| HOME | General Introduction Many people think of mathematics in terms of rules to be learned in order to manipulate symbols or study numbers or shapes in the abstract for their own sake. Mathematical theory does develop in the abstract; it need have not dependence on anything outside itself. The truth of the theory is measured by logic rather than experiment. However, one of its most valuable uses is in describing or modelling processes in the real world, and thus there is constant interaction between pure mathematics and applied mathematics. Mathematics may be considered as the very general study of the structure of systems. Since the study is unrelated to the physical world, rigorous formal proofs are sought, rather than xperimetnal verifications. Theory is presented in terms of a small number of given truths (known as axioms) from which the entire theory can be inferred. Thus, the aims are for generality in approach and rigour in proof, aims that explain the traditional concern of mathematicians for the unification of seemingly different branches of mathematics. As an example, Rene Descartes showed that geometrical figures could be described in terms of algebra, enabling geometric proofs to be established in terms of arithmatic, so that both generality and rigour were advanced. Applied Mathematics and Modelling There is no sharp boundary between the study of mathematical systems in the abstract (the field of pure mathematics) and the study of such systems to make inferences about certain physical systems that are described by the mathematical theory (the field of applied mathematics). In principle, any branch of mathematics may turn out to describe some physical, economic, biological, medical, or other system. Modelling a physical system consists of seeking a formal mathematical theory that conforms with the properties of the physical system. Often, as for example in computer simulation of space travel, the mathematical theories are very large and complex, but sometimes the model can be quite simple. Sometimes, known mathematics can describe and predict the behaviour of the sytem; at other times, the modelling can give rise to completely new branches of mathematics. Applied mathematics encompasses many specialized fields in which the relationships between the experimental findings and the mathematical theories are well established. Although the subject can include the application of statistical theory to such areas as sociology, the term is usually restricted to the application fo the methods of advanced calculus, linear algebra and other branches of advanced mathematics to physical and technlogical processes.
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