The Mathematical Experience
by Phillip J. Davis, Reuben Hersh
Average Customer Review:
Paperback (14 January, 1999)
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Editorial Review

We tend to think of mathematics as uniquely rigorous, and of mathematicians as supremely smart. In his introduction to The Mathematical Experience, Gian-Carlo Rota notes that instead, "a mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof ... is more often than not a way of making sure that our minds are not playing tricks." Philip Davis and Reuben Hersh discuss everything from the nature of proof to the Euclid myth, and mathematical aesthetics to non-Cantorian set theory. They make a convincing case for the idea that mathematics is not about eternal reality, but comprises "true facts about imaginary objects" and belongs among the human sciences. ... Read more

Reviews (13)

Informative and engaging
The authors deal with various important aspects of mathematics and about practising mathematics. They also deal with the philosophy of mathematics. By and large, they do it engagingly. Specifically, they tackle why mathematics seems to 'work'; how a mathematician actually goes about doing mathematics; they offer some light treatment of a few mathematical topics, and they illustrate mathematical thinking as well.

This book is best read by students thinking about choosing mathematics as a career, or even just as a field of study. Although, any layperson will come off with a greater appreciation of what mathematics is, and what mathematicians do.

Philosophy, History and Myths of Mathematics
The Mathematical Experience by Philip J. Davis and Reuben Hersh
1981 Houghton Mifflin Company, Boston

Is all of pure mathematics a meaningless game? What are the contradictions that upset the very foundations of mathematics? If a can of tuna cost $1.05 how much does two cans of tuna cost (Pg. 71)?If you think you know the answer, don't be so sure.How old are the oldest mathematical tables? What is mathematics anyway, and why does it work?Can anyone prove that 1 + 1 = 2?
This is a book about the history and philosophy of mathematics. I'm certainly not a mathematician, and there are parts of the book I will never understand, yet the balance of it made the experience well worth while.The authors presented the material so that it is interesting and (mostly) easily understood.They have a creative way of making a difficult subject exciting. They do this by giving us insights into how mathematicians work and create.They live up to the title making mathematics a human experience by adding fascinating history.Frankly I was shocked when they pointing out how even mathematicians have made questionable assumptions and taken some basic "truths" on faith.They show the beauty of math in the "Aesthetic Component" chapter. Ultimately the question that comes up again and again is the question of whether or not we can really know anything about time and space independent of our own experience to make an adequate foundation for a complete system in mathematics. If you have ever wondered about the world of mathematics and the personalities involved you might consider this book.If you are a mathematics teacher you should read this book. If you are a mathematician you could find it quite unsettling.
It contains eight chapters, each one broken up into many subtitles so if you do get bogged down in the mathematics it isn't for long. There are 440 pages.I'd like to see a much more complete glossary for people like me who need it.

Immerse yourself.
Back in the early 90's when I was an almost-penniless mathematics student I was standing in front of a bookshelf in my local bookstore and had to choose between this and Gödel, Escher, Bach. I chose this book and I still don't regret it. [I have also subsequently bought GEB :-)]
Driven by their obvious love of the subject, the authors do a credible job of tackling just what it is about mathematics that makes mathematicians love it so much, often to the bafflement of the rest of the world. A particular personal favourite is the series of four conversations between an "ideal mathematician" and, respectively, a University Public Information Officer, a philosophystudent, a positive philosopher and a sceptical classicist.
I would recommend this book to students of mathematics at any level beyond the elementary, especially those with an interest in the foundations of their subject. The authors do however acknowledge that some parts of the book will seem alien to the layman. ... Read more

Isbn: 0395929687
Subjects:  1. History    2. History & Philosophy    3. Mathematics    4. Mathematics (General)    5. Philosophy    6. Science/Mathematics    7. Study and teaching    8. Mathematics / History   

The Heart of Mathematics: An Invitation to Effective Thinking
by Edward B. Burger, Michael Starbird
Average Customer Review:
Hardcover (01 April, 2000)
list price: $69.95
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Reviews (10)

Math Appreciation, not computation or rigor
_The_Heart_Of_Mathematics_ was never intended as a traditional textbook to teach you how to calculate.If that is what you are looking for, you need a diferent book.Its value -- and this is the best book of its kind that I have found -- is in helping the reader gain an appreciation for mathematics.Its title could well have been _Math_Appreciation_ .It was most likely intended as a way of satisfying the "math requirement" for non-math majors who feel allergic to math.

I have read comments from several people debating the merits of this book.Perhaps it would help to use inject an analogy into the conversation.Suppose you wanted to learn (or teach) music.One teacher chooses to teach her students how to play the piano;
another has her students listen to CDs of great performances; another teaches his students how to read music; and another teaches the biographies of Beethoven and Mozart.Which of these teachers is right?Which kind of music do you want to learn?

The question itself is mistaken, if you think that it has exactly one correct answer.The best answer is: ALL OF THEM.The problem here is not in what one of these approaches will teach, but in what it omits.

Now, translating back from the metaphor: I want my children to learn how to compute AND how to love math.Which is right?Both of them.

This book shows you how to have fun with math.If you or your students end up learning something, and wanting to learn more -- that's the idea.

Learning To Think Mathematically
This is an excellent textbook in that its primary emphasis is on some of the great ideas in mathematics and effective thinking. Topics covered are numbers, infinity, geometry, topology, chaos and fractals, uncertainty, and decision making. The text is replete with myriad illustrations which assist the student in grasping key ideas. Text would be excellent for a liberal arts math course, an enrichment course for middle school and high school teachers, or as a course for advanced high school students. Comes with 3D glasses and exploration CD.
In summary, conveys the beauty of mathematics while teaching students to think.

Great adult self ed resource
The disturbing reviews indeed completely miss the point. The goal of this book is not to turn you into a mathematician. It is to help you appreciate what mathematics is.

I am planning on using this text for an adult self ed study group this fall. The goal is not to try to prove Cantor's method. You explore it and gain some understanding, but it isn't a mastery course that you come out of passing a test for, unless you are sitting in a classroom designed with that in mind, and the larger audience for this book is not in that narrow context. If you come out of it learning how to think mathematically, learning different ways to approach solving problems, learning that there is fun, beauty, art, order and sense to math, if you begin to *see* math in the world you live in, in nature, in ways you never noticed before - that is the goal. It is also threaded with history and the human drama that created math.

Both negative reviews were so poorly written and clearly missed the point that I dismissed them, but others I've recommended the book to have been confused, so I felt the need to respond.

I have also watched the video/DVD series these two authors put out through the Teaching Company, the Joy of Thinking, and I love what they are doing. Is every lecture perfect and resonating with everybody? No, but most resonate with most people. It certainly opened my eyes to things I never understood. Much of this book covers the same type of material.

Some people will find it more interesting than others, that is the nature of personal preference certainly. But the negative feedback indicates the book is flawed based on specific use in college classroom context, and it appears the reviewers did not understand the purpose of the book.

The four vs. five stars reflects the fact this is a first ed and could be just little more user friendly for lay people vs. college course users. I look forward to seeing the 2nd edition.

