A few years ago, I had the pleasure of reviewing George Szpiro’s book *Numbers Rule* for MAA Reviews. That book discussed the history of various mathematical theories of voting, from Plato through modern days. If you click the above link, you will see that I gave the book a very positive review, but commented that “there are certainly more rigorous and in-depth mathematical treatments of the issues that Szpiro discusses” and his book left me wanting more. So I was happy when I was asked to review the book *Mathematical Theory of Democracy* by Andranik Tangian. This book is much more technical and goes into more theoretical depth, and as such it will have a more limited audience, but it is worth seeking out if you are looking for this type of treatment.

*Mathematical Theory of Democracy* is divided into four parts. The first part is concerned with the history of democracy. There are chapters dedicated to Athenian Democracy, the Roman Empire, Italy in medieval times, the age of enlightenment, and modernity. The first three of these chapters read very much like a textbook on political philosophy, with nary an equation in sight. By the time Tangian writes of the Age of Enlightenment, mathematics starts to show up more directly and the author dedicates significant time to comparing voting methods that may be familiar to many mathematicians. More explicitly, Tangian compares voting methods that are Bordavian (comparing a slate of candidates on quantitative estimates of merit) with those that are Condorcetian (comparing pairs of candidates individually) from both a mathematical and a philosophical perspective, leading to some interesting contrasts between the two. The final chapter on modernity discusses some of the limitations of representative democracies, both in a mathematical sense with results such as Arrow’s Impossibility Theorem and in a more general sense — as the author writes, “the recent local wars under the banner of democracy in countries where its relevance is more than questionable only compromise the democratic idea.”

The second part of Tangian’s book is entitled “Theory” and has five chapters whose goals are to model democracy in various forms. The first of these chapters is about “Direct Democracy”, and develops several quantitative measures about Athenian-style representative democracies including the “popularity”, the “universality”, and the “goodness index”. He then uses these measures to evaluate some actual examples from history as well as give geometric interpretations and theoretical discussions on topics such as the efficiency of democracies.

Another chapter discusses dictatorships and democracies, and mathematically analyzes the circumstances in which it is beneficial to society to have a single person hold all the power. There are also two chapters on representative democracies, where Tangian analyzes in depth the German parliamentary elections and also looks at the formation and dissolution of party coalitions. He also writes at length about statistical tests on representative capacities and how likely a democracy is to follow a consensus public opinion. Unlike in the first part of the book, Tangian uses quite a range of mathematics, from geometry to combinatorics to statistics, in these chapters and at times the theory gets quite dense. There is a significant amount of data that the author analyzes, and he also proves a number of theorems in between lengthy philosophical discussions of democratic ideals.

The next set of chapters looks at various applications of the theory that Tangian has constructed in the earlier chapters, including public opinion polls, stock market predictions, and models of traffic flow. The book concludes with several appendices, mostly covering technical material such as Chebyshev’s Formula, Multinomial Sums, Probability Tables, and statistical significance. The fact that the author chooses to hide some of the most detailed mathematics in appendices throughout the book is just one sign that he is an economist writing for social scientists rather than for mathematicians. There are several other things in the book that may strike mathematicians as odd, such as non-standard notation and terminology, and several places where he could have greatly streamlined and shortened his proofs and computations if mathematicians were his main audience. But at the same time, Tangian does not shy away from writing about detailed mathematics and his choices, if not my exact cup of tea, are all justifiable and maybe even correct given this book’s most likely target audience.

The previous paragraph may make it come across as if the reviewer was not impressed by *Mathematical Theory of Democracy* so let me assure you that this is far from the truth. In this book, Tangian has managed to write about an incredibly wide range of topics in great mathematical, historical, and philosophical depth, and yet the book is very readable. It is not a casual read, but a reader who wishes to dedicate the required time and energy will learn quite a bit about Democracy, its history, its limitations, and its strengths.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College, whose interests range from cryptography to Galois theory to graph theory. He can be reached at dglass@gettysburg.edu.