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This classic remains one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. ... Read more

Customer Reviews (11)

A luminous beacon: Karl Popper's non-authoritarian system of knowledge
From the late 1960's to the mid 1970's I was a young person grappling with the question of what constituted truth.On the one hand, the conservative religious outlook of my rural upbringing was clearly ill-equipped to address the modern world.On the other hand, the Marxist views espoused by many of my peers were ridiculous.This book, however, and other books written by Karl Popper, cut though both of these views like so much paper.This book goes way beyond the high school notion of the scientific method.It exposes the intellectual poverty of approaches like astrology, Marxism, and Freudian Psychology, which offer an explanation for anything--but shrink from really putting themselves out there and daring to make a prediction whose failure will expose their shortcomings.Painful as it was at the time for me to realize that Christianity also fell into the same category as these other untestable conjectures, it showed me the way to a world view that respected no authority other than ideas that could be rigorously tested.

A must read
Any college level degree in any discipline is undeserved unless the student has understood the thesis of this book. It is long, sometimes difficult, and not always on the mark, but it is a must read to understand the nature of scientific knowledge. The gems are disperesed throughout, so one must be patient in discovering them.
The philosophy of science has been placed on a firm footing, and all pretenses to the notion that Science can discover Truth have been destroyed.

My conjecture on this book
Written with atypically clear prose and full of wit and insight.Sir Popper is truly one of the 20th century's greatest minds.

Conjectures and Refutations
This is Popper at his mature full authoritive best. This work connects his early classic on Open Society with the works of contemporary philosiphers as well as with the Greeks. Popper makes the connection with modern science and it's foundations from the ancients. Anyone how has failed to avail themselves of Poppers insights into knowing and learning is the poorer for it.

How do we know what we know? We don't, we only guess...
We guess. We make up a story that explains the phenomena we observe. Why is there night and day? Because the sun rises from the east and sets itself down in the west. Aha! Let's check that. Let's go east. After a while we realize that we're living on a sphere and so we know our story is wrong. The sun does not rise up and set. So what does it do? We're on a ball and the sun is above and it moves around us! Great! Problem solved. But wait! Let's check that. The stars also move. Let's plot their course, and then... and so on and so forth.

For Popper, this is how we've built up our picture of the world. We make bold conjectures to explain what we see see and then we check, or rather we establish a failure test. If the test is true, then our guess is false. But if we only test to see if a guess is false, rather than to try and prove that it's true, then how do we know our guess is true? Popper answers that we do not and cannot ever know that. Knowledge lies beyond our grasp, we can only seek knowledge without ever hoping to attain it.

This is the central theme of Conjectures and Refutations, which itself is a comprehensive overview of Popper's epistemological thought. The book is divided in two parts titled, you guessed it, I-Conjectures and II-Refutations.

Part I, Conjectures, comprises the first ten chapters. Popper begins with an overview of his thesis, then explores the nature of the problems that face people who think about the world and who act in it (chapter 2), he presents three other views of what we mean by knowledge (ch. 3), he describes tradition and the history of thought (chapters 4 to 6) and then turns to a critique of Kant, and of the effect that Newton's astonishingly successful theory of gravity, unrefuted for over 200 years, had on the development of western science (chapter 7). Popper then closes part I with discussions on what the difference is between science and metaphysics, on why mathematics works so well in describing the world, and finally on the nature of scientific knowledge, i.e. what do we mean by "knowledge".

In part II, Refutations, Popper does not present anything, he destroys existing theories which he feels are wrong. He returns to metaphysics and because it deals in existential statements that are irrefutable (e.g. you can't prove the Devil's existence or non-existence) he seeks to separate its pursuits from that of science. Science deals only in refutable statements.

Then Popper attacks other theories and problems. He gives a succinct summary of his attack on historicism, his name for the belief that History follows laws (e.g. Marxism) and that historians should be able to predict the future course of mankind.

In short, Conjectures and Refutations is a complete introduction to Popper's thinking. It sketches out all his thoughts on the social sciences and describes in fair detail his thinking on the development of science. He ends on a positive note. It may seem depressing that we can never obtain true knowledge, but we can certainly find sufficiently rich rewards in the pursuit of it. ... Read more

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A series of monastic notes, opinions, experiences, and reflections considers such issues as the ""death of God,"" politics, modern life and values, and racial strife. Reissue. ... Read more

Customer Reviews (10)

Not the place to start
If you are new to Thomas Merton, do not read this book first. There is lots of great writing in this book, but it is not an easy read. Almost everything here is excerpts from his journals or notebooks. Thus much of the material is unpolished and undeveloped. Rarely does the author stay on a single topic for more than four or five pages, and the new topic is frequently completely different.

Nevertheless, Merton's brilliance does occasionally burst forth. A great example is "In Louisville, at the corner of Fourth and Walnut, in the center of the shopping district, I was suddenly overwhelmed with the realization that I loved all those people, that they were mine and I theirs, that we could not be alien to one another even though we were total strangers...." which goes on for two pages, and is one of the finest descriptions of a mystical experience ever put down on paper. Because of passages such as this, I will never give up my personal copy of this book! The good writing is wonderful and, as a Merton fan for more than 20 years, I would not want to be without any of it. Yet, the best writing is mixed with other more ordinary, often repetitive, material. Probably because the book is not particularly well organized, there is no table of contents (although there is an index).

I read this book from cover to cover, rarely reading more than five or six pages during a sitting. It took me about six weeks. I recommend reading it the same way, cover to cover, so that none of the good material is missed. The book is best for people who have read other books by Thomas Merton. I recommend reading THE SEVEN STOREY MOUNTAIN, or NEW SEEDS OF CONTEMPLATION, or NO MAN IS AN ISLAND, or all three, or several others, before reading CONJECTURES.

One benefit of the unpolished nature of this material is that it allows the reader to begin to understand Merton as a person. So, after you read this book, if you really like it, you might proceed to read his journals and letters if you really want to understand this fascinating monk.

Conjectures
I bought this for someone.It is my understanding that this book is typically Merton as it is very "deep" in content.

Important Reflections in a greedy, Apathetic, and a Violent Age
Father Thomas Merton's (1915-1968) book CONJECTURES OF A GUILTY BYSTANDER was published in 1965 and is as timely now as when it was first published 45 years ago. Merton's book reflects serious thinking re the contemplative religious life, the false facade of our "civilization," the defiant refusal to think critically, and what Bonhoffer (1906-1945) called "cheap grace."

Merton began this book with thoughts re the Protestant clergyman Karl Barth (1886-1968) who recognized that sudden decline of manners, distortion of religious convictions, etc. Merton was clear that as Catholic and a Trappist Monk, he did not completely agree with Barth, but he thought Barth's concerns were serious and thoughtful. Merton, as readers may know, was also impressed with Buddhism and the work of D.T. Suziki (1870-1966). Merton was NOT a Buddhist, but he found "food for thought" in the work of the Buddhists and some Protestants. This may be akin to the comment made by St. Thomas Aquinas (1225-1274) who argued that truth is wherever honest men it.

As stated above Merton was a contemplative Trappist Monk. He DID NOT condemn the contemplative religious life, but he wrote some thoughtul criticisms. He agree that those who are Catholic religious should adhere to the rule of their order. However, he wrote that too many rules can make the contemplative religious life sterile and useless. He wrote about what some of the Buddhists call compassionate detatchment which Merton thought could turn into callousness. Metion suggested a careful balance.

Merton not only wrote about the Catholic cloistered life, he also had serious concern about the secular world of which he had little respect. Merton said that due to the worship of greed and unreasoned violence, one could not distingush "Christians" from atheists. Greed and materialism have destroyed serious religious convictions and values. Among the distortions that Merton noticed were the shallow views of God and political power. Merton wisely wrote that political power is transitory while God is permanent. Merton also noticed the obvious that political labels attached to religion undermined the true meaning of religion.Merton remarked that Americans were ruled by predjudice,weakness, clever but false political slogans, etc which enslave men and women. On the other hand,Ghandi and St. Francis of Assisi (1180-1226) were among the freest men in history. Basically, Merton thught men sold their souls to avoid facing truth and to evade honesty.

One of the reasons that Merton gave for the shabbiness of contempoary "civilaztion" was that men have been trapped by the corruption of language and have marginalized andcondemned such virtues such as compassion, mercy, thoughtfulness, etc. Men prefer violence and hypocrisy to shield themselves from uncomfortable truth. Merton DID NOT argue that freedom was perfect and was honest enough to admit that freedom has limits. He warned his readers not to be placated by illusions which can mean being manipulated by those with political power and wealth. One of the "traps" used by political hucksters is the cliam that God is here with them and not there among other nations and those who are "different." As an aside Merton was obviously Catholic, and Catholic menas universal-not national or local.

