Full Title: "Vitriolic, Spleen, and Invective: the Character and Background of an Academic Debate in the 17th Century."

A book review by Jef Raskin, published in: The European Legacy, Journal of the International Society for the Study of European Ideas.

Squaring the Circle: the war between Hobbes and Wallis  by Douglas M. Jesseph (Chicago: The University of Chicago Press, 1999) xiv + 419 pp. US $80 cloth, US $28 paper  ISBN 0-226-39899-4 (cloth)  ISBN 0-226-39900-1 (paper)

Squaring the Circle had captured my attention by the first sentence of the preface: "This book began as a footnote." I am a great lover of scholarly footnotes,  excursions where the author digs more deeply, shares an insight, provides context, comments obliquely, introduces extra evidence, or adds a wry touch of appropriately academic humor.  (see my parody on academic footnotes: An History of the Yarmulke) Squaring the Circle, is, indeed, an extended footnote to a historical dead-end: Thomas Hobbes's attempts to reformulate mathematics and to solve two of the three classic problems posed by the ancient Greeks.

The cast of characters is full of names familiar to those who know the history of the mathematics, science, and philosophy of 17th century Europe: Descartes, Galileo, Mersenne, Boyle, Cavalieri, Fermat,  Hooke, Huygens,  and Snell, to mention a few who haunt technical texts to this day. Hobbes interacted with them all. The scenes are played out against the dramatic backdrop of the Reformation,  and of men scrambling to keep their footing in the shifting sands of loyalties from the monarchy and Charles I,  to Cromwell, and then back to the monarchy and Charles II. The English Civil War was a time when religious belief was prescribed by the state and atheism was a punishable crime, of which Hobbes was often accused (and of which he accused his opponents). And much was outside the realm of scientific inquiry: "[Q]uestions of the magnitude and origin of the world are ... not to be settled by philosophers, but by those who are lawfully responsible [the English King, for example] for regulating the worship of God", said Hobbes, although here his interest is more political than philosophical.

Science has gained scope but lost flavor over the years, we no longer indulge as Wallis did in publishing "Due Correction for Mr. Hobbes; or Schoole Discipline, for not saying his Lessons right" and as Hobbes did replying with "Markes of the Absurd Geometry, Rural Language, Scottish Church-Politicks, And Barbarismes of John Wallis Professor of Geometry and Doctor of Divinity." Jesseph revives some of these old writings, so that we can see how it was done. To find anything today that descends to the vulgarities and scatological references they used, one has to go to the movies. I am sure that feelings run as deeply now as then, but we have purged our language and concealed the daggers we once carried openly at our sides.

At first, I thought that Squaring the Circle was about a mathematical dispute. It begins there, but it most certainly does not end there. You do not need a mathematical background to understand the book, though if mathematics comes easily to you, some details will be more vivid. One of the delights of this book is to be reminded of how the goals and methods of mathematics have changed.  Hobbes spent decades arguing that points without extent and lines without width were absurd, and tried to base his mathematics on a more physical foundation. Wallis argued that Euclid's definitions were cogent. A hundred years later, Kant could still argue the absolute necessity of Euclidean geometry. Now we take "line" and "point" as undefined, rule that our axioms are arbitrary (so long as they are consistent and lead to interesting results) and accept non-Euclidean geometries even in our descriptions of the physical world. While the old theorems remain true, mathematics itself has changed.

After coming to grips with the intertwining of politics, personality, religion, and philosophy, and reading on, I came to see the book as a tragedy: Hobbes's decline from respected scientist, geometer, and influential writer to a mathematical non-entity, confirmed in his folly, beyond logic and the help of his friends. The tale has a moral: to have lost the ability to say "I am wrong" means that you have also lost the ability to learn. Jesseph does not attempt to glorify Hobbes, but uses Hobbes's failures to put the successes of the time in relief. These were the days before Newton and Leibniz and, with hindsight, you can feel the calculus struggling to be born as Hobbes and his contemporaries flail at problems that are now regarded as trivial. The impossibility of "Squaring the Circle" (constructing, with compass and straightedge, a square of the same area as that of a given circle) was not to be demonstrated for another 200 years, and an understanding of the foundations of calculus and the ability to handle infinitesimals and infinity in a mathematical contexts was also far in the future.

Some of the disputes, such as where Hobbes argues for the primacy of geometry over arithmetic (he was eventually to regard all algebraic approaches to proof as inherently suspect) while Wallis argues that arithmetic has priority, now seem so pointless that it can be difficult to understand why they argued at all. But, as Jesseph said of Wallis, "There were few, if any, in England who had more reason to fear and detest Hobbes and his teachings." They disagreed on nearly every point, a basic one being whether sovereigns should hold religious authority or must bow to it. No compass can swing that arc, nor any straightedge show the way. Wallis might well have ignored Hobbes puerile mathematics if it were not a convenient tool for tarring Hobbes's reputation.

In the last chapter the book takes yet another turn. Here Jesseph takes up the cudgel himself and indulges in what I judge to be his real aim in writing the book: to state his case in a current academic dispute. If he does not do this as colorfully as was done in the 17th century, he does it with equal vigor. The last chapter is titled "Persistence in Error; Why was Hobbes So Resolutely Wrong." But only one and a half leaves are truly devoted to that theme. The greater part of the chapter argues against "the thesis that scientific controversy is nothing more than a cover for more fundamental differences of opinion on how best to achieve social order" and disputes the idea that mathematical and scientific correctness are contingent on having the backing of "the most, and the most powerful, allies." This last quote is from Shapin and Schaffer's book Leviathan and the Air-Pump (Princeton 1985) which Jesseph likes about as well as Hobbes liked Wallis (and vice versa). Instead of "shall the state have preeminence over religion", the question is "shall sociology have preeminence over mathematics."

If at times the book has been a bit of a slog through historical and mathematical minutiae (made accessible by Jesseph's clear organization and excellent writing style), here we are in contemporary debate, and the turbocharger has kicked in. Jesseph makes his case well, but you might discount my saying so because I was on his side long before I read this book.

The postmodernist view that political power has absolute sway over scientific correctness, like the ultrafeminist claim that a different set of theorems would have been true had most mathematicians been female, ignores the indisputable fact that our physics and mathematics work; the former is grounded in what is and the latter is based on the logic we have inherited from the world (see my essay "The Reasonable Effectiveness of Mathematics"). The kind of non-trivial changes in science that sociological factors might induce would, if applied, cause technological failure and thus expose themselves as false. Had Hobbes all the political backing in the world, his mathematics would have failed. His proofs were wrong, pi is not 3.2 (this startling result is a consequence of one of his circle squaring attempts, as Wallis was delighted to point out). No legislature can change the ratio of the circumference to the diameter of a circle. That the inverse square law holds for the brightness of a light seen at a distance is not subject to the dictates of a cabal, however powerful.

Instead of a book on a dispute over Hobbes's mathematics, which is what I thought I had in hand (and which would have been sufficient to make a good book), I find in addition a fascinating tangle of many threads of the intellectual life and political times of England and Europe in the mid-1600s, and an ending with a twist that reflects on a current academic debate. The dose of Latin and iota of Greek I absorbed in high school proved handy, especially in reading untranslated titles, but I would not have lost much without that. Squaring the Circle offers interesting discussion, lively characters, compelling analysis, and solid research so that while not a book for everyone, it should have considerably wider appeal than the mathematical audience for whom I originally assumed it was written.