my history of mathematics teacher at UGA has called this a definitive work. I ordered it as a supplement to the class... and from my reading of it, I can put my stamp of approval on it. It's good--mathematical but also historical; If it's not as delicious prose-wise as most history we have to forgive him. Those are not easy fields to try to shuttle between.
I will say that you should not expect a deep treatment of the math. If you are interested in something like 'the ontological evolution of the western idea of number' this is not a good place to look; if you want to watch calculus fall with a thud out of the churning events of the seventeenth century, practically pristine, then Kline will take you there and the ride is smooth and scenic.

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!

I found the book in college library. It is the best one on math history I have read.

I agree with the above review and would simply like to add my own thoughts. The book illustrates the fascinating way in which mathematics, society, religion, politics and of course physics have affected each other (it goes both ways!) through out the ages. Furthermore, the author nicely illustrates the processes by which people think and how those processes have also changed through the ages (i.e., The Age of Reason versus The Renisance). This book left me with real insights as to the nature and limitations of the current state of mathematics and physics. Things are not as they seem, my friend! Lastly, the author displays an appreciation for the humor and irony of the history which makes this book hard to put down at times. I never thought a math/history book could be a "page turner"... Read it.

In most mathematics classes, students are presented with a completed edifice, and given a floor plan to help them navigate the halls. While this approach works for many people, others need a little more basic information. In this book, Morris Kline builds the building, starting with the mud and straw of the bricks.

"Mathematics in Western Culture" shows that the history of mathematics is one of hundreds of years of people sitting in the sand, drawing shapes and lines, scratching their heads, and trying to figure things out. This is not necessarily Dr. Kline's intention for the book, but this is certainly one of the many messages to be derived from it.

A fascinating, exciting book which makes mathematics more understandable and accessible.

This book did my heart to shake. The emphasis in this book is on undergraduate mathematics education. What is mathematics ? It is a major branch of our culture,the backbone of scientific civilization,and the basis of our technology and insurance structures. Its value to the social sciences,biology,and medicine is also by no means negligible. Morris kline is speaking clearly why professor can't teach a student. I thank to God about this book.

I wish to point out first the positive aspects of the book. First of all, it should be noted that Morris Kline is one of the greatest mathematicians and now discusses a very important philosophical issue that is pertinent today.

Kline shows a great insight concerning the history of the development of mathematics, a recount of the problems that different mathematicians had throughout history, the way they pretended to solve the problem, their logical and illogical reasons for doing so. He at least defends himself very well looking to history to prove how uncertain mathematics is.

However, his book lives up according to a fallacy. Let's say that somebody thinks that certainty depends on a property "F" characteristic of some "a" mathematical system. Then the fact that up to that point it was believed by many people that F(a), then mathematics was certain, while when they discovered that it was not the case that F(a) then certainty of mathematics can no longer be established. An analogy with science will make clear the fallacy. Galileo insisted that the certainty of science on the universe depended greatly on the fact that the planets and stars moved in perfect circular orbits; Kepler on the other hand proved that the planets move in eliptical orbits. It would be an exaggeration to think, that the certainty of science is lost just because planets move in eliptical orbits.

Another problem is that he states that mathematics is also uncertain because the irrational reasons to admit certain mathematical entities or axioms. However, the *validity* of the axioms is what is at stake in mathematics, not the subjective reasons that somebody had to admit them. An analogy again with science can show this second fallacy. Some of the reasons Copernicus admited that the Sun was the center and not the Earth, was because the Sun was the noblest star, and because it would restore the perfection of the circles in which planets revolve, because it had been lost in the Ptolemaic geocentric view of the universe. Do these reason should really dismiss the validity of Copernicus' theory? No. The same happens with mathematics. The illogical reasons that somebody might have to discover something, is irrelevant concerning the validity and certainty of mathematics.

Also, there is the fallacy that because that there is a development of mathematics in one area that seems to be unorthodox at some moment, might compromise the certainty of mathematics. For example, he uses the development of "strange" algebras or "strange" geometries as examples of this. Non-Euclidean geometry doesn't invalidate Euclidean geometry, as Morris seems to suggest, nor does imply the loss of certainty of Euclidean geometry. It only means that Euclidean geometry is one of infinite possible mathematical spaces. Certainty is guaranteed in each one of them.

Also, he seems to use the word "disaster" concerning Godel's theorems. But it was a "disaster" only to *some* philosophical schools. Godel's theorems doesn't seem at all to imply the uncertainty of mathematics, since Godel himself believed in its certainty during his entire life. In fact, Platonist propoposals such as Husserl's, though Edmund Husserl posited the completeness of mathematics, his main philosophy of mathematics is supported *even after* Godel's discovery. The only thing refuted in his philosophy is the completeness of mathematics, but not his mathematical realism, nor his account of mathemathical certainty. Interestingly, Husserl is never mentioned in the book (just as many philosophers of mathematics ignore his philosophy).

