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"An engaging and unusual perspective on the no-man's land between mathematics and the law."—John Allen Paulos
From the impossibility of a perfectly democratic vote to a clarifying model for affirmative action debates, constitutional law professor and math enthusiast Michael Meyerson "provides an engaging and unusual perspective on the no-man's land between mathematics and the law" (John Allen Paulos). In thoroughly accessible and entertaining terms, Meyerson shows how the principle of probability influenced the ...
"An engaging and unusual perspective on the no-man's land between mathematics and the law."—John Allen Paulos
From the impossibility of a perfectly democratic vote to a clarifying model for affirmative action debates, constitutional law professor and math enthusiast Michael Meyerson "provides an engaging and unusual perspective on the no-man's land between mathematics and the law" (John Allen Paulos). In thoroughly accessible and entertaining terms, Meyerson shows how the principle of probability influenced the outcomes of the O. J. Simpson trials; makes a convincing case for the mathematical virtues of the electoral college; uses game theory to explain the federal government's shifting balance of power; relates the concept of infinity to the heated abortion debate; and uses topology and chaos theory to explain how our Constitution has successfully survived social and political change.
Copyright © 2002 Michael I. Meyerson.
All rights reserved.
(Healthy and Ill)
It reminds me of an answer given some years ago in the School at Oxford, when the Examiner asked for an example of a syllogism. After much patient thought, the candidate handed in:
"All men are dogs;
All dogs are men;
Therefore, All men are dogs."
This certainly has the form of a syllogism.... And it has the great merit ... that, if you grant the premises, you cannot deny the conclusion. Nevertheless, I feel bound to add that it was not commended by the Examiner.
The preceding is a lesson on the uses of logic, given by the Reverend Charles Dodgson, who is better known by his pseudonym, Lewis Carroll. The ability to think logically is essential for anyone who wants to understand the Constitution. Teachings from the world of mathematics present both the power of logic and the dangers inherent in utilizing it to confront difficult constitutional issues.
One of the foundations of the mathematical method is that knowledge leads to more knowledge. By the power of logical reasoning, simple truths can lead to countless conclusions of complexity and subtlety. In about 300 B.C., Euclid wrote the Elements, which has been called "the most influential textbook of all time." From a mere handful of axioms, Euclid was able to establish hundreds of theorems, encompassing all the important propositions of Greek mathematics. It is not so much the propositions, as the techniques of proof, which make his work of enduring importance.
The critical first step, of course, is the selection of axioms. An axiom can be defined as "a statement used in the premises of arguments and assumed to be true without proof." We must take the correctness of axioms for granted because we have to start somewhere. Some proposition must be the initial one, the one from which the others flow.
Similarly, we never are afforded the luxury of having all our terms defined. Definitions, after all, involve describing one term in reference to another. Unless we are willing to allow circular definitions, where two terms define one another, we must accept undefined terms. Euclid, for example, defined a point as "that which has no parts" but neglected to give a useful definition of parts.
Once we have decided what to take for granted, we prove the rest from that. Since undefined terms and unproven assertions are not likely to fill us with confidence, the power of a system is sometimes determined by how few axioms and undefined terms it contains. Hence, Aristotle declared that, "other things being equal, that proof is better which proceeds from the fewer postulates."
Choosing the right axioms is an art. Generally, the axioms should be simple and consistent with one another. They should be logically independent of one another, or else one would be more properly considered a theorem of the other. Finally, axioms "must be fruitful; like carefully selected seeds they must yield a valuable crop...."
The "crop" consists of theorems, which are defined as statements that are "derived from premises rather than assumed." Some theorems may be so obvious they could have been axioms; others may well be surprising and counterintuitive. For one example of the power of this method, see on the next page Euclid's proof that there exists an infinite number of prime numbers.
One of the most common forms of logical analysis involves the use of syllogisms, in which two statements (premises) are linked to form a conclusion. The classic (almost clichéd) syllogism is
1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.
Similar syllogistic forms can draw conclusions from statements that are less absolute but are sometimes, although not necessarily always, true:
1. Some members of the faculty can't be fired.
2. All people who can't be fired can be independent thinkers.
3. Therefore, some members of the faculty can be independent thinkers.
What is so powerful about this form of logical structure is that it leads to an unquestionable solution: If the premises are all true, then the conclusion must be true as well. Now, not any connection of statements will work. Obviously, if the logical argument is not well formulated, as in the story told by the Reverend Charles Dodgson at the beginning of this chapter, the result is laughable. But if the form is correct, the conclusion is inescapable.
