
The book as a whole is leavened by the author's deep feeling for and broad view of his subject. This is a welcome antidote to the conventional stereotype  not unknown even among teachers of the subject  of mathematics as the science of number and magnitude. If a substantial number of educators could be weaned away from shallow and outmoded views on the subject, our schools might not be faced with the paradox that as mathematics becomes more and more important, fewer and fewer students are expected to appreciate it. Is it asking too much to hope that books such as this will become required reading for all wouldbe curriculummakers?
This account of the contributions made by mathematics to the sciences and arts, philosophy and religion, unfolds like a splendid epic from Euclid's 'Elements' to Einstein's theory of relativity. Intended for the intellectually curious reader without a mathematical background, the book is always stimulating even when it makes severe demands upon the reasoning powers. Unfortunately, the arts are slighted, ...
The object of this book is to show that mathematics has been an important influence in the development of our Western culture. This might be relatively easy if the author were writing for mathematicians; but the foreword by Professor Courant implies that the book is addressed to the group of intelligent people who do not have a background of mathematical knowledge, and this makes the undertaking a formidable one.
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Many mathematicians are sceptical with regard to books about mathematics written by mathematicians for the nonmathematical reader. The book under review makes a strong case in support of its thesis, and will be of very considerable interest to mathematicians. The reviewer would be interested, however, to read an appraisal of the book written by one of the nonmathematical laymen to whom it is supposedly addressed.
It is a formidable project set by the author of this book: to defend the thesis that mathematics has been, and is, a major influence in the development of western culture, and to present his arguments to the general public. "Almost everyone knows that mathematics serves the very practical purpose of dictating engineering design. Fewer people seem to be aware that mathematics carries the main burden of scientific reasoning and is the core of the major theories of physical science ... has determined the direction and content of much philosophic thought ... has supplied substance to economic and political theories, has fashioned major painting, musical, architectural, and literary styles, has fathered our logic, and has furnished the best answers we have to fundamental questions about the nature of man and his universe. ... Finally, .. mathematics offers ... aesthetic values at least equal to those offered by any other branch of our culture." To argue such sweepstakes persuasively requires at least a wide understanding of the history, meaning, and implications of the various parts of mathematics. More than this, a prime requisite is a crusader's concern for its importance to the thinking world in general. Kline's project is far broader than a new exposition of mathematics as handmaiden, even as queen, of the sciences, far deeper than merely another popularized discussion of mathematics. Only long germination of the ideas and evidence from such diverse fields as poetry and economics, only meditation and self crossexamination, could prepare the author for the thesis. Only a flair for careful but artful writing could prepare the thesis for its public. The author qualifies on all points; the result is highly admirable.
"The object of this book is to advance the thesis that mathematics has been a major cultural force in Western civilization." This thesis has been so successfully defended that the reader should need no persuasion to agree with the last sentences of the book: "When we consider the number of fields on which mathematics impinges and the number of those over which it already gives us mastery or partial mastery, we are attempted to call it a method of approach to the universe of physical, mental, and emotional experiences. It is the distillation of highest purity that exact thought has extracted from man's effort to understand nature, to impart order to the confusion of events occurring in the physical world, to create beauty, and to satisfy the natural proclivity of the healthy brain to exercise itself. We, who live in a civilization distinguished primarily by achievements owing their existence to mathematics, are in a position to bear witness to these statements."
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While the reader of the book might not agree with the author's own philosophy, he is led to consider seriously the main theme that "the development of mathematical ideas and methods has determined the dominant attitudes toward nature, and as a consequence, toward religion and society." Professor Kline has done a real service for mathematicians and nonmathematicians in displaying the range and force of his subject.
Although the material in this book is arranged in a historical order, it is not a history of mathematics; the history (of which there is necessarily a great deal) is only incidental to the main motive, which is to discuss the cultural significance of mathematics in the development of Western civilization.
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In the reviewer's opinion, the author has well sustained his thesis, ... and has produced a work that no mathematician, and no scientist, can afford to ignore; and a similar remark holds for the layman who wishes to be wellinformed on the development of Western culture.
This is a companion piece to Kline's Mathematics in western culture, in which the author was chiefly concerned to show mathematics as a great creative art. Here his purpose is to exhibit its role as the prime tool in the search for scientific truth and natural knowledge. To him, there is no real distinction between these functions; neither can make a significant advance without the other.
