Customer Reviews

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

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As a mathematician, I did not need to read the first half of the book, which explains very clearly some of the concepts used in the meatier second half. But I was very impressed with the clarity of Ronan's exposition. One valuable bit of terminology that he uses is calling a group (a technical mathematical concept that is the central subject of the book, and which he explains with great lucidity) an "atom of symmetry". This is a perfect way to convey the meaning of a group, and give the lay reader an easy way to conceptualize it.

Besides explaining things in terms that any intelligent reader can understand without getting lost in details -- AND without blurring the truth, either (quite a feat!) -- Ronan gives an engrossing account of which mathematician had which insight, and discussed it with which other mathematician, etc., so that the way progress in math occurs is elucidated. I'm a mathematician who doesn't know a great deal about the main subject of this book, and can honestly say that I learned a lot of intriguing stuff by reading it.

The math described is very pretty. For those who understand the terminology, I'll mention that this book's main subject is the classification of the finite sporadic simple groups (and it is now known that there are exactly 26 of them in all). The largest and most complicated of these 26 is enormous, and known as The Monster, whence the title of the book.

Ronan also describes several loose ends -- bits of mathematics that are not well understood -- to further give the lay reader an accurate picture of how mathematicians and mathematics works.

Do not walk or run, but *skip* to your nearest book emporium and buy this book.

Disclaimer: I have never met the author, have no financial interest in the sale of this book, and the above is entirely my personal opinion.

The story of the `Monster Moonshine' is told eloquently and with great enthusiasm in this book, and gives to the curious reader the needed insight into both the relevance and the mathematical constructions needed to bring it about. To understand in-depth the Monster requires a highly advanced background in mathematics, and to understand its connection with physics requires even more. The book though is not written for professional mathematicians, but rather for the general reader, who may have heard about the Monster through the popular press. Even though the author explains the ideas very well, a general reader however may find the book tough going at times. Those readers who have at least a background in mathematics that could be obtained in a typical undergraduate curriculum could better appreciate it.


There are many parts of the book whether the author gives really good explanations and motivations for various mathematical concepts. One is where he introduces the concept of symmetry via solid geometry and the `Platonic solids', which allows a more straightforward comprehension for readers without extensive mathematical preparation. He also uses it to introduce the concept of `duality', which is actually something that even readers with a good background in mathematics will appreciate. Although he does not define what it means for objects to be dual to each other rigorously, he gives examples, and for the purposes of the book merely notes that such objects will have the same symmetries. Another one is the use of the Sam Loyd tile game to explain the difference between even and odd permutations. Still another is the introduction of Lie groups as being a generalization of Galois theory for differential equations.

The author also discusses briefly the life histories of the mathematicians involved in the relevant group theory including their idiosyncrasies and different methods for doing mathematical research (and also the famous fictional mathematician `Bourbaki' who in reality was a group of highly respected mathematicians). Readers curious about the publishing habits of mathematicians will find out, interestingly, that they usually publish alone, and when they do publish together there is no arguing about whose name comes first: the listing of names is done in alphabetical order. Also interesting is the discussion on the role of reviewers of the research papers that led to the Monster. Since only a tiny minority of individuals understood (or were interested in) the relevant constructions, the anonymity of the reviewers was essentially compromised. But this did not act as a retardant to the research, and these events are another strong argument against anonymous reviewing.

The author also makes strong commentary against the use of computers in doing proofs of mathematics. He insists on being able to check the papers by hand, and details a fascinating story about how complicated calculations that seemed to formidable to do without the assistance of a computing machine were actually accomplished by some of the mathematicians involved in research into the Monster. One can't help but be impressed by their achievements in this regard. However, proofs done by computing machines are just as good as those done by humans. In fact, one might argue that machine proofs are always better, since their logic is impeccable and the likelihood of committing mistakes is very small. In addition, the intermixture of colloquial language with mathematical symbolism that is typical of human proofs makes totally rigorous proof unattainable, if one insists on a strict interpretation of deduction.

