- Series: Dover Books on Mathematics
- Paperback: 656 pages
- Publisher: Dover Publications; Reprint edition (October 18, 2017)
- Language: English
- ISBN-10: 0486272397
- ISBN-13: 978-0486272399
- Product Dimensions: 5.5 x 1.2 x 8.5 inches
- Shipping Weight: 1.4 pounds (View shipping rates and policies)
- Average Customer Review: 7 customer reviews
- Amazon Best Sellers Rank: #2,029,858 in Books (See Top 100 in Books)
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The Development of Mathematics (Dover Books on Mathematics) Paperback – October 18, 2017
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Let me stress the historical order of mathematical results, that E.T. Bell presents. He tries to show what mathematics of the past had significance for later mathematics - the problems and the solutions. When students study mathematics today, they often get the feeling that all these theorems must of had some purpose in the past that's been lost. This book attempts to show and lay out these connections, both mathematical and cultural.
One can argue that he doesn't give enough to actually learn any of it; but, then again, he's covering all the mathematics up to his time in six hundred pages. The stories are skillfully woven throughout. There's few books in mathematics much less outside of that presents so many ideas and thoughts.
An intellectual problem is not knowing of the existence other worlds and ideas. This book gives the lay of the land and the philosophical lessons of mathematics throughout history.
As the keynote of the book Bell sounds an old quotation: "There is probably no other science which presents such different appearances to one who cultivates it and to one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is in the purple bloom of vigorous youth, everywhere stretching out after the attainable but unattained, and full of the excitement of nascent thoughts; its logic beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness."
To the student of mathematics the historical development of his subject appears all too inevitably as a wilderness, and moreover as an almost impenetrable one when the last century or two are approached. With research pressed in this time and at the present on many fronts by a vast number of investigators, with many different groups of these pursuing apparently quite diverse objectives, and with all of them changing their tactics and goals disconcertingly often, the residue of their attainments is a sweltering jungle indeed. Through this the present book lays a very welcome road. The typical and more significant trends and episodes are isolated, the genesis, growth and efflorescence of some of the concepts and methods, whose survival to the present is their guarantee of significance, are traced, and often their decadence in periods of sterile overelaboration is observed.
The book is not of the "popular" kind, as this term is generally understood, since it makes small effort to be intelligible to readers wholly uninitiated mathematically. Indeed, its appeal will probably be found to vary almost directly with the reader's mathematical attainments. The less trained will find much that is entirely narrative and non-technical, and will some-times find quite enlightening the concise but generally clear technical surveys that are given. The advanced student of mathematics and science will find much more to interest him, and will value the orientations which the book supplies. Professional mathematicians, even those who are themselves momentarily engaged in extending mathematical theories and their applications, will find the book a thoroughly worth-while reading of mathematical evolution. This is not to say by any means that they will in all instances read from the noted trends and related episodes precisely the same inferences as does the author. The better, perhaps, that in some cases they should not.
For the purposes of this review it is convenient to regard the book as falling into two parts, consisting respectively of the first six chapters, which treat of mathematics up to the year 1637, and the remaining seventeen chapters which terminate the discussion at the present time. The first part, which begins with a general prospectus, is given over thereafter to a review of mathematics in the ancient Babylonian and Egyptian eras, in the Greek period, in the dark age of Europe, through the Arabian epoch and the Renaissance. While completely nontechnical, even these chapters are not to be regarded as a historical text. There is not the customary cataloguing of names and facts, but rather a sort of running narrative commentary, of which a full appreciation will be somewhat conditioned upon the reader's previous knowledge of the history. Bell acknowledges these pages to hold in the main a collation of material from more or less familiar and classical works. These chapters appear to be by far the weaker part of the book; to be in fact a trifle pedestrian, though not always unprovocative. As is well known, iconoclastic tendencies are not invariably eschewed by Bell. The so-called debunking of tradition is often salutary. An excess of it, however, though it adds a sensational element to the reading, may in the case of immature or otherwise undiscriminating readers leave impressions that are not wholly fortunate or just. Enjoyable or regrettable, as the reader may find them, he will find here, and throughout the book, a sprinkling of the quips and sophistications which those who know Bell would rather expect, and some will perhaps deplore his occasional momentary lapses from a generally prevalent high scholarly objectiveness to the inclusion of less happy and rather discordant contemporary comment.
The peculiar contribution of the book is by all odds to be found in its second part. Here Bell's excellent qualifications for his task, which include a technical equipment beyond the range of the usual historian, and a literary facility far beyond the range of the usual mathematician, really come to bear. The wide gamut of topics discussed is perhaps best suggested by the chapter headings, which are the following: The beginnings of modern mathematics 1637- 1687; Extension of number; Toward mathematical structure; Arithmetic generalized; Emergence of structural analysis; Cardinal and ordinal to 1902; From intuition to absolute rigor, 1700-1900; Rational arithmetic after Fermat; Contributions from geometry; The impulse from science; From mechanics to generalized variables; Differential and difference equations; Invariance; Certain major theories of functions; Through physics to general analysis and abstractness; Uncertainties and probabilities.
It would be entirely impossible to abstract these chapters briefly. They should be read in their completeness. Mathematics and mathematicians live in them, and not infrequently lend themselves to genuine drama. The presentation of the whole is admirable. It is flowing and graceful and often characterized by a genuine and delightful humor. A feature which will be prized is Bell's almost invariable practice of labeling all investigators and notable publications with their nationality and dates.
The publishers of the book are to be thanked for an attractive and legible volume. Bell deserves recognition and high praise for such a significant work. Many the scientist who has come to realize, to his humility, that his vaunted work would in his absence have soon been accomplished by another. One may safely venture that no other would soon have written this book had Bell not done so.
I recommend that one read a more conventional history of mathematics (such as Boyer, Kline or Gratton-Guinness) before attempting this controversial one.
Be forewarned that Constance Reid, in her biography of Bell, points out errors in this book. I forgive Bell for those because no one person could possibly comprehend in detail all the abstruse mathematics which he covers relatively well. I recommend this book only to readers already somewhat knowledgeable in mathematics.