*Given that individual members of society have their preferences, how should we aggregate them to figure out what society as a whole prefers?*

*How should we divide financial resources, pieces of cake, or congressional representation between people in a way that is as fair as possible?*

These questions and similar questions related to social choice have many possible answers, and have been thought about for a long time by philosophers, political scientists, pundits, and, yes, mathematicians, with different groups each having their own take on the issues. The mathematics of voting is a subject that has been very well-tread in recent years, with books by mathematicians such as Brams, Straffin, Taylor, and (probably most famously) Donald Saari being just some of the books in the field that have been reviewed by your team of MAA Reviewers. Unsurprisingly, most of these books focus on the mathematics behind the mathematics of voting. A new book by George Szpiro, entitled *Numbers Rule: The Vexing Mathematics of Democracy from Plato to the Present* is also about the mathematics of voting and social choice, but takes a different approach, focussing on the history and the people who have brought us various mathematical developments in the field.

Each chapter of Szpiro’s book focusses on the contributions of one person (or, in a few cases, a small group of people) to social choice theory. The first set of chapters deals with voting, and how we combine individual preferences to figure out what preferences society as a whole may have. The book begins with a chapter entitled “The Anti-Democrat” about Plato’s idealized city, as described in *The Republic* and *Laws*, in which traditional democracy is replaced by an elaborate system of guardians and magistrates and Cnossians. Another chapter, “The Letter Writer”, looks at letters of Pliny the Younger, who was concerned that the outcome of a vote between three options did not best represent the will of the people, predicting some of the problems that third party candidates face in contemporary US Elections.

Later chapters (“The Officer” and “The Marquis”) look at people whose names will be familiar to anyone who has done reading on voting theory, Borda (of “Borda Count” fame) and Condorcet. While the mathematics discussed in these chapters is unlikely to be new or surprising to MAA members who read the book, Szpiro discusses many biographical details which may be. For example, did you know that Condorcet was a feminist who, despite feeling that women should not have the right to vote, did ask “Why should beings exposed to pregnancies and to passing indispositions not be able to exercise rights that no one ever imagined taking away from people who have gout every winter or who easily catch colds?”

Later chapters in the book shift away from the questions of voting and move to questions of congressional representation. In particular, how many congressmen should each state receive? The US Constitution is surprisingly vague on this question, specifying only that each state shall have at least one representative and that “the number of representatives shall not exceed one for every thirty thousand.” It is left up to congress to decide how many representatives there are in total, and how many should represent each state. Unsurprisingly, this vagueness has led to many conflicts over time, and in *Numbers Rule*, Szpiro discusses the various methods proposed by founding fathers such as Webster and Hamilton, as well as “The Ivy Leaguers” Walter Willcox, Joseph Hill, and Edward Huntington. Szpiro leads us through a series of congressional debates, special panels, and politicking, explaining along the way the mathematical problems and “paradoxes” that the various solutions lead to. These chapters have more numbers than the chapters on voting, and are slightly more technical, although the emphasis is firmly on the people and the storytelling.

The question of how we vote and the question of how to apportion seats in congress are both quite interesting and both have unsatisfying endings. As most mathematicians know, Kenneth Arrow proved in 1949 that there is no system of voting which satisfies a small number of conditions that just about anyone would want from a voting system. Similarly, Allan Gibbard and Mark Satterthwaite showed that any election can be manipulated if some voters misrepresent their preferences. More recently, Michel Balinski and H. Peyton Young have shown that there is no system of congressional apportionment which is both unbiased and avoids the so-called “Population Paradox” in which states lose seats despite a growing population. Szpiro discusses all of these theorems and how they muddy the waters and remove any hope of a wholly satisfying solution to the questions we started with. A final chapter, entitled “The Postmoderns” considers several systems that have been developed in light of these theorems, such as Switzerland’s model of apportionment and the single transferable vote model which is used in house elections in Australia as well as many city council elections here in the United States.

One nice feature of Szpiro’s book is that each chapter ends with a biographical appendix, typically a page or two describing aspects of the lives of the people he is writing about that do not pertain to social choice theory. Several chapters also contain mathematical appendices giving proofs of the more technical results discussed in the main text. These features, as well as Szpiro’s engaging storytelling, make for a very nice book. There are certainly more rigorous and in-depth mathematical treatments of the issues that Szpiro discusses, but for a reader who is primarily interested in learning some of the historical context of the characters who have contributed to the mathematics of social choice theory, it is hard to imagine a better book.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College. His main mathematical interests include number theory, algebraic geometry, and cryptography. He can be reached at dglass@gettysburg.edu.