Godel, Escher, Bach: An Eternal Golden Braid

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Twenty years after it topped the bestseller charts, Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is still something of a marvel. Besides being a profound and entertaining meditation on human thought and creativity, this book looks at the surprising points of contact between the music of Bach, the artwork of Escher, and the mathematics of Gödel. It also looks at the prospects for computers and artificial intelligence (AI) for mimicking human thought. For the general reader and the computer techie alike, this book still sets a standard for thinking about the future of computers and their relation to the way we think.

Hofstadter's great achievement in Gödel, Escher, Bach was making abstruse mathematical topics (like undecidability, recursion, and 'strange loops') accessible and remarkably entertaining. Borrowing a page from Lewis Carroll (who might well have been a fan of this book), each chapter presents dialogue between the Tortoise and Achilles, as well as other characters who dramatize concepts discussed later in more detail. Allusions to Bach's music (centering on his Musical Offering) and Escher's continually paradoxical artwork are plentiful here. This more approachable material lets the author delve into serious number theory (concentrating on the ramifications of Gödel's Theorem of Incompleteness) while stopping along the way to ponder the work of a host of other mathematicians, artists, and thinkers.

The world has moved on since 1979, of course. The book predicted that computers probably won't ever beat humans in chess, though Deep Blue beat Garry Kasparov in 1997. And the vinyl record, which serves for some of Hofstadter's best analogies, is now left to collectors. Sections on recursion and the graphs of certain functions from physics look tantalizing, like the fractals of recent chaos theory. And AI has moved on, of course, with mixed results. Yet Gödel, Escher, Bach remains a remarkable achievement. Its intellectual range and ability to let us visualize difficult mathematical concepts help make it one of this century's best for anyone who's interested in computers and their potential for real intelligence. --Richard Dragan

Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.

Book Description

Winner of the Pulitzer Prize, this book applies Godel's seminal contribution to modern mathematics to the study of the human mind and the development of artificial intelligence.

Customer Reviews:

Down the Rabbit Hole..........2007-05-18

This is a difficult book.

Difficult to read. Difficult to understand. And, I'm finding, difficult to review. What's it about? Good question. The author, himself, isn't very clear on this point, describing it as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll." I'm not sure I can do better than that. I will tell you this, however: if the book has a "point," it does seem to be that man's consciousness is ultimately mechanical and, therefore, that there is no reason that machines cannot finally be intelligent in the same sense that man is. (And, in fact, be as man in just about every internal way.)

While I take issue with this conclusion, and some of Hofstadter's reasoning along the way, I don't think that my debating his points is the basis on which a prospective reader should decide whether or not to pick up this book. Instead, the prospective reader should know: that this is a lengthy and deep work. It will take a *long* time to read properly, and most readers should not read more than a chapter a day. Many of the sections, and especially the various dialogues that preface the chapters, are quite clever. (These dialogues are usually between Achilles and the Tortoise, of Zeno's paradoxes, and their friends.) Some of the chapters grow incredibly technical. The subject matters vary, wildly and rapidly, and there will be points in reading where you will question your investment.

In the end, you will feel good for having pushed through the hard bits. It will coalesce, more or less, into a whole. Whether you finally agree with Hofstadter's conclusions or not, you'll have learned much and thought about important topics you might otherwise not have.

A good book, certainly not for everyone... but, if you're the "right" audience--someone deeply interested in questions of intelligence, mathematics, computer science and free will, and possessed of a bit of an ironic sense of humor--then this book cannot be recommended highly enough.

Five stars, for the work it represents, and the doors it opens to the reader.

Come one, come all.......2007-05-16

As you can see from other reviews, people tend to walk away from this book with a variety of different impressions. Math, Art, Logic, Philosophy, Human Perception and Thought, it has it all. This is second to the Bible in my collection as a book I've read multiple times and can still come back to a read again for even more insight and perspective.

Bound with the "braid"?.......2007-05-14

Can someone tell me, in plain English, what this book is about? On the little matter of determinism--is he for it or against it? He does not seem to have come to praise Godel, Escher, Bach for their strangeness but rather to bury strangeness and its resistance to materialism. He seems to be saying that strangeness is hardwired and can be programmed into a formal system by someone who sees it for what it is--in short, that computers will some day rise to the level of consiousness and self-reference. But wouldn't such a system be curved in upon itself and lack strangeness? If strangeness could be hard-wired into AI, would it still seem strange? Nothingness annihilates strangeness, but then the absense of strangeness is the actual limit of the theories of value seen in those who follow Heidegger. In order to eliminate the difference between soul and matter, they must give up the resistance of soul to the limitations of material existence; at which point "strangeness" becomes a matter of verbal virtuosity and conceptual sleight of hand. "Strangeness" becomes the same thing as cleverness. Or am I misreading this fascinating book?

Excellent book!.......2007-05-14

Hofstadter combines the awe in math, music, art, artificial intelligence, language and computers into one big book called GEB. Its takes the reader on an ecstatic journey with a clever use of parallels between the structure of math, music and finite but endless loops that appear in Escher's works. Dialogs between Achilles and Tortoise are very interesting.

"This sentence is false.".......2007-03-19

A simple example of recursiveness in music is the song "row, row, row your boat." The song becomes recursive as each new line is started when the original line makes it to "gently down the stream." In this way, we have a musical example of the artistic portrayals of Maurits Cornelius Escher whose paintings invariably fosuc on recursive visual themes such as two hands in the process of drawing each other.

In each case, the depiction challenges our ability to pidgeon hole the phenomenon we are examining. Which line is the harmony, which is the melody in "row, row, row your boat"? Which hand is drawing which in the Escher print?

Liguistically, the same effect occurs when we examine the statement "This sentence is false." Logically if we accept the statement at its face value being false then it becomes an accurate representation (in that it correctly asserts its falseness). On the other hand, we are also drawn to the conclusion that the statement is true (again because it is self referentially accurate).

Ultimately, we are forced to logically conclude that we can neither bracket the statement "This sentence is false" with either all true statements or all untrue statements. As indicated previously, like the song "row, row, row your boat" or an Escher painting, the sentence defies pidgeon holing owing to its recursive quality.

Back in 1931, Kurt Godel shocked the mathematics community with his assertion that mathematically consistent systems themselves necessarily produce formally undecideable propositions (the math equivalent of "This sentence is false"). At the time of presenting his paper, it was Godel's intent to demonstrate the unique nature of human intellect because if we can resolve undecideable propositions then there must be something unique to the process of human intellect.

While Godel certainly brought undeniable genius to the creation of his theorem, it doesn't follow that the theorem proves the uniqueness of human intellect. And the reason Godel's theorem doesn't prove the uniqueness of human intellect is because its logical limitations are our own.

Just as Godelian mathematics can't prove undecideable propositions, neither can we "prove" them.

However, we can "believe" undecideable propositions. (In this regard, two easy cases in point are Goldbach's conjecture -- that all even numbers are the sum of two primes -- and that parallel lines really are parallel.) In this way, Godel's theorem, in combination with modern research on artificial intelligence, shows that it is the emotive side of reason that defies the strict logical limitations of Godelian constructs.

These hard won discoveries have combined to make for some surprising findings.