Merton had harsh condemnation of contemporty society. He saw urban culture as one of useless "progress" and war. Merton warned readers that preoccupation with technology, war, etc. was an attack on God's Creation-nature. Merton rhetorically asked that if nature can be cruel, is nature as cruel as men. Merton cited the Psalms with their praise of God's Creation or nature. Basically, Merton thught that men and women have become so shallow that they are bored with God's Creation and have become intellectually and spiritually dead. Marton had no repsect for the Marxists. He condemned the Marxists because they hated men for what they supposedly are not what they ought to be. Merton had high regard for Sacred Music, and complimented Johh Nenry Cardinal Newman (1801-1890) for his appreciation of Sacred Music which is not only an escape from foolishness but is "uplifting" in praise of God.

As might be expected, Merton condemned the Cold War and had some amusing comments about it. When folks were installing bomb shelters, the stupid mentality that after a nuclear war, folks would emerge from their bomb shelters to renew "The American Way of Life." Merton considered the series of Berlin Crises as more sport rather than serious diplomatic issues. Men were too easily fooled by phony crises which Orwell (1903-1950) warned against in his book 1984.

Merton dealt with Catholic Theology as well as contempoary secular "culture." Merton had good material re St. Thomas Aquinas. Merton thought that Aquinas used philosophy and theology to open the world to men as God's Creation to be explored and enjoyed. Aquinas knew that his thinking was radical during his lifetime,but Aquinas thought his work was too important to be avoided. Aquinas and his mentor St. Albertus Magnes (1193-1280) appreciated the work of Aristotle (384-322 BC), but they NEVER made a cult of Aristotle's work. Merton critisized some to the Thomists for again closing the world by cliaming that Aquinas "was the last word"-a claim that Aquinas would have shunned.

Again, Merton DID NOT condemn the Catholic contemplative religious life. When secular critics argued that the cloistered religious life was not productive, Merton responded that those who made that claim approved of nuclear weapons and possibly the detruction of God's Creation and life itself. Such contradictory thinking was/is due the lack of respect for God, His Creation, etc. As Merton so aptly described men, they have become "human grasshoppers." Another amusing anecdtote that Merton made was whether men worship God or do they want God to bless nuclear weapons which would destroy God's Creation. In other words, war is an insult to God.

Merton was also critical of those who were obsessed about "saving their soul." Merton thought such concern would lead to inaction, apathy to problems, the denial of mercy and love, etc. Merton argued that God's power (whatever that is) was reflected in mercy, love, forgiveness, etc. Perhpas they late US Supreme Court Justice Oliver Wendal Holmes described it best when he wrote, "Those who worry that their soul is damned generally speaking have a soul that is not worth a damn"

As Merton noted too many people were/are impressed with ecnomic and political power. Merton noted St. Anselm's (1033-1107) remark that Christ refused power to save men and woman. Salvation, whatever that means, can be achieved by giving up power. Merton was concerned that what men should condemn are sin and evil. To attack others because they are"different" while not condemning evil and sin make men shallow and mean spirited.

The undersigned enjoyed this book which took time to read. This writer knows that others are more knowledgeable about Merton's books and will accept constructive criticism. This is a book to be read and reread to be digested and enjoyed.



James E. Egolf
July 23, 2010

Mature Merton
This is a collection of Merton's later writing and reflects a tolerance and recognition of other religious traditions that was lacking in his famous autobiographical work, the "Seven Story Mountain".This is Merton in his mature years, reflective yet critical and thankfully, less dense than some of his earlier writing.His commentary on the personalities and events of his day means that the work has an inevitably dated feel.But the perennial wisdom of the author guarantees a point of view that mainly transcends the issue of time.An essential work I would suggest, for any follower of the Merton canon.

Monk Shares Insights with a Busy and Violent World
Thomas Merton was a troubadour of contemplative life from America's Gesthemane Abbey.His books, including the famous autobiography 'Seven Storey Mountain' have made him one of the greatest spiritual writers from America.While this reviewer has read nearly a dozen of his sage works, 'Conjectures of a Guilty Bystander' is among his most colorful and engaging.Journal-like in its presentation and conception, Merton reflects upon the headlines of the Black Movement, Kennedy's assassination, and the Cuban Missile crisis with great depth and insight.The wisdom he provides doesn't date the topics he covers.'Conjectures' would be fine as a historical document, but his commentary provides more than an antidote for history repeating itself.There are also trappings--no pun intended--of his little anecdotes of the monk's life.His observations of new candidates and the liturgical calendar hold simple truths that we can embrace with the variety of seasons.

I would hope every Catholic, and every non-Catholic, would embrace this book.It straddles the value of Eastern spirituality and widens the scope of Catholic experience.While many conservatives embrace G.K. Chesterton and Peter Kreeft as beacons of light and truth, Thomas Merton expands the scope and splendour of that truth without contradiction.Personally, I loved the part where Merton talked about medeival "Passion" plays demonizing Jews.He railed against the practice, and I read it just when the controversy about 'Passion of the Christ' was brewing, just before its release. I could see why there was so much trepidation after reading his historical perspective. ... Read more

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Customer Reviews (25)

Great book for mathematically inclined readers
I really enjoyed reading this book - probably more than any other popular mathematics book.

As other reviewers point out, the book covers a lot of material in little space. For me, this level of detail was just right - specific enough to actually learn something but not enough to make it dry and boring. As a computer scientist, I am familiar with many topological and algebraic concepts. I can imagine the book being too dense for someone with little mathematical background, but I found it to be just perfect.

The main story is very well organized with the focus on topology nicely combined with relevant and interesting historical facts. The author stays on topic and does not go into contributions irrelevant to topology.

Poincare's Conjecture brought to light
This is for the most part a detailed, dispassionate account of the pursuit for a solution to Poincare's Conjecture which, although somewhat difficult to grasp for non-mathematicians, is nonetheless insightful and intriguing.The author conveys the excitement and profound consequences of solving one of the most important and difficult problems of the 20th and 21st century mathematics, giving context to the pursuit both historically and scientifically. Up to theclimatic account of the solution by an enigmatic reclusive russian whose resemblance to Rasputin is uncanny and the attempt by a pair of unscrupulous mathematicians to usurp his achievement, the book delivers on all levels to entertain and elucidate the importance of the conjecture and the people, some of whom devoted their lives, who became enthralled by it. A good follow up to this book which essentially is a biography of the reclusive Grigory Perelman is Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century

Wonderful popular math
DO does a very good job of explaining the basis of Poincare's famous conjecture and of Perlman's proof of it.As with all popular treatments of highly abstruse mathematics, the non-initiate inevitably reaches a point of conceptual overload.It simply is not possible to form a crystal clear idea of concepts under discussion absent a formal understanding of the mathematics itself.And so it was for me with TPC.That's to be expected, though.Popularizations are a species of translation.For the very strong impression he gave me of Poincare's conjecture and its relation to geometry, topology, and questions to do with the shape of the universe, I would rate DO a very capable translator.In the final analysis, I am only marginally better able now to describe the Poincare Conjecture and its proof than I was before I read TPC.But having read TPC I now have a much stronger notion of the ideas that surround the conjecture, and a deeper appreciation for the beauty of complex mathematics.TPC is a terrific book.

The Poincare Conjecture
This book is well written. A very complicated subject written in terms that average intellectual
people can understand.The subject matter deals with material that most people have never
contemplated.The Perfect Rigor is the story about Grigory Perelman, a mathematical genius who
recently solved the Poincare Conjecture.Anyone who reads The Poincare Conjecture will have a
great appreciation for the brilliance of Mr. Perelman.

