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Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present First Edition

by George Szpiro (Author)
4.3 4.3 out of 5 stars 12 ratings
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A lively history of the peculiar math of voting

Since the very birth of democracy in ancient Greece, the simple act of voting has given rise to mathematical paradoxes that have puzzled some of the greatest philosophers, statesmen, and mathematicians. Numbers Rule traces the epic quest by these thinkers to create a more perfect democracy and adapt to the ever-changing demands that each new generation places on our democratic institutions.

In a sweeping narrative that combines history, biography, and mathematics, George Szpiro details the fascinating lives and big ideas of great minds such as Plato, Pliny the Younger, Ramon Llull, Pierre Simon Laplace, Thomas Jefferson, Alexander Hamilton, John von Neumann, and Kenneth Arrow, among many others. Each chapter in this riveting book tells the story of one or more of these visionaries and the problem they sought to overcome, like the Marquis de Condorcet, the eighteenth-century French nobleman who demonstrated that a majority vote in an election might not necessarily result in a clear winner. Szpiro takes readers from ancient Greece and Rome to medieval Europe, from the founding of the American republic and the French Revolution to today's high-stakes elective politics. He explains how mathematical paradoxes and enigmas can crop up in virtually any voting arena, from electing a class president, a pope, or prime minister to the apportionment of seats in Congress.

Numbers Rule describes the trials and triumphs of the thinkers down through the ages who have dared the odds in pursuit of a just and equitable democracy.

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  1. ISBN-10
    0691139946
  2. ISBN-13
    978-0691139944
  3. Edition
    First Edition
  4. Publisher
    Princeton University Press
  5. Publication date
    April 4, 2010
  6. Language
    English
  7. Dimensions
    6.5 x 0.75 x 9.75 inches
  8. Print length
    240 pages
  9. See all details
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Editorial Reviews

Review

"One of Choice's Outstanding Academic Titles for 2010"

"Honorable Mention for the 2010 PROSE Award in Mathematics, Association of American Publishers"

"Mathematicians, economists, and political theorists have made their own attempts to elucidate the math of voting, and figure out better electoral systems. The story of these efforts is told in Numbers Rule. . . . Timely."---Anthony Gottlieb, New Yorker

"Clear and energetic. . . Szpiro charts this history selectively and with the use of major characters to render vivid a story of rival systems, which can easily degenerate into equations. He is a mathematician and uses tables to illustrate his arguments: but these are accessible to simple understanding. He is also a journalist and thus can tell a story."---John Lloyd, Financial Times

"Although voting problems manifest subtle mathematical complexities, Szpiro is an excellent communicator of mathematical concepts with a nimble ability to sidestep technical jargon. . . . An interesting, selective introduction into the complexities of voting reform."---Donald G. Saari, Times Higher Education

"A history of social choice theory, with much more detail (yet still readable) than one is used to receiving on this topic. I liked this book very much."---Tyler Cowen, Marginal Revolution

"I knew from reading Martin Gardner's columns that every voting system you can devise will occasionally turn up paradoxical results. . . . Szpiro walks you through the whole subject with very few equations."---John Derbyshire, National Review

"Engaging storytelling . . . 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, MAA Reviews

"In Numbers Rule, mathematician and journalist Szpiro presents a refreshingly different presentation of the mathematics of voting and apportionment. . . . The mathematical content is not trivial, and it is well written, very clear, and should be accessible to readers with an understanding of arithmetic and a willingness to play with numbers." ― Choice

"Highly entertaining. . . . Anybody who has ever decried election results will be fascinated in Szpiro's accessible explanations of the paradoxes and enigmas that occur in all methods of election, from electing a pope in Rome, to apportionment of seats in the Congress by our founding fathers to ensure justice for all, even the minority."---Phil Semler, Sacramento Book Review>

"The author skillfully placed the development and evolution of the Social Choice theories in a broad historical context. The book shines in weaving the emergent math theories with historical circumstances. . . . [E]njoyable and informative."---Alexander Bogomolny, Cut the Knot

"Szpiro's book is a highly recommended good read on the history of the problems, which could illuminate a seminar series on the issues."---Ron Johnston, Environment and Planning

"It is an excellent addition to a growing body of literature that aims to convey ideas from the mathematical sciences to general audiences. Moreover, Szpiro's book is unique among other offerings in the mathematical social sciences in that it focuses on the historical development of the field. The narrative is engaging, witty, and easy to read."---Jonathan K. Hodge, Notices of the AMS

"George Szpiro's Numbers Rule does not break any new ground in the field of social choice theory, but it is probably the most entertaining book one is likely to find on the subject. . . . [A]ll social choice theorists should read this book. . . .Szpiro's book is ideally suited to be a supplementary reading for graduate classes in social choice theory."---Justin Buchler, Public Choice