Probably the first among these most observable to the general public through the misconception of science fiction is that emotion somehow stagnates the operation of intellect. In this way, it was HAL 9000's personality as much as the creepiness of that personality that was surprising to 1968 movie goers watching "2001: A Space Odessy." As demonstrated in the movie, it was the fact of HAL's emotive connections with the ongoing actions of his crew that prompted "him" to formulate and act on plans.

Second, modern research has shown that human intellect is not best characterized as being a "blank slate" but rather a delicate combination of various systems that survey reality in the own ways. An easy example is the human eye which uses a combination of three different light cones to measure redness, greenness and blueness. It is the relative comparisons of these cone findings that nudges your visual perception to observe the color of an object. At the intellectual level, one system is entirely devoted to our understanding of artifacts. How do they work? How can they be modified for use in a situation? Another system comprehends animate creatures. Yet another system recognizes faces. Still another system is devoted to language acquisition.

And significantly all these systems acquire information emotively. We see the face of a parent and emotively appreciate it (unless we suffer from a particular cognitive disorder that has disabled our ability to do so as for example discussed by Oliver Sacks in his great book "The man who mistook his wife for a hat"). We remember a concept learned and emotively evaluate it. In this way, freedom, communism, taxes are not just intellectual constructs but ideas that spark real feelings on our part.

In creating Godel, Escher, Bach, Douglas Hofstadter displayed true genius in linking three domains wherein recursiveness seems to play such a pivatol role. As he indicated, they are three shadows cast from the same source.

In re-concluding this book, however, I couldn't help but think of other possible titles that could be added to a Godel, Escher, Bach type encyclopedia: "Phi, Di Vinci, Bach" -- the story of the "golden ratio" of phi which plays a role in Di Vinci's art work and as it so happens also in the music of Bach; "Pascal, State Lotteries, Happy Birthday" -- the story of Pascal's wager and how an appreciation of statistics will make us understand why states will never lose money running a state lottery for reasons akin to why relatively small groupings of people will have at least two that share the same birthday; and "Klein, Carroll, Kubrick" -- the story of Oscar Klein's bottle which can resort to the fourth dimensionj to fill itself up and how speculations by the physicist J Richard Gott suggest that Alice and all of us may have originallyu gone down the rabbit hole for a real space odessy through time itself.

The point here is not that Hofstadter was incorrect but (no pun intended) merely incomplete in his survey when he said that Godel's proof, Escher's paintings and Bach's music were but three shadows cast from the same source. The point here is that -- properly examined -- those three shadows, together with the encyclopedia I've suggested, would direct us not only to the origins of consciousness but also the origin of origins itself.

Why Beauty Is Truth: A History of Symmetry

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A history of symmetry
Delightful book
A well-written book for the non-specialist
Dissapointed
group theory

Why Beauty Is Truth: A History of Symmetry
Ian Stewart
Manufacturer: Perseus Books Group
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Binding: Hardcover

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ASIN: 046508236X

Book Description

An eminent teacher and writer explores an idea both simple and complex, both multidisciplinary and unifying--the story of symmetry.

At the heart of relativity theory, quantum mechanics, string theory, and much of modern cosmology lies one concept: symmetry.

In Why Beauty Is Truth, world-famous mathematician Ian Stewart narrates the history of the emergence of this remarkable area of study. Stewart introduces us to such characters as the Renaissance Italian genius, rogue, scholar, and gambler Girolamo Cardano, who stole the modern method of solving cubic equations and published it in the first important book on algebra, and the young revolutionary Evariste Galois, who refashioned the whole of mathematics and founded the field of group theory only to die in a pointless duel over a woman before his work was published.

Stewart also explores the strange numerology of real mathematics, in which particular numbers have unique and unpredictable properties related to symmetry. He shows how Wilhelm Killing discovered "Lie groups" with 14, 52, 78, 133, and 248 dimensions--groups whose very existence is a profound puzzle. Finally, Stewart describes the world beyond superstrings: the "octonionic" symmetries that may explain the very existence of the universe.

Customer Reviews:

A history of symmetry.......2007-08-07

This is an excellent book, although to fully understand it you need some good background in math and physics. It traces 4000 years of research in mathematics and physics, from Babylonic science (to whom we owe the sexagesimal system) to Ed Witten and superstrings. The thread of the story is symmetry, a concept that leads to group theory via the efforts to solve some the antiquity's problems (for example, the duplication of the cube) and the polynomial equations, specially the quintic. Although I am an avid reader of this kind of books I learnt quite a few things and others, although not new to me, I found were very well explained.

Among the first group, the cubic geometric solutions of Persian Omar in the 11th century, the name of Killing (the mathematician who classified simple Lie algebras in one of the most beautiful math papers, according to Stewart), the fact that Liouville rescued Galois papers from oblivion, the relation of octonions to string theory, Hamilton's carving of the fundamental relations of his quaternions in the Broome Bridge, the role of the exceptional Lie groups in physics, Witten's starting career as political journalist, etc.

Among the second: the description of gauge symmetries, the comparison between the unity of life and the unity of the fundamental forces, etc.

The reader will enjoy the well known story of how mathematicians were forced to use complex numbers in trying to apply the cubic formula and the fascinating life of Galois who so unhappily was killed in a duel at the age of 21, a duel that he had apparently exactly 50% chance of survival.

Stewart is critical of the anthropic principle, even in its weak form. According to him a sufficient condition should not be confused with a necessary condition and who knows in which exotic forms can complexity emerge. I think that we also should reflect on his suggestion that the search of a Theory of Everything is a residue of our monotheistic culture.

One of the main themes of the book is the unreasonable effectiveness of mathematics (a famous article by Wigner has this title) and the ethernal dilemma: is mathematics invented or discovered? The exceptional Lie groups seem to be put there by a deity. These are fascinating subjects and no definitive answers can be given.

One little criticism: Stewart does not distinguish properly hadrons and leptons and leds the uneducated reader to believe that all particles are either made of quarks or are gluons.

Delightful book.......2007-07-19

This book made math and its history extremely readable. Its core idea was symmetry and how it acted as the driving force behind many mathematical inspirations. Ian Stewart is a master writer and he proves himself again in this book. He defines symmetry not untill p.118, where he sees symmetry as a kind of "transformation" which when applied to a mathematical object preserves its structure. Then he explains these individual aspects of symmetry in relation to Galois' groups. Near the end of the book, he brought physics into the discussion, and showed how deep abstract sense of beauty also played a crucial role in developing physical ideas. To some, it may appear bizarre, as most of the book talks about mathematicians and their 'beauties,' and suddenly physics creeps in. But in hindsight, the sense of beauty and truth is never complete without the taste of reality. Physics serves that purpose. And so he ends:
"In physics, beauty does not automatically ensure truth, but it helps.
In mathematics, beauty MUST be true - beacause anything false is ugly."

A true ending to a beautiful book.

A well-written book for the non-specialist.......2007-07-16

Some of the reviews of this book seem to feel it doesn't present enough group theory. I think they are looking for a more technical book than Stewart meant to write, and so they are downgrading the book for reasons that are not fair to the book.

