A Tour of Mistakes

Abstract

In these pages,' Steven Lubet recently reviewed A Tour of Calculus, by David Berlinski.2 Inspired by both beauty of calculus and Berlinski's description of it, Lubet waxes poetic on many parallels between law and calculus. It is completely understandable--even admirablethat one might be led to ruminations on relationship between calculus and one's own discipline. There is little doubt that subject of calculus stands as one of great intellectual feats of Western thought. It has had profound implications for physics, engineering, economics and many other disciplines-so why not law? Alas, these philosophical musings would be more persuasive had Professor Lubet better understood what it was that he was writing about. Lubet's errors come in two types. first is just a misunderstanding of history, but it is a misunderstanding that unfortunately forms basis for an entire section of his review. second type of error is more fundamentally mathematical: he does not distinguish between a definition and a theorem. Just as Lubet draws legal lessons from calculus, we can draw legal parallels from his mistakes. While some of these might be comforting, others will be more unsettling. As noted by Lubet, development of calculus was done more or less simultaneously in mid-17th century by Sir Isaac Newton and Gottfried Wilhelm Leibniz. Leibniz based much of his development of subject on idea of an infinitesimal, a class of numbers that are smaller than any other number. According to Lubet, the `infinitesimals' turn out to be a futile fiction, notwithstanding Liebnitz's [sic] own endorsement of them. In 1734, Bishop Berkeley that they do not and cannot exist.3 Lubet goes on in Part III to draw a number of legal parallels to this discrediting of idea of infinitesimals. While legal conclusions he draws from these events may well be true, Lubet cannot base them on invalidity of infinitesimals: fact of matter is that Leibniz was right. To be fair to Lubet, ultimate vindication of Leibniz's belief in infinitesimals is hidden in a footnote by Berlinski: The development of [non-Archimedean] fields by logician Abraham Robinson in twentieth century has made possible development of calculus entirely along lines anticipated by Leibnitz [sic].4 Nevertheless, anyone with serious mathematical training would not have needed Berlinski's footnote; Lubet's error highlights danger of relying on secondhand knowledge of a field quite different from one's own. Moreover, culpability aside, Lubet has lost foundation for legal insights he draws from purported invalidity of infinitesimals. And what of supposed proof' of Bishop Berkeley? Berlinski writes that [w]riting in 1734, Bishop Berkeley wasted no time in attacking very idea of infinitesimals, and later says that [1]ooking backward, we can see that Berkeley was entirely correct,5 but never claims that Bishop Berkeley proved conclusively anything about existence of infinitesimals. Indeed, he couldn't have, since by appropriately generalizing idea of a number, Abraham Robinson was able to define them. Lubet should be more careful in using term proof' in context of mathematics. Lubet sees more parallels between computation of area under a curve and way that legal trials proceed by means of accretion of detail.6 Surprisingly, Lubet doesn't draw obvious parallel, that just as sum of more and more rectangles gives better and better approximations for area under a curve, as a trial proceeds evidence presented gives a better and better approximation of truth. He instead focuses on error in mathematical approximation: An integral combines rectangles until limit of error approaches zero, but error-zone never actually becomes zero. …