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Journey through Genius: The Great Theorems of Mathematics (Paperback) (平装)
by William Dunham
| Category:
Mathematics, Science |
| Market price: ¥ 168.00
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¥ 158.00
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MSL rating:
 Good for Gifts |
| MSL Pointer Review:
This book is an important read for a layman trying to get a better grasp on the actual historical building blocks of math. |
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AllReviews |
1 Total 1 pages 10 items |
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Kenneth (MSL quote), USA
<2007-01-30 00:00>
William Dunham has brought life to a subject that almost everyone considers dull, boring and dead. Dunham investigates and explains, in easy-to-understand language and simple algebra, some of the most famous theorems of mathematics. But what sets this book apart is his descriptions of the mathemeticians themselves, and their lives. It becomes easier to understand their thinking process, and thus to understand their theorems.
I am a layman with a computer science degree, and a layman's understanding of mathematics, so I am no expert! But I loved this book.
I found Dunham's description of Archimedes' life and his reasoning for finding the area of a circle and volume of a cylinder to be (almost!) riveting.
Dunham's decription of Cantor and his reasoning regarding the cardinality of infinite sets was fascinating to me. But most of all, I loved his chapter on Leonhard Euler. Having in high school been fascinated by Euler's derivation of e^(i*PI) = -1, I was even more amazed at the scope of this man's genius, and Dunham's description of his life.
The chapter on Isaac Newton is an especially good one as well.
Dunham smartly weaves these important theorems of mathematics into the history of mathematics, making this book even more understandable, and, dare I say it, actually entertaining!
This book is a gem, and for anyone interested in mathematics, it is not to be missed.
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A reader (MSL quote), USA
<2007-01-30 00:00>
Dunham has done an excellent job of taking us through the history of mathematics providing a context with the civilization of the time. He shapes his production around what he considers to be the great theorems of mathematics. The order of presentation is chronological. Early on we see great admiration for Euclid and his "Elements" as two of Euclid's theorems appear on the list, a proof of the Pythagorean theorem and the proof that there are infinitely many primes. Euler and Cantor are also honored with two theorems included among the collection. However there is more to Dunham's presentation than just the proofs. We find other related results by these masters and other great mathematicians that were their contemporaries. He shows reverence for Newton. Gauss and Weierstrass and others are mentioned but none of their theorems are highlighted. It is not his intention to slight these great mathematicians. Rather, Dunham's criteria seems to be to present the theorems that have simple and elegant proofs but often surprising results. His coverage of Cantor is particularly good. It seems that he is most knowledgeable about Cantor's mathematics of transfinite numbers and the related axiomatic set theory.
For a detailed description of the chpaters in this work, look at the detailed review by Shard here at Amazon. I found this book well written and authoritative and learned a few things about Euler and number theory that I hadn't known from my undergraduate and graduate training in mathematics. Yet I did not give the book five stars.
There are a couple of omissions that I find reduce it to a four star rating. My main objection is the slighting of Evariste Galois. Galois was the great French mathematician who died in a duel at the early age of 21 in the year 1832. Yet, in his short life he developed a theory of abstract algebra seemingly unrelated to the great unsolved questions about constructions with straight edge and compass due to the Greeks and yet his theory resolved many of these questions. I was very impressed in graduate school when I learned the Galois theory and came to realize that problems such as a solution to the general 5th degree equation by radicals and the trisection of an arbitrary angle with straight edge and compass were impossible.
Now, Galois theory is certainly beyond the scope of this book but so is non-Euclidean geometry and aspects of number theory and set theory that Dunham chooses to mention. He spends a great deal of time on Euclid's work and the various possible constructions with straight edge and compass. Also, in the chapter on Cardano's proof of the general solution to the cubic, he also presents the solution to the quartic and refers to Abel's result on the impossibility of the general solution to the quintic equation. This would have been the perfect place to introduce Galois who independently and at the same time in history proved the impossibility of solving the general quintic equation by radicals. Oddly Galois is never once mentioned in the entire book.
In discussing number theory and Euler's contributions, the theorems and conjectures of Fermat are mentioned. This book was written in 1991 and it presents Fermat's last theorem as an unproven conjecture. Andrew Wiles presented a proof of Fermat's last theorem to the mathematical community in 1993 and after some needed patchwork to the proof, it is now agreed that Fermat's last theorem is true. There are a number of books written on Fermat's last theorem including an excellent book by Simon Singh. It seems that Dunham's book is popular and has been reprinted at least 10 times since the original printing in 1991. It would have been appropriate to modify the discussion of Fermat's last theorem in one of these reprintings.
