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Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics
Mathematics is driven forward by the quest to solve a small number of major problems--the four most famous challenges being Fermat's Last Theorem, the Riemann Hypothesis, Poincaré's Conjecture, and the quest for the "Monster" of Symmetry. Now, in an exciting, fast-paced historical narrative
ranging across two centuries, Mark Ronan takes us on an exhilarating tour of this final mathematical quest.
Ronan describes how the quest to understand symmetry really began with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. Galois, who spent the night before he died frantically scribbling his unpublished discoveries, used symmetry to understand algebraic equations,
and he discovered that there were building blocks or "atoms of symmetry." Most of these building blocks fit into a table, rather like the periodic table of elements, but mathematicians have found 26 exceptions. The biggest of these was dubbed "the Monster"--a giant snowflake in 196,884 dimensions.
Ronan, who personally knows the individuals now working on this problem, reveals how the Monster was only dimly seen at first. As more and more mathematicians became involved, the Monster became clearer, and it was found to be not monstrous but a beautiful form that pointed out deep connections
between symmetry, string theory, and the very fabric and form of the universe.
This story of discovery involves extraordinary characters, and Mark Ronan brings these people to life, vividly recreating the growing excitement of what became the biggest joint project ever in the field of mathematics. Vibrantly written, Symmetry and the Monster is a must-read for all fans of
popular science--and especially readers of such books as Fermat's Last Theorem .
ranging across two centuries, Mark Ronan takes us on an exhilarating tour of this final mathematical quest.
Ronan describes how the quest to understand symmetry really began with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. Galois, who spent the night before he died frantically scribbling his unpublished discoveries, used symmetry to understand algebraic equations,
and he discovered that there were building blocks or "atoms of symmetry." Most of these building blocks fit into a table, rather like the periodic table of elements, but mathematicians have found 26 exceptions. The biggest of these was dubbed "the Monster"--a giant snowflake in 196,884 dimensions.
Ronan, who personally knows the individuals now working on this problem, reveals how the Monster was only dimly seen at first. As more and more mathematicians became involved, the Monster became clearer, and it was found to be not monstrous but a beautiful form that pointed out deep connections
between symmetry, string theory, and the very fabric and form of the universe.
This story of discovery involves extraordinary characters, and Mark Ronan brings these people to life, vividly recreating the growing excitement of what became the biggest joint project ever in the field of mathematics. Vibrantly written, Symmetry and the Monster is a must-read for all fans of
popular science--and especially readers of such books as Fermat's Last Theorem .
264 pages, Hardcover
First published January 1, 2006
15 people are currently reading
476 people want to read
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Displaying 1 - 30 of 39 reviews
May 25, 2009
The hero of this book, the Monster of Symmetry, is a giant snowflake that requires 196,884 dimensions to describe it. Rather than being a mere mathematical curiosity, its structure points to deep connections between symmetry, number theory, and mathematical physics. Ultimately it may help us come to understand the fabric of our universe -- or even the infinite multiverse.
This is the story of the quest to discover the Monster. Beginning in revolutionary France in the 1830s, it takes the reader up to the present day -- and beyond. Mark Ronan describes the driving determination, odd coincidences, and extraordinary people that were drawn into what became the greatest collaboration in the history of mathematics. He describes how the study of simple, regular shapes led to the whole field of simple groups, and the search for the basic nuts and bolts of symmetry. The hunt for the Monster began to classify all these nuts and bolts, the "atoms", of symmetry, and this bold project began to accelerate after World War II as new methods were devised to fill in the gaps in the classification, in the same way that chemists and physicists worked to fill out the Periodic Table of the Chemical Elements. The quest involves shapes in numerous dimensions, weird multi-crystals, and different ways of packing spheres that now underlie a good deal of modern technology. But some mathematical entities just don't fit the scheme, and the biggest, most bad-assed member of these outlaws was the Monster. As mathematicians drew ever-nearer to it, this dazzling entity's sheer scale and complexity became clear.
But mathematicians are still struggling to grasp the nature of the Monster. Given the extraordinary links between it and the deep physical form of the universe, it will become far more important in the future than we realize even now, a piece of 22nd- or 23rd-century mathematics arriving by chance in the 20th century, hinting at answers to questions not yet asked, which may not be asked for decades or even centuries. When we finally understand the nature of the Monster, we will probably thereby open up a whole new, far more profound vista of the nature of physical reality.
This is the story of the quest to discover the Monster. Beginning in revolutionary France in the 1830s, it takes the reader up to the present day -- and beyond. Mark Ronan describes the driving determination, odd coincidences, and extraordinary people that were drawn into what became the greatest collaboration in the history of mathematics. He describes how the study of simple, regular shapes led to the whole field of simple groups, and the search for the basic nuts and bolts of symmetry. The hunt for the Monster began to classify all these nuts and bolts, the "atoms", of symmetry, and this bold project began to accelerate after World War II as new methods were devised to fill in the gaps in the classification, in the same way that chemists and physicists worked to fill out the Periodic Table of the Chemical Elements. The quest involves shapes in numerous dimensions, weird multi-crystals, and different ways of packing spheres that now underlie a good deal of modern technology. But some mathematical entities just don't fit the scheme, and the biggest, most bad-assed member of these outlaws was the Monster. As mathematicians drew ever-nearer to it, this dazzling entity's sheer scale and complexity became clear.
But mathematicians are still struggling to grasp the nature of the Monster. Given the extraordinary links between it and the deep physical form of the universe, it will become far more important in the future than we realize even now, a piece of 22nd- or 23rd-century mathematics arriving by chance in the 20th century, hinting at answers to questions not yet asked, which may not be asked for decades or even centuries. When we finally understand the nature of the Monster, we will probably thereby open up a whole new, far more profound vista of the nature of physical reality.