The Kindle edition illustrations need work
The book is excellent and worth rereading.But Amazon should fix those illustrations which are too small that enlarge into pixelated nothings.Is geometry spelled with acute first e intentionally to denote the structure rather than the area of study? If so it isn't clear and I never saw it elsewhere.Does the printed book have this? And fix the years which start with i instead of 1. ... Read more

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The innumerate may initially find this undramatic material for a novel.Yet Doxiadis offers up a beautifully imagined narrative, which reveals ararefied world of the intellect that few people will ever enter, in whichnumbers are entirely animate entities, each possessed of "a distinctpersonality." Without ever alienating the reader, he demonstrates theenchantments of this art as well as the ambition, envy, and search for glorythat permeate its apostles. Balancing the narrator's own awkward move intoadulthood with the painful memories of his brilliant relative, Doxiadisshows how seductive the world of numbers can be, and how cruel a mistress."A mathematician is born, not made," Petros declares--an inheritance thatproves both a curse and a gift. --Burhan Tufail ... Read more

Customer Reviews (26)

Read it before starting your PhD
It is a very fascinating mathematical story about a bright mind who spent his life on a very difficult problem. The book is very well written and it gives a lot of mathematical details the most important ones about number theory Godel's theorem etc. If you are a PhD student you will identify the dilemmas you have encountered when hitting on a dead end in your research. If you are not a student you will still enjoy the endeavor in the mathematical challenges of the past century. I am in general a slow reader but I read it in one night.

well done novel that serves as light and entertaining reading about a mathematical mind obsessed with an unsolved math problem
Uncle Petros & Goldbach's Conjecture is a well written book about how a ambitious and brilliant mathematician gets caught up in the pursuit of an unsolved mathematics problem to the exclusion of all other aspects of life. The story is told through the eyes of his nephew who progresses from childhood to middle age during the course of the book. Awed and fascinated by his mysterious uncle, the boy becomes enamored of mathematics himself. Since I don't want to give away the plot, I will just say that the lives of the uncle and nephew are intertwined throughout the book in an exciting way that keeps you wondering what will happen until the very satisfying ending. I highly recommend this book both for the writing style of the author and for its subject matter.

I dont like this book
I want to make a short comment here:
1) This book is quite confusing. It mixed up the real story with the make up one.
2) Saying that Ramanujun thought Goldbach Conjecture is wrong was a huge mistake. Everyone believe that Goldbach Conjecture is true. And Ramanujun is a real genius. I cannot stand this make up part.

A charming small novel
Short, simple, and totally engaging.Just the thing to have with you on a trip, or to sit down with for a pleasant afternoon.

fun read for the mathematicallly curious
I picked up this book after watching the movie A Beautiful Mind -- it's a delightful, easy-to-read novel about an aspiring mathemetician who tries to get to know the truth about his eccentric mathemetician uncle.The book details the uncle's life-long struggle to solve Goldbach's Conjecture, which (for you non-mathemeticians, like me) posits that every even number greater than two is the sum of two primes.I felt like the novel showed a somewhat realistic perspective of a driven, obsessed scientist, and showed how the scientist's activities and behaviors affected and even alienated those around him.The protagonist is the only member of the mathemetician's family who bothers to get to know and appreciate him -- and that is a struggle, as the mathemetician often does things to alienate his nephew.Don't be put off by the math; it's actuallyq quite easy to read. ... Read more

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For over 100 years the Poincaré Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. In 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincaré Conjecture in the affirmative. This book provides full details of a complete proof of the Poincaré Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincaré Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate. The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincaré Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. In addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA). ... Read more

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A proof of the Poincare conjecture?
For those readers who sincerely want to understand the recent proposed proof of the Poincare conjecture due to the mathematician Grisha Perelman, this book is the only one so far that claims to discuss the proof in the detail required. If one is to judge a book solely by its introduction, then this one is off to a good start, for the authors give a quick overview of the strategy behind the proof along with a discussion of what background is needed to understand it. Those readers who come to the book with only a background in geometric topology will need to become familiar with various techniques and concepts in analysis and differential geometry. In the introduction the authors inform the reader that this background will be developed in later chapters, but to really understand the proof one needs an understanding of it that goes beyond the formal. The Ricci flow plays a central role in the proof, and the Ricci flow equation is presented in the introduction as a nonlinear version of the heat equation. The physicist reader will appreciate this description and perhaps wonder if it, along with the strategy of understanding the topology of manifolds that are "unions of epsilon-tubes and epsilon-caps" is best done in a quantum framework due to the its emphasis on distances and scale (the "capping" phenomena that is discussed in the introduction is somewhat reminiscent of the regularization and "smoothing" that goes on at small scales when studying quantum time evolutions or in the symplectic category the notion of capacity).

Readers who insist on constructive proofs every step of the way may be disappointed to learn that the authors make frequent use of non-constructive strategies. An example of this is the proof of non-collapse where constructions are made that contradict the maximum principle of Hamilton. Another example is the establishment of the existence of canonical neighborhoods for all points of large scalar curvature. A contradiction is derived by assuming that one can find a particular sequence of points of arbitrarily large curvature where there are no canonical neighborhoods.

Refreshingly, the authors do not hesitate to use diagrams in the book, increasing its didactic quality and making the mathematical constructions much easier to follow. A purely formal treatment would make the reading much more intense but no doubt readers would still draw their own diagrams if the authors chose to write the book in such a fashion. So many of modern mathematical papers and books are written this way, making their understanding much more time-consuming and making proof checking extremely difficult.

Chapter one is fairly standard background in the differential geometry of Riemannian manifolds and not all proofs are given. Readers are expected to either consult the references or fill in the details of the proofs themselves if so inclined. Of particular interest is the notion of an open cone over a Riemannian manifold and how to compare curvature in Riemannian manifolds (the proof of the Bishop-Gromov relative volume comparison theorem is omitted).

In chapter two the authors concentrate on the Riemannian geometry of manifolds of non-negative curvature. One of the more interesting questions in this context is to what extent a geodesic ray is minimizing, which for the case of zero curvature is illustrated by straight lines (viewed as limits of geodesics getting longer and longer). Busemann functions are introduced, which measure how far a point in a complete, noncompact, non-negative curvature manifold is out toward infinity in the direction of a geodesic ray, and the authors illustrate their most useful properties (such as Lipschitz continuity, which is crucial since distance is usually not a smooth function on Riemannian manifolds). Their definition of the Busemann function is different from some in the literature by a simple change of sign, which only has the effect of making the Busemann functions pointwise bounded below (instead of pointwise bounded above as some do). Proposition 2.3 is important, in that it shows that the Laplacian of a Busemann function is non-negative (i.e. that it is "superharmonic"). This property is used later to derive a maximum principle for complete Riemannian manifolds of non-negative Ricci curvature.

Also introduced in Chapter 2 is the Toponogov theory, which allows one to compare lengths in complete Riemannian manifolds with those in Euclidean space. Specifically one constructs a triangle in such a Riemannian manifold whose sides are minimizing geodesics and estimates one side in terms of the other two sides and the included angle. This is followed by a discussion of the `soul' of a Riemannian manifold. In Lemma 2.9 the authors do not explicitly construct the compact subset K but instead prove its existence by contradiction. In the proof of Corollary 2.11 the authors refer to the "ends" of a manifold before they introduce them in the next section, wherein they are defined in the usual way. Section 2.5 makes use of the fact, proved earlier, that the Busemann functions are superharmonic, in order to derive a `maximum principle' for connected Riemmanian manifolds. This in turn is used to prove the `splitting theorem', namely that a complete Riemannian manifold with non-negative Ricci curvature and two ends is isometric to a product of a compact manifold with the real numbers.

The most important part of Chapter 2 concerns the notion of a `epsilon-neck structure' on a Riemannian manifold and the accompanying notion of an `epsilon-neck', which from the definition and the author's remarks is essentially an extended round cylinder. The authors show that a complete, positively curved Riemannian 3-manifold cannot contain an epsilon-neck of arbitrarily small scale (with scale being inversely related to the Ricci curvature). In the proof of this assertion it is not clear how they substantiate expression (2.2) in the proof of Proposition 2.19. Also, no concrete examples of epsilon-neck structures are given but the diagrams in the introduction assist in their understanding.

The authors begin the study of Ricci flow in Chapter 3, viewing it first as a nonlinear version of the heat equation (for the Riemannian metric). This discussion is rather hurried, leaving the details of how to express the Ricci tensor in terms of local harmonic coordinates to the references. The use of harmonic coordinates guarantees that the Ricci tensor is only dependent on terms quadratic in the metric and its derivatives. In the discussion on shrinking solitons, the Lie derivative appears in expression (3.2) but is not introduced anywhere before then. The proof of local existence and uniqueness for Ricci manifolds is only sketched using the "DeTurck trick" which breaks the gauge invariance (under the diffeomorphism group) of the Ricci flow. The resulting flow, called the Ricci-DeTurck flow, is then strictly parabolic. A lot of space is then devoted to "index gymnastics" in the authors' discussion on the evolution of curvatures in an orthonormal frame. In addition, very detailed computations are given for how distances behave under Ricci flows and in estimation of the derivatives of the curvature under a Ricci flow. At this stage, the reader will have to take the authors word for the utility of these computations. The page space might have been used more productively to discuss epsilon-neck structures in greater detail than what is done so far in the book (with concrete examples given of these entities).

