Mathematics and the Legal Imagination: A Response to Paul Edelman

Article excerpt

I was very flattered to read Paul H. Edelman's review of my book, Political Numeracy. (2) He is a first-rate mathematician and legal thinker, so his kind words are very much appreciated and his criticisms are taken seriously. (3)

My goal in writing the book was to explore different aspects of the multifaceted relationship between mathematics and the Constitution. The Constitution itself contains many numbers, and the very heart of democracy, the concept of "majority rule", is an arithmetic concept. I wanted to examine the reasoning behind the framers' numerical choices--why 2/3 of the Senate is needed to ratify treaties; why slaves were counted as 3/5 of a "person"; and why we have two houses of Congress and 538 presidential electors. (4) I also wanted to explore the nature of logic, both the logic used in the presentation of legal arguments and the dangers that result from not questioning the fundamental postulates of one's own reasoning. On the whole, Professor Edelman has positive things to say about my approaches to these first two goals. (5)

My final goal was to explore how an understanding of various areas of modern mathematics could inform and improve our thinking about the constitution. This is the part of the book which receives Professor Edelman's strongest criticisms. He asserts that the use of what he terms "mathematical metaphor" is essentially a waste of time. As he put it, the more one knows about mathematics and the law, "the less persuasive these metaphors tend to be."

I suspect that Professor Edelman's background as a mathematician is preventing him from seeing that mathematics can trigger a non-mathematical imagination and create mental images that permit new ways of thinking about non-mathematical topics.

Mathematics is not simply a tool for resolving problems. Despite its reputation for being tedious, inaccessible, and boring, mathematics is actually a glorious way of thinking, with a deeply aesthetic quality. The poetry of mathematics can illuminate all manner of thought.

Consider, for example, this passage from Oliver Wendell Holmes, Sr.'s pre-Civil War essay, The Autocrat of the Breakfast Table: "All economical and practical wisdom is an extension of the following arithmetical formula: 2 + 2 = 4. Every philosophical proposition has the character of the expression a + b = c. We are mere operatives, empirics, and egotists until we learn to think in letters instead of figures." (6)

All thinkers, especially those engaged in legal analysis, can benefit from this admonition to reason abstractly, in the universal rather than the particular. Math does not serve as a mere "metaphor," but creates an effective means for reexamining one's thoughts.

Likewise, James Madison used a simple mathematical picture in Federalist No. 10, to describe how a national government minimizes the evils of majority factions:

  Extend the sphere, and you take in a greater variety of parties 
   and interests; you make it less probable that a majority of the 
   whole will have a common motive to invade the rights of 
   other citizens; or if such a common motive exists, it will be 
   more difficult for all who feel it to discover their own 
   strength, and to act in unison with each other. (7) 

Obviously, when Madison referred to a "sphere" he was not contemplating a literal geometric shape. Rather, he was creating a mental picture of a container increasing in size, to encompass a larger geographic area. Moreover, his basic point was inherently mathematical: The larger the voting population, the more difficult it is to maintain a permanent working majority.

I believe that what Laurence Tribe wrote about the legal import of modern physics holds for modern mathematics as well: "my conjecture is that the metaphors and intuitions that guide physicists can enrich our comprehension of social and legal issues. …