April 7, 2011
I loved this book! I tried to explain to a class I was teaching on Symmetry that I could not put it down. It was almost like a detective novel! The fact that it was probably the seventh or eighth book on symmetry I had read by then makes this all the more surprising and shows how well the author does his job.
The book describes how the largest ever mathematical proof was solved by hundreds of mathematicians over many, many pages of proofs. Sounds dry? It's not! This history goes back to the beginnings of group theory, and runs all the way through to quite recent times. It includes some very interesting discussions of the personalities involved.
Highly recommended!
The book describes how the largest ever mathematical proof was solved by hundreds of mathematicians over many, many pages of proofs. Sounds dry? It's not! This history goes back to the beginnings of group theory, and runs all the way through to quite recent times. It includes some very interesting discussions of the personalities involved.
Highly recommended!
January 15, 2015
On second reading, I've downgraded this book from five stars to four. Towards the end the maths dries up and the story becomes one of names, dates and places. I'm feeling brave enough to tackle some more technical details but they are lacking. What is the "next book up" on this topic?
April 22, 2017
Este libro consigue contar la historia de la clasificación de los grupos simples finitos de manera interesante, intrigante incluso (¡hay cliffhangers matemáticos!)... hasta sus dos o tres últimos capÃtulos, que tristemente se vuelven más secos y aburridos, a pesar de que el escritor estaba metido en el ajo y conoció de primera mano a los protagonistas principales. Por esto ha perdido la quinta estrella.
La narración versa fundamentalmente sobre las personas que desarrollaron la clasificación, pero hay matemáticas más o menos serias imbricadas en ella, con algunas fórmulas y conceptos desarrollados aquà y allá. He sufrido con la historia de Galois una vez más (y he visto una interpretación conspirativa de su muerte que es nueva para mÃ; ¿será cierta?), me he reÃdo de lo lindo con algunas de las ocurrencias de Lie y he aprendido quién es quién en la teorÃa de grupos del siglo XX. Es impresionante ver en particular cómo afectó a la comunidad matemática la subida al poder del partido Nazi, pues Alemania era uno de los focos principales del álgebra de la época; la anécdota sobre Noether y Witt es indignante, y el comportamiento de este último de lo más peculiar.
He echado en falta explicaciones matemáticas más detalladas (sobre las simetrÃas de los sólidos platónicos, por ejemplo), porque el autor decide, con valentÃa, meterse a explicar conceptos complicados y sin embargo parece que se arrepiente y los deja muy en el aire, vistos de refilón. Además algunas explicaciones me han parecido poco iluminantes (como las referidas a la relatividad). En cambio otras son bastante buenas. Además he aprendido por fin por qué el teorema de Feit-Thompson, que "lo único" que dice es que cualquier grupo simple no cÃclico tiene orden par, y necesitó más de 200 páginas para ser demostrado, está considerado una de las piedras angulares de las matemáticas del siglo XX (SPOILER: existÃa una técnica para clasificar grupos a partir del centralizador de una involución, y todo grupo de orden par posee una involución; por tanto todo grupo simple no cÃclico es suceptible de ser analizado mediante esta técnica).
Como libro de divulgación lo considero espectacular: quien no conozca cómo funciona la investigación matemática podrá experimentarla como si la viviera desde dentro, ayudado por comentarios muy esclarecedores sobre cómo funcionan la demostración de teoremas, la creatividad matemática, el trabajo colaborativo, la vida nómada de los académicos, la publicación en revistas revisadas por pares, la contribución de los alumnos de doctorado y el paso del corpus de conocimiento a las generaciones más jóvenes. Y todo ello con un tema de partida poco habitual. En este respecto es un libro único e imprescindible.
La narración versa fundamentalmente sobre las personas que desarrollaron la clasificación, pero hay matemáticas más o menos serias imbricadas en ella, con algunas fórmulas y conceptos desarrollados aquà y allá. He sufrido con la historia de Galois una vez más (y he visto una interpretación conspirativa de su muerte que es nueva para mÃ; ¿será cierta?), me he reÃdo de lo lindo con algunas de las ocurrencias de Lie y he aprendido quién es quién en la teorÃa de grupos del siglo XX. Es impresionante ver en particular cómo afectó a la comunidad matemática la subida al poder del partido Nazi, pues Alemania era uno de los focos principales del álgebra de la época; la anécdota sobre Noether y Witt es indignante, y el comportamiento de este último de lo más peculiar.
He echado en falta explicaciones matemáticas más detalladas (sobre las simetrÃas de los sólidos platónicos, por ejemplo), porque el autor decide, con valentÃa, meterse a explicar conceptos complicados y sin embargo parece que se arrepiente y los deja muy en el aire, vistos de refilón. Además algunas explicaciones me han parecido poco iluminantes (como las referidas a la relatividad). En cambio otras son bastante buenas. Además he aprendido por fin por qué el teorema de Feit-Thompson, que "lo único" que dice es que cualquier grupo simple no cÃclico tiene orden par, y necesitó más de 200 páginas para ser demostrado, está considerado una de las piedras angulares de las matemáticas del siglo XX (SPOILER: existÃa una técnica para clasificar grupos a partir del centralizador de una involución, y todo grupo de orden par posee una involución; por tanto todo grupo simple no cÃclico es suceptible de ser analizado mediante esta técnica).
Como libro de divulgación lo considero espectacular: quien no conozca cómo funciona la investigación matemática podrá experimentarla como si la viviera desde dentro, ayudado por comentarios muy esclarecedores sobre cómo funcionan la demostración de teoremas, la creatividad matemática, el trabajo colaborativo, la vida nómada de los académicos, la publicación en revistas revisadas por pares, la contribución de los alumnos de doctorado y el paso del corpus de conocimiento a las generaciones más jóvenes. Y todo ello con un tema de partida poco habitual. En este respecto es un libro único e imprescindible.
