Proving the Point: Connections Between Legal and Mathematical Reasoning.

• Author: Termini, Maria

1. Introduction

   Fermat's Last Theorem was a famous unsolved problem for three and a half centuries, and its solution made headlines even outside of mathematical circles. The Theorem takes its name from Pierre de Fermat, an avid amateur mathematician who died in 1665. (2) After Fermat's death, among his possessions was a mathematics text that included his notes related to the Pythagorean Theorem. The Pythagorean Theorem gives a formula for determining the lengths of the sides of a right triangle. If c is the hypotenuse of a right triangle--i.e., the side opposite the right angle--and a and b are the other two sides of the right triangle, then [a.sup.2] + [b.sup.2] = [c.sup.2]. (3) While the lengths of the sides of a right triangle are not always whole numbers, it is not difficult to find whole numbers for a, b, and c that fit the formula. For example, a, b, and c could be 3, 4, and 5, respectively, because [3.sup.2] + [4.sup.2] = [5.sup.2]. Another example is 5, 12, and 13. These are known as Pythagorean triples. (4)

   The Pythagorean Theorem and Pythagorean triples appear in Arithmetica, a book written in approximately 250 C.E. by Diophantus, a scholar in Alexandria. (5) Centuries later, while studying his Latin copy of Arithmetica, Fermat considered the Pythagorean Theorem and variations of Pythagorean triples. (6) Specifically, he wondered whether any solutions could be found for a, b, and c when they are raised to any power greater than two. (7) Are there any whole number solutions when a, b, and c are each cubed? When they are each raised to the fourth power or the fifth power? Fermat thought the answer to each question was "no." (8) In the margin of his copy of Arithmetica, Fermat stated that for any whole number n greater than two, there are no whole numbers a, b, and c that make the equation [a.sup.n] + [b.sup.n] = [c.sup.n] true. (9) Then, in Latin, Fermat noted: "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." (10) After Fermat's death, his son found and published this and other mathematical notes, but Fermat's proof of his "last theorem" was never found. (11) For decades, and then centuries, other mathematicians tried to recreate or discover the proof for themselves, without success. (12)

   In 1986, Andrew Wiles, then a professor of mathematics at Princeton University, set himself the task of proving Fermat's Last Theorem. (13) He worked on the project for seven years, mostly alone and in secret. (14) In 1993, Wiles announced a solution in a series of three lectures at Cambridge University, but the proof had to be reviewed by referees before publication. (15) As with everything related to Fermat's Last Theorem, the peer review process was not easy. During the review process, one of the referees discovered a problem with the proof. (16) Wiles went back to work to try to address the gap the referee had identified. (17) With another year of work, and in collaboration with his former student, Richard Taylor, Wiles was able to bridge the gap, though he nearly gave up in the process. (18) Ultimately, the proof was published as two separate but related papers, which together totaled 130 pages and filled the May 1995 volume of the Annals of Mathematics. (19)

   Andrew Wiles is a professional mathematician. His entire career has consisted of studying and teaching mathematics in academia. For Pierre de Fermat, on the other hand, mathematics was a hobby, though a very serious one; Fermat's career was in the law. (20)

   This connection between law and mathematics in Fermat's life is not as surprising as it might seem at first blush. Legal reasoning and mathematical reasoning are profoundly connected. Abraham Lincoln explained that studying mathematics helped him better prepare for his legal studies:

   In the course of my law-reading I constantly came upon the word demonstrate. I thought, at first, that I understood its meaning, but soon became satisfied that I did not.... At last I said, 'L[incoln], you can never make a lawyer if you do not understand what demonstrate means;' and I left my situation in Springfield, went home to my father's house, and staid there till I could give any propositions in the six books of Euclid (21) at sight. I then found out what 'demonstrate' means, and went back to my law studies. (22) scholarly collaborations. Id. at 155. Someone who collaborated with Erdos has an Erdos number of 1, someone who collaborated with one of Erdos's collaborators has an Erdos number of 2, and so on. Id.

