Numbers Rule

The Vexing Mathematics of Democracy from Plato to the Present

by George Szpiro

Numbers Rule by George Szpiro

Until about 20 years ago, the 1-2 math courses the Liberal Arts majors had to take as a requirement for graduation came traditionally from the Calculus sequence: Precalculus or Calculus I, at best. The dismal retention rate for Calculus topics led to a reevaluation of course offerings. Mathematics does not lack in more current and relevant topics that require minimum preparation and offer insight into the depth and beauty of the subject. Among these are the theories of Social Choice. Voting systems, the problem of seat apportionment in national assemblies, and power and ability to form coalitions among parties and popular representatives became a staple of Liberal Arts Mathematics. George Szpiro's recent book covers in depth the theories of voting and apportionment.

Szpiro's volume is far from being a textbook that usually would include a greater variety of topics, come in sizes that exceed 700-800 pages and at a hefty price in a vicinity of $100, if not more. The book may of course be used as supplement in a Liberal Arts course. Both instructors and students will find it a rich source of information, with a sensibly exhaustive coverage of main topics. But completeness of coverage is not the strongest point of the book. 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.

The book consists of thirteen chapters, of which the last presents the case studies of three democracies - Switzerland, Israel, and France - and the manner in which they cope with the difficulties inherent in the democratic process. Each of the other twelve chapters focus on a specific topic and on the personalities that made a significant contribution to its development. Except for the first chapter on Plato, I found the level of details fascinating. The author comes through as a lucid storyteller and an exceptional math expositor.

Besides Socrates and Plato in the first chapter, the author sets up encounters with Plinys, both the Younger and the Elder (on the background of the 79 AD Vesuvius eruption), the Spanish theologian and philosopher, Ramon Llull (Rymond Llully), the German cardinal Nikolaus Cusanus (with the infamous and embarrassing episode of the Papal Schism in the background), Chevalier Jean-Charles de Borda - a mathematician, engineer and a naval officer, who was responsible for the determination of the first unit of length. Chapter by chapter the reader learns of the life and discoveries of such luminaries as Marquis de Condorcet, Pierre-Simon de Laplace, C. L. Dodgson (a.k.a. Lewis Carroll), Kenneth Arrow, the 1972 Nobel Prize winner in Economics.

The author illustrates many a voting procedure and shows by example which ones are subject to vote manipulation (first noticed and apparently applied by Pliny the Younger) and which suffer from internal inconsistencies known as voting paradoxes. The voting strand culminates in the introduction of Arrow's Impossibility Theorem which asserts that, under reasonably natural assumptions, no voting system (with the exception of the dictatorial rule) may avoid all the vexing paradoxes - inconsistencies are inevitable. Arrow's Theorem is a finest kind of abstract axiomatic mathematics with important repercussions in economics and voting. For his mathematical work Kenneth Arrow was awarded the 1972 Nobel Prize in Economics.

Arrow' theorem shows both the power and the limits of mathematics. Like Gödel's Incompleteness Theorem, it employs powerful mathematical tools to shed light on what may or may not be expected of mathematics. The choice of a voting system is more a matter of consensus and politics than that of mathematics. G. Szpiro leads the reader of his book to the same conclusion concerning the second theme of the book, the theory and methods of apportionment.

In the United States, already with the first attempt at adoption of an apportionment method, the politics, not mathematics played the dominant role. President Washington has vetoed a bill passed by the House that would adopt Alexander Hamilton's method, which caused the House to accept the second method under consideration - that by the then Secretary of State Thomas Jefferson. (The latter it was noted favored the state of Virginia, the home state of both himself and Washington.) Instructively, it was the very first veto cast by a US president, and only one of two that G. Washington cast in all his tenure at the post.

In time, the infighting over the apportionment methods spread form the Congress to the academia. After the 1910 Census, the Congress failed to choose between two methods on the table: Webster's of equal (major) fractions and Hill's of equal proportions. With scientists getting involved in the squabble, the two methods changed the nomenclature, respectively, to Webster-Wilcox (WW) and Huntington-Hill (HH). Their proponents became divided between the "Cornell" and "Harvard" schools. (W. F. Wilcox was a Cornell professor of social sciences and statistics, while E. V. Huntington was a professor mathematics and mechanics at Harvard.)

Szpiro narrates the story of the often acrimonious argumentation in succulent detail showing his humorous side where humor is called for. As became obvious by early 1929, the feuding politicians and their academic counterparts were unable to reach an agreement. Congress then turned to the National Academy of Sciences (NAS) to help resolve the issue. Subsequently, the NAS study group that was comprised of professors from University of Chicago, Yale, Princeton and Johns Hopkins, came out with a report. "Sharp as razors, they deduced that it was necessary to reach a solution to the apportionment problem in whole numbers." The group had unanimously (although with a piquant side line) recommended the HH method for adoption. The method was eventually adopted but political consideration did not let the matter rest. In 1948, another NAS committee has been formed led by Luther Eisenhart (member of the previous study group). The committee included two other Princeton mathematicians Marston Morse and the illustrous John von Neumann. The committee stuck with the HH method. "This was not altogether mind blowing since the new committee was not about to disallow its forerunner, especially since one-third of the members--the committee's chairman Eisenhart--provided continuity."

The twelfth chapter of the book is devoted essentially to the 1982 result of Michel L. Balinski and H. Peyton Young. Balinski and Young proved that the only apportionment methods that avoid paradoxes are the so called divisor methods. Among the methods mentioned in this review, the Jefferson, WW, and its bitter rival HH all fall into this category. They also looked into the matter of fair representation: "a state should receive no more and no less than its fair share of seats." Depressingly, none of the divisor methods is assured of compliance with this requirement. A. Hamilton's method does, but it also leads to various paradoxes. Their third result concerns the possible tendency of a method to produce apportionment that consistently favors either small or large states. What they found was that "From among all methods that use divisors, the Webster-Wilcox method is the only one that is practically unbiased. This conclusion flies in the face of the recommendations by the two NAS committees.

"How about that? Thirty years after Arrow's Impossibility Theorem we are again left in a lurch. We posited just three humble prerequisites for a good allocation method: it should it unbiased, avoid paradoxes, and stay within quota. Is that too much to ask for?" Mathematics shows that the answer to that question is really, "Yes, it is too much." And this is the limit beyond which mathematics has no power. The matter should be settled by consensus, among the people and for the people.

To conclude, the social choice topics supply an excellent source for math practice as well as a for the demonstration of the power and weaknesses of mathematics. George Szpiro's book is a lucid and comprehensive account of two topics of broad interest and concern. It provides for a enjoyable and informative reading. Please have a look.

Numbers Rule: The Vexing Mathematics of Democracy from Plato to the Present, by George Szpiro, pp 226, Princeton University Press, 2010. ISBN 0691139946. List price $26.95. $17.79 at

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