April 9, 2011
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.
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.
April 6, 2024
Georg Friedrich Gauss once described mathematics as ‘the queen of the sciences�. That may be true (it is so, in my opinion) but it definitely doesn’t show when come to the field of popular science books. This might be related to another famous quote, this time by Stephen Hawking: “Someone told me that each equation I included in the book would halve the sales�.
Mark Ronan’s book doesn’t include many equations, but I’d ascribe the problems with math outreach in books to another factor, besides lay-frightening numerical and algebraic symbols. Most pop science books can shift the math to behind the stage, and focus on metaphors, graspable explanations, interesting material-world connections, and hero-scientist biography. Math books are more limited in this: they can’t really hide all the math away (the math is the point, after all). In interesting but abstract and technical developments, it can be very difficult to connect it to things an individual can grasp and see. Still, it is worth a try, and Mark Ronan has done it with this excellent little book, for which I can only pile words of praise. In Symmetry and the Monster, he has done a squaring of the circle, of sorts, creating a book about a very abstract and important mathematical achievement (the classification of Finite Simple Groups), and managing to make it both accessible and highly entertaining for the most part, modulo a little bit of effort and willingness on the part of the reader.
Small aside: it is perhaps a well-known but no sufficiently interiorized fact that the 20th century has been the greatest, most golden age of mathematics ever, easily putting all other centuries to shame. Advances in the science have been accelerating since at least Newton and Leibniz and the Scientific Revolution, which means the 19th century was a silver age, the 18th a bronze one. The great achievements of Classical Greece hardly warrant more than copper in comparison. This has also tended to happen in other sciences, with Physics as a case in point, but its achievements have been much more easy to popularize and write about - Einstein and Relative, the birth Quantum Mechanics, the making of the Atomic bomb� The latter, in fact, has an excellent book by Richard Rhodes, which with its well-crafted, almost literary narration and collective scientific authorship has really reminded me of Ronan’s.
So back to Symmetry and the Monster: this book attempts to trace the history and main developments in a mathematical quest that started in the 19th century and concluded as a massive collaborative work and one of the pinnacles of 20th century mathematics: the classification of Finite Simple Groups. It is a story that begins with the birth of modern, ‘Abstract� Algebra, which has little to do with what we’re taught in high school, and a lot to do with the study of abstract structures with properties. One of these is a group, which is a set of objects with one operation you can do on them. A good example is the integers under addition: if add three or more, it doesn’t really matter how you group the sums (associativity); if you add 0 to any, you get what you started with (identity); for any integer there’s another one that when you add them together, you get 0 (inverses); and adding any two numbers together gets you a third that is still in the list of integers (closure). All this might seem nitpicky or trivial, but it actually turns out to be tremendously useful, and to model a lot of things in the real world, like symmetries (which can be simplified, explained and studied as groups).
Now many groups have an infinite number of elements (like the integers under addition), but other are finite. And some groups can be ‘factored� (like numbers) into smaller, constituent groups. Groups that can’t be reduced any more are called simple groups, or as Mark Ronan prefers to call them in this book, ‘atoms of symmetry�.
From their appearance in the work of Evariste Galois, where they were invented as part of a proof of why 5th degree equations cannot be solved (like the lower-degree ones) using a simple algebraic formula, Ronan chronicles the development of their study, and the surprising discovery that finite simple groups could probably be classified into a list, that would include all their possible instantiations. On the way, we discover different types of groups (permutation, cyclic with prime order, continuous, or lie�) and the lives of the mathematicians who labored at them. We see how the project of making a table of the different families of finite simple groups develops, becoming a highly technical, demanding and flourishing branch of modern mathematics. We also see the discovery of ‘exceptional� groups which don’t fit into any of the categories, and the work to pokemon-like ‘catch them all�, prove their existence and properties and incorporate them into the table. The last and biggest of these sporadic simple groups is the Monster that gives its title to the book, and which has an added interest because of weird and unexpected connections it makes with completely different branches of mathematics (Number Theory) and the real world (Physics, String Theory).
Besides the discovery and the objects, the lives and anecdotes of the mathematicians working at the project are heavily referenced in the book, and add a non-trivial element to making it lighter and more accessible. I believe Ronan himself is a mathematician that did a lot of work in this area, so he knows many of the figures he talks about in the book (including the sadly deceased, maverick and magical genius John Horton Conway).
As I said in the second paragraph, this book is an excellent read, and I think is not too daunting even for a relatively lay audience. The mathematical mechanics are mostly only hinted at and/or summarized, and is mostly expressed in understandable language, with a couple of exceptions (I got a bit lost in the part explaining how you can use cross sections to discover other finite simple groups, along with some mention of ‘subgroups� of these which I didn’t really get). But small details aside, if you have a minimal interest in mathematics, you are likely to both enjoy and understand what this book talks about, and you will learn of a heroic epic that is at least as important as a feat of human science as the production of the atomic bomb. The Oppenheimers in this volume are much less known, much nicer and their work definitely less deadly, but in many other ways, it matches and surpasses the achievements of the Manhattan Project.
