Jurimath: The Flowering of an Ancient FieldMay 18, 2008
Philip J. Davis
In 1881, the future Supreme Court justice Oliver Wendell Holmes wrote,
"The life of the law has not been logic; it has been experience. The law embodies the story of a nation's development. . . . it cannot be dealt with as if it contained the axioms and corollaries of a book of mathematics."
The famous Judgment of Solomon can be said to deal with the fraction 1/2, but it also shows how the tension between logic and experience (i.e., what is perceived to be the common sense of the time and place) can lead to outcomes that are either disturbing or productive.
It is true that the law and mathematics are often at daggers drawn; I've heard it said that mathematicians and lawyers can't talk to one another. How does legal evidence differ from mathematical evidence? A strict mathematical constructionist (I am not one) might say that evidence is proof, and proof---tout court---is a sequence of logically connected statements that begin with basic assumptions and end with the presumptive assertion. In law, by contrast, de-fining admissible evidence entails the explication of many, many volumes and is a work in progress (consider, for example, the status of DNA as evidence). Of course, a one-sentence summary might state: Legal evidence is what the judge admits in evidence. Yet this implies---surprisingly---a convergence of the two views: The criterion used by the Clay Mathematics Institute for the awarding of its million-dollar millennium prizes, for example, is that a proof is valid if a qualified group of experts agree that it is. And thousands of pages of material accepted in the past as proofs are available to the experts as models.
A clever lawyer, in the presence of a nonmathematical jury, can make hash of a probabilistic argument. If the prosecution, backed by a statistical appraisal, says that a certain event has only one in a million chances of happening, the defense could claim that in the case at hand, the event did happen--and cite the analysis of Diaconis–Mosteller  showing that rare events happen more frequently than one might suppose. Adding to the ambiguity and confusion, "expert" mathematical witnesses have been known to testify on opposite sides of a case, reflecting disagreement within the scientific community.
Let's go back in time and look at the matter historically. Despite the striking differences in professional outlook, mathematics and the law have been mixing it up for millennia.
What is the origin of numbers? Though their onset is located in the "mists of time," the explanations that have been put forward range from the perception of similarities in the world of discrete objects, through the necessities of measurement and trade, and on to mysticism and even religion. (With respect to the latter, L. Kronecker (1823–91) declared: "God made the integers and all the rest [of mathematics] is the work of man.")
Histories of mathematics begin by displaying a variety of ancient number symbols and systems, but they do not and cannot really penetrate the primordial mist. Nonetheless, consideration of the development of such symbols is important, for some scholars base their opinion that (Western) writing developed in Sumeria on the way that the numbers of different sorts of things (e.g., jars, loaves, quantities of grain) were represented pictorially .
What is the origin of law? Though histories of mathematics seem not to entertain the possibility, I should like to advance the hypothesis that mathematics and the law developed simultaneously and with mutual interactions---or, at the very least, that there was an early marriage of the two. And for this mix I'll coin a term---"jurimath," whose meaning I interpret loosely.
Consider: Old Egyptian documents carry diagrams of geometric plane areas delimited by rope-stretchers to mark property boundaries, presumably for purposes of taxation (among other things).
In one of the oldest known codes of law, that of Hammurabi (c. 1810–1750 BC), one already finds a jurimath mixture in "governmental price controls":
"If a merchant gives grain or silver as an interest-bearing loan, he shall take 100 silas of grain per kur as interest [= 33%]; if he gives silver as an interest-bearing loan, he shall take 36 barleycorns per shekel of silver as interest [= 20%]."
In other portions of the code, Hammurabi mandates proper behavior and explicitly spells out punishments for infringements. Usually: Throw the miscreant into the river.
In Deuteronomy 19:14 one finds a plea for geometrical "invariance":
"Thou shalt not remove thy neighbor's landmark, which they of old time have set, in thine inheritance which thou shalt inherit. . . ."
Laws governing landmark removal are on the books in many places today.
Let's leave the Ancient World. It's a rare culture that does not concern itself with the questions "how many?" and "how much?" We in the 21st century are absolutely obsessed with these questions. Our obsession often plays out through law and jurisprudence. This route could be said to follow the vision of the famous attorney Gottfried Wilhelm, Freiherr von Leibniz (1646–1716), who seems to have dreamed of a formalization of mathematics/law that would make it possible to settle disputes by calculation. Law professor/combinatorialist Nicholas Bernoulli (1695–1726) discussed the question of how many years must elapse before a missing person can reasonably be declared dead and his estate divvied up---a problem that had been dealt with already by the Ancient World without the help of mathematical probability.