... Read more

Isbn: 1559534079
Sales Rank: 19832
Subjects:  1. General    2. Mathematics    3. Mathematics (General)    4. Science/Mathematics   

What Is Mathematics, Really?
by Reuben Hersh
Average Customer Review:
Paperback (01 July, 1999)
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Editorial Review

In What Is Mathematics, Really?, author Reuben Hersh proposes a philosophy of mathematics that he calls "humanism" and uses this philosophy to analyze age-old questions of proof, certainty, and invention versus discovery. He also surveys the history of the philosophy of math. Readers of all levels of mathematical experience will be stimulated by the fascinating and perspicacious discussions Hersh has to offer. ... Read more

Reviews (5)

At times full of empty rhetoric
This book is interesting, and has made me think hard about my own views. Mostly that is because I disagree with a lot of the beliefs touted by Hersh.

Hersh starts out with an approach to the hypercube, always a fascinating topic for me. Pages later he is trying, very unsuccessfully though he doesn't realize, to decimate the "old fashioned" views about numbers, physical objects, social conventions, and basically everything Intelligible. His arguments are so terrible they basically ruin the book. He uses every form of slimy rhetoric to be convincing. Read these pages with an open eye and you'll see how often he plays on the reader's biases and preconceptions to make his point seem clear.

For the rest, the book is quite inclusive. A lot of this is interesting. But all of these ideas have existed without the help of Hersh, who would seem to have problem accepting this fact unless we reduced the timespan of their existence to include Hersh's closest biological relatives, the rest of us humans.

A choppy rough draft in philosophy of mathematics
This book comes across as some kind of extended constructivist/pragmatist complaint. Disjointed in its execution, it gives the appearance of a bunch of lectures too-quickly thrown together. Some weak arguments appear here and there, a few even coming across as downright silly. Perhaps its because Hersh has a simplistic, even at times sophomoric understanding of philosophy. He also has the lazy-man's habit of quoting huge tracts of other peoples writings without giving any sort of application or interpretation. On the up side, the book does have an encyclopedic breadth, so it's not a complete waste of time, even given its weaknesses. I took down several references. Did I like the book? Yes. Hersh should have retained an editor, or perhaps spent another year tidying it up. One more thing: Hersh is very anti-theistic. He downgrades Platonism on the basis that nobody believes in God anymore. He really should get out more, or at least read some sociology. The vast majority of the human race and even westerners believe in God. Hence, maybe Platonism in mathematics isn't so crazy after all.

Really philosophy of mathematics
The book offers the best kind of live, seriously thought out, philosophy of mathematics--in real contact with mathematical practice and teaching. Hersh writes from a deep love of mathematics and a deep concern to make it accessible to others, and for him both of those motivate philosophic reflection on the nature of mathematics.

Hersh notes that mathematics is a social enterprise. People may pursue it alone in their rooms, and even do the greatest thinking that way (as Andrew Wiles did some great thinking in near secrecy on the way to proving the Fermat theorem). But what they think about is not their sole creation (witness the many enthusiastic citations Wiles gives to what he owes others). What we call "proofs" in actual practice are not complete deductions in formal logic, nor simply "whatever persuades you". They are reasonings that live up to a socially recognized standard.

Hersh believes, and argues, that students who understand the social nature of mathematics will approach it with more interest and less fear than those who think it is inhuman perfection. Actually, I think he is wrong about that. Students today generally believe literature is a social product, but they still too often think that "getting it" is an arcane and uninteresting skill of English teachers. But Hersh's view deserves careful consideration and you can learn from him whether you agree in the end or not.

I will also say that Hersh's descriptions of earlier philosophies of mathematics are not always historically very accurate. And though he has genuine concern to give sympathetic accounts of them (before giving his own refutation) he does not always succeed. But neither are his versions notably worse than the versions in other similar books. For accurate accounts of Plato or the 20th century giants Poincare, Hilbert, Brouwer, and so on, you have just got to read the originals.

Anyone interested in philosophic thought about math, and not just solutions to one or another specific technical problem in the philosophy of math, should read this book. But don't only read this one. ... Read more

Isbn: 0195130871
Subjects:  1. History & Philosophy    2. Mathematics    3. Philosophy & Social Aspects    4. Science/Mathematics    5. Study & Teaching   

Mathematics: The Loss of Certainty (Galaxy Books)
by Morris Kline
Average Customer Review:
Paperback (01 May, 1982)
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Reviews (12)

Great book by a great author
English:
This book isn't meant to be a mathematics book, still it offers a very good qualitative view of the problems it describes - at least as long as the reader has a not-zero competence in mathematics.
Don't forget what Kant wrote, in the introduction of his masterpiece "Critique of Pure Reason" i.e. "that many a book would have been much clearer if it had not made such an effort to be clear": there are topics that can't be explained in "too simple words".
There are a lot of divulging books that are not clear for competent reader and seem to be clear forinadequate readers: this is not the case of Kline books, which provides a interesting reading for an interested reader.
Italiano
Questo libro non intende essere un testo di matematica, ciò nonostante, offre un'ottima visione qualitativa dei temi che tratta - almeno se il lettore ha unacompetenza non nulla in matematica.
Non si dimentichi quello che Kant scrisse nel suo capolavoro "la critica della ragion pura", ovvero "molti libri sarebbero stati molto più chiari se non avessero voluto essere così chiari": ci sono argomenti che non possono essere spiegati in "termini troppo semplici".
Esistono molti testi divulgativi che non sono per nulli chiari per il lettore competente, e sembrano essere chiari per il lettore inadeguato: non è questo il caso del libro di Kline, che offre una lettura interessante per un lettore interessato.

Mathematical Uncertainty
A delightful and important book for all math enthusiasts.A must read for budding mathematicians.

This book authoritatively chronicles the gradual realization that mathematics is not the exploration of hard edged objective reality or the discovery of universal certainties, but is more akin to music or story telling or any of a number of very human activities.

Kline is no sideline popularizer bent on de-throwning our intellectual heros - he speaks knowledgeably from within the discipline of mathematics, revealing the evolution of mathematical thought from "If this is real, why are there so many paradoxes and seeming inconsistencies?" to "If this is just something people do, why is it so damned powerful?"

What is certainty, and how is it lost?
Clearly Morris Kline is an historical master, and his retelling of the story of mathematic is lush and partially rewarding. Initially, the book seems pitched toward the non-specialist, but as one goes deeper into Kline's story of mathematics, the more esoteric it becomes. Not having had math of any sort since high school, I found the story riveting and confirming of many inchoate intuitions, especially with regard to the rather counter-intuitive and historically controversial status of mathematical entities like irrational numbers, negative numbers, complex numbers, and infinitesimals, etc. However, the book becomes quickly inaccessible to the non-specialist after the introductory chapters, despite a valiant effort by the author. Moreover, the book was somewhat unconvincing in demonstrating its grandiose claims of the sort that (paraphrasing) there is no truth in mathematics, or that there is no justification for calculus or no factual evidence that supports the calculus, etc. His notions of truth, justification, evidence, and certainty seem reductively dependent upon the rather limited epistemic portrait of certainty to be gotten from ancient Greek ideals of unassailable first principles and the deductions gotten from them. Epistemology itself has an evolving story that must be taken into account, and the epistemic notions employed by Kline(i.e. truth, evidence, etc.) would have also benifited from historical elaboration since they are central to Kline's evaluations of mathematics at every point. Since we are not ancient Greeks beholden to these limited epistemic ideals, there is only a loss of certainty for us to the extent that we adhere blindly to self-evident first principles and deduction as the only norms that could confer epistemic values like certainty. Kline is persuasive in his arguments that alternate algebras and geometries are possible and useful, and one can hardly doubt that such alternatives lend themselves to a degree of modesty and potential relativity in mathematical claims to knowledge. Once one begins to admit other epistemic norms (like adequacy to empirical reality/experience or applicability to future problems, etc.) into the picture, one wonders if all the alternate algebras and geometries remain on equal footing. In any event, a degree of relativity in mathematical description and expression is not incompatible with a modest realism in mathematics. Nor are self-evidence and deduction the only norms for rationality, justification, evidence, warrant, or certainty. So long as one is cautious about the epistemic premises upon which the loss of certainty is predicated by Kline, the book is a thought provoking read! ... Read more

Isbn: 0195030850
Sales Rank: 66275
Subjects:  1. History    2. History & Philosophy    3. Mathematics   

Mathematics : The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library)
by Keith J. Devlin, Keith Devlin
Average Customer Review:
Paperback (01 February, 1997)
list price: $19.95
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Reviews (5)

OK
This book has a lot of good information along with some excellent illustrations.