Mark Ronan’s book doesn’t include many equations, but I’d ascribe the problems with math outreach in books to another factor, besides lay-frightening numerical and algebraic symbols. Most pop science books can shift the math to behind the stage, and focus on metaphors, graspable explanations, interesting material-world connections, and hero-scientist biography. Math books are more limited in this: they can’t really hide all the math away (the math is the point, after all). In interesting but abstract and technical developments, it can be very difficult to connect it to things an individual can grasp and see. Still, it is worth a try, and Mark Ronan has done it with this excellent little book, for which I can only pile words of praise. In Symmetry and the Monster, he has done a squaring of the circle, of sorts, creating a book about a very abstract and important mathematical achievement (the classification of Finite Simple Groups), and managing to make it both accessible and highly entertaining for the most part, modulo a little bit of effort and willingness on the part of the reader.
Small aside: it is perhaps a well-known but no sufficiently interiorized fact that the 20th century has been the greatest, most golden age of mathematics ever, easily putting all other centuries to shame. Advances in the science have been accelerating since at least Newton and Leibniz and the Scientific Revolution, which means the 19th century was a silver age, the 18th a bronze one. The great achievements of Classical Greece hardly warrant more than copper in comparison. This has also tended to happen in other sciences, with Physics as a case in point, but its achievements have been much more easy to popularize and write about - Einstein and Relative, the birth Quantum Mechanics, the making of the Atomic bomb� The latter, in fact, has an excellent book by Richard Rhodes, which with its well-crafted, almost literary narration and collective scientific authorship has really reminded me of Ronan’s.
So back to Symmetry and the Monster: this book attempts to trace the history and main developments in a mathematical quest that started in the 19th century and concluded as a massive collaborative work and one of the pinnacles of 20th century mathematics: the classification of Finite Simple Groups. It is a story that begins with the birth of modern, ‘Abstract� Algebra, which has little to do with what we’re taught in high school, and a lot to do with the study of abstract structures with properties. One of these is a group, which is a set of objects with one operation you can do on them. A good example is the integers under addition: if add three or more, it doesn’t really matter how you group the sums (associativity); if you add 0 to any, you get what you started with (identity); for any integer there’s another one that when you add them together, you get 0 (inverses); and adding any two numbers together gets you a third that is still in the list of integers (closure). All this might seem nitpicky or trivial, but it actually turns out to be tremendously useful, and to model a lot of things in the real world, like symmetries (which can be simplified, explained and studied as groups).
Now many groups have an infinite number of elements (like the integers under addition), but other are finite. And some groups can be ‘factored� (like numbers) into smaller, constituent groups. Groups that can’t be reduced any more are called simple groups, or as Mark Ronan prefers to call them in this book, ‘atoms of symmetry�.
From their appearance in the work of Evariste Galois, where they were invented as part of a proof of why 5th degree equations cannot be solved (like the lower-degree ones) using a simple algebraic formula, Ronan chronicles the development of their study, and the surprising discovery that finite simple groups could probably be classified into a list, that would include all their possible instantiations. On the way, we discover different types of groups (permutation, cyclic with prime order, continuous, or lie�) and the lives of the mathematicians who labored at them. We see how the project of making a table of the different families of finite simple groups develops, becoming a highly technical, demanding and flourishing branch of modern mathematics. We also see the discovery of ‘exceptional� groups which don’t fit into any of the categories, and the work to pokemon-like ‘catch them all�, prove their existence and properties and incorporate them into the table. The last and biggest of these sporadic simple groups is the Monster that gives its title to the book, and which has an added interest because of weird and unexpected connections it makes with completely different branches of mathematics (Number Theory) and the real world (Physics, String Theory).
Besides the discovery and the objects, the lives and anecdotes of the mathematicians working at the project are heavily referenced in the book, and add a non-trivial element to making it lighter and more accessible. I believe Ronan himself is a mathematician that did a lot of work in this area, so he knows many of the figures he talks about in the book (including the sadly deceased, maverick and magical genius John Horton Conway).
As I said in the second paragraph, this book is an excellent read, and I think is not too daunting even for a relatively lay audience. The mathematical mechanics are mostly only hinted at and/or summarized, and is mostly expressed in understandable language, with a couple of exceptions (I got a bit lost in the part explaining how you can use cross sections to discover other finite simple groups, along with some mention of ‘subgroups� of these which I didn’t really get). But small details aside, if you have a minimal interest in mathematics, you are likely to both enjoy and understand what this book talks about, and you will learn of a heroic epic that is at least as important as a feat of human science as the production of the atomic bomb. The Oppenheimers in this volume are much less known, much nicer and their work definitely less deadly, but in many other ways, it matches and surpasses the achievements of the Manhattan Project.
November 20, 2010
I have to give this one five stars, if for no other reason because it was the book I was reading when I finally started to understand group theory. It covered the history and importance of this branch of mathematics much the same as but in a more concise manner. Another difference is this book was thematically structured around the development of a particularly interesting artifact of group theory known as the Monster (this is actually what mathematicians call it by the way, due to its sheer unprecedented size). I think using a concrete example throughout the discourse was a good choice. Ronan also chose well to sometimes stop the progression and say "let's remember how we got here," proceeding to review the series of discoveries for a few paragraphs that led to a certain new advance.
I had been prepared to never understand group theory. For all I knew, it wasn't something that someone with a lack of formal math education (beyond calculus 20 years ago) was capable of learning from just reading books without a teacher. But somewhere halfway through this book things started to click into place a little bit. After that the book became immensely more enjoyable since I could follow along with the challenges and successes of the researchers with a significantly more understanding.
It is an interesting story, and something that surely will be of particular importance to understanding the future we will be experiencing in the coming decades. I don't ever want to be the typical post middle-age person who has checked out of modern times just because they couldn't keep up with the scientific advances, and I am more convinced than ever that a basic understanding of group theory is going to be necessary for any hope of hanging on.