The view of the Ricci flow as being essentially a heat equation for the time evolution of the metrics is continued in chapter 4, where the authors generalize the famous (strong) maximum principles for the heat equation to the case of Ricci flow: since the heat equation is parabolic, then positivity of its solution at the initial time guarantees that positivity for all times in the future. The authors prove the analog of this first for scalar curvature as a warm-up, and then prove the maximum principle for tensors. The gauge invariance of the Ricci flow equation is then broken in order to prove a maximum principle for the Ricci curvature. All of these considerations lead the authors to proving `strong' maximum principles for Riemann and Ricci curvature.

The proof of Theorem 4.18 is not clear at first reading since the authors seem to merely restate the hypothesis of the theorem instead of assuming that the conclusion is not true and deriving a contradiction. They should have perhaps informed the reader right away that they intend to use continuity to derive a contradiction by assuming flatness for a choice of time less than the final time. ... Read more

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Ladies: A Conjecture of Personalities is a book of voices. First Ladies between Martha Washington and Mamie Eisenhower tell their own stories—or, to be more exact, whatever they want—in their own words and in their own styles. Ladies: A Conjecture of Personalities crosses boundaries between fact, conjecture, and, most importantly, centuries. Through dialogue-boxes, the ladies talk to each other across eternity, where anything is possible. The modern First Ladies, from Mrs. Kennedy through Mrs. Clinton, participate in commentary. They talk to the reader and they talk amongst themselves. And they sympathize, empathize, and quarrel amongst themselves. They talk about their husbands, their children, the White House, and the times they lived in. And, of course, politics. It’s chatty. It’s catty. It’s fun. It’s informative. It’s a must-read for anyone interested in history. ... Read more

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A Delightful Story of the First Ladies of the Land
Lucy Hayes, wife of President Rutherford B. Hayes and Carrie Harrison, wife of President Benjamin Harrison, challenge the other First Ladies to speak in their own voices and write what it was like to be the wife of the President of the United States or whatever they want really. The book ends with Mamie Eisenhower, because we know about the later First Ladies. However, they are not left out, as they can't help interrupting the others' chapters. None of these ladies ever really learned to be silent!

What follows Lucy's challenge are twenty-nine chapters written in the Ladies' own voices. Feather has studied Presidential history for over thirty-five years and owns more than 1000 president-related volumes. She delved into the Ladies' lives and in her book they speak in their own voices with their own stories of experiences, hopes, dreams and thoughts.

Most of these women never intended to be First Ladies. Many found more sorrow than joy in the experience. The role of First Lady changed through the years as did the White House. Some were great entertainers and others wanted to stay to themselves. Some were ill or disabled. In this book, they speak from their own times, which influenced their lives differently than if they lived today. They usually talk about their whole lives, including the occurrences before and after they were First Ladies.

What is amazing is the diversity among these ladies. Their sorrows were great, as seventeen of them lost children, including Edith Roosevelt who lost sons to war. Four had husbands who were assassinated, and the Ladies tell of the scars these experiences left. Many wanted nothing more than to go home after their husbands' terms and sit with them on the porch of their home in another place and enjoy a simple life. Some did that. Many never got that opportunity.

They talk of feelings of wayward husbands, difficulties with in-laws, and their children's successes and failures. All of them are all delightful. They were all valuable people in the history of our country, some very much involved in the support of their husbands and their role as President, and others in the background, but all affected by their position.

Feather spent the first five years trying to make this work as a play. She says it was too confusing, and it was also about nine and a half hours long! The next two years were trying to figure out how to write it as a book. Once she got the basic idea of self-written chapters and dialogue boxes, the rest came a lot easier. Her viewpoints are fresh and unbiased, using the dialogue boxes where other ladies interrupt to add opinions and further information.

As a historical novelist, I truly believe in the value of learning history. Here the author teaches in such a delightful way, the learning comes with no effort. The reader learns more than dates and facts. Even if it is only conjecture, it is conjecture based on a great deal of study and thought.

History Through The Eyes ofWomen History Forgot
When I picked up Feather Schwartz Foster's book, "Ladies: A Conjecture of Personalities", I'm not sure what I was expecting.Short essays about some of the First Ladies -- woo, hoo. These women aren't "has beens", or even "almost has beens" -- they are the consorts of has beens.Who remembers Lou Hoover or Sarah Polk?How about Julia Tyler?Some of these names are so obscure as to have dropped completely out of public awareness.How interesting could they be?

Boy was I wrong!

First off, the book itself is creatively constructed -- each lady is given a chance to tell her story while the others comment.It's like a big hen party where the ladies mingle.I imagine them sipping Earl Grey and eating snickerdoodles in homey companionability.This approach makes even the dullest of the presidential wives amusing.We see them through their own eyes within the context of their own historical time -- but we also see them through the personalities, mores and historical perspectives of other First Ladies.

For example, Mary Lincoln, Jacqueline Kennedy and Pat Nixon's snotty remarks about Andrew Johnson's obnoxious behavior at his inaugeration are in sharp contrast to Eliza Johnson's insistence that her Andy wasn't a drunk.The technique demonstrates how blind we are to the failings of our loved ones -- and how quick we notice the faults of others.It allows characters long lost in the fog of time to become human again.

I have to say that I loved this book from the first page, but when the various ladies began to chime in with their own reminiscences, catty remarks, sympathies and empathies, I found myself reading -- and rereading, certain passages with new respect for the impossible circumstances these women found themselves in simply because they were married to presidents.

Clever, informative and amusing, "Ladies" is well worth the read.

Ladies: A Conjecture of Personalities, Wit & Wisdom
Never before have I enjoyed a romp through history so thoroughly.From the introduction by Lucy Hayes to the author's epilogue claiming fictional, yet plausible, conjecture on the part of the "voices" who wrote their stories, a full and satisfying story is told.The reader feels what our First Ladies are re-living as they tell about their place in history-or lack thereof, about their husbands and about the times in which they lived.

I got the distinct feeling several of our First Ladies knew they would have made better Presidents than their husbands did, if the times had allowed.In reality, only the times have changed; we humans are as raw or refined, as wild or as tame, as selfish or as giving, regardless of the century we inhabit.

Through the many friendships and rivalries, the politics, and the "place" each woman was expected to inhabit, we learn how they lived and loved.To fully appreciate these women who were our First Ladies, we must allow them to live in our minds, to breathe and function, to grieve and rejoice.Feather Schwartz Foster brings the stage and the characters to us.All we need to do is sit back and enjoy the ride!

If they only knew...
Feather Schwartz Foster, in her book Ladies: A conjecture of Personalities, has done exquisite research on the First Ladies of the past. It is a wonderfully unique idea to weave the lives, similarities and differences of this part of our history through each others experiences.
One can only ponder some of the ideas she presupposes. Having done a lot of research myself on many of the First Ladies, her "fiction" rings true-to-life. I was fortunate enough to spend one afternoon and evening with Ike and Mamie Eisenhower. Her description of Mamie and her eclectic tastes took me back to that memorable day in my own life.
Whether you are a history buff or not, this is a great read to tweak your curiosity.

Delightful compilation of what
"Ladies: A Conjecture of Personalities" is a book filled with the voices of America's First Ladies.From Martha Washington to Mamie Eisehnhower, these vibrant, lively women speak from the past as modern day First Ladies comment from the sidelines.

The "moderns," as Lucy Hayes comments, already had their say.

Author Feather Schwartz Foster brings to life these fascinating women through dedicated chapters, written in the form of a letter from each Lady.Customs, houseguests, dinner, chores - no part of a woman's life is ignored.

Though this is a work of fiction, it is very much based on facts and filled with vivid details taken from over a thousand presidential history books.Readers will be lost in the past and feel as though they have seen a glimpse of each personality portrayed.

This book is unique in more ways than one.Comments from the "moderns" are interspersed throughout as they speak to one another as well as to the letter writers.They argue, compliment, agree and turn up their noses.

Particularly poignant is Mary Todd Lincoln's chapter in which the wives of other assassinated presidents speak with sympathy and empathy, while other wives in turn speak ill of her appearance and state of mind, even going so far as to malign her beloved son.

The wives of our presidents, both past and present, inherited by marriage a position of influence.From slaves to maids, to leaders of other countries, diplomats and friends, the influence of a First Lady can be measured only by the legacy she leaves.

"Ladies" will absorb the reader's attention from the first page to the last.

Wise, naïve, elegant, crass, refined and catty; all personalities are represented in this unique ode to the women who stood by America's presidents.