It is undoubtedly a very good book for one who is a great lover of mathematics.It is not, however, a book that is difficult to put down.It did not captivate me and convince me about all I am missing by not being a math junkie.

A pleasure to read
The book is fun to read ,very informative and very clear.
It doesnt matter if you know some university mathematics or not ,anyhow you will find this book a pleasure to read ,and if you dont know nothing about mathematics it will change your perspective on the world. I suggest you will get a copy and read it.Excellent book!.

Imaginative, Engaging,Fascinating, Delightul
This marvelous book to explains to non-mathematicians the joy, beauty and power of mathematics. Each topic is presented in an original manner with alot of colorful illustrations to delight the eye and mind.Devlin shows how mathematical thinking is critical to our exploration of the world around us. This is one my top ten of all time list ... Read more

Isbn: 0716760223
Sales Rank: 449622
Subjects:  1. General    2. Logic    3. Mathematics    4. Philosophy Of Mathematics    5. Philosophy Of Science    6. Science/Mathematics   

Mathematics and the Search for Knowledge
by Morris Kline
Average Customer Review:
Paperback (01 November, 1986)
list price: $18.95 -- our price: $18.95
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Reviews (3)

A fine introduction, but...
This is a good, basic introduction to the history of math, but its promise of engaging philosophical issues falls short.It's not until Chapter XII that Kline really engages the philsophy of mathematics, but then attempts to cover a lot of ground in a short span, and the distinctions and groupings Kline forms of different thinkers' conceptions of mathematics leads to strange bedfellows; particularly Kline's claim that Wittgenstein is basically a mathematical empiricist and yet believes that mathematics is a human creation.To my mind there's an inherent contradiction there.Most bothersome, however is the almost entire lack of proper footnotes.Some of the most interesting quotes will be difficult (if not impossible) to find b/c Kline only tells us the author of the quote and oftentimes doesn't even include the author in the bibliography!In particular I'm thinking of a fascinating quote from Evariste Galois - Kline gives no indication of where I can find it!Still, I have to give this book a 4 just b/c it's a good introduction that reads rather quickly.

An excellent history of both Mathematics and Physics
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).

Excellent, esp. for non-mathematicians (like me!)
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 impressionstudents acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected aboutnotable phenomena and in some instances even contradicts perception.It isthe essence of our knowledge of the physical world.It not only transcendsperception but outclasses it."

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

If you are at alllike me, and suspect you might have missed something in your misspentyouth, get this book. ... Read more

Isbn: 0195042301
Sales Rank: 500586
Subjects:  1. Applied    2. History & Philosophy    3. Mathematics    4. Science/Mathematics    5. PHYSICS   

When Least Is Best : How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible
by Paul J. Nahin
Average Customer Review:
Hardcover (24 November, 2003)
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Reviews (2)

excellent - I want all his books
Finally, a solid book that challenges the lay reader just like the best math teachers do - by showing the elegance and power of mathematical reasoning.

This is top shelf material. Nahin is one heck of writer and must be one hell of a teacher! Bravo!

Already ordered his book on the history of imaginary numbers.

6 stars: ******

Off the Charts
Nahin's book is a tour de force about the deep intellectual threads that surround the notion of optimality.In physics, engineering, and mathematics, while touching on a wide range of applications, he asks over and over again:What is the optimal solution and why does it matter?Since I've spent most of my professional career thinking about optimality in one form or another, I was skeptical about how much new I would find in this book.But I was astounded to find something new and interesting on virtually every page.Some examples:

--Preface:Torricelli's funnel, which has finite volume and can be filled, but has infinite surface area and cannot be painted; and a slick proof that an irrational number raised to an irrational power can be rational.

--Chapter 1:An optimization problem that is not amenable to calculus, but whose solution can be discerned by some clever insight;an optimization problem that is amenable to calculus, but whose solution can be arrived at by algebra; and the use of the arithmetic mean-geometric mean inequality in optimization.

--Chapter 2:The ancient isoperimetric problem of Dido on maximal area, how it remained unsolved until modern times;the fact that there exists a figure in the plane whose area is equal to the area of the period at the end of this sentence and which contains a line segment one million light years in length that can be rotated 360 degrees within the figure (the shape of the figure is a little hard to picture); and the fact that there are two consecutive prime numbers the gap between which is greater than a googolplex (don't ask what they are).

--Chapter 3:Optimization problems involving the viewing of a painting, the rings of Saturn, folding envelopes, carrying a pipe around a corner in a hallway, the maximum height of mud ejected from a wheel, and other daily concerns.

--Chapter 4:Snell's law, the path of light, and the feud between Descartes and Fermat.

--Chapter 5:The power of the calculus, the aiming of basketballs and cannon, Kepler's wine barrel, United Parcel Service package size constraints, L'Hospital's pulley problem, and the geometry of rainbows.

Chapter 6:Galileo's work on the descent of a particle sliding along the arc of a circle; the discovery of the minimum-time brachistochrone curve by Jacob Bernoulli arrived at by an argument based on the path of light in a variable-density medium, his feud with Newton, and Newton's anonymously published solution to the problem; the isochronous property of both the circle and brachistochrone, which states that the descent time is independent of the starting location along the cure (a point mentioned in chapter 96 of Moby Dick and which left me wondering which paths are isochronous since a straight line is clearly not);the fact that the brachistochrone is about 1.5% faster than the circular arc and that a brachistochrone tunnel dug from New York to Los Angeles would entail a travel time of a mere 28 minutes assuming frictionless sliding and no propulsion; the fact that 45 degrees maximizes range of a golf ball but 56.466 degrees maximizes arc length; the Euler-Lagrange equation of the calculus of variations and its proof formulated by Lagrange at age 19;the hyperbolic cosine shape of the catenary loaded by its own weight as compared to the parabolic shape of a string under uniform loading; the rigorous solution of the isoperimetric problem by Weierstrass; and the theory of soap bubble shapes by Plateau who was blinded by an optics experiments he performed during his Ph.D. research; and a brief illustration of optimal control theory

Chapter 7:Hofmann's solution of Steiner's problem on minimum distance inside a triangle and its use by Delta Airlines to save money on its phone bill; the traveling salesman problem, linear programming, a tutorial on dynamic programming along with a brief bio of IEEE Medal of Honor awardee Richard Bellman with emphasis on the fact the IEEE is an engineering society.