I had been prepared to never understand group theory. For all I knew, it wasn't something that someone with a lack of formal math education (beyond calculus 20 years ago) was capable of learning from just reading books without a teacher. But somewhere halfway through this book things started to click into place a little bit. After that the book became immensely more enjoyable since I could follow along with the challenges and successes of the researchers with a significantly more understanding.
It is an interesting story, and something that surely will be of particular importance to understanding the future we will be experiencing in the coming decades. I don't ever want to be the typical post middle-age person who has checked out of modern times just because they couldn't keep up with the scientific advances, and I am more convinced than ever that a basic understanding of group theory is going to be necessary for any hope of hanging on.
May 20, 2019
With some warts, Ronan has nearly achieved the holy grail of mathematics exposition: (1) short, (2) readable, (3) about mathematical objects of significance and interest.
While the book has significant flaws, I give it 4/5 because the writing is clear enough for a non-mathematician, and there are so few of those in existence. Bad marks for supposed applicability of the ideas to the real world (including the usual demons of quantum physics, binary arithmetic = computers, and coding theory [which, see , users found gzip to be good enough and didn’t need the theoretically superior Leech lattice]), and nerd-machismo (Sophus Lie’s coke-bottle eyeglass rims are somehow muscular; chaired professors are cited for their intellectual immortality, when in reality most people don’t care at all about these 19th-century academic chairs or their works). Ronan also displays a scant understanding of history by, for example, using exactly one source for his entire chapter on Sophus Lie. But, whatever: it’s better that he clearly understands, and very clearly explains, the actually interesting mathematics --- which, in every other case, a non-mathematician would mis-explain.
If you wonder if mathematics is interesting or not, give Ronan a read. You should be able to understand the interesting structures he describes, and the amount of physics and Bell Labs mysticism is soft-pedalled much more than the rest of the genre.
While the book has significant flaws, I give it 4/5 because the writing is clear enough for a non-mathematician, and there are so few of those in existence. Bad marks for supposed applicability of the ideas to the real world (including the usual demons of quantum physics, binary arithmetic = computers, and coding theory [which, see , users found gzip to be good enough and didn’t need the theoretically superior Leech lattice]), and nerd-machismo (Sophus Lie’s coke-bottle eyeglass rims are somehow muscular; chaired professors are cited for their intellectual immortality, when in reality most people don’t care at all about these 19th-century academic chairs or their works). Ronan also displays a scant understanding of history by, for example, using exactly one source for his entire chapter on Sophus Lie. But, whatever: it’s better that he clearly understands, and very clearly explains, the actually interesting mathematics --- which, in every other case, a non-mathematician would mis-explain.
If you wonder if mathematics is interesting or not, give Ronan a read. You should be able to understand the interesting structures he describes, and the amount of physics and Bell Labs mysticism is soft-pedalled much more than the rest of the genre.
February 14, 2011
It was good, and it made me more interested in learning more about some of the sporadic simple groups (this is pretty close to my field of research).
The best points: it provides a good history of finite group theory and is an engaging read. I think that he also does a good job making the technical mathematics accessible to the layperson.
I have two complaints:
1. In order to retain my membership as a pedantic mathematician, I will point out that he says that he says the order of the alternating group on n letters increases "exponentially" as n gets larger. This is false, although I understand that people are starting to commonly mis-use the word "exponentially."
2. I lost track of the number of times Ronan says, "We will hear more about _____ later." It seems that this would be a result of poor organization, although the lives of these group theorists intersect so much that it would probably be impossible to avoid (although he could have simply let us find out that we will learn more about _____ on a couple of these occasions).
These are pretty minor points.
The best points: it provides a good history of finite group theory and is an engaging read. I think that he also does a good job making the technical mathematics accessible to the layperson.
I have two complaints:
1. In order to retain my membership as a pedantic mathematician, I will point out that he says that he says the order of the alternating group on n letters increases "exponentially" as n gets larger. This is false, although I understand that people are starting to commonly mis-use the word "exponentially."
2. I lost track of the number of times Ronan says, "We will hear more about _____ later." It seems that this would be a result of poor organization, although the lives of these group theorists intersect so much that it would probably be impossible to avoid (although he could have simply let us find out that we will learn more about _____ on a couple of these occasions).
These are pretty minor points.
August 30, 2017
Ronan paints a fascinating historical perspective of the mathematical developments that lead up to the discovery of the Monster, as well as the "Moonshine" connections to other domains (e.g. number theory, string theory). However, I felt that although Ronan's enthusiasm for the topic is supreme and his passion for explaining the mathematical process shines, the writing of this book lacked polish and the analogies used were not clear; as a math enthusiast, I was able to recall enough of the theory of finite groups to understand where he was trying to go with his analogies and to flesh out details that were glossed over. My interest in the subject has definitely been piqued, and I was able to find some papers which provide a more technical survey of this topic, such as Moonshine by Duncan, Griffin, and Ono.
July 16, 2018
A fun read that will likely get you at least somewhat interested in group theory. But the book uses analogies to talk about the ideas of the mathematical concepts rather than directly explaining them. If I were to explain something like addition in a similar manner that Ronan sometimes explains things in his book it might sound like this: it is an operation that takes two numbers and combines them to create another. It’s good because it doesn’t complicate things usually, but can end up sounding way too vague, leaving you just as confused as you would have been if he had just explained it.
But still, I recommend this book as it is genuinely interesting and you will learn about groups, their symmetry, the monster, and the stories of the mathematicians who discovered them. It’s a goof introduction to group theory and has driven me to want to find out more about the topic.
But still, I recommend this book as it is genuinely interesting and you will learn about groups, their symmetry, the monster, and the stories of the mathematicians who discovered them. It’s a goof introduction to group theory and has driven me to want to find out more about the topic.