The U.S. Constitution (Article I, Clause 3) contains a neat piece of jurimath:
"Representation and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers. . . ."
Because we prefer not to send fractions of an individual to represent us in the House of Representatives, a "best" integer solution to the apportionment problem has been made statutory.
In 1865–68, in the first case in America in which a probabilistic argument was presented as evidence, mathematicians Benjamin Peirce and his more famous son, Charles Sanders Peirce, the latter an applied mathematician and founder of the subject of semiotics, made use of statistical analysis in a case of presumed forged handwriting in a will .
Florence Nightingale, "The Lady of the Lamp," a student of James Joseph Sylvester in mathematics and a fellow of the Royal Statistical Society, employed hospital and morbidity statistics to suggest changes in sanitary practices; the ensuing precepts, which are still in place legally, have made us a healthier bunch. Modern laws against the use of lead or asbestos have affected many of us personally. Lead pollution resulted in the recent recall of objects manufactured in China and to demands by individual members of Congress that the head of the Consumer Product Safety Commission resign.
In recent years numerous issues involving mathematics and law have resulted in congressional consideration or have ended in court challenges. Writing in SIAM News some years ago ("The Courts Count," April 1994---well before the 2000 presidential election count was decided by the Supreme Court), I discussed the Supreme Court decision---really Solomonic---on the type of census figures that can be used: direct head counts or sampling-based adjustments.
Within everyday experience, money in its various aspects is the principal social mathematization in place, and the courts are so clogged with such issues that we are no more conscious of them as instances of jurimath than we are aware in our daily lives of the laws of gravity. In the past, apart from money, jurimath questions that involved or threatened to involve court or congressional actions appeared only sporadically; they are now arriving thick and fast. Off the top of my head, and with knowledge derived only from reading the daily papers, I can think of
- the legality of gerrymandering (political geometry),
- the difficulties associated with electronic voting machines,
- forensic medical evidence,
- the ethics of placebo use in medical testing,
- the appropriateness of the Electoral College system for electing U.S. presidents,
- limits on punitive damages that can be imposed,
- the prohibition or the countenancing of gambling systems,
- the ongoing controversy as to the implications of the bell curve vis-à-vis intelligence,
- the implementation of technologically sophisticated IDs.
As though these issues were insufficient for the point I wish to make, I sought out additional material and discussions in Jurimetrics, the oldest journal dedicated to law and science and one that, according to the Sandra Day O'Connor School of Law (Arizona State University) Web site, "provides scholars and researchers with a wealth of thoughtful articles and is frequently cited in opinions of state and federal courts, legal treatises, textbooks, and scholarly articles in a wide range of other journals." Contributors have included lawyers, statisticians, philosophers, psychologists, cognitive scientists.
On its Web site Jurimetrics maintains an index that conveniently separates out questions hinging on mathematical considerations. I picked up a few more issues there: When are racial and ethnic statistics relevant? When can causation be inferred from probabilities? What can be said about evidence obtained from large-scale screening of people? Can the distinction between games of skill and games of chance be clarified mathematically? What is the validity of the Bayesian probability methods (personal or community degree of belief) that are often invoked in jurimath arguments? How can lawyers present statistical scientific evidence in a form that judges and jurors can understand and evaluate?
So what is my point in all this? It would seem, first of all, that sometime after 1881 Justice Holmes changed his mind; in 1897, in The Path of the Law, he wrote:
"For the rational study of the law, the blackletter man may be the man of the present, but the man of the future is the man of statistics and the master of economics."
Secondly: Moviegoers will surely recall the word "plastics," whispered to the young man played by Dustin Hoffman as an up-and-coming field in The Graduate (1967). Four decades later, I would be tempted to whisper "jurimath" into the young graduate's ear, realizing that he would probably not know what I meant. The financial rewards in the practice of jurimath may not be top dollar, but the intellectual challenges strike me as tremendously stimulating.
 P. Diaconis and F. Mosteller, Methods for studying coincidences, J. Amer. Statist. Assoc., 84 (1989), 853–861.
 D. Schmandt-Besserat, Oneness, twoness, threeness, The Sciences, 27 (1987), 44–48.
 For a thorough modern analysis of the Peirce methodology: J. Amer. Statist. Assoc., 75:371 (September 1980), also available on the Web; and SIAM News, April 1994.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at firstname.lastname@example.org.