Feather Schwartz Foster's first foray into historical fiction is a stand-up clapping success.She is an accomplished song writer and has three decades of experience in advertising and public relations.Numerous links to her upcoming appearances and speaking engagements are listed, as well as links to interviews with Foster about "Ladies" [...]
... Read more

Product Description

The development of mathematical competence -- both by humans as a species over millennia and by individuals over their lifetimes -- is a fascinating aspect of human cognition.

This book explores when and why the rudiments of mathematical capability first appeared among human beings, what its fundamental concepts are, and how and why it has grown into the richly branching complex of specialties that it is today. It discusses whether the ‘truths’ of mathematics are discoveries or inventions, and what prompts the emergence of concepts that appear to be descriptive of nothing in human experience. Also covered is the role of esthetics in mathematics: What exactly are mathematicians seeing when they describe a mathematical entity as ‘beautiful’? There is discussion of whether mathematical disability is distinguishable from a general cognitive deficit and whether the potential for mathematical reasoning is best developed through instruction.

This volume is unique in the vast range of psychological questions it covers, as revealed in the work habits and products of numerous mathematicians. It provides fascinating reading for researchers and students with an interest in cognition in general and mathematical cognition in particular. Instructors of mathematics will also find the book’s insights illuminating.

Product Description
Michael O’Brien has masterfully abridged his award-winning two-volume intellectual history of the Old South, Conjectures of Order, depicting a culture that was simultaneously national, postcolonial, and imperial, influenced by European intellectual traditions, yet also deeply implicated in the making of the American mind.

Here O’Brien succinctly and fluidly surveys the lives and works of many significant Southern intellectuals, including John C. Calhoun, Louisa McCord, James Henley Thornwell, and George Fitzhugh. Looking over the period, O'Brien identifies a movement from Enlightenment ideas of order to a Romanticism concerned with the ambivalences of personal and social identity, and finally, by the 1850s, to an early realist sensibility. He offers a new understanding of the South by describing a place neither monolithic nor out of touch, but conflicted, mobile, and ambitious to integrate modern intellectual developments into its tense and idiosyncratic social experience.
Michael O’Brien has masterfully abridged his award-winning two-volume intellectual history of the Old South, Conjectures of Order, depicting a culture that was simultaneously national, postcolonial, and imperial, influenced by European intellectual traditions, yet also deeply implicated in the making of the American mind.

Here O’Brien succinctly and fluidly surveys the lives and works of many significant Southern intellectuals, including John C. Calhoun, Louisa McCord, James Henley Thornwell, and George Fitzhugh. Looking over the period, O'Brien identifies a movement from Enlightenment ideas of order to a Romanticism concerned with the ambivalences of personal and social identity, and finally, by the 1850s, to an early realist sensibility. He offers a new understanding of the South by describing a place neither monolithic nor out of touch, but conflicted, mobile, and ambitious to integrate modern intellectual developments into its tense and idiosyncratic social experience.
... Read more

Product Description
Before Pascal and Fermat's discovery of the mathematics of probability in 1654,how did we make reliable predictions? What methods in law, science, commerce, philosophy,and logic helped us to get at the truth in cases where certainty was not attainable? In TheScience of Conjecture, James Franklin examines how judges, witch inquisitors, and juriesevaluated evidence; scientists weighed reasons for and against scientific theories; and merchantscounted shipwrecks to determine insurance rates. Sometimes this type of reasoning avoidednumbers entirely, as in the legal standard of "proof beyond a reasonable doubt"; at other times itinvolved rough numerical estimates, as in gambling odds or the level of risk in chance events.

The Science of Conjecture provides a history of rational methods of dealing withuncertainty. Everyone can take a rough account of risk, Franklin argues, but understanding theprinciples of probability and using them to improve performance poses serious problems, thesolution to which we have only learned over many generations and after much trial and error.This study explores the coming to consciousness of the human understanding of risk. ... Read more

Customer Reviews (4)

Brilliant and Dazzling
Franklin has done a brilliant job of showing that the concept of probability usedby practically all civilizations throughout history[for instance,Hebrew(Talmud),Greek(Aristotle),Medieval(Scholastics),etc.],prior to Pascal's attempt to mathematicalize the meaning of the term probability so as to help out some French noblemen who were trying to find a way to get an edge in certain types of gambling ,was the inductive,qualitative,comparative concept based on recognizing probability as being a primarily nonnumerical,vague concept that could not be precisely defined.Its usage was built into the common languages that human beings developed over time.Franklin's book supports the original logical interpretation of probability first put forth in published form in 1921 in the A Treatise on Probability(TP) by John Maynard Keynes.Keynes's theory is a general theory which analyzes probability from the most general use of the term(qualitative,nonnumerical,comparative) to its most specialized forms(quantitative,numerical,interval,mathematical,statistical,frequency,subjective).There are a few places in this book where the author could have given a substantially better analysis.One place is where he discusses Keynes's concept of the weight of the evidence.He bases his discussion only on chapter 6 of the TP,ignoring Keynes's additional discussions in chapter 14,section 3 and chapter 26,where Keynes became the first scholar in history to both define weight,w,on the unit interval between 0 and 1,and to invent a decision rule,his conventional coefficient of risk and weight,c,which solves a number of the paradoxes of decision theory, as well as a number of the anomalies rediscovered by Tversky and Kahneman.Franklin also fails to point out Keynes's original development of an interval valued approach to probability in chapters 15 and 17 of the TP.The scholar who comes closest to Keynes in incorporating a clearcut role for a vague,comparative,qualitative understanding of probability is D.Ellsberg.

Extraordinarily lucid account of abstruse subjects
This is the sort of book whose indispensability creeps up on you: you start it without any idea that you'll require it to broaden your mind, but it insidiously works its magic. Totally unclassifiable -- it mixes the disciplines of history, mathematics, philosophy and jurisprudence -- it also happens to be a rivettingly lucid read, notwithstanding the outwardly abstruse nature of its materials.

The science of common sense.
This book is about common sense: the way ordinary people have reasoned about the world since the beginning of recorded history. It is a compendium of anecdotes, about anecdotal thinking. I find the insights engrossing, entertaining, and scholarly-if not scientific. This book hopes to rigorously analyze the processes that thinkers have followed throughout recorded history in order to reach rational conclusions. These processes are interesting in their history of use as official rules of thumb, but they are fatally flawed. The fundamental flaw is that the nonscientific processes are not reasoning- they are persuasion, as in rhetoric. Being nonscientific in nature,"The Science of Conjecture" is misnamed, but quite worthwhile to those of us who would like to understand the lawyer and jurors' mind.

A great read on the development of our modern thinking
If you read "Sophies World" by Jostein Gaardner and wanted something with more bite, this book is it. It's one of the few truly intellectual books I've read without being academic or boring. I had no idea how much we take certain things in our 21st century thinking for granted. One example is juries and innocence until proven guilty. The book is a marvelous history of legal and ethical thinking and how we came to civilized methods to deal with charges of guilt. It makes me aware of the manipulative power of different styles of logical arguments. Buts it's not only about law. The author explains why Islam is fundamental (God can't be wrong) so why bother considering pros and cons of situations. Christianity was lucky to have the reformation and counter-reformation to challenge why there is probability/chance or unknowing. There are great sections on scientific theory - reasoning for hard sciences like physics and astronomy. Why cannot astrology be a science? Because there are no hard rules; too much depends on the art or experience of interpreters who "explain" exceptions to rules, because so many situations don't follow their rules. The sections on soft science describe biology and medicine, and the impact of clinical trials. How did we arrive at "scientific thinking" to establish proofs? Its all here. I'm not into mathematics and the title sounded so boring to me - mathematics and before the 16th century ie Pascal. If ever there was a case for "don't judge a book by its cover" this is it. Its solid reading, but it is also deeply satisfying and fascinating in understanding a little bit more about how and why we think like do in the 21st century. As an aside the author is also a Latin scholar who translates many texts, correcting false interpretations. But he does it in subtle ways; nothing show-off. James Franklin dazzles us with his humility one moment and superb, accessible writing on complicated subjects the next moment. I never knew that "like" and "probably" were introduced from Greek. Medieval Europeans did not have sophisticated languages that included "like" or "probably" but with medieval enlightment they were introduced. What an impact these two words had. The author corrects cultural misthinking of how poor medieval thinking was. It was an explosion of brilliance in virtually a person's lifetime from 1150-1200. The Renaissance was mild in comparison. This book touches and explains our human development of consciousness and thinking in so many fields eg law, medicine, science, ethics. The author draws on Ancient Greek texts, Roman texts, the Talmud, Jewish philosophers, Islamic philosophers, Christian theologians and even Sanskrit writings.The subjects discussed heavily affect my daily life and thinking. Understanding a little bit of what we take for granted, makes me reconsider glib, slick arguments I'm confronted with in newspapers and television every day. If you buy the book, it's a great read over 1-2 months that can be savored and sipped like a great wine. ... Read more