December 27, 2018
Last three to four chapters are the most important chapters in this book. The rest are about mainly introducing the concept of group and Lie groups, with a slight of a monster. He tried to explain as best as he could about the concept of the simple group, but ended up repeating again and again that simple group is the group that is so "simple". This book basically tells you that there are mysterious connections between monster, modular form, and vertex operator algebra, to which the connection is called the Moonshine.
March 20, 2017
This is a well written book especially considering the abstract nature of the material. I personally would have liked a little more detail on things. Maybe a description of four group families that have a Euclidean description. Though I don't think this would have be good for most readers. Most of the time when I would wish he would explain something it would appear in a few pages. So he addressed most of my criticisms before the book ended.
December 20, 2019
A well written book in the sense of understandable to someone with some interest and understanding of mathematics and history. Abstract mathematics is a help. The book is an interesting story of connections. Sometimes it feels like things are glossed over and the book could have contained both more history and technical explanations (perhaps more illustrations and examples). Still, I found it an enjoyable read over 2 days and an introduction to an interesting phenomenon
September 1, 2019
This describes the journey from the origins of group theory to the discovery of "The Monster" - a gigantic multi-dimensional symmetry that completed the classification of the groups. The biographical details are fascinating and frequently very sad, particularly when they concern the destruction wrought on German mathematics by the Nazis.
September 6, 2023
A history of the classification of finite simple groups exposited in the general direction of non-mathematicians. Richly larded with history, or at least anecdotes, and agreeably excitable in its telling.
I didn't learn any maths along the way, but I also haven't read the appendices which may contain some.
I didn't learn any maths along the way, but I also haven't read the appendices which may contain some.
August 30, 2024
The fascinating history of a densely technical mathematical quest is told here in an approachable story. I did not follow every detail of the maths, but could follow the larger narrative. By making this the story of the people involved, Ronan provides a window into this world. Thoroughly enjoyable.
November 20, 2016
I enjoyed the birds-eye view of strange connections between areas within mathematics and physics. His explanations lost me a few times, but it was well worth the read.
April 10, 2019
Goes into great detail on the project to classify the simple groups.
July 14, 2023
Great read. Will write a review when I am free.
June 9, 2014
Too bad the book omits the obvious reference to "the shoulders of giants". What did I get out of the book? I found it fascinating to see how scientific discovery evolves in practice over centuries. It takes an oddball here and an oddball there to get the best results. Sometimes you go back to old books to find out what other have found out before you.
Ronan provides a partly first-person account of this endeavour called Monsters and Moonshine and, while doing so, gives some valuable insights into how mathematical discovery takes place: for instance, you rarely have a mathematical research team but do your research individually, which is a pain, while acknowledging any help you have received from for example your thesis supervisor; you have to come up with a theorem if you wish to be immortal, and so on. The material Ronan provides is anecdotal. Those passages are fun to read but do not amount to real eye-openers. The book is at its strongest when it takes the bird's eye view and recounts how an idea has evolved over time. A couple of times Gauss is quoted as making the author's fundamental point: "Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard." Ronan's sense of history is impressive.
As for the math itself, Monsters are objects that belong to group theory. Group theory studies algebraic structures called groups or, as Ronan explains, which "deals with the study of symmetry" on page 1. The young French mathematician Évariste Galois turns out to be our hero. He discovered simple groups of permutation in 1830. Ronan calls those simple groups of permutations "atoms of symmetry". "Simple" meant, according to Ronan, groups that could not be broken down (or "deconstructed" as he prefers to say) any further. So, a Monster is a group...yes, what is it exactly? An exceptional symmetry object, is the answer. The main quest in this book is to find those exceptional symmetry objects, of which 26 have been found so far. The second largest of them is called affectionately the Baby Monster. The largest is called the Monster. By "large" we mean that the Monster has a staggering number of dimensions: 196,883. I don't pretend to understand where exactly we are going with this but that is what we are told.
Moonshine is even more opaque. Moonshine is an attempt to explain some of the staggering correspondences that Monsters in group theory exhibit with number theory. Those correspondences are crystallized in an object in number theory, namely the number 196,884, which is almost the same as the number of the dimensions of the Monster. One is still at a loss to understand the connections, Ronan admits. He does not say much more. He does say, though, that string theory seems a promising home for both the Monsters and the Moonshine.
The implication is that this is all very cutting edge. Therefore, one has to forgive the obscurity. However, the reader cannot avoid the impression that the book would have benefited from more careful editing. Many things are repeated at least twice. When that keeps happening, you start feeling as awkward as trying to laugh when you hear the same joke twice. Except this book is not funny. It does not make you laugh.
To be fair, Ronan's excitement for his subject is contagious. I maintain that the book works best as a peek into the mind of someone who is discovering something that boggles the mind. How do you get your mind around something that boggles the mind? Maybe you don't. Hence the excitement, not only for the Monsters and Moonshine but for finding and pushing the limits of our minds.
Ronan provides a partly first-person account of this endeavour called Monsters and Moonshine and, while doing so, gives some valuable insights into how mathematical discovery takes place: for instance, you rarely have a mathematical research team but do your research individually, which is a pain, while acknowledging any help you have received from for example your thesis supervisor; you have to come up with a theorem if you wish to be immortal, and so on. The material Ronan provides is anecdotal. Those passages are fun to read but do not amount to real eye-openers. The book is at its strongest when it takes the bird's eye view and recounts how an idea has evolved over time. A couple of times Gauss is quoted as making the author's fundamental point: "Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard." Ronan's sense of history is impressive.