For a control audience, the connections between control and optimization are addressed by the lengthy discussion on the calculus of variations and the tutorial on dynamic programming.My only (minor) disappointment was the lack of more discussion about the nature of optimality in mechanics, that is, the least action principle, the specialization of Hamilton's principle to conservative systems.This underlying principle of mechanics is not, in fact, a statement of optimality but rather one of stationarity.

This book is clearly the result of immense effort.The author's notes suggest that most of the book was written in a single year, which is amazing.Not only are many topics covered, but mathematical details abound.The author, who is known for popular treatments of technical subjects (An Imaginary Tale:The Story of i, Dueling Idiots and Other Probability Puzzlers, The Science of Radio, Oliver Heaviside:Sage in Solitude, Time Travel), just seems to get better and better.

The book was produced with painstaking care.While there are surely errors somewhere, I spotted exactly zero.I would guess that the book has roughly half as many figures as pages, all drawn with great accuracy.To say the price of the book is reasonable would be an understatement.

Who might find this book of interest?The book is really a popular book of mathematics that touches on a broad range of mathematical problems associated with optimization.Some mathematical sophistication, and certainly calculus, is needed to follow the details.But much in this book could be digested by students in high school, even before calculus.The flavor and richness of the subject matter cannot help but whet the curiosity of neophytes.Undergraduate and graduate engineering students of all disciplines will find something that relates to their coursework. ... Read more

Isbn: 0691070784
Sales Rank: 57022
Subjects:  1. History    2. History & Philosophy    3. Mathematics    4. Maxima and minima    5. Science/Mathematics    6. Applied Science and Engineering    7. Mathematics / History    8. Physics   

Archimedes : What Did He Do Besides Cry Eureka? (Classroom Resource Material) (Classroom Resource Material)
by Sherman Stein, William Watkins, Jr.,, Andrew Sterrett, Frank A. Farris, Stephen B. Maurer, Julian Fleron, William A. Marion, Sheldon P. Gordon, Edward P. Merkes, Yvette C. Hester
Average Customer Review:
Paperback (15 June, 1999)
list price: $28.95 -- our price: $28.95
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Reviews (2)

Remembering Archimedes for more than his naked stroll
The thought of a man running naked through the streets shouting with joy over a physical and mathematical discovery is one to warm the hearts of all who value knowledge. When Archimedes experienced this flash of joy, little did he know that his actions would become the genesis of a legend that would last for thousands of years. However, he should be remembered for so much more than that and several of his significant mathematical contributions are explored in this book.
It is really amazing to realize how close he was to inventing calculus 22 centuries ago, which was 18 before Newton and Leibniz. With notation that was minimally expressive, he was able to solve problems using a technique that demonstrates at least a rudimentary understanding of the concept of a limit. While many different problems can be solved using calculus, it only takes one breakthrough solution to demonstrate how it can be applied to so many of the others. It can be plausibly argued that algebraic and decimal notations would have been the tools that would have allowed him to overcome those last barriers. One can only speculate on how that would have changed history.
The book is not exhaustive and no attempt is made to make it that. Ten of his most significant discoveries are presented and the solutions are those of Archimedes, although modern notation is used. While the proofs are generally easy to follow, one is often left in awe as to how he thought of how to approach some of these solutions. The explanations are succinct, yet thorough, which is the signature of a solid storyteller.
Given the answers to the question posed in the title of this book, one can pose another that logically follows. Was Archimedes the greatest mind of all time? If the legends are correct, then the answer is probably yes. However, even if the unconfirmed stories are false, the mathematical and mechanical discoveries should make him a legend for more than one short stint of becoming a 'natural man.'

Published in Journal of Recreational Mathematics, reprinted with permission.

Recommended for all mathematicians and scientists
The author's aim is to make what he views "as Archimedes' most mathematically significant discoveries accessible to the busy people of the mathematical community."In this he succeeds admirably.The book isnot only understandable by anyone who "recognizes the equation of aparabola," but is also very well written in a style that brings outthe beauty of the mathematical ideas discussed, as well as the power ofArchimesdes' creativity. As the author points out, the book treats mostof Archimedes' mathematical discoveries.The presentation cleverlyintegrates Archimedes' own writing with the author's modern explanation ofthe ancient discoveries.Frequently, before a main idea is introduced, aquotation fromArchimedes' own writing is presented in which the masterreveals his thinking about what he had accomplished in that particulartopic.

In addition to providing the scientific community with a detailedaccount of Archimedes' main mathematical discoveries and an insight intothe ancient master's thinking, this book, I believe, can be useful in theclassroom in a variety of ways.The most obvious use, of course, would bein designating it as a textbook or a reference in courses on the history ofcalculus or, more generally, on the history of mathematics.But it wouldalso make an excellent textbook for a course on axiomatic mathematics: thebook starts with a few axioms from which Archimedes had developed thetheory of center of gravity and used it throughout a good part of thematerial covered in the book, includingthe development of the volumes ofa paraboloid and a sphere and the theory of floating bodies.

In sum, thisis an excellent book that should be within reach of any person interestedin mathematics or science. ... Read more

Isbn: 0883857189
Sales Rank: 382827
Subjects:  1. Archimedes    2. Biography & Autobiography    3. Biography & Autobiography / Science & Technology    4. Biography/Autobiography    5. General    6. History & Philosophy    7. Mathematics    8. Mathematics (General)    9. Mathematics, Greek    10. Science (General)    11. Scientists - General    12. Biography    13. History    14. History of mathematics    15. Mathematicians    16. Mathematicians And Their Works    17. Mathematics / General    18. Science & Technology   

Mathematics and the Physical World
by Morris Kline
Average Customer Review:
Paperback (01 March, 1981)
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Reviews (5)

Bedtime reading for clever high school students
"Unfortunately the relationship of mathematics to the study of nature is not presented in our dry and technique-soaked textbooks", says Kline in his preface. This is his starting point for presenting mathematics as a product of scientific endeavour. The book covers a whole lot of classical mathematics and physics. It is extremely readable, with lots of pleasant discussions of both the actual mathematics and the history behind it. The prerequisites are virtually none. I would recommend this book to high school students who wish to understand the real reasons why, quoting from the preface again, "cold symbols and formulas" have come to be "regarded as of supreme importance in human thought".

Mathematics and the Physical World by Morris Kline
This work is an excellent reference for the history of
mathematics. It begins describing some ancient numbering
systems. i.e. The Hindus utilized negative numbers. There was
an evolution in geometry. The development and refinement of
curves were set forth into equations.Newton's laws were
formulated . i.e. F= MA
The motion of projectiles evolved into the use of the sine and
cosine to describe curvilinear motion. The laws of gravity,
motion and oscillations were refined further into a multiplicity of uses in mathematics and theoretical physics. Many of the fundamental
laws and processes of the earlier mathematics have evolved into
important applications in theoretical and practical engineering.
Examples are Newton's Laws, the Bernoulli equations and a host
of other scientific achievements.

AJourney InTime
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. ... Read more

Isbn: 0486241041
Sales Rank: 61298
Subjects:  1. General    2. History Of Mathematics    3. Humor    4. Science/Mathematics    5. Science / General   

Sacred Geometry: Philosophy and Practice (Art and Imagination)
by Robert Lawlor
Average Customer Review:
Paperback (01 March, 1989)
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Reviews (10)

An excellent presentation. Glad I bought it
Reading Robert Lawlors book took me out of a classroom and into a discussion of the origins of mathematics. Just enough details to all the material covered, making it a breath of fresh air to others stodgy presentations.

