This comprehensive history traces the development of mathematical ideas and the careers of the mathematicians responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the 19th century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.
Mathematical Thought From Ancient to Modern Times, Volume I
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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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42 of 42 people found the following review helpful
By Chan-Ho Suh
Format:Paperback
As one might expect from a 3-volume history, _Mathematical Thought_ is comprehensive; Kline covers basically all the important mathematical developments from ancient times (e.g. the Babylonians) until about 1930. Note that (as Klein himself mentions) the coverage of ancient mathematics, while taking up a good half of the first volume, is necessarily modest, and if that is the reader's primary interest, s/he would do best to seek out specific histories on the Greeks, Chinese, etc. [Kline gives several useful references, as always].
The reader interested in the 18th and 19th centuries will find plenty of food for thought. For example, the story of non-Euclidean geometry is covered well, and Kline does a good job of putting the discoveries in the light of the times. One notable thing I learned is that Lobachevsky and Bolyai were not the discoverers of non-Euclidean geometry, nor were they the first to publish material on that subject. Others before had expressed the opinion that non-Euclidean gometry was consistent and as viable a geometry as Euclidean (e.g. Kluegel, Lambert...even Gauss!) It remained for Beltrami to later show that if Euclidean geometry were consistent, so is non-Euclidean. Of course, important events like the invention of Galois theory are also mentioned. Really, if it's a major mathematical development before 1930, Kline will have it somewhere in these 3-volumes.
Incidentally, Kline advances the interesting theory that Lobachevsky and Bolyai somehow learned of Gauss' work on non-Euclidean geometry (which he kept secret and was not learned of until after his death) through close friends of Gauss: Bartel (mentor to Lobachevsky) and Bolyai's father, Farkas. [I understand that this theory has been shown false by recent research into Gauss' correspondence] Kline is careful to indicate it is only speculation by phrasing words carefully, e.g. "might have..." and "perhaps he..." I can appreciate Kline's various speculations and opinions, usually they are very interesting, and (at least in these volumes) he always does a good job of highlighting where the account of history ends and his ideas begins. Even so, luckily for those who like unbiased historical accounts, he inserts himself into the text rarely. This may surprise readers who have read his other books, like _Mathematics: the Loss of Certainty_. This history is a scholarly work, although one can't really say that about his other works.
Kline also writes quite a bit about the development of the calculus, as one should expect, given its major role in forming modern mathematics. I got a much deeper appreciation of calculus from reading various sections, which explained how this or that area was influenced or invented because of certain calculus problems.
I debated about giving this book 4 stars since there are a few minor flaws. One I've mentioned above; I think Kline should have kept his voice objective, instead of occasionally going into a little diatribe on his pet peeves. This is minor, since he doesn't do it too often, and I suppose he can be excused for being human. Another is that the index is rather weak. For a work of this magnitude, one expects that one ought to be able to find the phrase "hyperbolic geometry" in the index. Surprisingly one doesn't. "Non-Euclidean geometry" is there, but not the other phrase, which is synonymous and more common nowadays. There are other examples, but this is the one that comes to mind now.
Finally, I should add that I have not read every page of this history nor am I even close to doing that. I have read parts of all three volumes, and the quality seems consistent. That said, this is not a history one should read straight through. It is meticulous and well-documented, which can make for rather dry reading, so I suggest you do plenty of skipping around. I found (and will probably still find) Kline useful for helping me understand the context of the various mathematical concepts I was studying. Not only that, but I found his explanations of some topics to be even better than those in standard textbooks. Because of the insights I've gained, I've decided to overlook the little flaws, so...five stars!
9 of 12 people found the following review helpful
By S. Matthews
Format:Paperback
if flawed. Not only do you have to wade through the gentleman amateur flavour of the first couple of hundred pages or so, but Kline manages to describe William Hamilton as 'the greatest English theoretical physicist after Newton'; even an Irishman would concede that the greatest English theoretical physicist after Newton was Maxwell - Hamilton was third. However with the first impact tremors announcing the approach of Leonard Euler, when the technical issues start to thicken, things improve enormously. Kline is clearly in awe of Euler, and does a good job of communicating why awe is appropriate.
