Book review - Symmetry and the Monster, Mark Ronan

★★★★★

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.