Product Description
This book provides an introduction to a topic of central interest in transcendental algebraic geometry: the Hodge conjecture. Consisting of 15 lectures plus addenda and appendices, the volume is based on a series of lectures delivered by Professor Lewis at the Centre de Recherches Mathematiques (CRM). The book is a self-contained presentation, completely devoted to the Hodge conjecture and related topics. It includes many examples, and most results are completely proven or sketched. The motivation behind many of the results and background material is provided. This comprehensive approach to the book gives it a ``user-friendly'' style. Readers need not search elsewhere for various results. The book is suitable for use as a text for a topics course in algebraic geometry; includes an appendix by B. Brent Gordon. ... Read more

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This introduction to recent developments in algebraic combinatorics illustrates how research in mathematics actually progresses.The author recounts the dramatic search for and discovery of a proof of a counting formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices.While it was apparent that the conjecture must be true, the proof was elusive. As a result, researchers became drawn to this problem and made connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young; to partitions and plane partitions; to symmetric functions; to hypergeometric and basic hypergeometric series; and, finally, to the six-vertex model of statistical mechanics. This volume is accessible to anyone with a knowledge of linear algebra, and it includes extensive exercises and Mathematica programs to help facilitate personal exploration. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something unique within Proofs and Confirmations. ... Read more

Customer Reviews (3)

Fantastic!
Everyone wants to know what mathematicians do should read this book. Yes, only very few among people can get familiar or endure gorgeous complicated formulas here, But even scan or skim this book one will learn much morethan he predicted. The goal of this book is simple enough: To prove aconjecture (surely when proved, it becomes a theorem) But the structure anddetails are well selected and carefully designed. First it introduces andexplains the original problem, than talk about it's history, how it doesconnect with other branches of math, what these branches is about, and whatthe other mathematicians do on this conjecture, and at last the mostexciting: how these results all bringed together and then solve this20-year-long famous conjecture in combinatorics. There are exercises afterevery chapter. so this is NOT merely a history survey (as most books do).Indeed it is a textbook and contains some very excellent introductions tosome branches(for example, plane partition). And I think this is THEcorrect style to populating mathematics--- Do not afraid of formulas, justshow readers the signs, the terrible formulas, show them how and whatmathematicians study and think! Every student major in mathematics shouldread it, and it surely is a must-have for reseachers in combinatorics.

Amazing book on how mathematical research is actually done
This book tells the story of one of the moreimportant areas of research in algebraic combinatorics in the last few years.Not only does it tell the story of how the relevant conjectures were proposed, and how they wereeventually proved (including some of the blind alleys and several caseswhere surprising connections were discovered), but it also contains abeautiful exposition of the relevant mathematics, at a level which a readerwith no more than a reasonable background in linear algebra can follow. The history is fascinating and the mathematics is beautiful.If you wantto know what research in mathematics is really like, this is the book toread.

Massterpiece of Mathematical Exposition
Dave Bressoud tells, in his wonderful gripping style, the fascinating recent history of the proof of the alternating sign matrix conjecture, and the not-so-recent background that lead to it. The book can be read on manylevels, and is full of fascinating historical tidbits. This book is a mustto anyone who wants to know how math is actually done, and who wants toshare in the excitement of discovery. ... Read more

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"This is one of the best popular books on mathematics I have ever read. I recommend it to anyone interested in the fascinating problems of mathematics. The author has done a marvelous job explaining difficult mathematical concepts and making them accessible."
– Amir D. Aczel, author of Fermat’s Last Theorem

"No book in recent decades conveys more forcefully and beautifully the excitement of mathematical exploration than Dr. Szpiro’s work."
– Clifford A. Pickover, author of The Mathematics of Oz

"A gripping and intelligent account of the solution of one of the great problems of mathematics–older than Fermat, and just as baffling. Kepler’s Conjecture offers the nonspecialist genuine insights into the minds of research mathematicians when they are grappling with big, important questions. I enjoyed the book immensely."
– Ian Stewart, author of Flatterland and What Does a Martian Look Like?

Sir Walter Raleigh simply wanted to know the best and most efficient way to pack cannonballs in the hold of his ship. In 1611, German astronomer Johannes Kepler responded with the obvious answer: by piling them up the same way that grocers stack oranges or melons. For the next four centuries, Kepler’s conjecture became the figurative loose cannon in the mathematical world as some of the greatest intellects in history set out to prove his theory. Kepler’s Conjecture provides a mesmerizing account of this 400-year quest for an answer that would satisfy even the most skeptical mathematical minds. ... Read more

Customer Reviews (7)

It all started with "stacking cannonballs"
I would have flunked geometry had not my brainy sister been in the same class with me. But math never really phases me, so it was not with trepidation, but with wonder and curiosity that I read this book.

Now, maybe it took me longer to understand the nuances and maybe I had to skip a few formulas here and there.. So it did take me a while to finish this book, but I found pages to dog-ear and passages to underline here and there, and was quite taken up with and impressed by the ways in which "some of the greatest minds"..."helped solve one of the oldest math problems in the world"

Mathematical abilities of the reader notwithstanding, this is a fascinating, highly readable narrative of a centuries-old problem/challenge that has been "solved" and re-"solved" year after year -- and most probably will continue to be "solved" for eons afterwards.

Disappointing
There are several forgivable errors in this book that a more thorough proofreading would have fixed.

My problem, however, is with the many mathematical inaccuracies. I have read and enjoyed many popular books about mathematics. I realize that it is a difficult trick to write a pop math book that is both accessible to a large audience and still faithful to the spirit of its subject matter. If an author does not feel up to the task, they should just leave it for someone else.

This author tries to give the impression that he is letting us in on the clever ideas behind some difficult mathematics. It becomes clear, though, that he has not understood these ideas himself.

Kepler's conjecture is a subject that could have inspired a very good book. Hopefully someone else will write that book someday.

Very good but needed (more) proofreading.
This is an excellent book for the following reasons.
1. It provides a good historical review and an up-to-date description of a math topic with hardly any other books to meet the need. This book only covers the (2D&)3D versions of the kissing problem (proof that only 12 spheres can touch a central one) and the densest packing problem (which lacked a proof for 300 yrs. until recently). The only other books available are Zong and Conway & Sloane which are multi-dimensional specialized academic treatments - C&S being out of date and Zong only giving Hales' new proof superficial mention. Compared to the hoopla that greeted solutions to the 4-colur problem and Fermat's last theorem, solving Kepler's problem has gone almost unnoticed.
2. It is one of those all too rare "popular" books that isn't afraid to include a fair bit of mathematics, albeit mostly in appendices. This is mainly spherical trigonometry, but that's the nature of the problem and it gives a good idea of how Hales' elaborate proof procedure works, although for the voluminous ugly detail you need to consult Hales' website.
However, there are a lot of errors and misleading details such as;
p.25 Dodecahedron & Icosahedron labelled the wrong way round.
p.125 line 11 ratio of boxes wrong way round - can be corrected if the words "divided......by" changed to "divided.......into".
p.137 in the first paragraph the word "later" should be "after Blichfeldt" and the 2nd. occurrence of "Rankin" should read "Rogers & Lindsey".
p.222 Kelvin's tetrakaidekahedron isn't; it's Weaire and Phelan's structure.
p.232 footnote 7 "radius 3" should read "radius 4"
p.246 1st. equation - superfluous factor of ½ on l.h.s. and the "3" in front of r.h.s. sqrt. should be deleted. 2nd equation - quotient slash "/" should be deleted between 3 and sqrt. on r.h.s.
p.255/6 line 14 expression for discriminant (determinant) has signs reversed throughout & should be multiplied by -1. Also the same capital D is inappropriately used for both the discriminant and the first auxilliary variable and lastly the 3 squared signs are twice missing from a.b.c partway down p.256.
p.277 equation - It is obvious that the total score is not 7.99961 as Szpiro claims but needs to be multiplied by 0.0553736 to convert it to 7.99961 points, where points are defined 100 pages earlier on p.171.
Also there is some confusion over FCC & HCP where on p.23 it is claimed the two packings "are exactly the same" (they're not, but they have the same density), whilst on p.230 it is implied that FCC has greater density because Gold, Silver & Platinum use it compared to the HCP of Cadmium , Cobalt & Zinc.
Perhaps it's time for a revised paperback edition !!
However, contra the reviewer below, Szpiro does not say that Cohen showed there must be cardinals between countability and the continuum: he actually says "some other notion of cardinality must exist between...." which is acceptable if a little unclear. (His criticism of Szpiro's comment on Gödel is valid, however.)