As for the math itself, Monsters are objects that belong to group theory. Group theory studies algebraic structures called groups or, as Ronan explains, which "deals with the study of symmetry" on page 1. The young French mathematician Évariste Galois turns out to be our hero. He discovered simple groups of permutation in 1830. Ronan calls those simple groups of permutations "atoms of symmetry". "Simple" meant, according to Ronan, groups that could not be broken down (or "deconstructed" as he prefers to say) any further. So, a Monster is a group...yes, what is it exactly? An exceptional symmetry object, is the answer. The main quest in this book is to find those exceptional symmetry objects, of which 26 have been found so far. The second largest of them is called affectionately the Baby Monster. The largest is called the Monster. By "large" we mean that the Monster has a staggering number of dimensions: 196,883. I don't pretend to understand where exactly we are going with this but that is what we are told.
Moonshine is even more opaque. Moonshine is an attempt to explain some of the staggering correspondences that Monsters in group theory exhibit with number theory. Those correspondences are crystallized in an object in number theory, namely the number 196,884, which is almost the same as the number of the dimensions of the Monster. One is still at a loss to understand the connections, Ronan admits. He does not say much more. He does say, though, that string theory seems a promising home for both the Monsters and the Moonshine.
The implication is that this is all very cutting edge. Therefore, one has to forgive the obscurity. However, the reader cannot avoid the impression that the book would have benefited from more careful editing. Many things are repeated at least twice. When that keeps happening, you start feeling as awkward as trying to laugh when you hear the same joke twice. Except this book is not funny. It does not make you laugh.
To be fair, Ronan's excitement for his subject is contagious. I maintain that the book works best as a peek into the mind of someone who is discovering something that boggles the mind. How do you get your mind around something that boggles the mind? Maybe you don't. Hence the excitement, not only for the Monsters and Moonshine but for finding and pushing the limits of our minds.
May 1, 2009
I really enjoyed this book. I think it gave a very rich yet comprehensible picture of the history of finite group theory (concentrating on the classification of the finite simple groups), starting with a bit of pre-history from the ancient Greeks, but really starting with Galois' immortal work on , and tracing the the major players in the 19th century like Cauchy, Jordan, Lie, Klein, Burnside, and the 20th century, like Dickson, Thompson, Gorenstein, Conway, Fischer, Aschbacher, etc., with a special emphasis on the largest sporadic finite simple group, the Monster, and the still-mysterious connections with modular functions, vertex operator algebras, and string theory, collectively referred to as Moonshine.
In terms of reviewing the contents, I cannot hope to do better than Rob Griess's review, so I will link to it here:
I have personal experience with some of the people discussed in this book and I have heard many of the same stories that were told to him, and I can say that his retellings of the stories are very faithful to what I have heard; it is a valuable historical document of the personalities and happenings during the time of the classification.
One anecdote I wanted to amplify on was the one about the name "buildings". J.P. Serre related to a group of us at dinner once that they had thought about using the term "skeleton" instead (as discussed in the book) but then the elaboration of the metaphor produced "coffin", "ossuary", etc., which were quite morbid, so they chose "chamber", "apartment", "building" instead.
For more details, I would recommend T. Thompson's "From Error-Correcting Codes Through Sphere Packings to Finite Simple Groups" for a pleasant mathematical narrative covering a significant portion of the middle of this book, and Conway and Sloane's "Sphere Packings, Lattices, and Groups" for an encyclopedic survey of much of the material, particularly the topics related to the Monster (e.g., the Leech lattice, the 26-dimensional Lorentzian lattice, Conway's Monster construction, and so forth). Also, for the middle part of the story (Klein, Lie, Dickson, Weyl, etc.) I would recommend Hawkins' "Emergence of the Theory of Lie Groups", a comprehensive history of that era.
I perceived a few minor infelicities in the writing, such as some somewhat awkwardly interjected background mathematical anecdotes like the one about Emmy Noether and the university not being a bathhouse, and a tendency to introduce a topic with a "we'll discuss this later" quite frequently.
I am curious to compare my experience of reading this book to that of someone without training in group theory.
In terms of reviewing the contents, I cannot hope to do better than Rob Griess's review, so I will link to it here:
I have personal experience with some of the people discussed in this book and I have heard many of the same stories that were told to him, and I can say that his retellings of the stories are very faithful to what I have heard; it is a valuable historical document of the personalities and happenings during the time of the classification.
One anecdote I wanted to amplify on was the one about the name "buildings". J.P. Serre related to a group of us at dinner once that they had thought about using the term "skeleton" instead (as discussed in the book) but then the elaboration of the metaphor produced "coffin", "ossuary", etc., which were quite morbid, so they chose "chamber", "apartment", "building" instead.
For more details, I would recommend T. Thompson's "From Error-Correcting Codes Through Sphere Packings to Finite Simple Groups" for a pleasant mathematical narrative covering a significant portion of the middle of this book, and Conway and Sloane's "Sphere Packings, Lattices, and Groups" for an encyclopedic survey of much of the material, particularly the topics related to the Monster (e.g., the Leech lattice, the 26-dimensional Lorentzian lattice, Conway's Monster construction, and so forth). Also, for the middle part of the story (Klein, Lie, Dickson, Weyl, etc.) I would recommend Hawkins' "Emergence of the Theory of Lie Groups", a comprehensive history of that era.
I perceived a few minor infelicities in the writing, such as some somewhat awkwardly interjected background mathematical anecdotes like the one about Emmy Noether and the university not being a bathhouse, and a tendency to introduce a topic with a "we'll discuss this later" quite frequently.
I am curious to compare my experience of reading this book to that of someone without training in group theory.
