ࡱ > ~ { | } , bjbj . u u $ 2 8 7 < s $ x * y T & z $ D R T T y 2 r r r y r r r ^ | `i\s { H 0 x S| 2 @ | | ˒ r r x : MATHEMATICS, POLITICS, AND LAW
Philip J. Davis
Brown University, Providence, R.I., USA
Philip_Davis @ brown.edu
FOREWORD
The number of potential topics that could be discussed under this designation is vast. Taxes, federal and local, are perhaps the most immediate places where mathematics, politics and law affect the average person. Who gets taxed and how much ? Is a corporation to be regarded as an individual ? Consider the US Tax code which runs to runs to many, many thousands of pages. What person, interested in a particular area of the tax law could really be sure that he has comprehended the do's and dont's and possibilities embodied in the corpus ? Do you need to file the long form ? Etc. Computerized programs of a variety of qualities, invoking principles of artificial intelligence, have been written and used by the tax lawyers, accountants, H & R Blocks, (a firm that does tax preparation), the IRS personnel and the general public.
Omitting taxes, Ill mention six topics . But as one says: The Devil is in the Details (and Ive omitted details in this article). The six are:
(1) The U.S. Constitution (2) The Census of the U.S. (3) Reapportionament of the House of Representatives (4) Redistricting, Gerrymandering (5) Voter ID's (6) Voting Systems.
Politics moves steadily back and forth into law; clearly the US Supreme Court must ultimately get involved in deciding the legality of mathematizations. Thus, there is a newly created field of mathematics and law called jurimetrics. This is a subject that should be added to the roster of applications of mathematics and be taught in colleges and law schools.
The United States Constitution.
The Constitution of the United States is the basic law of the country. It organizes the federal government, describes the relationship between the government and the States, and asserts the rights and obligations of everyone within the jurisdiction of the United States.
A remarkable document, its Preamble sets forth its intent and purpose:
We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity do ordain and establish this Constitution for the United States of America.
Numbers abound in the text of the Constitution. One can easily find more than eighty numbers (not counting instances such as "Article 1, Section 1" Here are a few:
"The House of Representatives shall be composed of Members chosen every second Year "
" No person shall be a Representative who shall not have attained to Age of twenty five Years and been seven years a citizen of the United States"
"The actual Enumeration [i.e. the census] shall be made within three years after the first meeting of the Congress of the United States and within every subsequent Term of ten Years"
"The Senate of the United States shall be composed of two Senators from each State, chosen by the Legislature thereof for six years; each senator shall have one vote". "The Congress, whenever two thirds of both houses shall deem it necessary, shall propose Amendments to this Constitution.."
" neither shall any Person be eligible to that Office [the Presidency] who shall not have attained to the Age of thirty five years, and been fourteen Years a Resident within the United States."
"If any Bill shall not be returned by the President within ten days (Sundays excepted) after it shall have been presented to him, the Same shall be a Law "
Thus we see how simple numbers control the processes and the flow of lawmaking In the discussion that follows here, the mathematics becomes considerably deeper.
The Census of the United States.
This is carried out every ten years by the Census Bureau, a part of the Department of Commerce. This Department contains a Statistical Research Division that has issued A Bibliography of Selected of Statistical Methods and Development Related to Census 2000.
Who is to be counted and how is the count to be carried out ? For example, there are an estimated five and a half million American citizens living abroad. Are they to be counted and if so, how ? In 1999 the Supreme Court ruled that statistical estimations cannot be used to make the count more accurate. There are known to be undercounts of certain groups such as children or non-home owners. These could be estimated by statistical methods. The undercounted tend to vote Democratic and so the Republicans oppose sampling. At the moment, sampling is allowed for certain purposes and not allowed for others. There is much riding on the numbers and one of the most important effects is the reallocation of the numbers of individual State Representatives to the House of Representatives. The matter is very contentious.
Reapportionment of the number of Representatives a State gets after a census.
The Huntington-Hill method was adopted in the 1920s with a standardized House size of 435 members. E.V. Huntington (Harvard Professor of Mechanics) was early on interested in axiomatics and statistical theory It is also known as the Method of Equal Proportions. But the fidgeting of the numbers goes on and on.
Redistricting . Gerrymandering. Are there geometrical criteria for allowable political or educational districts ?
Gerrymandering is the practice of drawing district lines to achieve political gain for legislators by leaving out or including specific populations in a legislator's district so as to ensure his/her reelection. Gerrymandering is a particularly interesting problem, because, as far as I know, it is quite ill-defined, and at least in some cases specifically illegal; e.g. a State Legislature may not draw district lines so as to deprive a minority group of all representation.
Yet, gerrymandering for partisan purposes has been specifically declared legal by the Supreme Court; in fact, retired Justice Sandra Day O'Connor has said that a party in control of the legislature that failed to draw district lines to its own advantage was being irresponsible.
Algorithms, and. computer programs have been written for generating fair (ungerrymandered) district lines. As far as I know, these have been adopted nowhere for the obvious reason that no state legislature wants to cede this power.
Objective rules to create districts
Another means of reducing gerrymandering is to create objective, precise criteria to which any district map must comply. Courts in the United States, for instance, have ruled that congressional districts must be contiguous in order to be constitutional. This, however, is not a particularly binding constraint, as very narrow strips of land with few or no voters in them may be used to connect separate regions for inclusion in one district.
Some of the rules proposed are
Minimum district to convex polygon ratio.
Shortest splitline algorithm.
Minimum isoperimetric quotient defined as 4 pi x area/perimeter
Maximum compactness , with compactness of a shape often defined as 4 pi x area/square of perimeter.
ID's for voting.
About 30 States require a photo ID or "simple" ID. What mathematics underlies the creation and the use of officially issued IDs? Pattern recognition. Specifically, feature and face recognition. This is a vast and flourishing field of wide applicability.
Voting Systems.
E.g., proportional representation (PR) systems. This is opposed to "winner take all."
Axioms , often contradictory, have been proposed for desirable voting systems. In any case, in the US, winning an election doesn't depend on the popular vote, but on the number of electoral votes. The electoral votes of a State is defined as the number of its Representatives in Congress plus two for its Senators This issue came up in the 2016 Presidential Election which was won via the electoral despite an overwhelming popular vote. A hue and cry went up demanding that the U.S. Constitution be changed on that point. Changing the Constitution has been done, but it is a a very difficult and time consuming process. [In 1961, using the census numbers then available, the famous Hugarian-American mathematician George Polya determined the Minimum Popular Vote that can [theoretically] Elect the President. His answer: 22%. What would it be today?]
Envoi
Human opinion is often thought to be subjective while mathematics is thought to be objective. When push comes to shove, politics and human nature can override mathematical formulations.
References
Andreas Fellner, Game Theory and the U.S. Presidential Elections. On line.
Michael O. Finkelstein , Basic concepts of probability and statistics in the Law. Springer, 2009.
Jeff Suzuki, Constitutional calculus :The Math of Justice and the Myth of Common Sense. Johns Hopkins Univ. Press, 2015.
The Handbook of Face Recognition
I wish to acknowledge the help of Stefanie Han in the preparation of this article.
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