I reviewed a book by Mario Livio called "The Equation that Couldn't Be Solved," and gave it 5 stars. After reading this book, I almost want to go back and lower my rating of Livio's book, but of course, I shouldn't do that just because a better book has come out since. Livio's book concentrates on a shorter timespan than this, but both feature the same things -- mathematicians' attempts to solve equations of higher and higher degrees, from quadratics to cubics to quartics, and failure to find a solution to the quintic, only to find (due to the work of Abel and Galois) that it couldn't be done; and Galois' invention of group theory to make his proof, followed by other mathematicians' revelation that group theory is just what the doctor ordered to explain symmetry.

Stewart's book goes further back in time than Livio's, and also devotes more space to the modern uses of symmetry in physics. So it puts everything in more context. And, simply put, Stewart is a captivating writer. I enjoyed Livio's book, but I could hardly put down Stewart's. This book gets a high 5-star rating from me.

But it IS a book for the non-specialist. It isn't a course in group theory, or the Galois theory of equations; it is an attempt to give a non-mathematician some idea of these subjects. It should not be rated on a set of criteria that ignore what Stewart was trying to do. The negative comments really are unjustified; but yes, I'll warn you away from this if you expect it to teach you all the group theory you'll need to do particle physics, or crystallography, or any of the subjects that depend on group theoretic concepts of symmetry these days.

Dissapointed.......2007-06-18

This book had a wonderful review in Scientific American.

I am a Chemist with a fair amount of math. The major reason I was dissapointed is basically I did not learn anything mathematical. There were some fascinating biographies of physicists and mathematicians. I am not saying I did not learn anything because I know it all already. When there was a subject introduced that it did not know, it was introduced using analogies that really stretched what was going on - like building a multistory building.

It was a good read for the personalites involved, but really not a place to learn anything.

I would suggest the classic "Chemical Applications of Group Theory" for those really wanting to learn something.

group theory.......2007-06-09

good book, but there is little original in his presentation that is not available from other recent sources. I agree with other viewers that what is needed is a good book written as well as this on the subject of group theory in relation to particle physics, nuts and bolts of applied symmetry. Nothing on the market that I know of. Any suggestions?

Mirror Symmetry (Clay Mathematics Monographs, V. 1)

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Detailed overview of the subject

Mirror Symmetry (Clay Mathematics Monographs, V. 1)

Manufacturer: American Mathematical Society
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Binding: Hardcover

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ASIN: 0821829556

Book Description

This thorough and detailed exposition is the result of an intensive month-long course sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives. The material will be particularly useful for those wishing to advance their understanding by exploring mirror symmetry at the interface of mathematics and physics.

This one-of-a-kind volume offers the first comprehensive exposition on this increasingly active area of study. It is carefully written by leading experts who explain the main concepts without assuming too much prerequisite knowledge. The book is an excellent resource for graduate students and research mathematicians interested in mathematical and theoretical physics.

Customer Reviews:

Detailed overview of the subject.......2005-05-16

Mirror symmetry has become an established branch of mathematics and mathematical physics, and research in the subject has resulted in brilliant developments. This sizable book contains essentially some (polished) lecture notes of a seminar series in mirror symmetry that was given in the spring of 2000. This reviewer only studied Part 5 of the book, entitled "Advanced Topics" and so only that part will be reviewed here. In addition, space constraints then dictate only a small portion of this part can be reviewed. Needless to say, any reader who intends to tackle this book will need a substantial background in modern mathematics and advanced physics, and a sizable commitment in time. The time spent is well worth it though, as both the mathematics and physics behind mirror symmetry has to rank as one of the most fascinating research topics in the last two decades.

In the chapter entitled "Topological Strings" the authors consider the functional integration of worldsheet geometries. This project involves essentially the integration over the complex structures of Riemann surfaces. Referring to this procedure as "quantum gravity", they do not address it in-depth, but instead focus on the coupling of topological sigma models to worldsheet gravity, which is called `topological string theory' in the literature. The authors first consider the case where the target is a Kahler manifold whose first Chern class is zero, since for this case the quantum cohomology ring is less easy to obtain, i.e. it can obtain contributions from holomorphic maps of any degree. Even for the case where there is no coupling to gravity, the degree 0 contribution is related to the classical intersection number. The contributions from higher degree result in the deformation of the classical cohomology ring into the quantum cohomology ring. The authors then ask whether there are any other correlators that will give nontrivial (non-zero) invariants in genus 0. Posing this question leads to the WDVV equation and the genus 0 topological string partition function. The n-point correlation functions of topological strings can then be defined as the nth partial derivatives of this function. For higher genus cases, the correlators are all zero, but the authors show the connection between the higher genus partition function and holomorphic anomalies. The case of three-dimensional Calabi-Yau manifolds is special, if one concentrates on the integration over the complex structures of the worldsheet. When the complex dimension of this moduli space is 3(g-1) then there are isolated points where holomorphic maps exist. Defining a topological string theory for Calabi-Yau threefolds is straightforward, as the author shows, and proceeds analogously to the case of topological field theory. A measure is defined on the moduli space of Riemann surfaces of genus g that cancels the axial charge anomaly. A genus g (>1) topological string amplitude, which is a section of a bundle over the moduli space of Calabi-Yau manifolds, is then obtained from this procedure. Modulo the presence of holomorphic anomalies, the authors show that the definition of topological string amplitudes is consistent with the topological symmetry. The origin of these holomorphic anomalies is discussed in fair detail by the authors, having their origin in the boundaries of the moduli space.

The rigorous mathematical formulation of mirror symmetry is of course of great interest to mathematicians. Because of its origin in string theory and quantum field theory, mirror symmetry has not yet received this kind of rigor. Chapters 37 and 38 of this book discuss some of the approaches that attempt to put mirror symmetry on a more rigorous foundation. One of these involves the use of `derived categories,' an approach that was recommended by the mathematician Maxim Kontsevich. The discussion in these chapters takes place in the context of D-branes, and Kontsevich conjectures that mirror symmetry is the equivalence of two categories: the derived category of coherent sheaves, and the category of Lagrangian submanifolds with flat U(1) connections. Specifically the equivalence entails the equivalence between the bounded derived category of coherent sheaves or `B-cycles' and the category of A-cycles with compositions defined in terms of holomorphic maps from disks. This latter category is derived from the Fukaya A-infinity category, as is shown by the authors. They discuss in detail this category, being essentially a generalization of a differential, graded algebra, especially how to obtain the compositions. In chapter 37, the authors give an explicit example of the equivalence of these categories for the case of the elliptic curve. The elliptic curve is interesting in this regard in that it is its own mirror, i.e. the complex parameter is mapped to the complexified Kahler parameter by the mirror map.

The derived category has sometimes been a stumbling block to those who want to understand the Kontsevich conjecture. The authors do not attempt to give the reader the needed insight into this kind of category, but merely take it to be a collection of all holomorphic bundles and coherent sheaves. Sheaves in this category can be subtracted from each other using a map between them. Physically, this subtraction corresponds to the annihilation of branes and anti-branes via a tachyon. Derived categories though are straightforward to think about if one views them from the standpoint of algebraic topology. Derived categories are rich enough to include notions of localization and triangulated objects (i.e. "complexes") and maps (i.e. morphisms) between these objects. This is a kind of "homology" but what is of main interest are homotopies between the morphisms. The class of homotopic morphisms between two complexes forms an abelian group and one can then obtain a category consisting of complexes as objects and classes of homotopic morphisms as morphisms. A cohomology functor can then be defined on this category, along with graded objects and differentials between them. The homotopic category can be given a "triangulation" and morphisms in this category that give rise to isomorphisms in cohomology are given special status, called `quasimorphisms.' The localization of this category with respect to quasimorphisms is called a derived category.