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... the keen inquirer who wants to know why it is that mathematics is the key to our knowledge of the physical world but has no equipment beyond elementary mathematics (and there are many such people in the world today) will get a surprisingly long way in Kline's hands, and so may be inspired to learn some calculus from this book and pursue its applications elsewhere. Both of Kline's books can be recommended to those seeking mathematics for the layman.
According to the author, this book is intended as a textbook for a one year terminal course in mathematics for liberal arts students. Mathematics is approached historically through descriptions of scientific, artistic, and philosophical milieu from which mathematical concepts and procedures have arisen. Important questions of sciences such as earth measure, cosmology, gravitation, and electromagnetism, and of arts, such as perspective drawing and musical composition, are described in detail sufficient to motivate discussions of mathematical notions they generate: geometry, algebra, trigonometry, calculus, and so on. Throughout these discussions what the author calls "the ways of mathematics" recur as a theme. These have to do with precision, notation, number, notions of order, and other attributes of mathematical predisposition. Noneuclidean geometries are introduced and serve as a barrier that separates from the classical notions the more modern, such as abstract algebra, statistics and probability, and mathematics directed toward the social sciences.
This book was written to serve as a text in oneyear terminal courses for liberal arts students. Accordingly, it emphasizes not mathematical facts or techniques but rather the nature and history of mathematics, its interactions with the sciences, and its impact on philosophy, religion and the arts. Because of the enormous number of topics taken up, the discussion of most of them is inevitably rather superficial. ... the chief difference of opinion regarding the book will be as to whether its approach is the proper one to take with liberal arts students.
This monograph was inspired by unpublished lecture notes of the late Rudolf Luneberg on the foundations of geometrical optics, based on solutions of the electromagnetic wave equation. The present authors have greatly extended Luneburg's original conception of the subject ...
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This is a clearly and logically written book which emphasizes strongly the mathematical, as contrasted to the physical, aspects of the classical electromagnetic theory of light.
Much is expected of the author of the very wellknown 'Mathematics in Western Culture' and 'Mathematics and the Physical World' and in this latest work of Professor Kline's such expectation is fully justified. The subtitle of this text is "An Intuitive and Physical Approach" and with regard to an intuitive approach the author says:
"In my opinion, a rigorous first course in the calculus is ill advised for numerous reasons. First, it is too difficult for the students. Beginners are asked to learn a mass of concepts so subtle that they defied the best mathematicians for two hundred years. Even Cauchy, the founder of rigour, gave formulations that are crude compared to what the current rigorous presentations ask readers to absorb. And Cauchy, despite his concern for rigour, missed the distinctions between continuity and differentiability and between convergence and uniform convergence. Before one can appreciate a precise formulation of a concept or theorem, he must know what idea is being formulated and what exceptions or pitfalls the wording is trying to avoid. Hence he must be able to call upon a wealth of experience acquired before tackling the rigorous formulation."
This "wealth of experience" is provided by the present text. With regard to the physical approach (and it is in this respect that the present text differs from other available texts) the author says: "The relationship of mathematics to science is taken seriously. The present trend to separate mathematics from science is tragic. There are chapters of mathematics that have value in and for themselves. However, the calculus divorced from applications is meaningless." Again: "In this book real problems are used to motivate the mathematics, and the latter, once developed, is applied to genuine physical problems  the magnificent, impressive, and even beautiful problems tendered by nature." It is in this "physical approach" that this text has a very great deal to offer. The problems are introduced with great care and attention to detail and the necessary concepts (force, velocity, acceleration, word etc.) are developed as required. In solving the problems the student is led, via false starts, to see that correct methods are "almost always preceded by groping" and that this is the way mathematicians work.
It is refreshing to find that books are still being written on elementary calculus in which the subject is approached from an intuitive and heuristic standpoint rather than attempting to inculcate in the beginning student the deadly rigorous viewpoint of the sophisticated mathematician.
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For the purpose for which they were designed these calculus books by Morris Kline would be splendid texts for a beginning student who needs a book that is readable, interesting, and not too demanding in mathematical rigour.
Were it not for the large page size this would be an admirable bedside book, for its contents are extremely rich. They consist of 50 items reprinted from the 'Scientific American' and are hence informative and authoritative; the editor is Morris Kline, whose books on the cultural and sociological impact of mathematics are widely known and appreciated.