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Shard (MSL quote), USA
<2007-01-30 00:00>
Some books, such as Ball's and Beiler's seem to have sparked a life-long love of mathematics in practically everyone who reads them. "Journey Through Genius" should be another such book.
In the Preface, the author comments that it is common practice to teach appreciation for art through a study of the great masterpieces. Art history students study not only the great works, but also the lives of the great artists, and it is hard to imagine how one could learn the subject any other way. Why then do we neglect to teach the Great Theorems of mathematics, and the lives of their creators? Dunham sets out to do just this, and succeeds beyond all expectations.
Each chapter consists of a biography of the main character interwoven with an exposition of one of the Great Theorems. Also included are enough additional theorems and proofs to support each of the main topics so that Dunham essentially moves from the origins of mathematical proof to modern axiomatic set theory with no prerequisites. Admittedly it will help if the reader has taken a couple of high school algebra classes, but if not, it should not be a barrier to appreciating the book. Each chapter concludes with an epilogue that traces the evolution of the central ideas forward in time through the history of mathematics, placing each theorem in context.
The journey begins with Hippocrates of Chios who demonstrated how to construct a square with area equal to a particular curved shape called a Lune. This "Quadrature of the Lune" is believed to be the earliest proof in mathematics, and in Dunham's capable hands, we see it for the gem of mathematics that it is. The epilogue discusses the infamous problem of "squaring the circle", which mathematicians tried to solve for over 2000 years before Lindeman proved that it is impossible.
In chapters 2 and 3 we get a healthy dose of Euclid. Dunham briefly covers all 13 books of "The Elements", discussing the general contents and importance of each. He selects several propositions directly from Euclid and proves them in full using Euclid's arguments paraphrased in modern language. The diagrams are excellent, and very helpful in understanding the proofs. If you've ever tried to read Euclid in a direct translation, you should truly appreciate Dunham's exposition: the mathematics is at once elementary, intricate, and beautiful, but Dunham is vastly easier to read than Euclid. The Great Theorems of these chapters are Euclid's proof of the Pythagorean theorem and The Infinitude of Primes, which rests at the heart of modern number theory. Dunham obviously loves Euclid, and his enthusiasm is infectious. After reading this, it is easy to see why "The Elements" is the second most analyzed text in history (after The Bible).
Archimedes is the subject of chapter 4, and he was a true Greek Hero. Even if most of the stories of Archimedes' life are apocryphal, they still make very interesting reading. However the core of the chapter is the Great Theorem, Archimedes' Determination of Circular Area. His method anticipated the integral calculus by some 1800 years, and also introduced the world to the wonderful and ubiquitous number pi. The epilogue traces attempts to approximate pi all the way up to the incomparable Indian mathematician of the 20th century, Ramanujan.
Chapter 5 concerns Heron's formula for the area of a triangle. The proof is extremely convoluted and intricate, with a great surprise ending. It is well worth the effort to follow it through to the end. Chapter 6 is about Cardano's solution to the general cubic equation of algebra. Cardono is certainly one of the strangest characters in the history of mathematics, and Dunham does a great job telling the story. The epilogue discusses the problem of solving the general quintic or higher degree equation, and Neils Abel's shocking 1824 proof that such a solution is impossible.
Sir Isaac Newton is the topic of chapter 7. Rather than go into the calculus deeply, Dunham gives us Newton's Binomial Theorem, which he didn't really prove, but nevertheless showed how it could be put to great use in the Great Theorem of this chapter, namely the approximation of pi. Chapter 8 breezes through the Bernoulli brothers' proof that the Harmonic Series does not converge, with lots of very interesting historical biography thrown in for good measure.
Chapters 9 and 10 discuss the incredible genius of Leonard Euler, who contributed very significant results to virtually every field of mathematics, and seems to have been a decent human being to boot. Chapter 10, "A Sampler of Euler's Number Theory", is my favorite in the book. A large portion of his work in number theory came from proving (or disproving) propositions due to Fermat, which were passed on to him by his friend Goldbach. This chapter gives complete proofs of several of these wonderful theorems including Fermat's Little Theorem, all of which lead up to the gem of the chapter. Taken as a whole it is the kind of number theory detective work that has lured so many people into the field over the years. Chapter 10 is a mathematical tour de force.