Everything in this book is therefore interesting, but the author does not want to leave the reader with the impression that there is no further work to be done on the Monster. This work he says involves obtaining a real understanding of the mathematical constructions behind the Monster. Also, there are further "coincidences" of a number-theoretic nature that need elucidation (one of these, interestingly, involves the integer 163). These issues will no doubt motivate a few young mathematicians to investigate the Monster in even more detail. It will be interesting to see what they find.

"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

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?

The beauty of this book lies in its revelations about the world of mathematics. The Monster is a super-large `group' and the theory behind it appears to be a perfect vehicle for the revelatory purpose. Its arithmetic involves multiplying and dividing rows of numbers differing by one from each other, or others raised to simple powers. Such simplicity immediately disarms those who might think that mathematics is just ever more complicated arithmetic, as most past schooling might have suggested. Instead Ronan thrusts us into a realm where concepts of spatial relationships are explored. Forget three dimensions, how about six or eight or more? I confess I didn't understand every paragraph, but that doesn't matter. It is the journey that counts. And when you get to the end look at the glossary and the utter simplicity of the definition of a `group'. It should give you a sense of wonder that something so apparently straightforward has led human minds on the fantastic journeys laid out in this marvelous little pocket-sized monograph - and that that is mathematics.

The book outlines the players and events that lead to the "Classification Theorem" -- a 'collective' proof by various mathematicians that identified all the finite groups. The theorem is estimated to span 15,000 pages of dense technical writing over various journal articles. The surprise in the classification effort was that there exist 26 exceptional groups and Ronan titled his book after the largest of these -- the Monster group.

The book was a good short read. It took me a day to finish it. The people involved in the classification along with some hints on the technical issues involved form the backbone of the story. I found it entertaining but wish that it had covered the mathematics in greater depth (at the level of an undergraduate) and had a better bibliography to guide future readings of a curious reader.

While it is simple enough to conceive an object in one, two or three dimensions, adding just one more dimension can be mind-bending. The four dimensional cube - or tesseract - cannot be truly perceived, but we can at least get a glimmer of it when we look at its projection, which appears like a cube within a cube. Five dimensions are even harder to perceive. The Monster, the subject of Mark Ronan's Symmetry and the Monster, has 196,884 dimensions. It seems appropriately named.

What is the Monster, however? This takes a while to describe, and it all begins with the brilliant Galois, a mathematical genius who would be dead by 20 after being on the losing side in a duel. Galois would make some major strides in the field of algebra known as group theory. A group is really just a self-contained set of numbers (or other components) with an operation (such as addition) and certain properties (such as closure, the idea that when you do the operation on two members of the set, you get another member of the set; for example, with the whole numbers and addition, adding any two positive integers gets you another positive integer).

Groups can be both finite and infinite, and among finite groups, there are so-called simple groups (or what Ronan calls atoms of symmetry). These are not simple as in easy, but simple as they cannot be deconstructed into simpler groups, just as when you factor a number, you cannot factor any further when you reach the prime factors. Most simple groups fit into certain families, but there also 26 exceptional groups (or sporadic groups). Determining that the number was 26 and finding all these groups is what Symmetry and the Monster is all about. The final group would be the biggest, by far: the Monster.

Perhaps the best book dealing with the solution of a tough problem is Simon Singh's Fermat's Enigma, dealing with the proof of Fermat's Last Theorem. Ronan's book is not as easy of a read, but then again, he has a tougher row to hoe: while Fermat's Last Theorem is relatively easy to understand (though difficult to prove), the concept of symmetry groups is a bit more esoteric. Operating within this constraint, Ronan does a good job, writing clearly, with both a sense of history and sense of humor. This is not an easy subject to really grasp, but it may be ultimately rewarding to those who stick with it.