Finish, then publish (please)...
There is a gulf between the promise of this book and what it actually delivers.The idea of exploring the deeper meaning of geometry (there is some wonderful insight here), and the approach of making it a doing experience for the reader are both worthy of praise.Unfortunately, like many books of this type, it is also riddled by mistakes: vague and/or spurious conclusions/interpretations, sometimes confusing layouts and printing, typos, misdirections, and enough false statements and faulty calculations that my trust in this book was steadily replaced by skepticism and annoyance.It seemed as though I was doing as much proofreading as learning.Whether all of the fault lies with the author is a question.Perhaps proofreaders and editors are called upon to check material they don't fully grasp.Or perhaps no one concerned expects us to study these books too carefully.

Essential for the Freemason
This book is essential for the Freemason who wants to go beyond the "lectures". It is a hands on, easy to understand approach to geometry. ... Read more

Isbn: 0500810303
Sales Rank: 37588
Subjects:  1. General    2. Geometry    3. Philosophy    4. Philosophy Of Mathematics    5. Proportion    6. Science/Mathematics   

Trigonometric Delights
by Eli Maor
Average Customer Review:
Hardcover (30 March, 1998)
list price: $40.00
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Reviews (7)

I don't like Tea
Some people might say: "This book wasn't my cup of tea".I suppose I don't like tea then. Maor's book "may" be interesting to the more historically fixated, but being more interested in math, I found this book too light on proof and theory and more of an anecdotal acounting of the lives of mathematicians. If you're like me, you don't care if the Ambasador of Zanzibar created the double-angle equation, you just want the proof; the proof is lacking, therefore so is the book.(My apologies to the Ambasador of Zanzibar, it isn't my intention to implicate you in any double-angle scandal.) I often secretly read math outlines in history class; this is like reading an outline of history in math class. The font was terrific though!

The Good Parts Are Good!
On the whole, this was a pleasant read.I'll try to give a sense of where the highlights are and aren't, since the book could have used some more rigorous editing to make it more uniformly good.

The bits on the early history of trigonometry were fascinating.I particularly appreciated the clear and complete explanations of problems from the Egyptian Rhind papyrus and from cuneiform sources.

Not all of the later historical developments are equally worth our time.The sidebars on Viète, Lissajous, and Landau were particularly good, but the ones on Agnesi and De Moivre didn't add much.(This is unfortunate in the case of De Moivre, but I think a sidebar just can't do justice to so great a mathematician--the fun and beauty is lost when you try to squeeze the highlights together.)I agree with Maor that the big names should not be allowed to slide into oblivion, but in a book like this the subject matter should always pass the stricter test of what intrinsic "delights" it offers.

In this genre, the digressive nature of a "journey of discovery" is part of the appeal.But sometimes the thread connecting the episodes was hard to discern here.Chs. 7-8, 10-12 are tedious and feel like padding compared to the well-sustained interest throughout most of the book.

On the other hand, Ch. 14 ("Imaginary Trigonometry") is a masterpiece.With only a basic knowledge of how complex numbers work, readers can appreciate three beautiful examples of conformal mapping (w=sin z, w=e^z, z=w^2).These mappings are chosen and illustrated to your imagination much better than any of the visual exhibits in a standard applied math textbook like Greenberg's "Advanced Engineering Mathematics."

It's in the nature of such a book that sometimes the key problems presented are solved with the help of something that Maor thinks is too advanced or tedious to present to his audience.The result can be that the story of historical progress is obscured by these "rabbit out of a hat" moments.At least, I found that I had to stop and look up the missing pieces, in order to make some of the developments as impressive as they were supposed to be.(I also had to look up some "well-known theorems" in Euclid, read up on the background to Stirling's factorial approximation, etc.)

generous
This book is really interesting primarily for its information about the history of trigonometry. There's some interesting stuff about the ancient Egyptians, Babylonians and Greeks; and a lot of great stuff about early European mathematics; stopping around Euler's time.

I hadn't studied trig in about 8 years, and I thought this would be a good review. Boy, was I wrong! I needed to do the review and then study this book!

Anyway, if you're a fairly gifted high school trig student, this book will certainly liven up the subject for you. If you're a college math major, it will be easy reading, and certainly interesting. If you're a teacher, you might find something interesting to entertain your students. Otherwise, unless you really like math or are really good at it, this book will probably be really difficult for you.

When I was feeling lazy I kind of breezed through the dense equations and looked for the conclusions, but when I was diligent I could usually make sense of them. You can do as I did and you won't miss much. Really, the highlights of the book are the historical information, not the equations. But if you can appreciate the equations as well, then you'll probably really enjoy the book.

Of course this isn't a life-changing or eye-opening book, but I gave it 5 stars just so no one thinks there's anything wrong with it. ... Read more

Isbn: 0691057540
Sales Rank: 558670
Subjects:  1. History & Philosophy    2. Mathematics    3. Science/Mathematics    4. Trigonometry    5. History of Science and Medicine, Philosophy of Science    6. Mathematics / History   

A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science
by Michael S. Schneider
Average Customer Review:
Paperback (08 November, 1995)
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Reviews (34)

Worthless garbage
I got assigned this book as part of a class.The idea that geometry and philosophy had any ties were ridiculously explored in this book.It achives nothing, but on the plus side it isnt very long, giving lots of space for drawings and quotes that have nothing to do with the subject matter.

I tried to approach this book with an open mind, but all I got was a headache and innumerable brain cells to flee in the overhwelming horror of how awful this thing is.

Numbers are Alive!
I attended a Sacred Geometry workshop sponsored by Phanes Press/David Fidelerback in 1996 and had the good fortune to meet the author of A Beginner's Guide to Constructing the Universe.The inspiring manner in which he presented difficult to grasp concepts (difficult for an innumerate, right brained type such as myself) helped me to reconnect with the actual humane-ness of mathematics, something so neglected/discouraged in education on all levels these days.This book is exemplary in that it directly purveys the spiritedness of the author himself-his genuine enthusiasm for the archetypal topic at hand, in this case the numbers one through ten.A Beginner's Guide to Constructing the Universe certainly is the place to start the voyage to reclaiming the spirit and life hidden with mathematics. Number Crunchers take heed because numbers are alive!

Jaye Beldo: Netnous@Aol.Com

Awakened!
When i read the book, i was awakened! And many secrets were revealed that were protected by elite societies in the past. Numbers shape the world, and the book explains why. In fact, it's common sense when you come to think about it. All shapes are numbers with appearance. Numbers cannot be seen. But they are manifested in the square as 4, triangle as 3. And to realize that all shapes can be derived by the vescica pesces, is amazing. Two circles that overlap at their centers. It's a metaphore that teaches us people how to interact with each other -- in a way that two beings should touch each other's centers ( but this is not included in the book...just a thought ). ... Read more

Isbn: 0060926716
Sales Rank: 11613
Subjects:  1. Astronomy - General    2. Mathematics    3. Number Theory    4. Philosophy    5. Philosophy & Social Aspects    6. Science/Mathematics    7. Science / General   

An Imaginary Tale
by Paul J. Nahin
Average Customer Review:
Hardcover (24 August, 1998)
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Editorial Review

At the very beginning of his book on i, the square root of minus one, Paul Nahin warns his readers: "An Imaginary Tale has a very strong historical component to it, but that does not mean it is a mathematical lightweight. But don't read too much into that either. It is *not* a scholarly tome meant to be read only by some mythical, elite group.... Large chunks of this book can, in fact, be read and understood by a high school senior who has paid attention to his or her teachers in the standard fare of pre-college courses. Still, it will be most accessible to the million or so who each year complete a college course in freshman calculus.... But when I need to do an integral, let me assure you I have not fallen to my knees in dumbstruck horror. And neither should you."

