The Mathematical Sciences: A Report (1968) / Chapter Skim
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3 The Community of Mathematics
3 The Community of Mathematics
Pages 45-56
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From page 45... ...
Certain abstract mathematical constructions constitute a prelude to formulations of quantum theory. By the end of the nineteenth century and at the beginning of the twentieth century a vast mathematical apparatus was being used not only for dealing with problems in astronomy and in physics and in many branches of engineering, but also in sciences like chemistry, which are built upon theories in 45
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From page 46... ...
A more basic difference between mathematics and experimental sciences, including even the most theoretical branches, lies in the extent to which mathematical research originates within the body of mathematics itself. Physicists, chemists, biologists, and psychologists are more or less directly concerned with observable phenomena.
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From page 47... ...
In this same memorandum he mites that: Twenty years ago, the typical mathematical preparation of the future engineer consisted of a review course in trigonometry and analytic geometry and a formal course in calculus aimed at proficiency in handling routine problems. Today, the review course has disappeared from the curriculum while
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From page 48... ...
So have its main sources of in spiration: the external world and its own internal structure. Mathematics done for its own sake is traditionally designated as "pure mathematics" and mathematical investigations aimed at increasing our understanding of the world are classified as "applied mathematics." Such a division of mathematics into pure and applied, however, is difficult to maintain; the origin of many important mathematical ideas can be ultimately traced to applications, and,
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From page 49... ...
For example, Hermann Weyl contributed in equal measure to the theory of groups as a pure mathematical discipline and to the effective uses of this theory in the theoretical constructions of atomic physics.
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From page 50... ...
Intellectual curiosity and intellectual excitement are the main motivating forces behind research in all mathematical sciences, as they are in all sciences in general. The thrill of recognizing a pattern in a seemingly chaotic situation and of reducing a large number of apparently unrelated phenomena to a single simple principle are again characteristic of all sciences; but in the mathematical sciences, and especially in core mathematics, the part played by such considerations is so predominant that some mathematicians consider mathematics to be as much an art as a science.
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From page 51... ...
We shall return to this question. Those intellectual efforts coming half way between core mathematics, on the one hand, and the sciences and technologies to which mathematics is applied, on the other, are often referred to as "applied mathematics." This is accurate only if it is understood that these efforts include not only the traditional parts of mathematics concerned with applications in physics, chemistry, and engineering
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From page 52... ...
Like all good mathematics, good applied mathematics is original and imaginative in the invention and use of its concepts and in its tentative modeling constructions. Its chief distinction from pure mathematics, which shows self-motivated progress along dimly discernible natural paths of growth towards intellectually satisfying goals, lies in adding to this conceptual activity a deep concern for the world of outer experience and a ready interest in problems beyond the confines of mathematics.
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From page 53... ...
It is said that during the recent past mathematicians have alienated themselves from the mainstream of scientific development. One thing modern mathematicians tend to overlook is that the giants of former days were all actively interested in physics as well as mathematics.
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From page 54... ...
. There have been contemporary mathematicians with interests as catholic as those of the greatest men of the nineteenth century for example, Hermann Weyl, Wiener, and Von Neumann or Gelfand and Kolmogorov but it is unreasonable to expect more than a few such men in a generation.
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From page 55... ...
On the other hand, this danger is acute as far as the interfaces between core mathematics and the applied and partly mathematical sciences are concerned and even more so between the mathematical sciences as a whole and the various users of mathematics. What is needed is a conscious effort by highly qualified people to overcome this "communication gap" by specially written books and articles, by interdisciplinary conferences and courses, and by other appropriate means.
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From page 56... ...
Its recommendations are addressed to the American Mathematical Society but may provide more general guidance. During the past decade, the mathematical sciences have made strenuous and successful efforts to establish communications between the investigators and teachers of mathematics at various levels.
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