Not Even Wrong: The Failure of String Theory And the Search for Unity in Physical Law

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The Fall of Strings
Woit dissects "the only game in town"
Right on the button
contains interesting information on physics as well as criticism of String Theory
A Good Proposal for Using Government Funds More Effectively

Not Even Wrong: The Failure of String Theory And the Search for Unity in Physical Law
Peter Woit
Manufacturer: Basic Books
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Binding: Hardcover

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ASIN: 0465092756

Book Description

Has physics gone off in the wrong direction? Peter Woit presents the other side of the growing debate on string theory--arguing that it's not even science

At what point does theory depart the realm of testable hypothesis and come to resemble something like aesthetic speculation, or even theology? The legendary physicist Wolfgang Pauli had a phrase for such ideas: He would describe them as "not even wrong," meaning that they were so incomplete that they could not even be used to make predictions to compare with observations to see whether they were wrong or not.

In Peter Woit's view, superstring theory is just such an idea. In Not Even Wrong, he shows that what many physicists call superstring "theory" is not a theory at all. It makes no predictions, even wrong ones, and this very lack of falsifiability is what has allowed the subject to survive and flourish.

Not Even Wrong explains why the mathematical conditions for progress in physics are entirely absent from superstring theory today and shows that judgments about scientific statements, which should be based on the logical consistency of argument and experimental evidence, are instead based on the eminence of those claiming to know the truth.

In the face of many books from enthusiasts for string theory, this book presents the other side of the story.

Customer Reviews:

The Fall of Strings.......2007-10-06

String theorists have so far been unable to use their results to predict new experimental findings. This book and Smolin's 'The Trouble with Physics' both attempt to document this failure of string theory. Smolin's book is better, but a tougher read. But this book is not bad, and you may want to read them both.

Woit dissects "the only game in town".......2007-10-02

"The fundamental problem with string theory is that, as far as its central goal of unifying physics goes, over the last nearly 25 years it has not only not made any progress toward explaining anything about particle physics, but, quite the opposite. Everything that has been learned about string theory makes it more and more clear that the original hopes for getting unification this way were just misguided and can't work. The derivative here is the wrong sign." Peter Woit, posted on his weblog September 13, 2007.

Some readers may think that this book gets off to a painfully slow start, given the author's long telling of the history of particle physics, particularly as regards work done with particle accelerators/ colliders. But stay with it [it's worth it!]. Woit holds degrees from Harvard and Princeton (PhD, theoretical physics) and has taught both mathematics and physics at Columbia. He happily describes himself as a mathematician, in large part because that is indeed the career he has chosen but also in large part because he is obviously disgusted with the current state of theoretical physics--in so far as the superstring/ M-theory disciples of Witten have abandoned anything resembling orthodox science. Woit shows no hesitation in acknowledging Witten's great genius, but unlike most theorists of recent decades he is not interested in worshiping at Witten's feet, no matter what the cost. And Woit isn't just some disgruntled nay-saying spoilsport (I can't strictly judge the psychological state of someone I don't personally know, but he doesn't strike me in this way at all). His concern is that there are other prospects for a unified theory that have been summarily brushed aside by the popular mantra that "string theory is the only game in town." [Federal] research funding, positions of influence notably including department chairs, academic and research hirings, increasingly all have played what we are told (by string/brane theorists themselves) is "the only game in town." But after three decades of glowing hype, this "game"--superstrings/'M-theory'/'brane-world'--has failed to move forward. It has essentially demonstrated that it cannot move forward in any scientific sense.

"Superstring theory is to a large degree thought of by mainstream physicists as mathematics and by mainstream mathematicians as physics, with each group convinced that it makes no sense within their frame of reference but presumably does within someone else's." pg 204

Like so many other armchair theorists, I've read and enjoyed books like Greene's `Elegant Universe' and Hawking's glossy `Universe in a Nutshell'. But any astute reader has to notice that no real connection is made between what we are told are compellingly "beautiful mathematics" and the physical world we can examine, and, given a sound theory, even interrogate, to any degree at all. It is particularly instructive to consider strings/ brane-world from the critical perspective of pure mathematics, i.e., Woit's perspective in this volume. It seems that the abstract equations ARE strangely "beautiful" UNTIL the math must be patched to conform to a universe with precisely three large spatial dimensions; as soon as we are forced to manipulate the additional dimensions, the beauty of the mathematics begins to fade. That `beauty' has been fading for 20 years at this writing. Woit finds the equations of strings/branes to be growing uglier at every turn. After decades of contortion, strings/branes are ever becoming less beautiful than advertised. And, as Woit briefly explains with stark, non-glossy frankness, strings/branes are NOT the only game in town, or at least they shouldn't be.

Many mathematicians would cautiously agree that strings/branes make for interesting mathematics--if they can be divorced from physics. There may be sufficient cause to suggest, as some have, that strings/branes make for interesting theology as well. But are strings/branes SCIENCE? At this late date it seems that the answer is probably `no'.

"If someone could come up with a legitimate, distinctive, testable prediction of string theory that gave even the correct order of magnitude for some experimental result, that would be a huge breakthrough."

As I have suggested to anyone willing to listen, read Peter Woit's thought-provoking book, especially if you've read Greene, Hawking, and/or Susskind.

Right on the button.......2007-08-30

What can I say? Like Peter Woit, I am a recovering mathematician, and this book has given me hope. "Not Even Wrong" carries my highest recommendation, especially for those empirically-inclined investigators who have become demotivated by the crisis in science. One proviso, though - don't read it in isolation. Essential companions are Lee Smolin's "The Trouble with Physics", and my own impassioned plea "The Virtue of Heresy - Confessions of a Dissident Astronomer". The Virtue of Heresy: Confessions of a Dissident Astronomer

Hilton Ratcliffe
Astrophysicist

contains interesting information on physics as well as criticism of String Theory.......2007-07-11

As most people know Not Even Wrong is critical of String Theory. What I did not realize going into the book was the detail it delves into in describing events leading up to String Theory, especially events having to do with the Standard Model. I think the book is worth reading just for this information which spans half of the book. I don't know whether String Theory is right or not as I am not a physicist but I do believe that criticism is a healthy thing and that the author is clear in his criticism of String Theory. I would have wished for more information on other Unified Theories but he devotes only one chapter to this. I guess, as the author points out, there is only one game in town and it is String Theory.

A Good Proposal for Using Government Funds More Effectively.......2007-06-23

Woit's book will be very helpful to technical people who do not work daily in the field of physics but want to remain up to date on the progress of this field. Woit's conclusions and recommendations will be widely accepted. Interestingly, in my book review of Leonard Susskind's book on The Cosmic Landscape in December 2005, I said, "I hold hopes for physicists but not much for strings." I made this statement because the length of a string is divisible and cannot be modeled by a zero-point. So, string theory was completely wrong on day one.

