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

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

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

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


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.


It may come as a surprise to many readers that our democratic institutions and the instruments to implement the will of the people are by no means foolproof. in fact, they may have strange consequences. One example is the so-called Condorcet Paradox. Named after the eighteenthcentury French nobleman Jean-Marie Marquis de Condorcet, it refers to the surprising fact that majority voting, dear to us since times immemorial, can lead to seemingly paradoxical behavior. I do not want to let the cat out of the bag just yet by giving away what this paradox is. Suffice it to say for now that this conundrum has kept mathematicians, statisticians, political scientists, and economists busy for two centuries—to no avail. Worse, toward the middle of the twentieth century, the Nobel Prize winner Kenneth Arrow proved mathematically that paradoxes are unavoidable and that every voting mechanism, except one, has inconsistencies. As if that were not enough, a few years later, Allan Gibbard and Mark Satterthwaite showed that every voting mechanism, except one, can be manipulated. Unfortunately, the only method of government that avoids paradoxes, inconsistencies, and manipulations is a dictatorship.

There is more bad news. the allocation of seats to a parliament, say to the U.S. Congress, poses further enigmas. Since delegations must consist of whole persons, they must be integer numbers. How many representatives should be sent to Congress, for example, if a state is due 33.6 seats? Should it be thirty-three or thirty-four congresspeople? Simple rounding will usually not work because in the end, the total number may not add exactly to the required 435 congressmen. Alternative suggestions have been made, in the United States as well as in other countries, but they are fraught with problems, some methods favoring small states others favoring big states. and that is not the worst of it. Under certain circumstances, some states may actually lose seats if the size of the House is increased. (This bizarre situation has become notorious under the designation Alabama Paradox.) Other absurdities are known as the Population Paradox and the New State Paradox. Politicians, scientists, and the courts have . . .

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