The last 2 chapters handle Cantor's work in the "transfinite realm", and should certainly serve to expand the mind of any reader. By the time you finish, you'll have an idea about the twentieth century crisis in mathematics, and its resolution, and what sorts of concepts are capable of making modern mathematicians squirm in their seats. Dunham does a beautiful job of demonstrating Cantor's proof of the non-denumerability of the continuum. At this altitude of intellectual mountain-climbing the air is thin, but it is well worth the climb!
In brief, "Journey Through Genius" might almost be considered a genius work of mathematical exposition. I can think of few authors more capable of conveying the excitement and beauty of mathematics, as well as an appreciation for the sheer enormity of the achievements of the human mind and spirit.
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A reader (MSL quote), USA
<2007-01-30 00:00>
There is a reason all the reviewers for this book have given it 5 stars. It is simply a wonderful book.
This is the kind of book that would make almost anybody learn to love mathematics. Although I myself have always liked mathematics, I can heartily recommend this book to anybody with just a passing interest in the subject.
Some of the theorems that are discussed could be somewhat involved. However, those who are not interested in the details can skip them without compromising their enjoyment of the book. William Dunham seamlessly weaves a story of these wonderful mathematical geniuses, the times they lived in, their motivations and last but not least, their theorems. A great mathematical epic unfolds as you move from one era to another and from one genius to another. The author's love for the subject is obvious and a lot of it is bound to rub off on you too.
Recommended without reservations.
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Joseph Kim (MSL quote), USA
<2007-01-30 00:00>
As a high school math teacher, I found Dunham's book perfectfor filling what is sadly lacking in math textbooks-the idea thatreal people have contributed to the progress of mathematics. Dunham makes it clear that mathematics is not simply calculation, but an exciting journey of discovery. I have included the book in my Advanced Mathematics courses for several years now, and my students always name it as one of the best parts of the class. The other day, one of my calc students corrected an underclassman's pronunciation of Euler, commenting, "he was great enough that we should pronounce his name correctly." Journey Through Genius is where he encountered Euler's greatness.
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Joseph Horton (MSL quote), USA
<2007-01-30 00:00>
The author modestly refers to the journey as that of others. In fact, writing his book was an expression of his own genius. True, he chronicled great mathematicians' works, but to weave things together as he did required a spirit kindred to their own. The book looks at the lives and works of several mathematicians. Results are put into sharp perspective. Unlike anything that I had read before, Dunham describes the reasons that things were of interest, rather than merely saying what the results were. Instead of only giving the result, he derives it, so that the reader has a visceral grasp of what's going on. I'm waiting for the sequel.
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Stephen Pletko (MSL quote), Canada
<2007-01-30 00:00>
A highlight of this easy-to-read book, by Dr. William Dunham, is that it explores twelve "great theorems of mathematics" that have occurred throughout history. (A theorem is a statement or proposition that is proved by logical reasoning from given facts and justifiable assumptions.) Dunham calls the theorems he explores "mathematical masterpieces" or "mathematical landmarks." He tries to get the reader to appreciate the "creativity" evident in them.
Four of the theorems presented were formulated in very ancient times. (For example, two were formulated circa 300 BC.) One was formulated in the first century, another was formulated in the sixteenth century, and two each were formulated in the seventeenth, eighteenth, and nineteenth centuries respectively.
What factors influenced Dunham to choose his particular theorems that "represent the best of mathematics" or what he calls the "Mona Lisas or Hamlets of mathematics?" Answer: (1) the theorems had to be insightful (2) they had to be formulated from history's leading mathematicians and (3) they had to be important (that is they had to resolve long-standing problems in mathematics or generate even more profound future questions).
The theorems themselves come from different branches of mathematics, that is, a variety are presented. As Dunham states: "The propositions of [this] book come from the realms of plane geometry, algebra, number theory, analysis, and [set theory]."
In general, to understand these theorems requires high school algebra and geometry. Some trigonometry and elementary integral calculus would also be useful.