Woit gave me a clear view of the histories of particle physics, strings, and the standard model. In Ch. 6, my mind became glued to the Yang-Mills theory and the new behavior named `asymptotic freedom.' This new behavior is consistent with other theories: (1) the infinite gap that separates a creator God from the universe; (2) the Riemann hypothesis on prime numbers; (3) the true atoms (Leibniz's monads); (4) Cantor's transfinite number; (5) and the origin of inertia of Bernard Haisch (see `The God Theory). So, Woit is right. It is time for physicists to return to basics and The Standard Model. But, they might also consider the reality of an active God.

Average customer rating:

fails to help
Good Book
A Geometrical Journey

Symmetry, Shape and Space
L.Christine Kinsey , and Teresa E. Moore
Manufacturer: Key College
ProductGroup: Book
Binding: Hardcover

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Accessories:

ASIN: 1930190093

Book Description

This text is suitable for introductory students, perhaps in programs such as education, art and architecture. The text contains some traditional material from geometry as well as more innovative topics. Throughout the text, the authors place strong emphasis on pedagogy, hands-on model building, a guided discovery method of learning, etc. Much of the material is written in such a way that it can be used in the classroom for enrichment projects, by prospective mathematics teachers.

Customer Reviews:

fails to help.......2007-09-10

Sorry, any text book without a glossary fails its primary goal of making its information accessible.

Good Book.......2007-02-08

The book does a good job of reintroducing basic geometry to the more remedial college mathematician. It also does a fine job of introducing higher order geometry. However, the best quality of the book is its ever present agenda to relate geometry to real life situations. This helped to give me a reason to study geometry for practical purposes. Too bad my professor is a douchebag.

A Geometrical Journey.......2005-12-03

This book will well serve any student who wants an interesting, visual approach to mathematics. A plethora of topics are explored including constructions, tessalations, other dimensions-the fourth dimension, polyhedra, three-dimensional symmetry, spiral growth, shape, graph theory, and topology.
All topics are presented, for the most part, in an intuitive, visual manner making it much easier for the average reader to grasp the concepts being presented. Students are required to analyze patterns thereby enhancing their analytical and visualization skills. Each chapter is presented in a stand alone manner. The book is well written and replete with numerous high quality drawings. The text would well serve
any instructor who is presenting a course for liberal arts students, a course in mathematics for teachers, or an enrichment course for high school students. The book would also provide an excellent, self-study guide for high school students who are interested in a gentle guide to mathematics outside the standard high school: algebra, plane geometry, and pre-calculus curriculum. An excellent and extensive biography is provided for the reader who wants to delve deeper into any of the many topics covered in the text. Overall, an excellent introductory text to the many facets of geometry.

Fearless Symmetry: Exposing the Hidden Patterns of Numbers

Average customer rating:

Let me Inject Some Reality into Discussion
From the Earth to the stars
outstanding
Rare - a well written book about math
Definition Dump from Freshman Abstract Algebra Course

Fearless Symmetry: Exposing the Hidden Patterns of Numbers
Avner Ash , and Robert Gross
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0691124922

Book Description

Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them.

Hidden symmetries were first discovered nearly two hundred years ago by French mathematician Évariste Galois. They have been used extensively in the oldest and largest branch of mathematics--number theory--for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination.

The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.

Customer Reviews:

Let me Inject Some Reality into Discussion.......2007-08-05

In spite of some of the comments posted already and in spite of what is on the book's back cover - this is a math book - this is a serious math book. I personally don't see that average person getting anything out of this if they hadn't had say Linear Algebra in particular. Calculus is not required but higher alegra is.

The reason I bought this book is that I read Ian Stewert's book on Symmetry and Beauty and found it lacking as it was not very mathematical.
I was not dissapointed in the level of math in this book. If anything, I got overwhelmed by the end.

I call this type of book "drill deep" but not wide. I like that idea.

The author's have a real ambitious goal. It's laid out on pages 11 and 12:
"in this book we explore ..representations...we consider sets, groups, matrices and functions between them. We show you in detail in one particular case that we develop throughout the book that sets us to our goal: mod p linear representations of Galois groups."

THIS IS THE GOAL OF THIS BOOK. They are not kidding this is what the book sets out to do and I belive accomplishes.

The authors are true to this goal in the "drill deep" mode. Example: Chapter 2 is Groups - not everything about Group Theory is presented but enough that is needed for the rest of the book. In a similar manner one chapter is on so called reciprocity laws. Chapter 4 is on Modular Arithmetic a crucial aspect to this book.

One prior reviewer indicated that each chapter is far more difficult than the last; this is sortof the general tenure of the book - but with exceptions if you know that material. Example, Chapter 5, Complex Numbers, for me was a relief sandwiched in between Modular Artimetic and Equations and Varieties. I can attest that for the subject "Complex numbers" - that they treated it at a relativley elementary level and focused on just those aspects needed later on. I am sure that for all subjects like "Quadratric reciprocity" that was the case. However, if you hadn't been exposed to quadratic reciprocity and Legendre symbols it is a tough slog.

For me the high point of the book was Chapter 8, I felt that I understood the difficult concept of the the Absolute group of the field of algebraic numbers by the end of the chapter. It is an infinite group that only elements can really be enumerated - Identity and complex conjugation. It fills in some (but not all) of the points in the number line between the group of rational numbers and the line with no gaps the field of real numbers.

Chapters 13 to 22 my ability to follow went way downhill and I just skimmed to get some highpoints.

I might return to this book in the future. I like the idea of not having to learn every aspect of something like alebraic ring theory , then every aspect of permutation theory etc. but just learning enough to accomplish some higher level of understanding like ultimatley how Fermat's Last Therom was solved.

I would recomend Stwert's book on Symmetry and Beauty first if you feel you want a more general understanding of this subject as opposed to a real math book which this is.

From the Earth to the stars.......2007-07-29

The book has a goal which is very difficult to reach: introducing people without every mathematical background to the contemporary research in Galois Theory, Number Theory and Diophantine Geometry. In such a situation it is always very hard to choose the best proofs to be written in the book, the best examples and the best way... Maybe, if was the author, I would have made more proofs in the second part of the book, and have chosen other examples for this part. However --- it is so light to criticize --- and the author achieved his goal in proportion of at least 80%, which is not less for an impossible goal!

outstanding.......2007-06-27

This is a very good introduction to arithmetic that everyone wanting to be initiated in this important branch of mathematics should read.

The authors achieved something remarkable: they were able to communicate with accuracy the deepest concepts of arithmetic without the boring style of many mathematics textbooks. The book is very engaging, with nice reflections about the nature of mathematical thought, as well as the motivations behind the concepts.

The authors managed to have a gradual build up of difficulty of topics all the way
to the proof of Fermat's last Theorem. Unlike other introductory texts that let you down
because in their effort to be more engaging, end up too elementary, this one is perfectly balanced. I will also recommend the "Calculus Gallery" as a second outstanding book introducing Analysis.

It would be great if other branches of mathematics like mathematical physics, algebra, mathematical informatics etc had the privilege of such well balanced and insightful introductions.

Well done!

Rare - a well written book about math .......2007-05-30

Unlike most math books, Fearless symmetry is well written. Key concepts from prior chapters are reemphasized in subsequent chapters so readers are less likely to get lost. This is the first book on groups and representation theory that made clear sense to me. I can see where galois therory is going and now have an understanding of the basic form of the proof of Fermat's last theorem.