Though the book is certainly instructive and Morris shows his knowledge of history of mathematics, due to these fallacies, he never proves his case.

I want to start by saying that I agree with all of the positive reviews of Morris Kline's book (from what I can tell, all but one person gave this book high marks). Morris Kline is indeed a great mathematician as well as a great writer/expositer of his chosen field. This book -- which explores the philosophical ramifications of mathematics via its history and argues for the essentially uncertain nature of mathematics -- is definitely a great book for math fans out there.

What has motivated my review is the one negative review by Sr. Barbosa of Puerto Rico (or so he claims). It is a sad but true fact that people who give their opinions on the web do not always give fair and reasonable opinions and/or are motivated by ulterior motives. Sr. Barbosa's review seems to fall into that category.

First, Barbosa attacks Morris Kline (he's got some nerve doing that) for Prof. Kline's supposed lack of understanding of mathematics. This frivolous insult is so ridiculous that it isn't necessary to discuss it further.

The negative review further contends that mathematics really is not uncertain. Sr. Barbosa, in that line of thought, also says that Kurt Godel didn't really believe in his own famous theorem! (Or at least that's the only way one can interpret Sr. Barbosa's statements.)

Even layman that are familiar with popular works on mathematics -- *Godel, Escher, Bach*, *Godel's Proof*, etc. -- realize that mathematics as a formal, axiomatic system has been PROVEN (for all time) to be incomplete and inconsistent ... i.e., "uncertain." These ideas have been further amplified by the works of Alan Turing and Alonzo Church (the Halting Problem) as well as Gregory Chaitin (Algorithmic Information Theory -- along with Andrei Kolmogorov and Raymond Solomonoff). In fact, Chaitin has proven that the natural number system -- ie, the counting numbers (1,2,3,...) -- is itself random (i.e., uncertain).

If that was not enough evidence in favor of Morris Kline (and contra Sr. Barbosa), then consider quantum physics and chaos theory. Both of those fields add further fuel to the idea that nature itself is uncertain. If nature is uncertain, then why shouldn't math (which often elegantly represents nature) be uncertain? Sr. Barbosa winds up looking foolish for arguing that Copernicus and other great thinkers of physics can be used to support Sr. Barbosa's views. On the contrary, physics seems to support Morris Kline.

In short, Morris Kline's book does a valuable service by looking at how mathematics has hisorically developed in an uncertain manner in order to further highlight the uncertainity in mathematics that has been logically PROVEN by others. Shame on Sr. Barbosa and others who constantly write misleading, unfair, and irrational reviews of books that can lead customers astray and unfairly malign quality work.

I wouldn't normally write a review of any book, but this book is really good (read the other reviews if you don't believe me), and I felt I had to write something. I highly recommend it for anyone who has ever wondered about the nature of mathematics.

I have always been fascinated by mathematics, but doubts started creeping into my mind about it when I was taught about the calculus, and all of a sudden, I began to question whether this was reality I was being taught, or just some convenient invention. After all, zero divided by zero doesn't make sense, and the idea of the "ultimate limit" seemed to be a trick, or dangerously close the Infinite, which is isn't much easier to swallow either.... Many years of engineering didn't make me feel any more comfortable, although clearly, it worked!

On reading this book, to my surprise (and somewhat to my consolation), I realized that even the great Newton and Leibniz did not justify their thoughts on this in a totally logical way, even though they helped to invent it.

Which makes you wonder...why does the physical world seem to follow mathematical patterns (or does it really...)? And did the thinkers justify their "laws" of mathematics and establish them beyond any doubt? Did "constructive intuition", whatever that might be, play the most important role in the creation of mathematics?

You may not get all the answers to these questions in this book (you won't get it in any other book this side of the universe), but you will certainly get a very thorough, deep and entertaining discussion these and many other questions you may not even have thought of. It is almost like being in a room with all these historical figures and listening to them arguing it out!

Best part is, the book is quite cheap! You'll like it!

This is by far my favorite calculus text. The selection out there ranges from cookbook (Stewart and Anton), to elementary (Adams), to quite advanced (Apostol and Spivak). This book really doesn't fall into any of those categories. Proofs are based on heuristic arguments rather than strict adherence to rigor, but this doesn't mean that the book is "dumbed down." Most people go through proofs in Apostol and wonder what they just read, whereas those in Kline greatly enhance the reader's ability to learn the material. Kline may sacrifice rigorous formalism for increased understanding, but most students of calculus will consider this a very good trade off. If you are looking for a theorem-proof, theorem-proof, ad infinitum treatment of calculus, this is probably not your book. If you are looking for a way to really learn the subject from a very gifted teacher, developing your mathematical and physical intuition in the process, then Kline is the best text you can get.

