Political Numeracy: Mathematical Perspectives on Our Chaotic Constitution.

• Author: Edelman, Paul H.
• Position: Book Review

POLITICAL NUMERACY: MATHEMATICAL PERSPECTIVES ON OUR CHAOTIC CONSTITUTION. By Michael I. Meyerson (1). W. W. Norton and Company. 2002. Pp. 9, 287.

Can mathematics be used to inform legal analysis? This is not a ridiculous question. Law has certain superficial resemblances to mathematics. One might view the Constitution and various statutes as providing "axioms" for a deductive legal system. From these axioms judges deduce "theorems" consisting of interpretation of these axioms in certain situations. Often these theorems are built on previously "proven" theorems, i.e. earlier decisions of the court. Of course some of the axioms might change, and occasionally a theorem that was once true becomes false; the former is a common feature of mathematics, the latter, though theoretically not possible in mathematics (since a theorem is by definition true) has been known to happen in mathematical practice as well. (3)

So maybe mathematics can help law scholars. That is certainly what Michael Meyerson believes. His new book is "premised on the belief that there are many legal ideas that can be explained or clarified by mathematics." (p. 47) He presents an extended set of examples to illustrate how mathematics and mathematical thinking can be useful in understanding legal issues. Some of his examples are persuasive indeed. Others are less compelling. In this review I will describe some of his examples and assess how much the mathematics really adds to the legal analysis.

WHAT CONSTITUTES MATHEMATICAL THINKING?

Before exploring Meyerson's examples it is worthwhile to consider what, exactly, it means to think mathematically about the law. There are a number of different things one might mean by this, although Meyerson treats them all the same. The persuasiveness of his applications of mathematics to legal analysis depends on which of these interpretations is being employed.

One can break down mathematical thinking in the law into three types: general logic, technical mathematics, and mathematical metaphor. By general logic I mean the use of standard deductive reasoning that might be taught in a formal logic class. For example, such claims as "If A implies B and B implies C then A implies C" or "If P implies Q then Q being false implies that P is false" fall into this category. The syllogisms that Meyerson discusses (p. 25), such as

1. All men are mortal.

2. Socrates is a man.

3. Therefore, Socrates is mortal. are also examples of what I call general logic.

The second class of mathematical thinking is what I will refer to as technical mathematics. Here theorems from mathematics are employed to provide a solution to a legal question. Technical mathematics is illustrated by Meyerson's discussion of the mathematical methods employed in the apportioning of the House of Representatives (p. 82). What distinguishes this class of thinking from general logic is the level of mathematical sophistication that is being employed.

The final class of mathematical thinking is mathematical metaphor. In this style of thinking the ideas of mathematics are used as a source of inspiration but not put to any technical use. So when Meyerson discusses "constitutional topology" (p. 134) he does not intend to give a formal definition of the Constitution as a topological space, but rather to use the ideas of topology to give some informal description of certain properties of the Constitution, to wit:

Imagine the federal government as a sphere containing all of the powers granted by the Constitution. Within that sphere, however, there exists a hole, consisting of the powers that are, in the words of the Tenth Amendment, "reserved to the States." Over time, the size of the hole has grown and shrunk relative to the size of the sphere, but the hole must remain if the Constitution's topological structure is to remain intact (p. 138). Clearly Meyerson does not believe that the Constitution is a sphere in any meaningful sense, but he finds the analogy of the sphere and the use of the language of topology illuminating. (4)

The remainder of this review will discuss Meyerson's uses of each of these types of mathematical thinking. Meyerson's use of general logic is impeccable, but I will question if its use is distinctly mathematical. His use of technical mathematics is the most interesting and persuasive part of the book. In addition to describing some of his applications, I will suggest some further applications of technical mathematics. Finally, I will cast doubt upon Meyerson's use of mathematical metaphor in understanding legal issues.

