The Nature of Mathematics

Anuraag Agarwal 12A/BMc TOK 3/5/07

Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its basic interest. The essence of mathematics lies in its beauty and its intellectual challenge. This essay is divided into three sections, which are patterns and relationships, mathematics, science and technology and mathematical inquiry.

Firstly, Mathematics is the science of patterns and relationships. As a theoretical order, mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world. The abstractions can be anything from strings of numbers to geometric figures to sets of equations. In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world.

A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced. Mathematicians are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest richness or complexity but on the contrary, in finding the greatest economy and simplicity of representation and proof. As mathematics has progressed, more and more relationships have been found between parts of it that have been developed separately. These cross-connections enable insights to be developed into the various parts; together, they strengthen belief in the correctness and underlying unity of the whole structure.

Mathematics is also an applied science. Many mathematicians focus their attention on solving problems that originate in the world of experience. They too search for patterns and relationships, and in the process they use techniques that are similar to those used in doing purely theoretical mathematics. The difference is largely one of intent. In contrast to theoretical mathematicians, applied mathematicians, in the examples given above, might study the interval pattern of prime numbers to develop a new system for coding numerical information, rather than as an abstract problem. Or they might tackle the area/volume problem as a step in ...

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The results of theoretical and applied mathematics often influence each other. Studies on the mathematical properties of random events, for example, led to knowledge that later made it possible to improve the design of experiments in the social and natural sciences. Conversely, in trying to solve the problem of billing long-distance telephone users fairly, mathematicians made fundamental discoveries about the mathematics of complex networks. Theoretical mathematics, unlike the other sciences, is not constrained by the real world, but in the long run it contributes to a better understanding of that world.

Secondly, because of its abstractness, mathematics is universal in a sense that other fields of human thought are not. It finds useful applications in business, industry, music, historical scholarship, politics, sports, medicine, agriculture, engineering, and the social and natural sciences. The relationship between mathematics and the other fields of basic and applied science is especially strong. This is so for several reasons, including the following:

Science provides mathematics with many interesting problems to investigate and solve, while mathematics provides science with powerful tools for analyzing data.

Mathematics is the language used in science. For example Newton’s second law is a=F/m. This is not simply a shorthand way of saying that the acceleration of an object depends on the force applied to it and its mass; rather, it is a precise statement of the quantitative relationship among those variables.

Mathematics and science have many features in common. They both with the development of powerful electronic computers are able to use technology to open up new fields of investigation.

Mathematics and technology also have a quite a strong bond between them. As we can see the invention of computer included a lot of mathematics, as it is needed for computer programming. It also has use in mobile phones and other electronical devices.

Lastly, using mathematics to express ideas or to solve problems involves at least three phases:

representing some aspects of things abstractly
manipulating the abstractions by rules of logic to find new relationships between them
seeing whether the new relationships say something useful about the original things

Mathematical thinking usually begins with the process of abstraction, which is noticing the similarity between objects or events. The aspects that these objects or events have in common can be represented by symbols (numbers, letters, other marks, diagrams, geometrical constructions, or even words). The circle is an abstraction derived from human faces, flowers, wheels, or spreading ripples. In addition, the letter A may be an abstraction for the surface area of objects of any shape, for the acceleration of all moving objects, or for all objects having some specified property. Abstractions are made not only from objects or processes; they can also be made from other abstractions, such as different kinds of numbers (Odd, even, real, integer, etc). Such abstraction enables allow mathematicians to focus on many different areas of mathematics instead of sticking to the same ones. There is so much mathematics in nature, which people don’t notice. The resulting outcome from abstractions is usually very good, provided that care is taken not to ignore features that play a significant role in determining the outcome of the events being studied.

After the abstractions have been made and the symbols have been chosen, those symbols can be combined in various ways according to precisely defined rules. Sometimes an appropriate manipulation can be identified easily from the intuitive meaning of the constituent words and symbols; at other times a useful series of manipulations has to be worked out experimentally or through trail and error. Typically, strings of symbols are combined into statements that express ideas or propositions. For example, the area of a square is A=s2. This equation tells us that the area directly depends on the length of the side and nothing else. In a sense, then, the manipulation of abstractions is much like a game: Start with some basic rules, then make any moves that fit those rules, which includes inventing additional rules and finding new connections between old rules.

In conclusion, the nature of mathematics is very unique and as we have seen in can we applied everywhere in world. For example how do our street light work with mathematical instructions? Our daily life is full of mathematics, which also has many connections to nature. Abstractions from nature are one the important element in mathematics. Mathematics is a universal subject that has connections to many different areas including nature.

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