Definition Dump from Freshman Abstract Algebra Course.......2007-05-16

The book is simply a compilation of definitions from a second rate textbook for Abstract Algebra 101 pasted together with high-five chit-chat from the mathematics-is-really-neat school of math edutainment. There are only a handful of illustrations in the book, none of which have anything to do with symmetry or patterns of numbers. The book Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity by M.R.Schroeder is a much better choice for the subject.

Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics

Average customer rating:

Slightly too dumbed-down
A most beautiful book
Symmetry
A mystery in 196,883 dimensions
An interesting Historical Review of Symmetry in Maths

Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics
Mark Ronan
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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ASIN: 0192807226

Book Description

Mathematics is being driven forward by the quest to solve a small number of major problems - generating excitement in the mathematical world and beyond. Four famous challenges have been Fermat's Last Theorem, the Riemann Hypothesis, Poincare's Conjecture, and, now, the quest for the 'Monster' of Symmetry. It is this latter that forms the topic of this book. Although its roots go back much further, the quest to understand symmetry really begins with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. He used symmetry to understand algebraic equations, and he discovered that there were building blocks or 'atoms of symmetry'. Most fit into a table, rather like the periodic table of elements, but there are 26 exceptions. The biggest of these was dubbed 'the Monster' - a giant snowflake in 196,884 dimensions. At first the Monster was only dimly seen. Did it really exist, or was it a mirage? Many mathematicians became involved. The Monster became clearer, and it was no longer monstrous but a beautiful form that pointed out deep connections between symmetry, string theory, and the very fabric and form of the universe. The story of the discovery involves some extraordinary characters, and Mark Ronan brings these people to life, and recreates in accessible language the growing excitement of what became the biggest joint project ever in the field of mathematics - the hunt for the Monster.

Customer Reviews:

Slightly too dumbed-down.......2007-10-02

According to the blurb on the back, the American Mathematical Monthly described this book as "truly a page-turner". I have to say it is not.

Mark Ronan's task is to take us through the history of group theory culminating in the recently-completed project to classify the finite simple groups. This has taken decades of work by large numbers of highly-skilled mathematicians, with proofs so long and abstruse that there is a genuine concern that no future generation of mathematicians will be able to comprehend them.

How do you communicate this to a lay audience? The key decision for the writer is to gauge his audience. Ronan's view is a readership which knows no group theory. He therefore can't even define a simple group: "a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself" - Wikipedia.

The reader, lacking help in engaging with the subject matter, is instead entertained by concise and amusing mini-biographies and anecdotes about the many participants in the quest. Ronan is a little dry as a writer, but in general this works well enough, although he is too indulgent of such monstrous personages as Sophus Lie. The final milestone in the classification project was confirmation of discovery of the mathematical Monster, the largest of the 26 sporadic groups. This was big news even on conventional news outlets, such as the BBC.

In conclusion, this book will work for mathematicians who know some group theory and who like the historical context spelled out. I don't think many people not educated in mathematics will make it through to the end. With this in mind, Ronan could have profitably added a chapter at the beginning (or even an appendix) where he took the reader through normal subgroups, quotient groups and on to simple groups. He would then have been able to use correct terminology (his own merely irritates) and the journey would have been a lot more satisfying. Perhaps for the second edition?

A most beautiful book.......2007-07-06

Since the first page up to the last, this extraordinary book keeps you in suspense.
It start telling about an extraordinary coincidence, found by two matematicians, that keeps you reading and interested; and in the last page ends with the same coincidence... But this time you know that this "coincidence" could be one of the most important misteries of the universe....of the human mind...and humankind.
It is a history about the questioning in the aparently game-like properties of space.
it is a history of the findings of many people, from ancient history, passing through all the years and up to the present time. From many cultures and languages. All trying to get answers to the riddles of form and symmetry.
You just need to have the knowledge of high school mathematics in order to understand the book. And to have an inquiring or philosophical mind.
The only formula that appears in the book is the solution of an equation of grade two. Also there are big numbers written as products of other numbers with exponents.
Also you have to know what is the meaning of a prime number.
Abstract mathematics will take now a new outlook to you.
Perhaps it is not so "abstract" after all.

Symmetry.......2007-06-18

Humans have shown a fascination for symmetry from the earliest times--from cave drawings to universal symmetric symbols as crosses, spirals, etc.. Symmetry is seen in ancient designs, architecture, calligraphy. From the developments in understanding the mathematics of symmetry seen in ancient civilizations--Egyptians, Greeks, Chinese, Hindus--the trend has grown with a greater understanding provided thru mathematical descriptions. "Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics" by Professor Ronan outlines in graphic clarity and drama the development of mathematical group theory, starting with Galois in the 19th century to the most recent, stunning achievement of those brilliant mathematicians who reached their crowning success in decoding "The Monster"--a group of near-unbelievable complexity. If one wishes to find intellectual stimulation and a glimpse into what may be the farther reaches of reality, I recommend highly that you obtain a copy. The writing is superbly clear and spare in its use of technical terms--all of which are clearly explained when they are used. Understanding the concepts helps, in my opinion, to develop ones rigor of thinking. As a psychiatrist, I am interested in the possible role that understanding of "The Monster" may play in giving us more expanded ideas about consciounsess itself. There are other applications of this knowledge to such areas as string theory. Professor Ronan is to be highly commended for having provided us with the means not only of understanding the essence of what may be the greatest intellecutal achievement known, but, also, the means of understanding more about those remarkable mathematicians in their humanity as well as in their brilliance and diligence.

Leonti H. Thompson, M.D.

A mystery in 196,883 dimensions.......2007-04-11

"The Monster" is an abstract mathematical object, dimly seen even by its discoverers. It sits at the heart of the Classification Theorem (the "Enormous" theorem), a proof in 15,000 pages with contributions from over 100 mathematicians. The Monster has tantalizing connections to algebra, number theory, and seemingly every other field that people have examined closely enough. It's driven an amazing amount of innovation in many areas, and has arguably changed the definition of mathematical proof.

Since it exists only in such rarefied atmospheres, The Monster itself is accessible to only the most diligent of seekers. Still, Ronan has done a fair job of explaining what this beast is, and why this unique object deserves researchers' attention, all in non- (or slightly-) mathematical language. More than that, Ronan has given biographical sketches of some of the remarkable characters that contributed to its study. Evariste Galois was one, that hot-headed, romantic, and tragic figure who spent the last night of his life scribbling his thoughts, before dying in a duel. Ronan also sketches the life of Sophus Lie (rhymes with "bee," not "buy"), a brooding Norwegian giant, and others from the earliest records of mathematics to the current day.

Along the way, Ronan touches on so many fields of mathematics that it's a wonder the Enormous Theorem could ever have been written: combinatorial design theory, finite fields, geometry, number theory, and lots more. Even more fascinating is how the fields morph into one another at their least-understood edges.

In any objective sense, it's a story of bookish people, each working quietly to create one of the bricks or beams that went into the Enormous edifice. Mathematicians are human, though, and as passionate as anyone else when they've devoted their lives to something. The words of this book are about mathematicians and their math. The spirit of this book, however, is about that passion, about the towering achievements that characterized what's known about The Monster, and about the thrill of discovery that so clearly remains to future researchers.