It is nevertheless fortunate that the history of mathematics, unlike that of science, is a discipline essentially invulnerable to whiggish prejudice.
It is nevertheless fortunate that the history of mathematics, unlike that of science, is a discipline essentially invulnerable to whiggish prejudice.
42 of 69 people found the following review helpful
Format:Paperback
Morris Kline's "history" is a disappointment. I have no doubt that Kline knows his mathematics, but he either does not know his history, or prefers to distort it so that it fits into his preconception of that history. To furnish an example: on page 181 Kline writes "In 529...Justinian closed all the Greek schools of philosophy...Greek scholars left the country and some for example, Simplicius - settled in Persia."(!) What Kline omits is that after a very short stay in Ctesiphon, Simplicius (and the other philosophers) returned to Greece. This is known from the Byzantine historian, Agathias', "Histories": "Priscianus of Lydia's Solution ad Chrosroem", (Chrosroes being the Persian king) which recounts a philosophical debate in Persia. Further, Justinian did not close centres of Greek thinking, but specifically those that were pagan; Alexandria's academy, it must be remembered, remained open as it was led by Simplicius' adversary the Christian Philoponus. The allusion by Kline that the Greek mathematical past was rejected is even more bewildering when the building of the Agia Sophia (built during Justinian's reign) is considered. This building was designed by 2 classically trained mathematicians (Athemius of Tralles & Isodore of Miletus) using the mathematical principles of antiquity which were still extant, known and in use (in the Greek east)!
On p. 197 Kline writes "The significant contribution to mathematics that we owe to the Arabs was to absorb Greek and Hindu mathematics [and] preserve it." Amazingly, his main reference for this chapter is O'Leary's "How Greek Science Passed to the Arabs". O'Leary makes it incontrovertibly clear that the "translations" of various Greek mathematical & scientific works paraphrased into Arabic came directly from Byzantium. Byzantium does not get a mention in Kline's book until later. It was not the Arabs who preserved Greek material, but the Greeks themselves! Further, his claims on Christian denunciation of pagan thinking might be true in the Roman Latin west, but in the Greek Byzantine east, Greek past achievements were a source of pride! (Anna Comnena's "Alexiad" is a good indication of the "pagan" aspects of Byzantine civilization. References to Homer alone - out of all other "pagan" authors - outnumber all biblical references.). In the Greek east, as a counter-point to the Latin west, St Basil decreed in his "Discourse to Christian Youth on the study of the Greek Classics", that "pagan" literature should be referenced as an aid to understanding scriptures. This text justified studies of the "pagan" past in Byzantium and proved invaluable when the Italians "discovered" it during the renaissance (this "discovery" was made by Leonardo Bruni in the 15th century).
On p.206 Kline, does not even seem to realise that Greece & Byzantium are part of Europe... Or rather, it is inconvenient for him to mention this without having to reorganise his premise... And so, on that page, he writes "...since the Arabs did have almost all the Greek works, the Europeans acquired a tremendous literature." The problem here is that the Arabs only ever paraphrased Greek technical manuals - not literature; Homer, Hesiod, Greek historians (eg Plutarch, Herodotus, Thucycides, etc) Greek playwrights (eg Aristophanes, Aeschylus, Euripides, Sophocles, etc), were totally unknown to Arabs. What is even worse, in O'Leary's book (which to reiterate is one of Kline's references), O'Leary wrote "...the Greek writers who influence the oriental world were not the poets or historians, or orators, but exclusively the scientists..." p. 1.
Of equal interest is what Kline writes on pp. 189-190: "About the year 1200 scientific activity in India declined and progress in mathematics ceased." If the reader were to read another book, one on art for instance, "Hindu Art and Architecture" (George Michell, Thames & Hudson), they would read:
"At the very end of the twelfth century northern India was overwhelmed by Muslim invaders...virtually all temple building came to a halt..." One wonders why the destroyers of Hindu mathematics are given credit for the preservation of this mathematics?
This book would have been a great introduction had Kline not had an obvious agenda.
I would recommend any of the following books (although they require a bit more reading):
1/ Otto Neugebauer "The Exact Sciences in Antiquity";
2/ Thomas Little Heath (his histories of Greek mathematicians/mathematics);
3/O'Leary's "How Greek Science Passed to the Arabs"; and
4/ Jacob Klein's "Greek mathematical thought and the origin of algebra"
as better introductions to this period of mathematical thought.