Nahin is a professor of electrical engineering at the University of New Hampshire; he has also written a number of science fiction short stories. His style is far more lively and humane than a mathematics textbook while covering much of the same ground. Readers will end up with a good sense for the mathematics of i and for its applications in physics and engineering. --Mary Ellen Curtin ... Read more

Reviews (31)

Many interesting aspects, but inconsistency a big detraction
Nahin's book has many amusing and interesting aspects, but it suffers from an overall lack of focus and consistency:

1)Is it math history (as the title suggests) or math exposition (as the preface suggests)?It is much more of the latter, and while there are enjoyable bits of each it serves neither one extraordinarily well.

2)Is it for a gifted high school student (as he alludes), or a practicing engineer/scientist/mathematician?He painstakingly belabors some simple things (definition of electrical current, etc.), yet at other times races through much deeper concepts (Green's Theorem, etc.).Without at least integral calculus, and better yet a few courses beyond that, much of the book would probably be very frustrating and/or inaccessible.For those with this background the painstaking elementary explanations are in the way.

3)Is it intended to be rigorous, pragmatic, or somewhere in between?This varies wildly from one topic to the next, to the point where both the careful reader and the casual follower are sure to both be left shaking their heads.

One other minor criticism: while his non-stuffy approach to this topic is at first refreshing, the overly informal style and excessive amount of first-person commentary (and attempts at humor) can grow annoying.

With these caveats, there really are some entertaining historical perspectives, some thought provoking approaches and derivations, and some nice tie-ins of different problems in engineering and mathematics.It makes for a good bedtime read for one with enough mathematical background and a willingness to forgive some trespasses.

Wish more books like this
Inspiring!
Explaining the true physical meaning of an imaginary real quantity and showing its real imaginary applications.

somewhat dense and no problems to solve by the reader
This book is well written, but, it does feel like the venerable professor took his lecture notes and strung them together, but dear me, he left out problems for the reader; this to me is a cardinal sin when it comes to expository math.

Maybe the professor could create a website with problems + solutions related to the subject matter - give us puzzle people a chance at solving at least a few problems on our own. ... Read more

Isbn: 0691027951
Subjects:  1. History & Philosophy    2. Mathematical Analysis    3. Mathematics    4. Number Theory    5. Numbers, Complex    6. Science/Mathematics    7. Theory Of Numbers    8. History of Science and Medicine, Philosophy of Science    9. Mathematics / History   

Mathematical Mysteries: The Beauty and Magic of Numbers
by Calvin C. Clawson
Average Customer Review:
Paperback (01 January, 2000)
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Reviews (6)

Study Guide
I was a student of Mr. Clawson's, and highly recommend any student of his to use this book as a study guide.

Superb book by a highly intuitive author
This book is superbly written by a highly intuitive author. The author knows how to connect to his intended readers. Book is highly informative and the equations are well defined and explained. Once you read a few pages, you are hooked. I just hope that Mr. Calvin C. Clawson will write more books on Mathematics, he will be doing a great service to would-be-mathematicians like me. More power to you sir!

Great introduction to Number Theory
Clawson does an excellent job of introducing the reader to a variety of number theory topics.With each topic, he provides enough information to understand the idea and appreciate its implications without being overly technical or tedious.Suprisingly, an advanced understanding of math is not required to enjoy this book.If you have an interst in number theory and need a starting point, this is the book. ... Read more

Isbn: 0738202592
Sales Rank: 260830
Subjects:  1. History & Philosophy    2. Mathematics    3. Number Theory    4. Science/Mathematics   

Journey Through Genius: The Great Theorems of Mathematics
by William Dunham
Average Customer Review:
Paperback (01 August, 1991)
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Editorial Review

In Journey through Genius, author William Dunham strikes an extraordinary balance between the historical and technical. He devotes each chapter to a principal result of mathematics, such as the solution of the cubic series and the divergence of the harmonic series. Not only does this book tell the stories of the people behind the math, but it also includes discussions and rigorous proofs of the relevant mathematical results. ... Read more

Reviews (45)

Just a pleasure to read.Truly inspiring.
This is my third read of Dunham's Journey Through Genius and I am still learning new stuff from this book.The author has hit upon just the right blend of math and history to make this a gem.There is enough mathematical rigor here to not render the work banal and as just another "popular math" book.Reading this book I am reminded of Kline's Mathematics And The Physical World except that this text is less wordy and more focussed on seventeen landmarks of mathematics.

Some theorems and proofs I have to read four or five time to really get the ideas behind them but the effort is worthwhile.

If you want to make your 2005 summer reading more rewarding and challenging than usual get this book.I have had this volume since 1992 and feed upon it at least twice a year.My guess is this book will become a classic if it has not already. A jewel that will become a true friend.

The BEST Book I Ever Read On Mathematics
There are lots and lots of books written on mathematics claiming to target mass audience and containing none to negligible real "mathematics". Yeah, I'm talking about those funny stupid books which keeps talking about math for 400 pages but shy away from putting one real equation or proof. Well, this book is different and if you ask me, it's the best book on mathematics I've came across so far. It's the collection of some of the cleverest not-too-obvious theorems derived from the scratch with really fluid explanation and plenty of diagrams. One of the coolest thing about this book is that it first gives you a historical preview of the problem which is usually gets really interesting and pretty fun to read, specially all those tid-bits about the people involved. So by the time you reach to the proof, you know why it was a hard to do thing and you can fully appreciate the clever twists and turns in the proof. You can literally enjoy it like some murder mystery thriller. The book is written with loads and loads of infectious passion for mathematics. If this is the way math textbooks are written, there would have been far more people with passion, love and deeper understanding of mathematics.

Very good
This was my introduction to some of the rare geniuses of Mathematics, I was fascinated with the likes of Issac Newton and some of the others I was just learning about. The basic concept in to highlight some of the greater theorys that scholars could rack there brains on. I'm a FEMALE ... never to comprhensible in the feild but none the less recognize the beauty and prosperity of mathematics, great introduction; ... Read more

Isbn: 014014739X
Subjects:  1. Biography    2. General    3. History    4. Mathematicians    5. Mathematics    6. Science/Mathematics   

Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem
by SIMON SINGH, JOHN LYNCH
Average Customer Review:
Paperback (08 September, 1998)
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Editorial Review

When Andrew Wiles of Princeton University announced a solution of Fermat's last theorem in 1993, it electrified the world of mathematics. After a flaw was discovered in the proof, Wiles had to work for another year--he had already labored in solitude for seven years--to establish that he had solved the 350-year-old problem. Simon Singh's book is a lively, comprehensible explanation of Wiles's work and of the star-, trauma-, and wacko-studded history of Fermat's last theorem. Fermat's Enigma contains some problems that offer a taste of the math, but it also includes limericks to give a feeling for the goofy side of mathematicians. ... Read more

Reviews (216)

Peak into Mathematical History
I found this book very interesting.Even though, the book concernsfermat, you come across references to works of other great Mathematicians.