However, this book is much more than just theorems. It details the history of important events in mathematics and puts these theorems in historical context. As well biographical information is given not only of those who created these "great theorems" but of other significant geniuses whose mathematical insight was essential to the development of these theorems. At the end of each chapter is an "Epilogue," usually addressing an issue raised by a great theorem.
A main feature of this book is that some of the actual writings of those who created the theorems are included or the actual writings of those who observed these geniuses are included. Here is one of my favorites from an observer:
"He always kept close to his studyes, very rarely went a visiting, & had a few Visiters...I never knew him [to] take any Recreation or Pastime, either in Riding out or to take ye Air, Walking, Bowling, or any other Exercise whatever, Thinking of Hours lost, that was not spent in his studyes...He very rarely went to Dine in ye Hall...& then, if He has not been minded, would go very carelessly, with Shoes down at Heels, Stockins untiy'd...& his Head scarcely comb'd."
Illustrations can be found throughout this book. They greatly aid the discussion.
Finally, mathematician and philosopher Bertrand Russell once said:
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the...trappings of painting or music, yet sublimely pure, and capable of stern perfection such as the greatest art can show."
This was my reaction to the great theorems in this book. I also experienced a sense of personal satisfaction that I could comprehend the works of these masters!
(first published 1990; preface; acknowledgements; 12 chapters; main narrative of 285 pages; afterword; notes; references; index)
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S. Shah (MSL quote), USA
<2007-01-30 00:00>
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. |
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Vineet Bansal (MSL quote), USA
<2007-01-30 00:00>
With all the new age pseudo-science books around - books that gloss through a lot of ideas (light cones, chirality, Godel) without a systematic explanation and without going in sufficient depth for you to actually understand the concept based on your high school science, it was a pleasant surprise when I chanced upon 'Journey through Genius' by William Dunham.
The great theorems of mathematics are discussed, in a mathematical sense, not in a layman's language that has never made sense to me. (mass "curves" space-time! - what does that mean??). It's a perfect blend of math and history - enough history for you to appreciate the impact of the theorem in question, and the requisite math to prove the point.
It's very successful at whetting your appetite to go out and start reading Euclid or Russell right away. For example, some of the seemingly baffling problems mathematicians have been wrestling with for around 2500 years - I had no idea about even the existence of those problems.
Did you know that given a circle drawn on the ground, an uncelebrated ruler and a compass (the idea here is, you don't 'know' yet that the area of a circle is pi*r*r, or for that matter, even what pi is), it is impossible to draw a square (with an arbitrary level of precision) with the exact area as the circle? (Unless it happens by accident of course). Apparently mathematicians tried to do it for over two centuries; until it was proved it cannot be done. (by proving that pi is transcendental, an issue I hope to understand some day).
This book is amazing. It makes you appreciate the logical process math takes to build upon a minimal set of postulates. For people who are seeking an escape from their daily routine of screen gazing and rediscovering the joys of pure science, this is a perfect starter.
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Paul Pendergrass (MSL quote), USA
<2007-01-30 00:00>
I am a recent college graduate with a BS in Math. I went through college simply as I could. I chose math because it was typically easy to me, but I never really tried to get any true grasp on it. I only began to understand the most basic concepts in math when I taught Algebra I for a semester to high school freshmen. That experience motivated me to develop a better understanding of the most basic concepts in math. Jump forward a year, and I am now about to enter grad school for applied math. After reading this book and soaking it in, I feel like I have the tools to be very successful now. I will take classes with a very different mind set. I am very excited to get started next week. I tell you that I am motivated so much that I am going to retake a few undergraduate math courses for my own sanity. Maybe I'm crazy. I have never been so excited about math or viewed it as a beautiful art. It is pure. Euclid's proof of the Pythagorean Theorem truly humbled me. I had viewed it before, but had never been motivated to try to understand it. That is where Dunham steps in. He creates such an excitement about it that it forces the reader to read and reread the proof. I have actually worked it out 3 times for myself to ensure that I truly get it. It feels powerful. Furthermore, I'm only about half way through the book. It is going to take me a while though. I have to convince myself that I can reproduce everything before I move on to the next chapter. Maybe I'm being hard on myself. For those of you who seem to suggest that if you took high school algebra and geometry everything is peaches and cream.... I don't get it. To read this book and really soak it in, one has to have the foundations of a true understanding of high school algebra and geometry. That is a different animal. It is easily attainable with the proper focus though. I ask that you read this book and make the world a bettter place! |
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