I've taken math through Calc through high school and college. No book has done a better job of explaining how it works than this one. All math books should be written like this one. Clear and practical. If there's something you need to know, he tells you exactly what it is. Such as, if you need to know something from trigonometry, he explains it well enough that you will either understand it or know what to look up in another book.

However, do not think this book needs other books in order to get through it. You should have already had trig and algebra and geometry. But even if you've forgotten it all, this book will get you through to the end.

This book doesn't have all those eye catching color visuals, but what makes this book great is its content and way of explaining concepts of calculus. There's a lot of reading which might seem cumbersome at first glance, but ultimately it's worth every second spent on reading. In many calculus book I've seen, there's a theorem, proof of theorem sometime, few examples and then exercise. This book more emphasizes more on discussing theorems and different concepts, proofs of theorems and why, how they work. The book is not very rich in examples, but once you get the concept, you'll be able to do any example. Also the examples that are given, explained thoroughly. So except for the quantity of examples, I have nothing else to complain. Thanks to Doverpublications for making such book affordable ...

I cannot imagine why someone of Morris Kline's stature would allow a publisher to print his work in print so small that nobody but an 18-year old can see to read it. (I'm 62). It was published by Dover Books and they should know better!! Nevertheless, Dr. Kline let them get away with it and there's no chance of me selling my copy back, not with 24 copies available used. Wonder if that's because nobody can see the print? -Dr. William E. Chauncey

Kline, a noted historian and educator of mathematics, wrote a book that stands the test of time. This isn't of much use to anyone with high-school math who doesn't care to know why math is the way it is. For everyone else, this is a good book. Solutions to problems at the end of the book are very handy. I recommend this book along with Timothy Gowers's "Mathematics: A Very Short Introduction".

A must have for the mathmatically curious. The subject seaquence is laid out in a logical order. Beginning with the premises of inductive vs. deductive reasoning, basic algebra, geometry, and the Calculus. This is not a good book for becoming proficient in sepcific areas of mathematics, but offered for me at least, a logical reference point for approaching the core sujects. I highly recommend this book for self-study.

Kline's book is a surprisingly accessible history of math, equations and all. It's a perfect balance for those who know their history and want to know more about the math behind scientific ideas, and those who know their math and want to delve into the practical applications of mathematical ideas.

I loved the examination of how Eratosthenes calculated the circumference of the earth pretty accurately for a guy without so much as a telescope. I had terrific "aha" experiences reading the book, and the students I tutor in math are getting a bunch of interesting background stories with their algebra lessons.

Galileo's and Newton's calculations are a great way for the theoretically inclined to get their feet wet in physics. For all of us mathematicians who actually thought math developed in a vacuum, this book has excellent examples of how inextricably linked math, science and history are.

What a journey! This book will never age with time. A must read for those interested in the humanistic value of a subject concider cold and forbiding by some who are disallusioned about what mathematics really is and its purpose in the history of mankind. A book that could only have been written by Morris Kline,an educator who saw the beauty of the subject. I can say no more.

This book is geared to the general reader who has a cursory knowledge of mathematics. The chapters are organized around physical phenomena and the math behind their explanation. The result is a charming and VERY useful book. I have the 1970 edition which is quite worn from frequent use. The chapter titled, Differential Equations - The Heart Of Analysis, is exceptionally beautiful and pertinent. Reading this book is akin to a treasure hunt. There is page after page of mathematical discovery. Reading the chapter on Motion Of Projectiles made me terribly angry at the banal way in which this topic is handled in high school texts. Things such as quadratic equations and the law of gravitation are explained very well. I sincerely believe that this book should be a required text for High School math students. Highly recommended. The Dover edition is very affordable so even if it means foregoing a meal, do it. Buy this book! Well worth your time.

By reading this short book, you will absorb a good foundation in both Mathematics and Physics. You will also acquire an infinite respect for Newton, Maxwell, and Einstein. In all cases, these geniuses developed theories regarding natural phenomena that often could not so readily be observed (if at all).

Morris Kline's thesis in this extraordinary book is stated clearly in the final words of his preface, and then presented through a historical survey throughout. Here are the key words, "Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it."

As far as I'm concerned, Kline makes his case. And I am one of those who received the erroneous impression in school that he mentions -- of course, I never managed to pay much attention in math classes, but that was only partially my fault.

If you are at all like me, and suspect you might have missed something in your misspent youth, get this book.