Review

"'Which candidate is the people's choice?' It's a simple question, and the answer is anything but. In Numbers Rule, George Szpiro tells the amazing story of the search for the fairest way of voting, deftly blending history, biography, and political skullduggery. Everyone interested in our too-fallible elections should read this book."―William Poundstone, author of Gaming the Vote: Why Elections Aren't Fair (and What We Can Do about It)

"Writing a book for a general audience on voting and electoral systems is a daunting task, but Szpiro succeeds admirably. He completely avoids technical jargon and focuses on the most important scholars and results in the field. This book fills a gap in the existing literature."―Hannu Nurmi, author of Voting Procedures under Uncertainty

"Numbers Rule focuses on key figures in the development of democracy and on the mathematics of voting, elections, and apportionment that they developed. Szpiro pays particular attention to the paradoxes that arise, and discusses them through examples."―Steven J. Brams, New York University

"Numbers Rule is very thoroughly researched and quite well written. The story Szpiro tells is both important and interesting. The most significant contribution this book makes is in the detailed history that it presents. It will have broad appeal."―Alan D. Taylor, coauthor of Mathematics and Politics

Product details

  • Publisher ‏ : ‎ Princeton University Press; First Edition (April 4, 2010)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 240 pages
  • ISBN-10 ‏ : ‎ 0691139946
  • ISBN-13 ‏ : ‎ 978-0691139944
  • Item Weight ‏ : ‎ 1 pounds
  • Dimensions ‏ : ‎ 6.5 x 0.75 x 9.75 inches
  • Customer Reviews:
    4.3 4.3 out of 5 stars 12 ratings

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Customer reviews

4.3 out of 5 stars
4.3 out of 5
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12 global ratings

Top reviews from the United States

Eric Mayforth
5.0 out of 5 stars A Fun Study of the Mathematics of Democracy
Reviewed in the United States on August 10, 2010
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In the United States, we now and then debate the merits of the Electoral College, but few people in this country pay attention to how the 435 congressional seats are apportioned to the 50 states after each decennial census. George Szpiro takes up this topic and other election-related problems in "Numbers Rule."

Szpiro describes how democracies from ancient Greece to the twenty-first century have dealt with the issues involved in making representation and elections as just as humanly possible. He describes how methods used to choose between multiple candidates progressed from those used to elect abbesses in the Middle Ages to those used in France in the eighteenth century, and shows the odd effects that can result when a third candidate is inserted into a previously two-man race.

This book was, appropriately enough, released in a year ending in '0', given that 2010 is a census year--the task of congressional apportionment will begin again soon. Szpiro recounts the intense debates between advocates of different apportionment methods in the early years of the republic and recalls many of the conflicts in later decades between states over the final representative apportioned. The author describes many of the mathematical issues that result, including the Alabama, New State, and Population Paradoxes--he shows mathematically how a state can, incredibly, lose a representative when the size of the House of Representatives is increased by one.

One trail that Szpiro did not go down involves the effect of an increase in the size of the House on presidential elections. Many people over the years have called for an increase of the size of the House of Representatives to anywhere from 600 to 1000 seats--in very rare instances this would be enough to change the result of an extremely close presidential election. Had the House contained, say, 870 seats instead of the 435 that it actually contained for the 2000 election, Al Gore would have won even without carrying Florida.

Szpiro reports the opinions of mathematicians concerning whether multi-candidate elections and congressional apportionments can ever be made completely fair, and provides brief biographical sketches of many of the mathematicians who dealt with these problems. The author closes by discussing election problems encountered in recent decades in Switzerland, France, and Israel.

"Numbers Rule" is a great study of the mechanics needed to put democracy in place and shows that they are not foolproof--one is reminded of Winston Churchill's assertion that "Democracy is the worst form of government, except for all those other forms that have been tried from time to time."
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W.J. Walter
3.0 out of 5 stars The Text is Incompetently Formatted
Reviewed in the United States on February 17, 2012
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My comments do not concern the substance of this text but rather its presentation. The very first word of the book (on the first page of the Preface) is "It". Unfortunately, the upper-case 'I' is separated from the lower-case 't' by a blank line. This same problem recurs (throughout the book) whenever a word is italicized or presented as a link, and, consequently, the text appears ragged and sloppy. Since this is a problem that can be easily fixed, one can hope that the publisher will put out a corrected version and instruct those of us unfortunate enough to have purchased this disgraceful mess as to how we can download the corrections.