//wiredweird

An interesting Historical Review of Symmetry in Maths.......2007-03-12

From what I have read so far it is very interesting. The history of how symmetry has become so important in various branches of mathematics has been quite eye opening. There are a number of small errors which did puzzle me at first. For example, the number of axes of rotational symmetry for a cube is 24, but not because there are 6 faces that can be each rotated 4 time as described by the author.
Also, for myself, I would have liked a little more theory and explanation of various Hypotheses: this may come later in the book.
Apart from these small gaps I would recommend this book to anyone interested the place mathematics has in the universe.

Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5) (Mathematical World)

Average customer rating:

the best math textbook I've ever had
An excellent primer on abstract algebra

Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5) (Mathematical World)
David W. Farmer
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

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ASIN: 0821804502

Book Description

In most mathematics textbooks, the most exciting part of mathematics--the process of invention and discovery--is completely hidden from the reader. The aim of Groups and Symmetry is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: Its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you.

Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirement.

Customer Reviews:

the best math textbook I've ever had.......2006-02-10

This book was the foundational textbook for a 100-level class in symmetry at my university. I recommend it highly to anyone who wants to get a better feel for what mathematicians actually do and think about and work with. Folks who never got into the higher math classes often have a different idea of what mathematics is all about than mathematicians. At the level of introductory algebra and geometry and even some calculus, math education often seems to be mainly about memorizing formulas and recognizing in which situations to apply them. That's an important thing to learn, but it is not useful for imparting an idea and a feel of the field of mathematics as a whole. Farmer's book brings home the understanding that mathematics is, at its heart, about patterns and that mathematics is not so much about memorization and application as it is about discovery.

The level of mathematical understanding required to get something useful out of this book is low. I believe the professor required beginning algebra as the prerequisite. If you can count to six, recognize the difference between a square and a pentagon, and understand that variables like n, m, or x can be used as substitutes for numbers then you probably have enough mathematical sophistication to work your way through this book and gain insights into the beauty of higher math.

An excellent primer on abstract algebra.......2002-09-23

Groups are the first structures encountered in abstract algebra and form the foundation for most of the others. Fortunately, they are also the easiest to physically represent, so in some sense they are the most concrete. In this book, groups are introduced as the motions and structures of geometric figures, so the presentation is largely by diagram rather than formula. Very little previous knowledge of mathematics is required and after reading the book, you will have a solid understanding of what a group is.
The first topic is the moving of a complete figure to a different location of the plane defined by a grid of points. By keeping the figure rigid and fixed in orientation, a set of legal moves is defined. After that, some of the rules are relaxed and that allows for additional moves to be added. Exercises and problems are put forward here and throughout the book, and with the accent on figures, often give the appearance of a game.
The next steps are then to allow for all possible rotations, translations and reflections of the objects, using these to explain the structure of a group. This is an effective way to introduce group theory, and is how I will do it if I teach abstract algebra again. Permutation and plane tiling symmetry groups are then introduced and examined, and their relationship to the previous groups discussed, which introduces the concept of isomorphism.
Basic group theory is something that everyone can understand, as humans have a natural affinity for patterns and recognizing them despite "trivial" alterations. This book is an excellent primer on group theory and I strongly recommend it to anyone either learning or teaching abstract algebra.

Published in Journal of Recreational Mathematics, reprinted with permission.

Average customer rating:

A Fine book with a few permutations

"Don't cry, I need all my courage to die at twenty."...Galois

A lively read for a wide audience

very accessible introduction to group theory and it's history

Sinusoidal

The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
Mario Livio
Manufacturer: Simon & Schuster
ProductGroup: Book
Binding: Hardcover

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ASIN: 0743258207

Book Description

What do the music of J. S. Bach, the basic forces of nature, Rubik's Cube, and the selection of mates have in common? They are all characterized by certain symmetries. Symmetry is the concept that bridges the gap between science and art, between the world of theoretical physics and the everyday world we see around us. Yet the "language" of symmetry--group theory in mathematics--emerged from a most unlikely source: an equation that couldn't be solved.

Over the millennia, mathematicians solved progressively more difficult algebraic equations until they came to what is known as the quintic equation. For several centuries it resisted solution, until two mathematical prodigies independently discovered that it could not be solved by the usual methods, thereby opening the door to group theory. These young geniuses, a Norwegian named Niels Henrik Abel and a Frenchman named Evariste Galois, both died tragically. Galois, in fact, spent the night before his fatal duel (at the age of twenty) scribbling another brief summary of his proof, at one point writing in the margin of his notebook "I have no time."

The story of the equation that couldn't be solved is a story of brilliant mathematicians and a fascinating account of how mathematics illuminates a wide variety of disciplines. In this lively, engaging book, Mario Livio shows in an easily accessible way how group theory explains the symmetry and order of both the natural and the human-made worlds.

Download Description

"What do the music of J. S. Bach, the basic forces of nature, Rubik's Cube, and the selection of mates have in common? They are all characterized by certain symmetries. Symmetry is the concept that bridges the gap between science and art, between the world of theoretical physics and the everyday world we see around us. Yet the ""language"" of symmetry--group theory in mathematics--emerged from a most unlikely source: an equation that couldn't be solved. Over the millennia, mathematicians solved progressively more difficult algebraic equations until they came to what is known as the quintic equation. For several centuries it resisted solution, until two mathematical prodigies independently discovered that it could not be solved by the usual methods, thereby opening the door to group theory. These young geniuses, a Norwegian named Niels Henrik Abel and a Frenchman named Evariste Galois, both died tragically. Galois, in fact, spent the night before his fatal duel (at the age of twenty) scribbling another brief summary of his proof, at one point writing in the margin of his notebook ""I have no time."" The story of the equation that couldn't be solved is a story of brilliant mathematicians and a fascinating account of how mathematics illuminates a wide variety of disciplines. In this lively, engaging book, Mario Livio shows in an easily accessible way how group theory explains the symmetry and order of both the natural and the human-made worlds. "

Customer Reviews:

A Fine book with a few permutations.......2007-09-08

If you are not a mathematician (and I am not), but have an interest in the subject, and a working knowledge of some elementary ideas, this is a terrific book. It has the easiest explanation of symmetry/Galois groups, etc., of any of the books I have tried on the topic -- oh sure, it rambles (as the severe critics here say) -- but try and find some other book on the subject that doesn't immediately drop you far beyond your depth. Livio has a knack for very, very clear explanations and great metaphors (permutations and probability are discussed in terms of finding a mate). I recommend it highly, especially if you can get it with one of Ian Stewart's books on the same topic.

