Journey through Genius: The Great Theorems of Mathematics by William Dunham (PDF)

Description

Like masterpieces of art, music, and literature, great mathematical theorems are creative milestones, works of genius destined to last forever. Now William Dunham gives them the attention they deserve.Dunham places each theorem within its historical context and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics. A rare combination of the historical, biographical, and mathematical, Journey Through Genius is a fascinating introduction to a neglected field of human creativity.“It is mathematics presented as a series of works of art; a fascinating lingering over individual examples of ingenuity and insight. It is mathematics by lightning flash.” —Isaac Asimov

User’s Reviews

Editorial Reviews: Amazon.com 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. Review “An inspired piece of intellectual history.”— Los Angeles Times“It is mathematics presented as a series of works of art; a fascinating lingering over individual examples of ingenuity and insight. It is mathematics by lightning flash.”— Isaac Asimov“Dunham deftly guides the reader through the verbal and logical intricacies of major mathematical questions, conveying a splendid sense of how the greatest mathematicians from ancient to modern times presented their arguments.”—Ivars Peterson, author of The Mathematical Tourist From the Back Cover A rare combination of the historical, biographical, and mathematicalgenius, this book is a fascinating introduction to a neglected field of human creativity. Dunham places mathematical theorem, along with masterpieces of art, music, and literature and gives them the attention they deserve. About the Author William Dunham is a Phi Beta Kappa graduate of the University of Pittsburgh. After receiving his Ph.D. from the Ohio State University in 1974, he joined the mathematics faculty at Hanover College in Indiana. He has directed a summer seminar funded by the National Endowment for the Humanities on the topic of “The Great Theorems of Mathematics in Historical Context.” Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐In a phrase, this is one of my favorite books on mathematics. I read it first when it was recommended by my Calculus I professor and thought it was great. I read it again when I took a course in the history of mathematics and thought it was brilliant. Now it remains one of my favorites and I return to it regularly for discussion of some remarkable theorems and the great minds who produced them.One of the first questions anyone might have before reading a book about mathematics is what level of mathematical sophistication is required on the part of the reader. In this case, the reader can feel pretty safe. While these are real and deep mathematical theorems, their proofs only require high-school level mathematics. In the vast majority of cases, the reader familiar with basic algebra and a little bit of geometry will have no trouble following the discussions. One theorem (Newton’s approximation of pi) requires a little bit of integral calculus and another (the discussion of some of Euler’s sums) requires a smidge of elementary trigonometry. In both cases, the author holds the reader’s hand through the discussion so even if you haven’t taken a course in trigonometry or calculus, you’ll still be able to follow most of the conversation.In fact, even if you don’t really have a lot of algebra and geometry, the bulk of the book will still be accessible to you. The majority of the text is a history of mathematics wherein the author discusses the context and importance of the theorems and some biographical details of their discoverers. While I find the recreations of the proofs themselves to be perhaps the most interesting part, the reader with a general interest (even if that interest is not supported by mathematical skill) will find the book fascinating. For those of us who do have some knowledge of mathematics, though, the recreations of the theorems presented in their historical context provides a rich and inspiring series of vignettes from the history of mathematics.This brings us to another important point. While this is a book about the history of mathematics. it is not *a* history of mathematics, and the theorems selected are not the only “great” theorems of mathematics, but a cross-section thereof. Many readers of sufficient mathematical background may quibble over the inclusion of some theorems at the expense of others–personally I would like to have seen more from combinatorics–but no one can deny that these theorems are remarkable in their elegance and in their importance in the development of mathematics from the Ancient Greeks to the very end of the nineteenth century.It might be helpful to know what theorems are actually included in the book. Aside from a handful of lemmas and minor results presented before or after each of the “Great Theorems,” the book consists of a single major result per chapter. They are:*Hippocrates’ quadrature of the lune*Euclid’s proof of the Pythagorean Theorem*Euclid’s proof of the infinitude of primes*Archimedes’ determination of a formula for circular area*Heron’s formula for triangular area*Cardano’s solution of the cubic*Netwon’s approximation of pi*Bernoulli’s proof of the divergence of the harmonic series*Euler’s evaluation of the infinite series 1+1/4+1/9+1/16+…*Euler’s refutation of Fermat’s conjecture*Cantor’s proof that the interval (0,1) is not countable*Cantor’s theorem that the power set of A has strictly greater cardinality than AEach of these theorems is surrounded by the historical discussion that makes this book a triumph not merely of teaching a dozen results to students but of actually educating students on the human enterprise of mathematics. It is not only interesting but, I think, important to be reminded of the human side of a field as abstract as mathematics, and Dunham bridges the mathematical and the biographical with remarkable dexterity. It is useful for the student of mathematics to understand that Cantor’s work on the transfinite was resisted by the mathematicians of his day just as much as students struggle with it when they’re exposed to it in today’s lecture halls. It might further be useful to know that, perhaps partly due to his demeanor and perhaps partly due to the attacks on his work, Cantor spent much of his life in mental hospitals–and yet, despite his unhappy life his work has achieved immortality as one of the great developments in mathematical history.I can’t recommend this book highly enough for the mathematician, the math student, or the merely curious. In fact, I recommend reading it twice. First, just read it straight through and enjoy the story of mathematics told through these vignettes. Then read it again with pencil and paper in hand and work through the theorems and proofs with the author as your guide. You’ll come away with a much deeper understanding of and appreciation for these great theorems in particular and mathematics in general.