A great guide to scientific achievement
Singh provides an excellent reconstruction of the events leading up to and involving Andrew Wiles enormous solution to Fermat's "most difficult problem in history".Scientific achievement most often requires huge effort and concentration, at levels hard to imagine.Singh takes us through the heights of amazing rewards, and the depths of self-doubt and depression.Wiles goes through both, and unhappily, many other mathematicians (tackling Fermat's Enigma or not) have ended with the later.

Singh backs the reader up in history to Pythagoras, starting with that most important, and yet relatively simple mathematical proof.He strolls us past Euclid, and (unfortunately) steams through progress made by Indian and Muslim mathematicians in the middle-ages.Singh provides enough math to grab at the reader, some in the appendixes (more involved), and enough in the text to lay the ground work for Fermat. The 17th century injects renewed life into science and math, and into Singh's narrative.The issues involved with Fermat's "final theorem" may be simple enumerate, but not to understand.Singh once again gets bogged down, at this point in trying to bridge sophisticated concepts of modern math into a base of knowledge through which the reader canconnect to Wiles achievement in 1994.Once that base is formed, and Singh moves to Wiles, the energy of the book increases by leaps and bounds. The book is very engaging in the last 3rd, where we can finally glimpse the incredible struggle, lasting over 8 years, that Wiles went through to prove Fermat right.

This book gets very high praise for linking incredibly complex science to the humanity of the people working with it.Readers will gain a fascination for Wiles' effort and achievement, and perhaps learn to want more.

Surprisingly suspenseful
As a non-mathemetician, who hit his own personal wall in freshman year's intermediate calculus, I confess that I do have a strange fondness for texts in pop-mathematica.Which, as I've found out, are something of a mixed bag.And understandably so - the genre demands both clear and concise writing *and* an ability to undertand topics that are, to say the least, arcane.Some texts have pulled it off.Others have failed.

This one, happily, pulls it off.Which actually came as a bit of a surprise to me.I knew of Fermat's Last Theorem before tackling this text, but I frankly didn't see why everyone got so damned worked up over it, aside for 300+ years of failure in proving the thing.But the nice thing about Singh's work is that he makes you understand *why* the theorem is actually relevant outside of the rarified circles of number theorists.And he does it with a clear, lucid style (due partially, I don't doubt, to oversimplifications and ommisions, but that's the ultimate fate of all texts in this genre).

As I said, I hit my own mathematical wall in intermediate calculus.But I was still able to grasp onto pretty much everything that Singh deigned to explain.(And I assume that what he left out - i.e. what the hell modular shapes actuallly are - he did so for a good reason).The proofs he presented, both in the text and in the multiple appendixes, were, if not simple, understandable and even enjoyable to puzzle my way through.

But the greatest triumph of this text is that Singh has made the quest to crack this theorem actually *exciting*.Which I didn't really thing possible.But, when you get to the last frew chapters and watch Wiles struggling to prove the theorem and then to patch the quite possibly devestating hole in his first proof - this book becomes quite a compelling little page turner.

As I'm not a mathemetician, I can't speak to the truth/falsity of some of the more complex notions in the text.And, since some of these ideas are really only expressible via formulas and arcane notation (see the first page of Wiles's proof), I'm sure that something has been lost in the translation into simple English.But that aside, this is quite the entertaining read. ... Read more

Isbn: 0385493622
Subjects:  1. Fermat's last theorem    2. History & Philosophy    3. Mathematics    4. Science/Mathematics    5. Mathematics / History   

e: The Story of a Number
by Eli Maor
Average Customer Review:
Paperback (04 May, 1998)
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Editorial Review

Until about 1975, logarithms were every scientist's best friend. They were the basis of the slide rule that was the totemic wand of the trade, listed in huge books consulted in every library. Then hand-held calculators arrived, and within a few years slide rules were museum pieces.

But e remains, the center of the natural logarithmic function and of calculus. Eli Maor's book is the only more or less popular account of the history of this universal constant. Maor gives human faces to fundamental mathematics, as in his fantasia of a meeting between Johann Bernoulli and J.S. Bach. e: The Story of a Number would be an excellent choice for a high school or college student of trigonometry or calculus. --Mary Ellen Curtin ... Read more

Reviews (41)

A must for students of Mathematics
This book was written well, and every student interested in Mathematics or pursuing a career in engineering or the sciences should read this.You really don't need to be a math genious to enjoy this book.I would recommend that high school Math teachers and even college professors assign a little reading each day of the history of their profession.This is one of those history books.

Great book Great author - Worth Reading!!
This is great book. Besides `e' it covers all the history and good stories about calculus. I did not get bored at all. This explains all the difficult concepts with great detail and fun to read. I never got bored. Maor does wonderful job of bringing together maths, fun and history.
From Napier to Newton he covers everything. It gives the insight to the common used notations today. This books is collectors item.

Never a boring moment.
How much have computers changed our lives?John Napier spend 20 years from 1594 to 1614 performing calculations for his logarithm tables.Today, that entire body of work is easily reproduced in minutes, using Microsoft Excel.But Napier'sinvention quickly spread around the world, creating a calculation revolution that empowered grateful scientists with speed they could only imagine before.I suppose it was the greatest computation breakthrough since the abacus.

From Napier forward, the story of e proceeds, eloquently recounted by Maor.There is not a boring moment in the book.
... Read more

Isbn: 0691058547
Subjects:  1. History & Philosophy    2. History Of Mathematics    3. Mathematics    4. Number Theory    5. Science/Mathematics    6. History of Science and Medicine, Philosophy of Science    7. Mathematics / History   

A History of the Circle: Mathematical Reasoning and the Physical Universe
by Ernest, Jr Zebrowski
Average Customer Review:
Paperback (01 September, 2000)
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Reviews (5)

Zebrowski's History of the Circle is a great fun book.
This is not a text book, but an fun read that discusses
interesting topics from physics and math.
It explains things so well that I wish I had Zebrowski as a professor or could find textbooks written by him.
I think the lengthy first review below as well as examples from the other reviews give a good idea of the contents of this book.
It's a joy to read, and to use when explaining things correctly to others.

interesting but ...
An interesting read which could have benefited from the
attention of a diligent editor.Sadly, historical details
are often muddled and in the later chapters the author
indulges in some freewheeling speculation regarding high
energy physics while quoting undergraduate level journals
as sources.
Nevertheless, all of the subject matter and most of the
text is well worth a look.

Looking For Pi Info? Its Not Here.
The first chapter BREIFLY addresses pi, so this not a good source for those of you that may have drawn the same conclusions about the title that I did.However, if you've ever wondered why there are 360 degrees in a circle and how that relates to time or other interesting trivia, this is a great source.I do recomend it to anyone with an intrest in the basic concepts of Physics which somehow work themseves in everywhere.The title would be more acurate if it removed "A History of the Circle" and just left it with "Mathmatical Reasoning and the Physical Universe" because it lacks far to much of the first subject. ... Read more

Isbn: 0813528984
Sales Rank: 284062
Subjects:  1. Geometry - General    2. History & Philosophy    3. Mathematics    4. Science/Mathematics   

Zero: The Biography of a Dangerous Idea
by Charles Seife, Matt Zimet
Average Customer Review:
Paperback (05 September, 2000)
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Editorial Review

The seemingly impossible Zen task--writing a book about nothing--has a loophole: people have been chatting, learning, and even fighting about nothing for millennia. Zero: The Biography of a Dangerous Idea, by noted science writer Charles Seife, starts with the story of a modern battleship stopped dead in the water by a loose zero, then rewinds back to several hundred years BCE.Some empty-headed genius improved the traditional Eastern counting methods immeasurably by adding zero as a placeholder, which allowed the genesis of our still-used decimal system. It's all been uphill from there, but Seife is enthusiastic about his subject; his synthesis of math, history, and anthropology seduces the reader into a new fascination with the most troubling number.