GENERAL LOGIC

Meyerson begins his foray into mathematical applications by considering the axiomatic nature of much of legal argumentation. He starts by analyzing the logical structure of the Declaration of Independence (p. 28) and continues through a dissection of the logical foundations of Korematsu v. United States. (5) Along the way he finds an opportunity to discuss Euler's proof of the infinitude of prime numbers and Lindemann's proof that it is not possible to square a circle. (6)

The early part of the chapter demonstrates the influence that formal deductive proof had at the time of the Founders and how the structure of a formal proof was consciously mimicked by them in the Declaration of Independence. He goes on to show how formal deductive reasoning is evident in many Supreme Court opinions. I found this historical section particularly interesting.

Meyerson wants to make some stronger points as well. His main point is that all formal logical arguments start from the assumption of certain axioms. If these axioms are false, then the validity of the subsequent deductions is irrelevant to the truth of the final conclusion. By subjecting legal arguments to a close logical reading one can uncover the axioms underlying the argument and thus better understand the nature of the argument. He applies this technique to notorious Supreme Court decisions such as Dred Scott (7) and Korematsu.

Meyerson's reminder that assumptions, both stated and unstated, are critical in deductive logic is an important one. It is often the first place to look in analyzing an argument. At the foundation of mathematical thinking is the ability to understand the difference between hypotheses and theorems, and so this is a fine beginning to Meyerson's task of demonstrating the importance of mathematical thinking to legal analysis.

It would have been helpful, though, if Meyerson had been more careful about two things. First, just because an argument is logically flawed, either because an assumption is false or because of some other logical error, it does not follow that the conclusion is false. Many scholars have critiqued the logic of Roe v. Wade (8) but still support the result. (9) So Meyerson's assertion that "Obviously, if the logical argument is not well-formulated, ... the result is laughable," (p. 25) is simply not correct. It is one thing to say that the conclusion is not proved, quite another to claim that the conclusion is itself false.

The second point about which Meyerson could have been more careful is his use of the term axiom. Early in his discussion of formal logic Meyerson defines an axiom as "'a statement used in the premises of arguments and assumed to be true without proof.'" (10) He then notes that a set of axioms "should be simple and consistent with one another," as well as "logically independent." (p. 24) With these definitions, then, it is incorrect to assert that an axiom is false within the logical system that it defines.

But Meyerson later forgets his own definition when he discusses Dred Scott. He notes that Taney's decision rests on the axiom that African-Americans are inherently inferior to whites, and remarks that "One lesson from Dred Scott is that if you start with an incorrect axiom, you are unable to reason intelligently." (p. 35) But this is a mischaracterization of what Taney did. He reasoned quite intelligently from the axioms that he started with. What Meyerson really means is that if Taney had chosen a different axiom, say the inherent equality of African-Americans and whites, then he would have come to a different conclusion.

While Meyerson has demonstrated that general logic is important in legal analysis, it leaves open the question of whether it should be considered uniquely mathematical. If so, then any discipline that proceeds in the western intellectual tradition can be said to benefit from mathematical thinking. That conclusion undercuts the novelty of Meyerson's claims.

These quibbles aside, Meyerson's beginning is a good one. He reminds us that it is important to make clear what one is assuming and what one is concluding in any argument, but particularly in legal ones, where it is easy to leave the hypotheses unstated. The more transparent the logic, the better it can be assessed.

TECHNICAL MATHEMATICS

It is in chapters 2-4 that Meyerson is best able to support his claim that mathematics can illuminate legal thinking. In these chapters he examines voting rules--including ruminations on the benefits of the electoral college--considers the super-majority aspects of the Constitution, and discusses the difficulties in apportioning seats in Congress. With interesting mathematics to be discussed and interesting law to ponder, this is the best part of the book. The only thing wrong is that there is not enough of it. Meyerson missed opportunities to put his material in a broader context and to include some additional mathematics, some even considered by the Supreme Court itself.

The theme of these chapters is voting...