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1. Introduction
Everyone is familiar with the power of mathematics to solve problems in physics.
Though Galileo is recognized more as a physicist than a mathematician, he was
a professor of mathematics at the University of Pisa (1589-1591) and the University
of Padua (1592-1610). Isaac Newton (1642-1727) makes any short list of both
the greatest physicists and mathematicians of all time. Other mathematicians
who made significant contributions to mathematics and physics include Leonard
Euler (1707-1783), Laplace (1749-1827), and Gauss (1777-1855).
Mathematics has also had an important role to play in chemistry, geology and
biology but what about mathematics and political science? Has mathematics had
significant applications in political science? I believe so and in my discussion
here I will deal with mathematical approaches to voting and elections. Contributions
of mathematics to voting began earlier than many people realize. During the
period of the French Revolution, two fascinating people with talent in mathematics,
the Marquis de Condorcet (1743-1794) and Jean de Charles Borda (1733-1799),
raised important ideas related to voting systems. Others who made contributions
to mathematical ideas that involve elections include Charles Dodgson (1832-1898),
Duncan Black (1908-1991), Kenneth Arrow, and John Kemeny (1926-1992), and Steven
Brams. Dodgson was a professional mathematician at Oxford, in addition to being
the author of Alice in Wonderland. Duncan Black was an economist who revived
interest in using mathematical tools to study voting systems. Black's book The
Theory of Committees and Elections revived significant interest in using mathematical
tools to study election questions. Arrow, though he taught in economics departments,
began his academic career as a mathematics major. Kenneth Arrow won the Nobel
Memorial Prize in Economics in 1972 in part for the insight he obtained into
group decision making processes in his 1951 doctoral dissertation.
 Marquis de Condorcet |
 Jean Borda |
 Charles Dodgson |
 John Kemeny |
If one is to do a mathematical analysis of any subject, one has to carefully
examine phenomena related to what one is investigating and make simplifying
assumptions, to construct what today are called mathematical models. Voting
is carried out in a surprisingly large array of situations: selection of candidates
for municipal, state, and national elections; votes that legislators make when
choosing among alternative courses of action; decisions by economic planners
about what course of action to take; selection by judges of the winner for a
skating competition; selection of a movie for best film of the year; or selection
of what should be served at the company picnic. What are the salient phenomena
involved in elections and voting? Elections require voters and alternatives
to choose from (typically people, but there are many other possibilities). To
express voter opinions about the alternatives requires a ballot of some kind.
After the voters make their judgments on the alternatives (candidates), it is
required that some decision method be used to arrive at the winning candidate,
winning candidates, or a collection of selected alternatives.
There are many interesting aspects of elections that probably will not play
a part in a first pass at using mathematics to study elections. Should felons
be restricted from voting? Should people who can not be present when the voting
is to take place have a way to cast a ballot in some other way? Are the machines
(or physical mechanism) currently used for voting the best choice possible?
(Best choice from what point of view?)
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