Suboptimal packing
I've read and enjoyed many "mathematics for the non-mathematician" books. For example, Four Colours Suffice by Wilson, and Flatterland by Stewart, both of which I recommend. I regret to say that I cannot heartily commend Kepler's Conjecture by G. Szpiro.

It has good intentions. Kepler's Conjecture is a math problem with history galore, which Szpiro recounts. It is a problem easily understood by TC Pits - (The Celebrated Person In The Street).

The problem. We want to store as many balls of uniform size (e.g. soccer balls) as possible in a large storage container. Is the most efficient packing the "obvious" one? The obvious one, not being so obvious as everyone believes, is the one used to store cannonballs adjacent to cannons - nestle each ball into the indentation provided by three mutually adjacent balls, and repeat as necessary. The result - you've seen it many times - is known as the hexagonal closest packing, or HCP.

Mathematicians idealize this problem by using a very large room (i.e. Euclidean 3-space) and identical spheres.

The problem started with wanting to efficiently store cannonballs in the hold of a ship, or on the deck of a ship. For balls on a brass monkey (yes, this is where the saying originated) there is a simple formula. For the more general problem of Euclidean 3-space, the venerable CF Gauss proved that HCP is the most efficient lattice packing in the early 1800s.

But Kepler's conjecture was only recently solved by Thomas Hales, and the mathematical community hasn't exactly accepted his proof. What did Hales do that Gauss didn't? Well, Gauss proved a result for lattice packings; Hales did not assume the spheres were constrained to have their centres at lattice points.

Szpiro does tell the story, with generous dollops of colourful history. So what's not to like? He gets too many details wrong. His writing isn't polished, let alone scintillating. His mathematical excursions are seldom proofs, and often incorrect if they are meant to be proofs. He chastizes mathematicians who claim more than they prove, so I am holding him to the standard he asserts mathematicians should meet.

I don't expect from Szpiro the details that your math teachers expected in your homework or exams, and I don't expect a mathematical treatise. But I do expect his explanations to be correct except for the details, and I believe that, as often as not, they are not correct.

Having said some rather cutting things about Szpiro's book, I'll now darn him with vague praise. If you know a teenager fascinated by sphere-packing, sure, let him/her read this book. It won't hurt them, and if they aren't mathematically precocious, they'll likely enjoy it. If they are mathematically mature enough to see the flaws, they may enjoy the book despite the flaws.

I expected more than Szpiro has to offer, and I don't think I have unreasonable expectations. I've seen too many books in the genre that meet my expectations to offer praise where I don't feel it is deserved.

On the flip side, I've seen worse. I just can't commend them to you wholeheartedly.

fascinating
An extremely interesting combination of mathematics and history. A book that will appeal to everyone. Szpiro shows off his flair and knowledge with a light humourous touch. Thoroughly enjoyable! ... Read more

Product Description
Starting with an explanation of what 'proof' means to a mathematician, this student-friendly introductory text is aimed at undergraduates in mathematics and related disciplines. It shows students how to read and write mathematical proofs and describes how mathematicians investigate problems and formulate conjecture. Students develop their skills in logic by following precise rules, and examples and exercises relating to discovery and conjecture appear throughout. The author also covers mathematical concepts such as real and complex numbers, relations and functions and set theory. ... Read more

Customer Reviews (7)

The Only Book of Its Kind
I cannot BELIEVE this book is not 5 Stars.This is the most complete book of its kind.I used this book for my "introduction to proofs" class when I was an undergrad and I can tell you this book has played a large part in my mathematical life.Before you can learn mathematics, you really need to know how to prove things at a fundamental level.This book will give your students the proper tools to enter upper division mathematics.I must say that when i took this class with the author it was, and is still, one of the hardest classes I have every had, however, I would not have made it to graduate school otherwise.The only books I have brought to grad school have been Analysis by W. Rudin and This book.If you think you want to be a Math Major, do yourself a favor and buy this book......

This BOOK SUCKS!!
This book really sucks. One of the worst books I've got in my collection of math books

I Wish this had been out when I was a student!
This is the most USEFUL undergrad math text I have ever seen.
It covers the essentials of modern mathematics: proofs, logic, sets, number systems. I especially value the discussion of logic, sets, proof strategies, and relations.

I would have preferred more discussion of boolean algebra and lattices, but that is an idiosyncrasy of mine. I also wish that it would include the killer explanation of bi-in-sur-jection, and of into/onto maps that I have sought for decades. (The best of the extant lot is in Kolmogorov & Fomin). This description should also include iso-mono-anti-tone.

I am not competent to judge the freedom from mathematical error.
But the exposition never seems muddled to me, and is nearly always clearer than other texts, which often grandstand when discussing deep math.

Review by a professor of math and computer science
Mathematics consists of calculations and proofs. Elementary mathematics consists mainly of calculations, and students often have difficulty when they advance to the point where proofs become important.This book is intended to help students develop the ``mathematical sophistication'' they will need in advanced courses.That sophistication involves concepts and skills from three areas:logic and proofs,sets and functions,and the basic number systems used in mathematics.

The first four chapters of the book are devoted to logic and proofs;the next three to sets and functions; and the last three to number systems. Everything that should be in such a book is included:propositional and predicate logic, proof by induction,Russell's paradox, functions as sets of ordered pairs, the concept of cardinality;examples of rings and fields, the completeness axiom for the real numbers;complex numbers as pairs of real numbers.

Dr. Wolf has brought to this book a lively wit, twenty years of teaching experience with the target audience, and the acumen and scholarship of a highly-trained mathematician and logician.The book thus entertains and educates, without sacrificing accuracy or precision. The twenty years of experience, for example, is highly visible in the section on "Hints for Finding Proofs". The scholarship is visible in the "Suggestions for Further Reading" at the end of each chapter. The wit is visible in the examples. The scholarship and experience are both visible in the selection of exercises.

The subjects of proofs and their logical foundations have challenged the minds of some of the world's deepest thinkers.Both the difficulties of the subject, and its beauties,are extraordinary. This book will help the reader to appreciate the beauties and overcome the difficulties.

This book is extremely confusing
Chapter 1 seemed to have nothing to do with the rest of the text.Playing Nim and being told the scientific, legal and philosphical ways to prove things had nothing whatsoever with learning how to prove things in mathematics.

The truth tables were helpful, but the "or" table needed more explanation from the beginning.

2.1 ex 5d: it's confusing to students to have "trick questions" when learning a concept for the first time."Tricks" should either be saved for more advanced students or not used at all.The many incorrect statements, proofs and such throughout the book violate the "never write down the wrong way to do something rule".It was extrememly confusing.The defininitions for converse etc. were clear until ex2 p. 29 where I got completely lost.Tautologies were helpful.

E.S., E.G., U.S., and U.G. were only muddying up the waters and since other texts don't use them, they seemed pointless.Ways to use/prove tables were helpful.Induction was very confusing to start with, especially when using summations, but then seemed to clear itself up in later sections.

Get rid of Naive Set Theory.It's wrong and beginning set theory students are only misled by being shown the wrong way first!The union of the power set section needs a clear example as to what the union of a power set is!

I'm not sure that beginning students are ready to critique quite so many proofs, especially the "borderline" ones.

Figure 7.11 is completely lost any effect on me.I don't understand it at all.

A glossary in the back of the text would also be useful. ... Read more

Product Description

Eugène Charles Catalan made his famous conjecture – that 8 and 9 are the only two consecutive perfect powers of natural numbers – in 1844 in a letter to the editor of Crelle’s mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it.

Catalan’s Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The first few sections of the book require little more than a basic mathematical background and some knowledge of elementary number theory, while later sections involve Galois theory, algebraic number theory and a small amount of commutative algebra. The prerequisites, such as the basic facts from the arithmetic of cyclotomic fields, are all discussed within the text.

The author dissects both Mihailescu’s proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof of Thaine’s theorem is a little class field theory used; it is hoped that this application will motivate the interested reader to study the theory further.

Beautifully clear and concise, this book will appeal not only to specialists in number theory but to anyone interested in seeing the application of the ideas of algebraic number theory to a famous mathematical problem.

Product Description

The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

Product Description
A detailed description of a most important unsolved mathematical problem-The Goldbach conjecture-is provided. For graduate students, lecturers and researchers in number theory and mathematical history. ... Read more

Customer Reviews (2)

Very good book
This book is very good for people who would be interested in solving this tough math problem.