I'm sorry to see that this book is now available only through out of print searches, but it's abridged version, "Mathematics for the Nonmathematician" by Morris Kline, is still in print and available from Dover publishing. One can see mathematics through the lens of the search for truth and meaning in the world in this book. It explains how the Greeks sought to understand nature through mathematics and how later developments in math led to people's dominance over nature. It also contains Francis Bacon's warning that one must obey nature to make use of it. It relates math to painting, music and philosophy. I can't say enough about how wonderful it is to understand math in this way.

It's only drawback is its dated attitudes about society and "barbarian" cultures. It doesn't explain the difference between "reason" and "empirical knowledge" though it mentions these as proof that modern ideas are "better" than old ones. The abridgement was published in 1967, so you can imagine how 50's the attitudes are. But if you can get past some minor rubs, it's so liberating to know how math developed. It helps to explain the why in math that goes unanswered so often.

I didn't like it. No siree, Bob. I didn't like it at all. Don't buy it. Just Trust Me.

When I was at Hi-School, here in Spain, in the United States of Europe, our teacher used to mention this book to us lots of times. Later on, when at University, I left no stone unturned at several University libraries till I found a copy of the book, a Spanish edition. I read the book just in one session, due to how interesting I found it. Professor Morris Kline (he taught at New York University, I think I remember)shows throughout the book the rare ability to absolutely master every mathematical concept he talks about and at the same time being able to see those concepts with fresh eyes, as if they were new for him, as if he himself were a teenager encountering those ideas for the first time and trying to come to grips with them. True that modern math is far more abstract and powerful than what was the knowledge body of mathematics say in the 17th century for instance. But at the same time, emphasis in detaching ideas from any connections with the physical world, abstracting as an end per se, and letting 'rigorousness' and formalism prevail over intuition, has led some areas of modern math to something which looks like an esoteric exercice consisting of sipping through the symbols in a book. Kline makes for instance the point that too much insistence on 'rigorousness' is equivalent to finding snakes under the jewels, and says that when a mathematician does not care any more or is not sensible to problems such as the movements of planets around a star, the behaviour of density waves (sound) in a cavity, the movement of a mass hanging from a spring, etc., then mathematics are close to over. He says, for instance and amongst many other things, that when mathematics -within a determined mathematical field- go too far away from the physical concepts that may have inspired them and grow out of themselves for too long, then only the fact of that task being in the hands of men with an extremely developed intuition may prevent that mathematical field from finally becoming barren. I found it very interesting how many hypocrisies Morries Kline pointed out in the modern style of teaching mathematics. Some teachers use intuitive, supposedly 'non-rigorous' methods to work out things IN their heads, but try to transmit it to the students in a modern, 'rigorous', and absolutely non-pedagogical way. A double language that usually will have coupled with it an equally twofold moral -hypocrisy-. They think clear in their heads but offer muddy explanations to the students, demoralizing them and making it look as if perfectly assembled and refined mathematical ideas were actually coming out of their heads instantly. It is all vane presumption that does not good to actual teaching. The final victims are the students. This may be one of the most important factors in accounting for the superiority of Japanese students over American ones in several math contests and comparisons held through the years. I too think that intuition is paramount in math and that 'rigorousness' is secondary when confronted to that. Think of Thales from Miletus, Aristarchus, Archimedes, etc. They lacked a good mathematical formalism, they did not even have a positional number system, and even so they worked wonders (Archimedes' laws of statics, mechanics, etc.) and realized by pure intuition things that were only rediscovered more than two thousand years later (say for instance that the planets all revolve round the sun, that stars are other, distant suns, etc.). Nowadays, on the contrary, we seem to have too much formalism, even loads of it, but not that great intuition of the ancient scientists. Sometimes I wonder what Newton or Archimedes would be able to do if they had a period of updating, a Pentium and some other of the resources we have today. Ha ha ha! Kurt Gödel also said once that whereas along the last three centuries abstract mathematics have experienced huge progress, the solution of complicated numerical problems that can be stated in a handful of symbols of elementary arithmetics (consider for example Goldbach's conjecture, or the now 'solved' Fermat's Last theorem)is very backward. This too comes to support the comments made above. I definitely think that Mr Kline's criticisms must be taken very seriously, at least for the sake of giving children a better, or at least decent, mathematical upbringing, instead of killing vocations even before they are born. He speaks along the book with astounding clarity and honesty. Somehow he looks like the character in that telltale, saying that the king is naked whilst almost everybody else -blind people obviously would not see it- too realizes it but does not dare claiming it out 'because no one else does', 'because it goes against tradition', out of cowardice or due to any other obscure reasons. For the good of future generations, for a better education, this book should be read by every physicist and specially by every mathematician, let alone those mathematicians who also have to 'teach' their discipline. I would have given this book six stars if it were possible. Pity that it is currently out of print. I went trough hell to find a copy, but all the pains were worthy by far. Definitely a must for any mathematics library.