August 1, 2017
It is very difficult to read a mathematical book with a good balance between explaining the maths and not delving into difficult details. Each reader has a different preference. This book didn't achieve that balance for me at all. It explains too clearly the very easy things such as how to know the order of a polynomial, giving examples and so on, but anything more difficult than that is not explained at all: not even attempted. As a result I did not get a feeling for the subject except it being very difficult. I would have loved an explanation of Galois' work which started it all, and is probably explainable to a non mathematician, but I did not understand it. From that point onwards, having not understood the foundational work, it was completely impossible to follow the rest. I would have been much happier if the author had attempted to give explanations of the difficult concepts.
November 15, 2009
As the book's subtitle promises, this book is about a quest--meaning, I had not sufficiently appreciated before I picked it up, a great deal of narrative about how mathematicians have to come to know what they know. As a non-mathematician, I did not find this at all helpful. The idea of multidimensional symmetry, for obscure reasons, fascinates me, but it is also intensely abstract and conceptually frustrating. The biographical details of the many people who contributed to group theory and such, especially in considering their false starts, only confuses things. Moreover, Ronan never quite gets beyond this mode to really distill the essence of the theory in its mature form, which is what I really want to understand. This book will probably be of much more utility to readers who already have a better grasp on the math and would like to learn more about the theory's human history, rather than someone like myself who is a nonspecialist interested in learning about symmetry and the Monster.
October 19, 2009
The subject matter of Ronan's "Symmetry and the Monster" is fascinating: Symmetry mathematics, an accidental quest of discovery that lasts for centuries, tie-ins with String Theory, Quantum Physics and the j-function. Still, though, I would not recommend this book if this is one's initial foray into studying any of these subjects. Ronan, while he writes gently and does explain some concepts, is first and foremost interested in the mathematical discoveries that led to the uncovering of the Monster Group. However, Ronan will bombard the reader with certain concepts that seem to appear in brain-aching clusters; paradoxically, Ronan will present a concept's name then write, "But more on this later..." as he continues the biography. Overall, it's an incredibly interesting book. Just do your homework before you go read it.
October 9, 2017
Interesting experience reading. The book had explanations of how groups worked. I grasped each one loosely, then kind of forgot what I had grasped as I read on. I had a kind of "" style grasp of the explanations as I went into the next explanation. So now I have a vague sense of what groups are (and of how the various structures leading up to the Monster are), which was more than I had before. It's like, I understand, but I don't.
It's good to have a taste of the mathematicians' vibe and "way of moving". I found myself thinking in terms of "These people / things are all different but I can take them to be the same in certain important respects", the same move the early group theorists made to invent group theory.
It's good to have a taste of the mathematicians' vibe and "way of moving". I found myself thinking in terms of "These people / things are all different but I can take them to be the same in certain important respects", the same move the early group theorists made to invent group theory.
December 17, 2024
I feel like I have been fairly generous with my rating system so I have adjusted it a bit. (3 stars is still a good book to me, just not up there as a favourite)
This book is a history about group theory, which, prior to reading this, was completely unknown to me with the exception of a brief mention on how it was used to prove the impossibility of quintic equations in another book. Getting a run down of the key discoveries and developments towards finding the Monster group sparked an interest I did not know I had for this subject.
The book does explain some concepts by way of analogy (finite simple groups being analogous to atoms, the table of finite groups being the periodic table, etc) which I felt didn't have to be sustained throughout the whole book.
Some of the explanations went way over my head, but I've come to expect that with the first reading of any book of this type.
This book is a history about group theory, which, prior to reading this, was completely unknown to me with the exception of a brief mention on how it was used to prove the impossibility of quintic equations in another book. Getting a run down of the key discoveries and developments towards finding the Monster group sparked an interest I did not know I had for this subject.
The book does explain some concepts by way of analogy (finite simple groups being analogous to atoms, the table of finite groups being the periodic table, etc) which I felt didn't have to be sustained throughout the whole book.
Some of the explanations went way over my head, but I've come to expect that with the first reading of any book of this type.
October 29, 2008
Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, By Mark Ronan. Oxford University Press, 2006. Note: I am also posting (probably several times) about this book and others like it on .
This is an excellent way for the non-mathematician to learn about what is going on in the attempt to classify symmetries by discovering all the finite "simple groups". The last one found was the Monster Group and the classification was completed in 1982. This book is full of fascinating information about how this came about and the tantalizing connections between physics and the Monster that have been discovered since.
This is an excellent way for the non-mathematician to learn about what is going on in the attempt to classify symmetries by discovering all the finite "simple groups". The last one found was the Monster Group and the classification was completed in 1982. This book is full of fascinating information about how this came about and the tantalizing connections between physics and the Monster that have been discovered since.
January 12, 2012
Good introduction on symmetry and group theory, with sufficient breadth though not depth. However, the writing style is quite 'vexing' sometimes, especially on the part where the author kept repeating "we shall meet with this later" or "we will deal with this in the next chapter" etc. too many times. It at one juncture really made the reader 'fed up'.
Nonetheless, the book is broad enough to let readers know what is at stake and what are the things covered in those topics, though the exposition is not clear in explaining some basic concepts, or some explanations are too much simplified that the essence is probably lost. It is still a good book, though not an excellent one admittedly.
Nonetheless, the book is broad enough to let readers know what is at stake and what are the things covered in those topics, though the exposition is not clear in explaining some basic concepts, or some explanations are too much simplified that the essence is probably lost. It is still a good book, though not an excellent one admittedly.
December 18, 2015
Informative, though very dense and perhaps beyond my reach. Math, unlike physics where I have more experience, is much tougher to grasp conceptually, and I feel I would've benefitted from more exploration on how the great monster of symmetry underpins our physical universe, though perhaps that's simply not known yet. A good read, less interesting to me than the Language of Numbers, but definitely gives one the same inspiring sense that there is a very complex and fascinating order to things that numbers can uncover in extraordinary ways.
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