The Mathematical Sciences: A Report (1968) / Chapter Skim
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6 Examples of Mathematics in Use
6 Examples of Mathematics in Use
Pages 101-116
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From page 101... ...
Accordingly, an exhaustive review of the penetration of mathematics into various areas of human endeavor would require volumes. In this chapter we describe a few typical examples: physics a science completely mathematized almost from its very inception; engineering design a fully mathematized technology; mathematics in the newer environmental sciences specifically, numerical weather prediction; economics in which the penetration of mathematics is about a hundred years old; the technology of management and operations in which mathematization is a World War II development.
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From page 102... ...
As one reviews the development of physics through the centuries, starting from the early studies of astronomy and Newtonian mechanics, proceeding through the nineteenth century formulation of electromagnetic phenomena and of the theory of heat and thermodyr~amics, and then to the modern development of relativity, quantum mechanics, and high-energy physics, one is struck with the increasingly abstract and sophisticated nature of the mathematical concepts that it was necessary to introduce for the description of natural phenomena. Such observation was undoubtedly behind the remark of the late British physicist, Jeans, that God is a mathematician.
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From page 103... ...
Be this as it may, it has been repeatedly demonstrated that a sense of form and an appreciation of elegance, abstraction, and generalization, which are the hallmarks of good mathematical development, are often also the characteristics of the new breakthroughs in physical insight. In fact, what one refers to as physical ideas often derive from properties of abstract mathematical concepts, which turn out to have widespread and deep-rooted applicability in natural phenomena.
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From page 104... ...
The invention of calculus, of differential geometry, of the ergodic theory, all represent mathematical developments stimulated by physical problems. We quote from a recent report by physicists (reference 1 1, page 162~: Through centuries of intimate contact, theoretical physics and mathematics have interacted strongly to their mutual benefit.
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From page 105... ...
Prediction of satellite orbits, guided and controlled, is another activity involving careful mathematical formulation and extensive numerical calculations. Once a computational program is well designed, the work is routine and
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From page 106... ...
:Fluid mechanics, one of the best-established areas of applied mathematics, is important in several engineering fields. The noise produced by jet aircraft and the shock waves associated with supersonic flight must be understood in detail before remedies and improvements can be suggested and designed; combustion instability leads to critical problems in the development of rockets; and systematic progress in minimizing the devastating effects of tornadoes cannot be expected until the understanding of atmospheric dynamics has greatly improved.
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From page 107... ...
The identification of communication engineering and mathematical logic as two systems that are both concerned largely with the manipulation of arbitrary symbols according to formal rules, made possible by information theory, tremendously broadened the horizon of the communication engineer and at a stroke opened to him vast areas of mathematics as a source of ideas for particular ciphers and coding schemes. It may happen that the same engineering objective may best be served first by one physical technique and then by another, generating new mathematical problems as they evolve.
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From page 108... ...
The guidance material consists of large-scale wind and weather patterns over the entire northern hemisphere. The basis for this is the approximate numerical solution on large electronic computers of hydrodynamic and thermodynamic partial differential equations constituting a mathematical model for the behavior of the atmosphere.
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From page 109... ...
It was not until the 1930's and early 1940's that empirical observations began to approach the frequency and detail needed for successful attempts at weather prediction on a mathematical basis. In particular, the new upper-air observational network developed during the 1930's elucidated the dynamics of the so-called jet stream, a great meandering river of air, five to eight miles high and hundreds of miles wide, which loops completely around the northern hemisphere at middle latitudes.
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From page 110... ...
Many aspects of the numerical techniques are still highly unsatisfactory, however, imposing severe limitations on the kinds of simulation that can be attempted. It is often difficult even to distinguish between distortions introduced by the numerical methods and those resulting from deficiencies in the mathematical model.
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From page 111... ...
Education in economics is now highly mathematically oriented. In most major departments in the United States, all economics PhD's are required to learn calculus, selected topics in advanced calculus, the elements of linear algebra and probability, statistical inference, and econometrics.
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From page 112... ...
The so-called "collective risk theory" focuses attention on the distributions of total claims of an insurance company at the end of a specified period of time, so that a reasonable judgment might be made about appropriate limits of retention or bounds of acceptable adverse fluctuations. These are essential con · .
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From page 113... ...
The exact distribution of total claims of an insurance company has been studied analytically for a variety of assumptions. Ingenious numerical approximations have been developed, and more recent broader analytic-numerical studies of total claim distributions have been made, relying extensively on computer · .
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From page 114... ...
At a leading business school (Harvard) , which is not "mathematically oriented" and where no such requirements are imposed, about 75 percent of the entering students have at least two years of college mathematics, several elective courses requiring that degree of mathematical sophistication are given, and there is a sizable group of faculty members who have PhD degrees in mathematics or applied mathematics.
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From page 115... ...
A comprehensive survey of this held can be found ire reference 13. Another important example of the penetration of mathematical methods into hitherto unmathematized areas is in the young science of mathematical linguistics, which applies mathematical methods arid the mathematical way of thinking to the study of living languages.
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From page 116... ...
116 The State of the Mathematical Sciences formal or numerical manipulations, what can probably be neglected, and what is surely negligible. It is not easy to teach these things explicitly; they are usually learned by experience in doing and thus come to depend on at least some facility with the manipulations concerned.
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