The correctness of the form can be tested by viewing the syllogism in abstract form. The abstract quality of this system permits the substitution of symbols for the words in each syllogism, which leads to their universal acceptability.
For example, the syllogism about Socrates can be restated as
1. Every M is a P.
2. S is an M.
3. Therefore, S is a P.
The independent-minded faculty syllogism would be restated as
1. Some M are P.
2. Every P is an S.
3. Therefore, some M are S.
It does not matter what the M, P, or S represents. Each of these syllogisms is correct. This abstraction not only gives logic its strength, it also reveals its ultimate weakness. In the words of one of the leading logicians of the twentieth century, Bertrand Russell,
Pure mathematics consists entirely of such asservations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true.... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Both for mathematicians and for others who use logical reasoning, the danger is glaring: if your initial axioms are incorrect, then your conclusions are not guaranteed. You may be arguing perfectly logically but still end up with a ridiculous conclusion because of an initial poor choice of axioms. Thus, some have remarked that "Logic is the art of going wrong with confidence."
Without question, the most successful application of logical analysis to the axioms of political science is the Declaration of Independence, that "famous 'mathematical' document." The Declaration of Independence is a paradigm of deductive reasoning, where political axioms are announced, then specific facts that show the applicability of those axioms are described, and, finally, the conclusion of independence is proclaimed.
In Thomas Jefferson's library was a book by a John Harris, entitled Lexicom Technicum, which defined axiom as "such a common, plain, self-evident and received Notion, that it cannot be made more plain and evident by demonstration." The ringing phrase "We hold these truths to be self-evident" is really a statement that the "truths" are to be considered the political equivalents of Euclid's axioms.
There were four such "truths" in the Declaration:
1. "[T]hat all men are created equal"
2. "[T]hat they are endowed by their Creator with certain inalienable rights; that among these, are life, liberty, and the pursuit of happiness"
3. "that, to secure these rights, governments are instituted among men, deriving their just powers from the consent of the governed"
4. "[T]hat, whenever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute a new government"
There is no need to supply any proof for these "truths." They are the "Elements" of our political system.
Jefferson then presented a long list of facts, demonstrating that the first two axioms had been violated by "a long train of abuses and usurpations...." According to the third axiom, governments are instituted to prevent such violations. According to the last axiom, when such violations occur, the people have the right to institute a new government. "We," the Declaration concludes, "therefore, solemnly publish and declare, that these united colonies, are, and of right ought to be, free and independent states." Q.E.D.
In his defense of the Constitution, Alexander Hamilton also relied on the power of axiomatic analysis. In Federalist 31, Hamilton explained that "In Disquisitions of every kind, there are certain primary truths, or first principles, upon which all subsequent reasonings must depend." He stated that this was true not only of "maxims in geometry," but with "these other maxims in ethics and politics ... [such as] the means ought to be proportioned to the end [and] that every power ought to be commensurate with its object...."
For example, he argued that because of its responsibility for guaranteeing national defense and "securing the public peace against foreign or domestic violence," the federal government needed the unqualified ability to raise money through taxation so as to perform its tasks. Hamilton criticized the opponents of the Constitution who denied the logical conclusions that flowed from the basic political principles: "The obscurity is much oftener in the passions and prejudices of the reasoner than in the subject."
Hamilton acknowledged that "it cannot be pretended that the principles of moral and political knowledge have, in general, the same degree of certainty with those of the mathematics." Nevertheless, he maintained that the major limitation on the use of political logic was the greater difficulty in which the results of its syllogisms were accepted than in the ethereal world of mathematics: "The objects of geometrical inquiry are so entirely abstracted from those pursuits which stir up and put in motion the unruly passions of the human heart, that mankind, without difficulty, adopt not only the more simple theorems of the science, but even those abstruse paradoxes which ... are at variance with the natural conceptions which the mind ... would be led to entertain upon the subject."
[W]e must never forget, that it is a constitution we are expounding.