   Lincoln's idea--to study mathematical reasoning in order to become better at legal reasoning--might seem strange, especially to those lawyers who do not like math. But law and mathematics have much in common, starting with the basic organizational structure of written analysis in each field. The logical structure of written proofs in upper-level mathematics bears a close resemblance to the basic organizational structure in legal writing: Issue, Rule, Application, Conclusion--often referred to by its acronym IRAC. With a proof, as in IRAC, a writer starts by telling the reader where the writer wants to go (I); then lists the rules or theorems already known (R); applies the known rules to the facts or "givens" (A); and reaches a conclusion based on that reasoning (C). (23)

   In addition to using similar organization, both legal and mathematical analysis use the same types of reasoning, including deductive reasoning, inductive reasoning, and arguments in the alternative. Written legal analysis also has similar purposes to those of written mathematical analysis. Both forms of writing aid the author's thought process and are used to convince the reader that the author is correct, to extend the body of knowledge within the field, and to teach those new to the field.

   While a comparison between law and mathematics may seem inapt because law is less determinate and less certain than mathematics, a closer look at mathematics reveals this lack of certainty is yet another point of similarity between these two fields. (24) Mathematics, far from revealing some universal truths, can only establish truths based on the assumptions made within the mathematical system being used. Similarly, lawyers cannot find "the law" by looking for universal truths, but instead must consider the laws as they exist within a given legal system. (25) Furthermore, just as existing enacted law and common law cannot address every possible scenario that may come before a court, it is impossible to develop a robust mathematical system in which all mathematical statements can be proven to be true or false.

   This Article proceeds as follows. Part II explores the similarities between mathematical analysis and legal analysis, including their similarities of organization, purpose, and types of reasoning. Part III addresses potential differences between law and mathematics, concluding that those differences are not as great as they appear. Part IV considers explanations from cognitive science about the effectiveness of these forms of written analysis. Part V draws lessons for legal analysis in light of the comparison to mathematical analysis.

2. HOW IS LEGAL ANALYSIS LIKE MATHEMATICAL ANALYSIS?

   Written legal analysis and mathematical proof are similar in several ways. In addition to organizational parallels, they also use similar reasoning and share many comparable purposes.

   A. Organization

   1. Mathematics

      Most high school students in the United States learn the "two-column proof' in geometry class. Two-column proofs provide a structure in which students can work through and explain their reasoning when proving a theorem. A two-column proof typically begins with the "givens"--the facts that should be assumed--and a statement of what will be proven. After that, the proof is laid out in two columns. The left-hand column contains numbered statements, which show the proof writer's reasoning, starting with the given facts and building on those facts one step at a time by applying rules to the facts. For each statement in the left-hand column, the right-hand column contains the "rule" that was applied to reach the statement. Each rule must be a given, a definition, a property, (26) a postulate, an axiom, (27) or a previously proven theorem. (28) The proof proceeds step-by-step, ending when the fact to be proven is the final statement in the left-hand column and it is justified by a valid reason in the right-hand column. Figure (1), below, shows an example of a simple two-column proof. (29)

      After geometry, many students never see mathematical proofs again. The proof, however, is an important part of advanced mathematics. In upper level mathematics courses--and with mathematicians generally--written proofs in paragraph form are preferred over two-column proofs.

      In many respects, these "paragraph" proofs are like two-column proofs. They typically begin with the givens and with a statement of what is to be proven, and move on to a series of statements and the justifications for those statements, building logically to the conclusion. (34) The main distinction is the form of the statements and justifications of each step of the proof. Instead of listing those steps rigidly in a two-column format, mathematicians usually explain themselves using complete sentences and paragraphs. (35) Figure 2, below, shows an example of a "paragraph" proof.

      Figure 2: Proof Theorem: If x and y are even integers, (36) then the sum of x and y is even. (37) Proof. Suppose x and y are even integers. (38) Since x is an even integer, there must be an integer n such that x = 2n. (39) Similarly, since y is also even, there must be an integer m such That y = 2m. Thus, x + y = 2n + 2m = 2(n + m). Since n and m are both integers, n + m must also be an integer. Then, since x + y = 2(n + m) and n + m is an integer, x + y is even. (40) 2. Law

      This basic proof structure--starting with the thing to be proven, stating rules and applying them...