A Tour of the Calculus of Justice

A TOUR OF THE CALCULUS. By David Berlinski. New York, New York: Vintage Books, 1995. Pp. 331. $14.00. The problems are solved, not by giving new information, but by arranging what we have always known. Ludwig Wittgenstein1 I. David Berlinski's A Tour of the Calculus2 has been called "a love letter, or song of praise, that makes difficult theory seem elegant, seductive and desirable."3 It is that, but it is also much more. Berlinski reveals, stepby-step, the manner in which calculus, through a system of numbers and equations, gives us the power to represent time and space in a real world.4 If anything, the author is too modest. Once understood, calculus does more than describe time and space. It allows us to see the past and predict the future. Using calculus, we can determine where things have been and we can discover where they are going. Needless to say, this is what trial lawyers do as well. Just as calculus is an intellectual tool that uses numerical notation to symbolize time and space, so is the common law trial a device to recreate what happened, where, and when. As I will explain shortly, a mathematical function draws a picture of events in a way that allows them to be disassembled and reassembled until they make useful sense. A trial does the same thing, using rules of procedure and evidence in the place of numbers. II. As every schoolchild no longer knows, the calculus was devised simultaneously and independently by Sir Isaac Newton and Gotfried Liebnitz in the winter of 1665-66. Such was the pace of academics in the Seventeenth Century that it was not until 1687 that Newton finally got around to publishing his Philosophiae Naturalis Principia Mathematica, commonly known as the Principia, and then only at the urging of the astronomer Edmund Halley. It is far less known, though pretty evident once you think about it, that the calculus of Newton and Liebnitz was only a beginning, just the first pass at the fully developed discipline that we know today. As Berlinski puts it, "what Newton and Liebnitz saw in the harsh but thoroughly unfocused dare of their own genius, they never clearly or completely understood."5 In this sense, the calculus is again similar to the jury system, which has been refined and reinvented-if never quite perfected--constantly in the many centuries since its origin. One of Berlinski's great contributions, then, is to convey an understanding of the historical development of the calculus. At every stage he shows how ideas rose and fell, how theories were polished and sometimes replaced, how new thoughts led to newer ones. One good example of this progression is the concept of instantaneous velocity. Speed seems obvious. If you know how long it took to get somewhere, then you can figure out how fast you were going. But this understanding of speed really only yields an average. It compresses a series of speeds into a single number, thereby blurring acceleration and deceleration, rates of change, and even the occasional frolic or detour. It is no trick to measure speed over a defined interval. But instantaneous speed-speed at a single moment, with no passage of time now there is another question entirely. Here is the conundrum, what Berlinski sees as the paradox of speed.6 In the pre-calculus conception, speed is determined through the division of distance by time. The standard notation, therefore, is III. There are two lessons here for law and lawyers. First, systems of thought are always imperfect, though they mature and develop over time. Today, at the end of the twentieth century, it seems achingly evident that Liebnitz was barking up the wrong tree in his search for infinitesimal numbers. How can a number that isn't zero be zero?15 But neither Liebnitz nor Newton, geniuses for all ages, ever came to this realization. It fell to their successors-mortals standing on the shoulders of giants-to make the necessary breakthroughs. …