Geometry. Off-shore we have the beautiful islands of Statistics (think of Mauritius) and Computer Science (think of Madagascar). Other islands include Logic and Set Theory and Number Theory. In these mathslands there are many beautiful places, vintage tourist attractions, some only recently "discovered" have names like Fractals and Chaos! Let us now take a closer look at some of the individual countries. Firstly, Algebra. This land has very close cultural ties with the Arab countries of North Africa and the Middle East. The word Algebra itself comes from the Arabic al-jabr, which means putting together different parts, or a reunion of parts (perhaps like fitting together pieces in a puzzle). In Algebra the main economic activity is the study of number systems. There are entities with names like integers, vectors and matrices and rules like addition, subtraction, multiplication, etc and living in Algebra you learn how to apply these rules to the entities mentioned earlier to come up with new entities which you then export to other mathslands. For example, given the entities 2 and 3 and the rule "minus" you may deduce that 3-2=1. You may then export the new entity 1 to Statistics land where it plays an important role. For instance, the statisticians, the name given to the inhabitants of this land, believe that the probability of any event occurring cannot be greater than this entity. Algebra or the study of Algebra dates back centuries and one of the earliest books on the subject was written by Diophantus of Alexandria, Egypt, in the third century! A book with the Arabic title ilm al-jabr wa'l-mukabala by a Persian mathematician named Abu Ja'far Muhammad ibn Musa, popularly known as al-Khowarizm which means the man of Khowarizm (Khowarizm is now called Khiva in Uzbekistan in the former USSR), seems to have popularized the word al-jabr. The word Algebra first appeared in the year 1551 in a book called Pathway of Knowledge written by Robert Recorde, an Englishman who was at that time teaching at the University of Cambridge.

Analysis is another massive country. The backbone of its economy is what is called Functions. These are simply rules that assign a fixed output to a given input. A simple example of a function is f(x) = x2 called a squaring function. To visualize what this function says; think of a magician with an empty basket. You put two oranges in his basket and a moment later the magician pulls out four oranges from the basket. If you put in three oranges you get nine back and so on. Basically this is a country full of magicians and black magic! Then there is Geometry. The inhabitants of this land are architects and surveyors. They study shapes of objects and space. They are interested in shapes like squares, circles etc and their properties. This special province is usually called Euclidean Geometry. In another province, called Topology, the people are more interested in those geometrical features which do not change when an object is twisted, stretched or deformed in any way. Mind you, the natives of these lands can deal with objects in more than the usual two or three dimensions. Four or even higher dimensions is not unheard of. A brief mention of Number Theory. This is the dominion of spies! For instance they study the properties of prime numbers and their factors so that they can write secret codes to hide information or try to break into other peoples computers and data bases to steal economic, military or other such secrets. This is called cryptography but is only one of the many fascinating things they do. Finally, Computer Science. This is a country whose ideology is not clearly defined. It is perhaps like Turkey, not too sure where its future (and past!) lies, East or West? It is said that Computer Science is the science of software or the study of Algorithms. The word Algorithm is derived from the Arabic al-Khowarizm, the name of the famous Uzbekistani mathematician mentioned earlier. Basically an algorithm is a method or procedure one uses to solve a given set of problems. Thus for instance, presented with a set of numbers 2, 4, 6, 8, 10 ... we can work out the principle of the succession of these numbers. The principle seems to be "Add 2 to the previous number to get the next number". This principle is then called an Algorithm. The problem lies in deciding the exact nature of the relationship between software and mathematics. For example, is software like mathematics or is it in fact mathematics? According to David Gelernter (an American professor of Computer Science at Yale University) there are two schools of thought on this issue. There are those who believe that writing a computer program and constructing a mathematical proof are equivalent and interchangeable activities. Thus the programming languages (tools used for writing software) should be defined mathematically using a series of equations. The other school of thought contends that designing a program is very different from doing a mathematical proof; in fact it's more like designing a car. You need a different set of technical skills. It is clear therefore to this group that programming ought to be defined in clear simple English and not in a series of mathematical equations. Well, I do hope you enjoyed your guided tour of Mathsland and hope you will decide to come back for a further visit or perhaps settle here permanently. It is a land full of promise and many more uncharted territories, waiting to be discovered by you.