Why is the Constitution so short? Because in many ways the Constitution provides only the axioms of our system of government, which we then are required to interpret and explain. According to Chief Justice William Rehnquist, in interpreting the Constitution, the Supreme Court tries to "discern among its 'essential postulates,' a principle that controls the present cases." This concept is captured in John Marshall's famous explanation of why interpreting the Constitution is a different venture than other forms of legal interpretation. Marshall stated that any document that contained all of the details of a government's structure and power would "partake of the prolixity of a legal code, and could scarcely be embraced by the human mind." Thus, by its very nature, a constitution requires that only the "great outlines should be marked, important objects designated," so that the minor ingredients "which compose those objects be deduced from the nature of the objects themselves."
Many of the great opinions penned by Marshall were styled deliberately as Euclidean proofs. In McCulloch v. Maryland, Marshall laid out his argument that states lacked power to tax a federally incorporated bank, as follows:
1. "This great principle is that the constitution and the laws made in pursuance thereof are supreme ... and cannot be controlled by [the states]."2. "From this, which may almost be termed an axiom, other propositions are deduced as corollaries ... that a power to destroy, if wielded by a different hand, is hostile to and incompatible with these powers to create and to preserve."3. "That the power of taxing it by the States may be exercised so as to destroy it, is too obvious to be denied."
In Gibbons v. Ogden, Marshall essentially apologized for his extended "proof" that federal law preempted a state-authorized monopoly to run a ferry in New York waters:
The Court is aware that, in stating the train of reasoning by which we have been conducted to this result, much time has been consumed in the attempt to demonstrate propositions which might have been thought axioms.... The conclusion to which we have come depends on a chain of principles which it was necessary to preserve unbroken; and, although some of them were thought nearly self-evident, the magnitude of the question, the weight of character belonging to those from whose judgment we dissent, and the argument at the bar, demanded that we should assume nothing.
One of the greatest examples of the use of logic to reach a conclusion of constitutional interpretation is Oliver Wendell Holmes's defense of "the marketplace of ideas." His oft-quoted analysis includes not only two competing syllogisms but reference to a mathematical proof as well:
Persecution for the expression of opinions seems to me perfectly logical. If you have no doubt of your premises or your power and want a certain result with all your heart you naturally express your wishes in law and sweep away all opposition. To allow opposition by speech seems to indicate that you think the speech impotent, as when a man says that he has squared the circle, or that you do not care whole-heartedly for the result, or that you doubt either your power or your premises. But when men have realized that time has upset many fighting faiths, they may come to believe even more than they believe the very foundations of their own conduct that the ultimate good desired is better reached by free trade in ideas—that the best test of truth is the power of the thought to get itself accepted in the competition of the market, and that truth is the only ground upon which their wishes safely can be carried out. That at any rate is the theory of our Constitution. It is an experiment, as all life is an experiment.
As with Euclid's proof of the infinitude of primes, Holmes begins by presuming the opposite of what he intends to prove. He states that censorship seems to be logical and that one "naturally" would want to sweep away opposition. That certainly would be the case, he says, if the only reasons for allowing opposition were that (1) the opposing speech was "impotent," that is, unable to prevail; (2) the speech was on an unimportant topic; (3) those in power doubted their ability to enforce a ban; or (4) those in power lacked faith in their own position.
But, Holmes says, history has shown that the ideas about which people were completely confident often turned out to be incorrect. (Note that Holmes uses volumes of history in his page of logic.) Holmes hardly needed to reference the calamitous event of his lifetime, the Civil War, in which so many were willing to die over their "faith" in the correctness of slavery. Therefore, he says, it is logical to believe in the premise that the free sharing of ideas will lead to the "truth" rather than in the competing premise that the "truth" is possessed by any one individual, even oneself. And this, he adds, is the "theory" of the Constitution.
Holmes concludes his analysis by highlighting the significance of using political, rather than mathematical, axioms. There never can be the same certainty in politics as there is in mathematics. The effectiveness of the marketplace of ideas is not a guarantee, he says, but merely "an experiment."
Excerpted from POLITICAL NUMERACY by Michael I. Meyerson. Copyright © 2002 by Michael I. Meyerson. Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
|Introduction: The Ugliest Number in the Constitution||16|
|1||Logic (Healthy and Ill)||23|
|3||The Positive Value of Consensus||71|
|4||The First Veto||82|
|5||What Does Equality Equal?||91|
|6||Game Theory and the Constitution||109|
|8||Infinity and the Constitution||148|
|9||The Incomplete Constitution||162|
|11||The Mathematics of Limits||209|
|12||The Limits of Mathematics||217|