⭐I had to buy this book for college. I enjoyed it and learned so much, that I bought two more of them as gifts for fellow math nerds. I cannot overstate how amazing this book is. If I would change a few things–there are not enough stories about women in mathematics and non-western mathematicians. An average high school students would have a difficult time with the contents, I believe.

⭐This is a wonderful book. People with a basic grasp of math who are open to the idea that math might be beautiful will be rewarded. But I have a PhD in math and thoroughly enjoyed it, and learned some things along the way. (Because math is taught very ahistorically, Chapter 1 was entirely unfamiliar to me).These are *not* “*The* Great Theorems of Mathematics,” as the subtitle suggests, but they certainly are “Great Theorems of Mathematics.” Most “Great Theorems” are too technical to be presented in a book of this sort, but Mr. Dunham has done a wonderful job selecting theorems that can be proved with a minimum of prerequisites. In some ways this is a more challenging task than choosing the “greatest” theorems.My main reservation is the fact that at times the proofs get more ponderous than necessary, and can wind up obscuring the simplicity and elegance of the mathematics. The most glaring example is the already-noted proof of Fermat’s Little Theorem (p. 226-9). The proof is incomplete, and presented in a very obscure way. The key fact, that (a+b)^p = a^p + b^p (mod p) follows easily and beautifully from the binomial theorem, so a complete proof could be given quite straightforwardly. I had the sense that some of the other theorems could have been presented somewhat more cleanly as well.The story behind Bernoulli’s proof of the divergence of the harmonic series is enjoyable, but Bernoulli’s proof is complex and unmotivated. Happily Mr. Dunham presents the beautiful proof Nicole Oresme from the 14th century. It is superior to Bernoulli’s in every way: shorter, more elegant, and more illuminating, since pursuing his line of thinking makes it clear that the series grows as the log of the number of terms. So it’s hard to see why Bernoulli is getting high marks for this particular proof, though he is overall a towering figure in the history of mathematics.Really, all my complaints are nit-picking. This is a wonderful book.I do want to defend Mr. Dunham from one of the other reviews: Euclid can prove (in modern language) that the area of a circle divided by the radius squared is a constant, and he can prove that the circumference divided by the diameter is a constant. But Euclid didn’t show that these are the *same* constant, and that is why Archimedes result can fairly be seen as “greater” than Euclid’s. Not that those theorems of Euclid’s were slouches by any means.

⭐This book has a quite interesting structure to it. Each chapter revolves around a specific mathematical theorem. The author then gives some backstory to the field of mathematics at the time, and then to the individual who happens to make a proof of this theorem. Then he presents the proof while trying to stay close to the original arguments, just presented in a somewhat modern way.I really liked this book. Especially interesting was the history and learning about these amazing intellectual giants. Even if you’re not great at mathematics, I would recommend this book. If you’re really weak at mathematics, I would possibly even recommend just skipping the proofs and simply reading the history. That’s okay!For me, the first few chapters were the least interesting because they’re mainly about geometry, a branch of mathematics that interests me less (Just personal preference). But even these chapters were great and the rest were just excellent!Highly recommended.

⭐This book was highly recommended to me after a chance encounter with a stranger on a group walk turned up a quite unexpected common interest in the history of mathematics. At first glance I was slightly disappointed in this slim paperback – it was first published in 1990 and some of the twelve topics didn’t look as if they’d be that interesting. But every chapter turned out to have interesting stories, well told. A worthy addition to other books of this genre, eg. by John Derbyshire (Unknown Quantity) and Marcus du Sautoy (The Music of the Primes).

⭐This is a fantastic book which traces the great concepts and inventions of Mathematics. If you are interested in the development of mathematics via the defining moments in it’s history this book should delight.Though I don’t considder myself to be particularly able in maths I do find it a very interesting subject so I was pleased with the way this book was written. It has about the right balance of discussion, history and examples (quite a few but the author makes it easy to follow the logic) – I thouroughly reccomend it!

⭐This is a thorough work about significant history in mathematics. It uses no advanced maths, and is easily readable: almost as quickly as a novel. Indeed I was amused that the author described Heron’s proof of his famous formula as a work similar to that of Agatha Christie, with disparate strands of investigation coming together in a denouement as brilliant as those of the great Poirot himself.

⭐My first copy was a present and I loved it from the first page. The historical thread means that it can be enjoyed as a narrative while skimming the surface of the maths; subsequent readings (preferably with pen and paper to hand) are enormously rewarding to those prepared to follow the clear mathematical expositions. The maths should be accessible to anyone who paid attention at 16+ exam level.William Dunham is a ‘must read’ author for anyone with an interest in this discipline.

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