Absolutely the best book on this subject
Now Goldbach conjecture is a $1Million question.This book is the best possible book to learn about the subject with the available progresses.The author himself has proved towards the finalgoal. Highly recommended. ... Read more

Product Description
Ragnarok. Armageddon. Doomsday. Since the dawn of time, man has wondered how the world would end. In The Last Three Minutes, Paul Davies reveals the latest theories. It might end in a whimper, slowly scattering into the infinite void. Then again, it might be yanked back by its own gravity and end in a catastrophic "Big Crunch." There are other, more frightening possibilities. We may be seconds away from doom at this very moment.

Written in clear language that makes the cutting-edge science of quarks, neutrinos, wormholes, and metaverses accessible to the layman, The Last Three Minutes treats readers to a wide range of conjectures about the ultimate fate of the universe. Along the way, it takes the occasional divergent path to discuss some slightly less cataclysmic topics such as galactic colonization, what would happen if the Earth were struck by the comet Swift-Tuttle (a distinct possibility), the effects of falling in a black hole, and how to create a "baby universe." Wonderfully morbid to the core, this is one of the most original science books to come along in years. ... Read more

Customer Reviews (16)

An understandable explanation of what the fate of the universe will likely be
This book is somewhat of a companion to the book "The First Three Minutes" by Steven Weinberg. In that book, Weinberg describes the physical conditions of the universe in the three minutes after the big bang winked it into existence. Davies distills current quantum and cosmological thinking down into an understandable explanation of what are the likely end scenarios for the universe.
In general there are two main endings postulated for the universe. The first is that it will continue to expand and all energy concentrations will eventually be dissipated to achieve what is literally universal thermodynamic equilibrium. The second is that the universe will cease expanding and start to contract. Once the contraction starts it will accelerate until all matter, which is the entire universe, will contract down into a singularity of infinite density. At this point time and the laws of physics will no longer exist.
Which end scenario is the eventual result is based on the density of mass in the universe, less than a certain amount and it is universal expansion, more than a certain amount and contraction is certain. At this time, the amount of mass in the universe does not appear to be enough to force a contraction; however cosmologists are finding additional mass all the time.
Where Davies diverges from other authors writing in this area is in his discussions of ways to create new universes and how advanced species could literally use the creation of new universes to be immortal. It was a fascinating thought because it is based not on speculation but on the highest levels of current cosmological thought. That section alone made reading the entire book worthwhile.

Escape from Yourself and Join the Stars
An elegant book on the ultimate fate of the universe - cosmological speculations based upon well known hard physics - and all very exciting stuff extremely well written, and easily grasped if one remembers any high school science at all.
All the subject matter concerns a physical scale (large and small), a time scale (long and short), and a temperature scale (hot and cold) of such stupendously extreme parameters that reading becomes a terrific brain stretching exercise that delightfully removes one from the tiny doings of one's own overly personalized and petty mind circus.

The human end and not so much the universe's
I am not a scientist, and do not understand technical scientific matters in a deep way. For better descriptions of what this book is about in scientific terms I recommend the reader of this reviewlook at the other Amazon Reviews.
I am the kind of reader who reads the scientific material in order to use it as basis for understanding certain fundamental questions. Science provides the ' truth' and my own thought the conjecture.
However what happens when the Science itself is conjecture, and what we are dealing with is alternative theories? From what I understand the most accepted view of the Universe's ' end' in scientific terms is its continuous expansion. Thus a number of readers on Amazon have indicated that the ' contraction scenario 'or ' big crunch scenario ' for the world's end is not held today by the great majority of Astronomers.
My question and here I share Davies' concern is not with ' the end of the Universe' the ' last three minutes as it were of everything' but with the ' ultimate fate of mankind'. And I wonder if the kind of material presented here brings to a greater understanding of that.
On the one hand it is possible to suggest that Mankind is such a small part of the Universe, and has been in existence for such a small interval of time that its continued existence through the kinds of times and distances the Universe is likely to exist through seems extremely unlikely. Here it is possible to speak about various scenarios of ' colonization' and the human transformations involved in them. Speculations in other words.
But then too since Mankind is now rapidly developing ' machine - intelligences' that may in some way become ' independent others' the question of ultimate human fate is connected with our own efforts here. In other words we might not wait until the last three minutes of the universe but might through our own ' creative means' put ourselves to sleep.
All this of course raises questions which are ultimately religious or spiritual.
And that question too I think should be asked in relation to ' the final time' of the universe.
All this leads me to the thought that the real subject we should be thinking about is the subject of God at the beginning or before the beginning, and God at the end or beyond the End. And what that means for us. And what we mean for God?
And how we are to make sense of this universe, and whether or not we can at all.
This is to say that the kinds of information Davies presents do not help me so much in my own quest to ' understand the ultimate meaning of who I am, who we are, and what my and our purpose is here on this earth and in this Universe.
For all those we now living love will end long before the physical Universe does.
Is there some other meaning for all of us only God knows ? Is there another universe next door or up above, or in some other way of perception and knowing we cannot possibly grasp?

Mind Expanding... (or is it contracting?)
This book overviews the current major ideas of cosmology and sets them against a logical background of ultimate fate. What happens to the universe ultimately depends upon whence we came. Davies takes several theories and then logically extrapolates the possible fates of the universe.

One of the things to keep in mind is the fact that "infinity is a long time" and this reoccuring theme is central to the ideas he develops. If the universe is expanding, what eventually happens to matter? If the universe is contracting, what will eventually happen to matter? Where does matter come from? Can matter be created or destroyed over infinite amounts of time?

Such ideas explored are the steady state theory, the expanding universe and "cold heat" death. The contracting universe and the eventual ceasing of all time, matter --- everything... The oscillating universe where matter can be created from "nothing" .

Some of the info is a little dated -- I suppose this applies if you are a graduate student in the Astrophysic department of Cambridge University. But for the average bloke with an interest in cosmology, one need not be worried about reading "old" materials. In fact the popular science, current considerations about the universes initial inflation stage -- that fraction of a secong when expansion and matter may have formed --- is well described and should serve as an intro to other reading.

The one thing that I really like about Davies is that his writing is clean and does not become a political tract: eg. Dawkins, Pinker and Dennet.

This trend towards writing "polically" based appreciations of scientific theory is based upon two things in my estimation: 1) the rise of the irrational, Voodoo Science and stark raving mad religious fundementalists --- scientific authors often rightly feel that they are fighting a rear guard action against the forces of darkness, and: 2) amazing egos that need to be assuaged (Dawkins and Dennet) so they feel that they must always address all potential attacks, however inconsequential, to defend their "good name."

Davies is clean and can present contrasting and even illogical ideas (Bede's "Darwin's Black Box") in a non-political way -- and still make the guy look irrelevant to modern science.

That is why, along with Matt Ridley, Davies is the best writer in popular science, worthy to assume the mantle of Carl Sagan.

A bit dry in the middle..
The book starts with what I might consider a fake-out:Davies starts by recounting a situation where some asteroid has been found to be on a collision course with Earth and the final minutes in our existence considering that we know we're going to be hit by a "global killer".While this is interesting, it's a bit of fiction.We were scared for a bit that later this century we would be hit by something heading our direction, but it was found to be missing us by just a little bit later.After considering the possibilities and probabilities of these happening for some pages, he notes that even if we do die this way, it's not exactly the last three minutes of the universe, just life on earth. Going on, he discusses the possibility of heat death, seemingly unavoidable by the second law of thermodynamics and something which depressed scientists to no end after they found it out.He also covers the possibility that the universe may stop expanding and start contracting at some point in the future.

Davies seems to work very hard to make the material not as dry as a AA member at a monastery by connecting most of the theory to what would actually happen, assuming that human life exists at that point.Unfortunately, the evaporating power of the material seems to take over, and I couldn't really get through this book all the way without forcing myself through long sections on black holes that I really didn't care much about.After the long discussion of black holes and how we could possibly get energy out of them stops, Davies got to the meat of what I was actually looking for: heat death or contracting universe.The last third of the book was actually much easier to read than the middle and much more intersting than most of the rest.Contracting and "Bouncing back" universes are discussed along with an actually interesting tangent about artifically creating universes by tricks with false vacuum.One other thing I really like about this book - Davies seems to go out of his way to make sure you know where to look up more information about the situations he talks about - even without resorting to looking at the notes in the back.

The book is actually better than many I could have read on the subject, and did increase my knowledge of the possibilities for the ultimate fate of the universe fairly extensively.It definitely gets my recommendation for geeky reading over the summer, at least if you can get through to the really interesting parts.At 176 pages, it is actually more reading than it looks like at a paltry paperbook size.If it weren't for the dry section in the middle, it wouldn't be B grade material. ... Read more