This book is a ripping yarn - I couldn't put it down. My wife asked when I came to bed at midnight : "Maths porn again darling? " Although I have done some group theory, my knowledge was nowhere near enough to make any meaningful attempt to understand the detail of the Monster project and like many others, it remained an intractable beast that others were battling with. Ronan explains in a high level way the history to the Monster starting with Galois and working his way through the historical development. He peppers the account with all manner of interesting observations about the participants which are revealing in ways that one does not often find in maths books. For instance, there is a revealing comment on page 152 about someone of the stature of John H Conway who confessed he "felt like a fraud" in giving talks early on in his work on Monster. It seems a graduate student asked him the obvious question namely " How do you now that your new group can't be decomposed into something simpler?" Maths is an unforgiving business.


Mark Ronan who has worked with and/or knows most of the heavy hitters in the field has done a wonderful job explaining the history of what is an extraordinary undertaking not only in purely intellectual terms but also in personal terms. The sociological dimensions of this immense task are reflected in all manner of small and large stories. Thus John H Conway bargains with his wife to have blocks of time away from the 4 kids so he can crack some problems and he manages in 12 ½ hours to prove something important about the Leech Lattice. That set him up for life. The proofs in this field can be hundreds of pages long - one by Mason is 800 pages long and has not been published. This itself imposes huge strains on referees. The classification task (which I had read about but had no detailed knowledge of what was involved other than a vague idea it was the equivalent of the 30 Years War) demonstrates what a small group of intensely committed people can do. What they were doing was to provide a set of knowledge that subsequent mathematicians could understand given that the barriers to entry to the detailed knowledge are so high.

At a purely personal level one has to marvel at how some of the people concerned threw their lot in with this "monstrous" task. Every budding PhD students knows that problem selection is important and it does not pay to spin one's wheels forever on some obscure problem.

There are some truly astonishing connections revealed in this book. The connection between the number theoretic j function and the character set of the Monster (see pages 192-193) is remarkable but then there is the even more remarkable connection between light rays and the Leech Lattice (see page 224).

Mark Ronan has done a great service to all those who have served and still served in the battle with the Monster. Most of the main workers in the field are no longer with us so Ronan's book provides the general community with some sense of their achievements.

For those interested in Lie Theory may I suggest John Stillwell's accessible book "Naïve Lie Theory" as a starting point. He strips a way a lot of the overheard that makes Lie Theory so daunting.

Peter Haggstrom
Bondi Beach
Sydney, Australia

One of the greatest achievements of the 20th century mathematics has been the classification of the finite simple groups. Groups are mathematical objects that tells us about symmetries, and like many other mathematical objects they are relatively easy to describe, but can be fiendishly difficult to fully understand. Sometimes understanding comes from a single brilliant insights by an incredibly gifted individual, and these individuals become part of the mathematical lore that can even touch upon the popular imagination. However, most of the time these days the game of mathematics has become complex enough that it can become increasingly difficult for any individual to fully contribute to on its own to the full problem. Professional mathematicians don't mind this at all: they thrive in collaborations and feed off of each other's work and enthusiasm. The collaborative nature of mathematics is at full display when it comes to the classification of finite simple groups, an effort that spanned hundreds of articles in scientific journals between 1955 and 1983. I have always been curious to find out more about this enterprise, and this book does a remarkable job at presenting it to the general reader. It is comprehensive without becoming technically hard to follow. Anyone who has ever taken a college level mathematics course should be able to read it without much difficulty, although some basic understanding of group theory and modern algebra would be great bonus. The book also doesn't dumb down mathematics to the point that it becomes irritating for those who have some mathematical sophistication, so even professional scientists and mathematicians can find it very informative and a rewarding read.

And if you are curious, the Monster from the title refers to the special simple finite group that has been one of the most fascinating mathematical objects discovered so far.

This is an interesting read in general but the author doesn't include enough examples to illustrate idea (e.g. some graphical examples of different rigid geometric transformaion of solids will be great at the beginning of the book). The author also introduce mathematical concepts without enough explanation. While some of the concepts are simple enough to be understood without clarification, some of the more complicated ones in the later chapters are not. So the readers who are not already familiar with the subjects might find it difficult to follow the author's arguments.