- Paperback: 390 pages
- Publisher: Oxford University Press; 1 edition (March 1, 1990)
- Language: English
- ISBN-10: 0195061357
- ISBN-13: 978-0195061352
- Product Dimensions: 8.9 x 0.8 x 5.9 inches
- Shipping Weight: 1.6 pounds (View shipping rates and policies)
- Average Customer Review: 16 customer reviews
- Amazon Best Sellers Rank: #300,543 in Books (See Top 100 in Books)
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001: Mathematical Thought from Ancient to Modern Times, Vol. 1 1st Edition
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"I have always had great regard for this book as the one which relates the development of modern mathematical ideas in a readable fashion."--Michael F. O'Reilly, University of Minnesota in Morris
"Outstanding scholarship and readability. One of only a couple of books available in English for in-depth historical studies at the fourth year/graduate level."--Charles V. Jones, Ball State University
"The consistently high quality of presentation, the accuracy, the readable style, and the stress on the conceptual development of mathematics make [these volumes] a most desirable reference."--Choice
"Without a doubt a book which should be in the library of every institution where mathematics is either taught or played."--The Economist
"What must be the definitive history of mathematical thought....Probably the most comprehensive account of mathematical history we have yet had."--Saturday Review
About the Author
Morris Kline is Professor of Mathematics, Emeritus, at the Courant Institute of Mathematical Sciences, New York University, where he directed the Division of Electromagnetic Research for twenty years.
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This work was written in a more innocent time. It is not the result of in-depth historical scholarship, but of compilation of information. As such it often has a rather sophomoric ring to it. Consider for example this passage:
"It was Wallis and Newton who saw clearly that algebra provided the superior methodology [to geometry]. Unlike Descartes, who regarded algebra as just technique, Wallis and Newton realized that it was vital subject matter." (p. 392)
No further evidence or references are given for this claim here. It is simply stated as if it was a straightforward fact. In my opinion this vague and simplistic statement is half vacuous and half false, but let's put that aside and look only at Kline's text. Indeed, in the very same paragraph we also read:
"It is true that excessive reverence for Newton's geometrical work in the Principia … caused English mathematicians to persist in the geometrical development of the calculus. But their contributions were trivial compared to what the Continentals were able to achieve using the analytical approach." (p. 392)
So even though Newton "clearly saw" that algebra was the "vital subject matter" of mathematics, his magnum opus nevertheless conveyed the exact opposite sentiment, namely an emphasis on geometry, whence analytical mathematics was advanced most by precisely those who did not see clearly its significance. A serious attempt at coherent historical understanding would attempt to resolve this apparent paradox; a sophomoric compilation of "facts" would not. Kline does not.
It is much the same with Kline's claim that Descartes "regarded algebra as just technique." Here too we find Kline stating virtually the exact opposite in other places:
"Algebra, for Descartes, precedes the other branches of mathematics. It is an extension of logic useful for handling quantity, and in this sense more fundamental even than geometry." (p. 280)
This is a rather anachronistic point of view, which also mars Kline's very poor account of Descartes's Geometrie, where we read this confusing nonsense:
"Descartes rejects the idea that only the curves constructible with straightedge and compass are legitimate and even proposes some new curves generated by mechanical constructions. He concludes with the highly significant statement that geometric curves are those that can be expressed by a unique algebraic equation … All other curves … he calls mechanical." (p. 312)
On this account it indeed seems that "algebra, for Descartes, precedes the other branches of mathematics," but in reality it is the exact opposite: algebraic curves are accepted as geometrical (i.e., exact and rigorous) by Descartes precisely *because* they can be generated by his construction method. Thus Descartes does not "propose some new curves generated by mechanical constructions"---on the contrary, he proposes constructions which he considers *geometrical*, and thus *not mechanical*, which he claims generates not "some new curves" but rather *all geometrical curves*, a class which, he claims, turns out to be coextensive with the class of all algebraic curves.
This is all essential to the intrinsic logic of Descartes's works, but it is incidental to it as precursor of the modern analytic geometry of today, which explains why Kline's discussion is so hopelessly botched.
This is a typical instance of Kline's anachronistic perspective: Kline is reliable-ish when it comes to precursors of modern ideas, but often very inadequate in terms of holistic and sympathetic understanding of past mathematics when the original motivations and contexts differed greatly from our own.