"Don't cry, I need all my courage to die at twenty."...Galois.......2007-02-09

When I came across this book,I thumbed through it and the figures that jumped out at me were a collection of things,mainly about mathematics,puzzles and other things that interest me. I graduated in Electrical Engineering nearly 50 years ago,and have had a lifelong interest in Mathematical Recreations and Puzzles of all sorts. Granted most of the Mathematics I studied has long since left me mainly because of lack of use.However,the lore,beauty,mystery and fascination of Mathematics has remained. A lot of the Mathematics discussed in this book falls into what I think of as Theoretical rather than Applied Mathematics;and then there's that whole area of Recreational Mathematics.
I have read all the other reviews here,and basically agree with all of them.Taken together they do a good job of telling what the book is about and the Mathematicians who searched for those elusive solutions.In fact,there is so much that could be covered that it would take many volumes to even only scratch the surface.
I don't know if I really "know" much more about Group Theory and Symmetry than when I started ,but I still found it a fascinating read. Kind of like a 5-day tour of Europe-Been there,done that,but do I "know" Europe?
Like I said,other reviews have pretty well covered the book;so I won't repeat.
However; I would like to point out a couple of things.
In chapter 6,the 15-Puzzle is discussed. This is one of the all time greatest puzzles.It has interested me for years. If you would like to know more about it,I strongly recommend you read "The 15 Puzzle" by Jerry Slocum and Dic Sonnefeld.After you see this book ,you'll probably agree it is one of the world's most interestting puzzles;and what a history and legend it has. I posted a review of it here on Amazon on June 6,2006.
If you haven't noticed ,the information on this book has a section "Inside the Book" and in this section under "text stats" ,it shows this book has a Fog Index of 16.2. A search on the net will show how it is calculated. It takes a sample of text,and by looking at the lengths of sentences,number of multiple syllable words,paragraphs,and so forth comes up with a number that shows how difficult it is to comprehend. 16.2 is a fairly high level; and that combined with the theoretical math concepts;there is lttle wonder tht many would find this a fairly difficult book to read.Of course,I'm referring to the Mathematical concepts as opposed to the Biographical information.
The author must have done a tremendous amount of research in writing this book, and in the extensive Notes and References he provides a huge amount of information for the reader who wishes to pursue anything further

A lively read for a wide audience.......2007-01-07

Symmetry is the topic of Mario Livio's THE EQUATION THAT COULDN'T BE SOLVED: HOW MATHEMATICAL GENIUS DISCOVERED THE LANGUAGE OF SYMMETRY, and will make an involving read for those involved in either science or art. Mathematicians solved algebraic equations until they came to a stop with the quintic equation, which resisted solution until two mathematical geniuses independently discovered it couldn't be solved using the usual methods. This account of 'group theory' explains both the concept of symmetry and the evolution of its foundations, and makes for a lively read for a wide audience from physicists and science majors to students involved in the arts.

Diane C. Donovan
California Bookwatch

very accessible introduction to group theory and it's history.......2006-09-05

The equation that couldn't be solved is about the history of group theory. The stories of two of it's early contributors Abel and Galois is told in detail. In addition the author provides an accessible overview of group theory. The specific equation that couldn't be solved is the quintic, which cannot be factored in general. That means that while there are specific examples of polynomials with a factor of x raised to 5 or greater that can be factored there is no general formula like the quadratic eqauation that can factor all quintic or higher polynomials. Although originally used to study factoring, group theory has evolved to be about many other things including the mathematical concept of symmetry. Symmetry arises in many parts of mathematics and science so it is very imporant. I came away from this book with a knowledge of the history of group theory and a smattering of knowledge about group theory and it's applications. I highly recommend this book to those people, like me, who are interested in mathematics and would like to peek under the surface to see what it is all about.

Sinusoidal.......2006-08-03

As I was reading this book, my interest level ebbed and flowed; it was like that all the way to the end. For the most part, the book held my attention, but often I got the feeling that the author was straying from the symmetry theme a bit too much. So this is very much a book of peaks and valleys (hence the title of my review).

The most fascinating parts of the book for me are certainly the masterfully-written biographies of Abel and Galois, and the author's discussion of the cubic equation and its gradual solution by dal Ferro, Fiore, Tartaglia, and Cardano. The author is totally smitten by the figure of Galois; he is described in wonderful detail, warts and all: the young hot-tempered revolutionary romantic who let politics consume him, largely as a result of the terrible misfortunes he endured in his personal and academic life, the tragic duel that resulted in his death from peritonitis, and the inexhaustible legacy of group theory which he bequeathed to the world.

The discussion about groups is first-rate. Permutation groups are rightly emphasized. I regret that he does not dwell on group theory more. But for the author to include a detailed description of normal subgroups is a mark of his willingness not to underestimate his readers. There is also a clear and concise discussion of the quantum world, relativity, and even string theory. But on the downside, there is the author's tendency to write in a sort of blithe, whimsical, off-hand manner that can grate on the nerves after awhile.

I was struck on occasion by fascinating statements that had never occurred to me before, such as: "All the electrons in the universe are precisely identical in terms of their intrinsic properties; there is no way to distinguish one from the other." And: "The chief reason we can interpret relatively easily observations of galaxies ten billion light years away is that we find that hydrogen atoms there obey precisely the same quantum mechanical laws they obey on Earth." And: "Female orgasm seems to be less about bonding with a great person than about a cold Stone Age evaluation of the mate's genetic endowment." This last revelation may have far-reaching effects, in the amount of mail which our esteemed author has received from female readers!

In summary, this is a very wide-ranging book, perhaps a little too wide-ranging, and not quite as polished and elegantly written as I would have liked. But it is still a fascinating exploration of group theory and its applications for anyone with a curious and open mind.

Deep Down Things: The Breathtaking Beauty of Particle Physics

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Particle Physics Made Easy
A Review From a Non-Physicist
It really is "Breathtaking"
A 'big-picture' conceptual guide to the Standard Model
Best popular particle/quantum physics books I have read

Deep Down Things: The Breathtaking Beauty of Particle Physics
Bruce A. Schumm
Manufacturer: The Johns Hopkins University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 080187971X

Book Description

A useful scientific theory, claimed Einstein, must be explicable to any intelligent person. In Deep Down Things, experimental particle physicist Bruce Schumm has taken this dictum to heart, providing in clear, straightforward prose an elucidation of the Standard Model of particle physics -- a theory that stands as one of the crowning achievements of twentieth-century science. In this one-of-a-kind book, the work of many of the past century's most notable physicists, including Einstein, Schrodinger, Heisenberg, Dirac, Feynman, Gell-Mann, and Weinberg, is knit together in a thorough and accessible exposition of the revolutionary notions that underlie our current view of the fundamental nature of the physical world. Schumm, who has spent much of his life emmersed in the subatomic world, goes far beyond a mere presentation of the "building blocks" of matter, bringing to life the remarkable connection between the ivory tower world of the abstract mathematician and the day-to-day, life-enabling properties of the natural world. Schumm leaves us with an insight into the profound open questions of particle physics, setting the stage for understanding the progress the field is poised to make over the next decade or two.

Introducing readers to the world of particle physics, Deep Down Things opens new realms within which are many clues to unraveling the mysteries of the universe.

Customer Reviews:

Particle Physics Made Easy.......2007-03-02

This book should be a must read for anyone that tries to understand particle physics. I've been looking for something like this for a long time. The Standard Model is explained with great skill and clarity, and with minimal use of math. This is not a mathematical book, but where minimal mathematics becomes necessary (group theory), it is introduced with the assumption that the reader knows next to nothing (which was my case) and developed to the point where, combined with physics, it makes sense. Most of the math only requires logic, not computations, and all you are required to memorize are a few rules -- conventions -- that only take a couple of lines. Beautiful.
The author limits himself to what is known and generally agreed about particle physics. The limits of the theory are also very well explained, but no significant steps into the unknown are made, which I think it is a good thing for once.
If you like Brian Greene, Michio Kaku, Lisa Randa