On p. 197 Kline writes "The significant contribution to mathematics that we owe to the Arabs was to absorb Greek and Hindu mathematics [and] preserve it." Amazingly, his main reference for this chapter is O'Leary's "How Greek Science Passed to the Arabs". O'Leary makes it incontrovertibly clear that the "translations" of various Greek mathematical & scientific works paraphrased into Arabic came directly from Byzantium. Byzantium does not get a mention in Kline's book until later. It was not the Arabs who preserved Greek material, but the Greeks themselves! Further, his claims on Christian denunciation of pagan thinking might be true in the Roman Latin west, but in the Greek Byzantine east, Greek past achievements were a source of pride! (Anna Comnena's "Alexiad" is a good indication of the "pagan" aspects of Byzantine civilization. References to Homer alone - out of all other "pagan" authors - outnumber all biblical references.). In the Greek east, as a counter-point to the Latin west, St Basil decreed in his "Discourse to Christian Youth on the study of the Greek Classics", that "pagan" literature should be referenced as an aid to understanding scriptures. This text justified studies of the "pagan" past in Byzantium and proved invaluable when the Italians "discovered" it during the renaissance (this "discovery" was made by Leonardo Bruni in the 15th century).
On p.206 Kline, does not even seem to realise that Greece & Byzantium are part of Europe... Or rather, it is inconvenient for him to mention this without having to reorganise his premise... And so, on that page, he writes "...since the Arabs did have almost all the Greek works, the Europeans acquired a tremendous literature." The problem here is that the Arabs only ever paraphrased Greek technical manuals - not literature; Homer, Hesiod, Greek historians (eg Plutarch, Herodotus, Thucycides, etc) Greek playwrights (eg Aristophanes, Aeschylus, Euripides, Sophocles, etc), were totally unknown to Arabs. What is even worse, in O'Leary's book (which to reiterate is one of Kline's references), O'Leary wrote "...the Greek writers who influence the oriental world were not the poets or historians, or orators, but exclusively the scientists..." p. 1.
Of equal interest is what Kline writes on pp. 189-190: "About the year 1200 scientific activity in India declined and progress in mathematics ceased." If the reader were to read another book, one on art for instance, "Hindu Art and Architecture" (George Michell, Thames & Hudson), they would read:
"At the very end of the twelfth century northern India was overwhelmed by Muslim invaders...virtually all temple building came to a halt..." One wonders why the destroyers of Hindu mathematics are given credit for the preservation of this mathematics?
This book would have been a great introduction had Kline not had an obvious agenda.
I would recommend any of the following books (although they require a bit more reading):
1/ Otto Neugebauer "The Exact Sciences in Antiquity";
2/ Thomas Little Heath (his histories of Greek mathematicians/mathematics);
3/O'Leary's "How Greek Science Passed to the Arabs"; and
4/ Jacob Klein's "Greek mathematical thought and the origin of algebra"
as better introductions to this period of mathematical thought.
Citations (learn more)
This book cites 90
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books
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- History of Mathematics, Vol. I (Dover Books on Mathematics) by David E. Smith on 11 pages
- A Short Account of the History of Mathematics (Dover Books on Mathematics) by W. W. Rouse Ball on 10 pages
- History of Mathematics (AMS Chelsea Publishing) by Florian Cajori on 9 pages
- A History of Mathematics, Second Edition by Carl B. Boyer on 8 pages
- The Mathematics of Great Amateurs by Julian Lowell Coolidge on 5 pages
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- Mathematically Speaking: A Dictionary of Quotations by C.C. Gaither on 13 pages
- The Milky Way Galaxy and Statistical Cosmology, 1890-1924 by Erich Robert Paul on 4 pages
- Using History to Teach Mathematics: An International Perspective (Anneli Lax New Mathematical Library) by Victor J. Katz on 4 pages
- A History of Analysis (History of Mathematics, V. 24) by Hans Niels Jahnke on 4 pages
- Modern Algebra: An Introduction by John R. Durbin in Front Matter, page 51, and page 136
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