The author, who has made valuable contributions to applied mathematics, is widely known for his outspoken opposition to the mathematical curricula in the schools. Here he successfully undertakes a formidable project: to survey, at a level comprehensible to a graduate student, the major developments in western mathematics from its beginnings in Mesopotamia until approximately 1930.
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... one of the most impressive features of this exposition is the writer's proof that the history of mathematics is part of history itself. For the first half of the book (until ca. 1800; after that time, space prevents an adequate treatment) he describes convincingly the social and religious conditions under which mathematics developed. He does not fall into the trap, common in histories of science, of talking about the relation between science and theology through the Renaissance as "warfare". This doctrine, which we inherit from the period of the Enlightenment, and which my generation was taught as fact, has undergone a thorough reexamination and has been found wanting.
It is the first general history which begins to reflect the actual development of mathematics, and is by far the best yet to appear.
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... nothing can, or should, dispel the fine impression that this book leaves. I am still amazed by the amount that Kline has achieved.
This is a superb book and in an ideal world all mathematics teachers (if they were still needed) would be presented with copies. ...
... the range of the book and its price (although this is by no means unreasonable when one considers what one is getting) effectively rule it out of consideration as a classroom text. I hope, though, that Professor Kline and his publishers can be persuaded to produce certain sections in paperback form. The book is too well constructed to permit the mere extraction of odd chapters, but not too much work would be needed to produce paperbacks on, say, 'Series', 'Differential equations' and 'The growth of abstract algebra' which would prove most valuable for use with undergraduate courses.
As readers of his earlier works well know, Morris Kline writes in an attractive, vigorous, and hardhitting manner, and he would be far from flattered by total agreement with his views. His persuasiveness notwithstanding, not every mathematician will accept some unmeasured statements on the role of applied mathematics in the development of his subject.
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Kline's Mathematical Thought is a sound and authoritative account of the development of mathematics with emphasis on that portion of its history since 1800, which is most difficult to present ... On the whole we are deeply indebted to Morris Kline for so effective a completion of so ambitious a project.
The first, hardcover, edition of this book of more than 1200 pages came out in 1972. Now it is republished, unchanged, in three handsome paperback volumes, which makes this impressive work, result of a lifetime of critical study and research, by the now emeritus professor of New York University, again available. The book contains a wealth of information concerning the mathematics of more than two millennia, from the Babylonians to the present century. The wellknown and appreciated attempts by the author in previous books to humanize the subject of mathematics are also characteristic for this book.
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Not only students of the history of mathematics can profit by Professor Kline's extensive and critical knowledge, but also students of more modern mathematics in specialized fields. Where the treatment of older mathematics is sometimes (but not always) a bit sketchy, it is in the more modern sections that we discover the particular strength of the book. Here and there we find passages that are mildly controversial, but this adds to the fun of reading the book.
Professor Kline presents a bold, comprehensive, and long overdue critique of the modern mathematics curricula for the elementary and secondary schools. Among the various groups of teachers, supervisors, publishers, etc., that are responsible for the development of the new curricula, the mathematicians are the main target of Kline's criticism: "The professional mathematicians are the most serious threat to the life of mathematics, at least so far as the teaching of the subject is concerned."
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The book is well documented. It is lucid, humorous, and its many analogies and excursus enhance its readability. The author obviously aims at generating controversy. He thus leaves himself open to criticism. In his "proper direction for reform" he does not offer novel solutions. For example, the mathematics laboratory which is here advocated has for some time been an essential component of the modern curricula.
Morris Kline has for many years been the acknowledged champ of the critics of the new math. In 'Why Johnny Can't Add' he collects his criticisms under one cover. His writing is clear and simple, though somewhat repetitious. It can be goodhumoured, and it is frequently biting. His personal opinions are expressed in strong terms, he is unafraid of going out on a limb, and he quotes thinkers such as Poincaré and Felix Klein to back him up.
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In a stinging chapter on the deeper reasons for the new math, the author excoriates the pure mathematicians for their mathematical narrowness, ignorance of science, and lack of interest in the psychology of learning. This chapter will stimulate thought and raise hackles. The author concludes with his own recommendations for reform. Briefly, mathematical education should be broad rather than deep, and the basic approach at all levels should be intuitive, and motivated by applications. The author would make a stronger case if he came up with some suggestions in detail, such as an actual course outline or some sample pages of text, including applications that the student can understand.
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