Why did the Church reject the use of zero?How did mystics of all stripes get bent out of shape over it? Is it true that science as we know it depends on this mysterious round digit?Zero opens up these questions and lets us explore the answers and their ramifications for our oh-so-modern lives. Seife has fun with his format, too, starting with chapter 0 and finishing with an appendix titled "Make Your Own Wormhole Time Machine."(Warning: don't get your hopes up too much.)There are enough graphs and equations to scare off serious numerophobes, but the real story is in the interactions between artists, scientists, mathematicians, religious and political leaders, and the rest of us--it seems we really do have nothing in common. --Rob Lightner ... Read more

Reviews (88)

It would have been better without the hyperbole
This book is about the history of zero, from ancient times to modern concepts.It's quite interesting and encompasses a lot of mathematics and philosophy as well as a bit of physics.

Although the book reads well, is nicely documented, and extensively researched, the author has a style that I found aggravating; his frequent use of poetic hyperbole.This limits the book's value for someone unfamiliar with basic concepts in mathematics and physics.

I'm not sure why Seife choose this style.There seems to be a movement (hopefully short lived) among science writers to dress up science and mathematics in poetic, flowery language.Whatever the reason, science has good reason to use strict meanings for words and a disciplined approach to scientific concepts.When authors poetically use words in technically incorrect ways they can make the prose pretty, but they often create confusion.

For example, Saif says "Zero and infinity are eternally locked in a struggle to engulf all the numbers.Like a Manichaean nightmare, the two sit on opposite poles of the number sphere, sucking numbers in like tiny black holes." [p. 145]

From a mathematical point of view this is pure gibberish.If one's intent is to educate others about mathematics, such poetic hyperbole is not only useless, but counter productive as well.For folks who don't already know a bit about mathematics, Seife's book is as likely to confuse as to educate.For those who already understand the concepts, the poetry might be pleasing, but from an educational point of view the hyperbole found throughout this book is a definite stumbling block.

Another problem I had with this book is the way Seife misstates some key aspects in modern science.For example, on page 171 he asserts the classical definition of a vacuum: "The vacuum, by definition, has nothing in it - no particles, no light, nothing."He then describes the quantum mechanical view of the vacuum, and the zero-point energy.Part of this explanation includes a nice description of the Casimir effect [p. 172], which is a measurement of the literal existence of the "virtual" particles predicted by Quantum Mechanics.What these experiments show is that these "virtual" particles actually exist, and can be detected by the force they exert on closely spaced metal plates.This is actually a beautiful example of how science changed our concept of the vacuum.Classically, we thought of the vacuum as having "nothing in it," but Quantum Mechanics tells us that the classical vacuum cannot exist.But even after his nice explanation of the Casimir effect, Seife goes and spoils it with this absurd statement:

"Casimir realized that he had felt the force of nothing." [p. 172] "This is the force of the vacuum, a force produced by nothing at all.This is the Casimir effect."

It's as if someone asserted that the space around us has "nothing in it," and then rejoices when the wind touches his face, and runs off spouting "I've felt the force of nothing."What the Casimir effect teaches us is that what we thought was "nothing at all" really is something, and that calling them "virtual" particles is just as silly as early mathematicians who called the square root of negative numbers "imaginary."

There are other mistakes as well.For example, on page 178 he says: "The speed of light is the ultimate speed limit; you cannot reach it, much less exceed it.Nature has defended itself from an unruly zero."

But this simply isn't true.Even a casual reader knows that the statement "you cannot reach it" is wrong.After all, photons travel at the speed of light all the time.Furthermore, scientists have known for years that, given the right materials, both the phase velocity and the group velocity of light can exceed the speed of light in a vacuum [Optics and Photonics News, June 2002].All this is consistent within the framework of relativity, but Seife's hyperbole is likely to mislead the novice.Indeed, recent experiments showing these phenomena have resulted in all sorts of pundits on the Internet claiming that relativity had been falsified.

By getting all wound up with poetic hyperbole about nature "[defending] itself from an unruly zero" the author has, I fear, unwittingly contributed to the confusion of non-scientists about the science of relativity.

I don't mean to give the impression that this is a bad book.I actually found most of it readable and pleasant.I enjoyed the historical aspects and appreciated how the author illustrates the influence of philosophy, and especially religion, in either advancing or retarding cultural acceptance of the concept of zero.I thought he did a particularly nice job of explaining the development of the calculus, and how the concept of zero played its part.As usual, the primary distractions were related to his use of poetic hyperbole, as well as careless analogies.For example, on page 126 he writes:

"... using calculus was as much an act of faith as declaring a belief in god."

This absurd statement was almost certainly made without thinking.After all, even though early mathematicians could not explain why the calculus worked - at least not with rigorous logic - they could demonstrate that it *did* work.Furthermore, anyone could use it.A person didn't have to believe in calculus or work themselves into an emotional frenzy to calculate the volume of a sphere.The same cannot, of course, be said of god.

This could have been a really great book.The subject matter and story of zero are fascinating.Unfortunately, Seife uses too many analogies that are either poor, extreme, or misleading.And his persistent tendency toward exaggeration was a real distraction for me.For these reasons I'd not recommend the book to someone not already somewhat knowledgeable about mathematics and physics - I think it would be too confusing.For those who can read between the lines of poetic hyperbole, though, I think the book is worthwhile.

Amazon doesn't allow "zero" stars
In fact, the author goes on so many fanciful tangents, I was actually surprised that he didn't mention that zero stars are not allowed in the Amazon rating scheme, because it would crash the whole system. And that's pretty much the tone of this breezy math/science/history book: The number zero or the concept of zero or the acknowledgment of zero challenges, frightens, destroys, and generally wreaks havoc wherever it appears, and, at the same time, the number zero is necessary and important, both in math and in the real world. Frankly, I was disappointed in this little book, though at times it can be charming and there is some interesting history. The book just didn't live up to the hype about the "danger" of zero. I wanted excitement and adventure; I got the history of an interesting number. Maybe I would have enjoyed it more if I had "zero" expectations.

"O" ( know "0")
If you think Zero is nothing then you might be right till you understand that nothing took centuries for man to discover and more centuries will go before we completly understand it. This book is a great work from Charles which goes from 300 BC to big bang and "Black Holes". I believe his great work could have been extended to ohter sides of zero like why is zero written as "O' ... why are heavenly bodies round and why mathematically area of circle is greater than Square which is greater than a trianglewhile the circumfrence of circle could be equal to parameter of Square which could be equal to parameter of a triangle.... as we know this is because the figure with highest number of sides will have maximum area and in case of circle the number of sides is infinite ... nothing has infinity inside !! The spiritual aspect too of zero needs more investigation ... All the best to the readers of this book for it has so many thought provoking information.. enjoy "Zero" and you might discover nothing with infinite hue of colors... ... Read more

Isbn: 0140296476
Subjects:  1. Arithmetic    2. Biography & Autobiography    3. Biography/Autobiography    4. History & Philosophy    5. Literary    6. Mathematics    7. Philosophy & Social Aspects