I should note that while I have verified that the problem I describe above appears in both the Kindle app for the PC and the Kindle app for the iPad, I do not know if it also shows up on the screens of devices made by Amazon.
3 people found this helpful
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Kindle Customer
4.0 out of 5 stars Thorough Treatment
Reviewed in the United States on December 6, 2018
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Exhaustive discussion of the problems of democratic decision making. Possibly too detailed for some readers. Much history and mathematical results.
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Michael T.
5.0 out of 5 stars Outstanding Historical Perspective on Voting
Reviewed in the United States on September 6, 2010
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This is a wonderful, historical look at voting systems. While voting might appear to be straightforward (we do it all the time), there are great difficulties in defining what the right winner is when there are more than two candidates. The standard "one with the most votes" (plurality) election is not particularly appealing since it is easy to split the opposition by adding candidates and having a candidate with very little support be the winner. This book traces understanding of these issues back to the Greeks and continues through the "dark ages", when clerics were particularly interested in how to elect Popes and abbots, through the Napoleonic period of Borda and Condorcet, up to the current era of impossibility theorems and computational complexity. The exposition is not mathematical (equations are exiled to chapter appendices) but it is analytical in the sense that concepts are clearly defined and the results are fully explained. Examples are interleaved to aid in understanding.

This book is highly readable and hits all the highlights. The exposition of the period from 1200-1450 was particularly interesting to me, since it is much less known than the relatively well-known French period.

If you have any interest in learning about why voting and apportionment are not straightforward, and want a readable, history-oriented book on approaches to these problems, I highly recommend this book.
7 people found this helpful
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George Poirier
5.0 out of 5 stars Troubling Issues in Seeking Fairness in Elections
Reviewed in the United States on May 11, 2011
The intent of elections in a democratic society is normally to reflect the choice of a majority of the voting public. However, as is brilliantly shown by the author of this fascinating book, the reality of the matter can be far different and surprisingly complicated. In prose that is clear, friendly, lively and often quite captivating, the author guides the reader through the logic and mathematics (arithmetic, really) of the voting process through the ages. From the time of Plato to the present, we meet the many thinkers who have tried to make the voting process as fair as possible. The author clearly explains each of the proposed methods and both illustrates and lucidly compares them - their pros and cons - with useful examples, complete with relevant tables. As a bonus, a couple of chapters are devoted to the methods of fairly allocating seats in Parliament - another area that contains unexpected complications and surprises.

Although I found parts of the first chapter rather boring (i.e., analysis of Plato's views as translated from his writings), the pace rapidly picks up with each succeeding chapter, quickly making the book hard to put down. I found the inclusion of biographical appendices on several key individuals, as well as the couple of mathematical appendices, to very nicely complement the main text. This book can be enjoyed by anyone, especially those concerned about the fairness of our election process.
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Top reviews from other countries

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Save the Planet. Stop Climate Change!
4.0 out of 5 stars very good
Reviewed in Germany on July 7, 2012
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This is a book for everyone who wants to know what a mathematical analysis of voting and election procedures can reveal.
The mathematics in the book is quite accessible and entwined with political history. Biographical sketches of important figures are also included.
The author happens to be a mathematically trained political journalist, so the book is readable, usually albeit not perfectly accurate on the mathematical facts and the political context is shrewdly explained.
A more readable or more stimulating introduction to the vexing mathematics of democracy is hardly imaginable.
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Henri Schmit
3.0 out of 5 stars I sistemi elettorali visti da un matematico
Reviewed in Italy on May 2, 2013
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Szpiro è un matematico e giornalista americano che scrive fra l'altro articoli di una colonna mensile di matematica sulla famosissima NZZ (il più importante giornale svizzero). in Numbers Rule l'autore presenta alcuni temi di logica e di matematica elettorale, spazzando da Platone a Kenneth Arrow, passando per Llull (meritevole per aver anticipato Condorcet/Copeland), Cusano (meritevole per aver anticipato de Borda), de Borda, Condorcet (decisamente sottovalutato dall'autore che non sembra comprendere il quadro di filosofia giuridica e politica delle soluzioni tecniche e delle formulette), Laplace, Lewis Carroll, la storia epica dell'apportionment negli Stati Uniti (raccontata molto bene e senza tecnicismi inutili), la teoria della scelta collettiva razionale e il teorema di possibilità di Arrow, le formule di ripartizione proporzionale in uso in Europa e il metodo biproporzionale proposto da Balinski e da Pukelsheim che emergono come i massimi esperti (della matematica) dei sistemi elettorali. Si tratta di un racconto gradevole e preciso al quale manca però il contesto logico e giuridico che governa la teoria (e la scelta) dei sistemi di voto e che i matematici finora hanno sempre mostrato di non sapere (o non voler) affrontare. Forse è meglio così. Mi spaventa l'aritmocrazia elettorale, nuovo velo dell'ignoranza.
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