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Numbers Rule

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

George G. Szpiro
Copyright Date: 2010
Pages: 248
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  • Book Info
    Numbers Rule
    Book Description:

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

    eISBN: 978-1-4008-3444-0
    Subjects: Mathematics, Political Science

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
    (pp. ix-xiv)
    (pp. 1-21)

    Plato, son of Ariston and Perictione, has been called the greatest of Greek philosophers by his admirers and chastised as the worst anti-democrat by his detractors. Socrates’ most brilliant student, Plato devoted his life to studying and teaching, to exploring the meaning of life, inquiring into the nature of justice, and pondering how to be a better person.

    His real name may have been Aristocles with the nickname Plato—meaning broad—given to him because of his wide forehead or because of his wide-ranging intellectual pursuits. He was born in 427 BC in or near Athens. Plato had two brothers,...

    (pp. 22-32)

    Like the administrators in the ancient Greek cities, the civil servants of the Roman Empire were concerned with governing well and dispensing justice. The magistrate and state official Gaius Plinius Caecilius Secundus, generally known as Pliny the Younger, raised a profound question about the proper way to vote on a particular issue. Born in AD 61 or 62 in what is now the Italian city of Como, Pliny had lost his father, a landowner, while still a child and was brought up by his mother. The main influence on his education was his mother’s brother, Pliny the Elder, a Roman...

    (pp. 33-49)

    In the Athenian Assembly the choices to be made were usually of the form yes/no, for/against, guilty/innocent. The decision-making process usually worked all right, because deciding between two options presented no special problems; a simple majority vote does the trick. And whenever an official had to be chosen from among more than two candidates, the selection was usually deferred to chance, or god, by drawing lots. With time it became apparent, however, that choices between more than two alternatives were inevitable, and electors were often hard put to agree on a winner. Reluctant to let the lot—or fortune, or...

    (pp. 50-59)

    After Llull’s foray, no further progress was made in the theory of voting and elections for over a century. Then, in 1428, the German student Nikolaus Cusanus happened upon one of Llull’s documents in Paris. He found the text sufficiently interesting to make a transcript and take it home. But he did more than reproduce the text for his own use. A few years later he improved upon Llull’s method, thereby establishing himself as the second pioneer of the modern theory of elections.

    The young man was born as Nikolaus Krebs in the German town of Kues in 1401, 102...

    (pp. 60-72)

    The eighteenth century was a period of enlightenment throughout the Old and New World. France, the United States, and Poland granted themselves constitutions. Nations were in upheaval as their citizens started demanding equal justice for all, showing concern for human rights, and calling for a regulation of the social order. At the same time, demands for quality government arose and the question of how officials were to be elected to high positions became important again. In this atmosphere two eminent French thinkers appeared on the scene. One was a military officer with numerous distinctions in land and sea battles. His...

    (pp. 73-88)

    Scholarly debate in the French capital, with its newspapers, publishing houses, academies, andsalontradition, was always very lively. It was no different with Borda’s voting scheme. As could be expected, his proposal of assigning points, or m-units, to preferences did not go unchallenged. The challenger came in the form of a nobleman, who was Borda’s junior by ten years. His full name was Marie-Jean-Antoine Nicolas de Caritat, Marquis de Condorcet.

    Born in 1743 in Ribemont, Condorcet was the only child of an ancient family of minor nobility. His father, a cavalry captain, was killed during a military exercise when...

    (pp. 89-99)

    When writing his essay in 1785, Condorcet was apparently already aware of Borda’s contribution of 1781. He admitted as much in a caustic footnote in which he acknowledged that the existence of Borda’s paper had been pointed out to him by friends, at a time when his own paper was already being printed. Somewhat patronizingly, he claimed that he would not have known anything about the paper save for the fact that some people had mentioned it to him. As is now believed, however, Condorcet was being less than truthful. In 1781, he was the perpetual secretary of the Académie...

    (pp. 100-118)

    The theory of voting and elections was not in a satisfactory state at the beginning of the nineteenth century. Majority votes fail to take into account the electors’ preferences beyond their top choice, and when the lesser choices are taken into account, cycles appear. The Borda count could result in the election of a candidate whom nobody really wants, while a Condorcet winner, who would beat every competitor in two-by-two showdowns, is not always guaranteed to exist. The theory was in a quandary, but further advances had to wait until an unlikely chap appeared on the scene. He was born...

    (pp. 119-133)

    With this we leave the irksome subject of voting and elections for a while to consider a different field of mathematical conundrums that plague democracies throughout the world. It is the problem of allocating seats in a parliament. Everybody would like to assign the number of delegates that a geographical region or a political party sends to the legislature in a fair and equitable manner. Unfortunately we shall see that the questions that will be raised are quite a bit as annoying, perplexing, and sometimes counterintuitive as the problems and paradoxes that occur when voting for a proposal or electing...

    (pp. 134-164)

    Frustrated and dispirited, politicians looking for solutions—not always mathematical, as often as not political—to the seemingly intractable problems of apportioning seats in Congress finally turned to a professional, Walter F. Willcox. Willcox, professor of social science and statistics in the department of philosophy at Cornell University, had been active in the census of 1900 and later became the Census Bureau’s chief statistician for population. He was instrumental in raising the debate about apportionment from the lowlands of politics to the realms of science.

    The Census Bureau had been established very recently. At the outset, in 1790, the first...

    (pp. 165-186)

    We now leave the matter of apportionment for a while and return to the troublesome problem of electing a leader. Remember Condorcet and his paradox? And how Lewis Carroll wrestled with it? Well the problem did not go away. Nor did it mellow with age. If anything it became more vexing. Enter Kenneth Arrow, Nobel Prize winner of economics in 1972 and one of the most important economists of the twentieth century.

    An outstanding graduate student at Columbia University in the late 1940s, Arrow was thinking about his doctoral thesis. It was an exciting time for budding economists, observing and...

    (pp. 187-201)

    We return to the frustrating subject of apportionment. In the preceding chapter I recounted that Kenneth Arrow proved that any election method that satisfies reasonable conditions of rationality—like avoiding cycles—is either imposed or dictatorial, and that Allan Gibbard and Mark Satterthwaite showed that any democratic election method can be manipulated. This chapter will, unfortunately, be the bearer of further bad tidings: a fair and true allocation of seats in Congress is also a mathematical impossibility.

    With the size of the House fixed at 435 in 1912, the Alabama Paradox no longer loomed. And after the inclusion of Alaska...

    (pp. 202-214)

    In this last chapter I will describe case studies of how three different countries wrestle with apportionment and elections in light of the impossibility theorems. Every representative democracy must select delegations for legislative assemblies, composed of integer numbers of parliamentarians. The delegations represent geographical regions or political parties. Some countries have come up with unique propositions; others still experiment with adequate, if not ideal, ways of allocating seats in parliament. By way of example I will mention two countries: one of the older democracies, Switzerland, founded in 1291, and one of the newer ones, Israel, created in 1948. Finally, I...

    (pp. 215-218)
  18. INDEX
    (pp. 219-226)