Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (33 page)

The Leviathan. From the frontispiece to the 1651 edition of
Leviathan
. “Non est potestas super terram quae comparetur ei,” declares the text surrounding the crown: “There is no power upon the Earth that compares to him.”
(British Library Board / Robana / Art Resource, NY)

As an exercise in political theory,
Leviathan
is as bold a tract as ever was, and rightly deserves its high place in the annals of Western thought. But does Hobbes’s prescription, for all its brilliance, describe any actual historical state? There have been, no doubt, regimes that strove for the ideal, most infamously the mighty totalitarian states of the twentieth century, among them Hitler’s Germany and Stalin’s Soviet Union. These, too, were states in which the people were (in principle, at least) united in the person of their leader, whose will spoke for the entire nation. Like the Leviathan, they, too, tolerated no dissent, considering it an offense to the national will (in Germany) or the forward march of the proletariat (in Russia). But even these dark regimes of the last century never achieved what Hobbes had in mind. For one thing, they both had to deal with real dissent, however sporadic or muted, whereas Hobbes virtually defined dissent out of existence. On a deeper level, both the German Nazis and the Soviet Communists saw themselves as standing for some higher ideal, whether the mystical destiny of the German nation or the world revolution that would lead to a classless utopia. Their totalitarian states were a means to accomplish these higher goals.

The Hobbesian Leviathan is something else: No higher truths—religious, mystical, or ideological—exist outside the law laid down by the sovereign, and anyone who claims that there are such truths is designated an enemy of the state. The Leviathan is the be-all and end-all, and exists for itself alone. As a bulwark against chaos, it is absolutely necessary, but it stands for nothing, and steadfastly denies that any higher ideals or purposes even exist. All that counts are the state, the sovereign, and his will.

Hobbes’s proposed solution to the crisis of the 1640s made his reputation as an original thinker, but it also put him at odds with his contemporaries—not only the Parliamentarians, whom he despised, but also the Royalists, whom he intended to support. There was, to be sure, much in
Leviathan
that appealed to the king’s men, most particularly the insistence on a strong central government led (preferably) by one man, and the condemnation of Parliamentarians and all dissenters as traitors to the commonwealth pursuing their own selfish interests. But there was also much in the book to trouble dedicated Royalists. A legitimate king, after all, rules his lands and people by divine right; he is God’s chosen, and can be replaced by no other. The Leviathan, in contrast, rules not by right, but as a practical necessity to prevent civil war. Anyone, in principle, can fill this role, as long as he is capable of defending the land and preserving the peace. If a king fails to do this, then he could in principle be replaced by someone else—as in fact happened in England. Some Royalists became so concerned about this that they accused Hobbes of having written
Leviathan
in support of Cromwell’s Protectorate, not of the king. This was not true—
Leviathan
was published two years before Cromwell became lord protector—but it was not unreasonable:
Leviathan
is in truth not a defense of legitimate monarchy, but an argument for a totalitarian dictatorship. Add to that the fact that
Leviathan
managed to offend all clergymen, both Anglicans and the court in exile’s French-Catholic hosts, and it becomes clear why this supposedly Royalist tract did not endear Hobbes to the king’s followers. Instead of being celebrated as a Royalist standard-bearer, he was relieved of his tutoring duties, banished from court, and soon found himself, ironically, back in revolutionary England.

Much to Hobbes’s disappointment, there seemed to be no takers for
Leviathan
’s extreme prescription for ending the civil war. But Hobbes knew he was right, regardless of what others thought. All that was needed to overcome their skepticism, he concluded, was to spell out his entire philosophy more clearly and comprehensively. Once he explained, step by step, precisely how he had arrived at his conclusions, the doubters, he was sure, would have no choice but to accept his political prescriptions. Now, Hobbes had never intended his political tracts
De cive
and
Leviathan
to stand alone. As conceived during his long, silent decades on the Cavendish estates, they were supposed to be the final part of a complete philosophy that would cover all facets of existence. The political crisis caused Hobbes to write the last part first and rush it to publication, in the hope (soon disappointed) that it would end the war and restore the king. Now that he was back in England, and working to buttress his case, he decided to go back and write the first two parts.

In 1658, Hobbes published
De homine
, a tract on human nature that confirmed his reputation for misanthropy but, compared to
Leviathan
, received little attention. His core principles, however, were laid down in
De corpore
(“On Matter”), which came out three years earlier.
De corpore
is a dense, technical book aimed at professional philosophers and is concerned with such abstruse matters as whether universals truly exist and whether matter is nothing more than extension. It has none of the lively imagery or fiery rhetoric of
Leviathan
, and probably had only a fraction of its readership. Nevertheless, it was
De corpore
, not
Leviathan
, that embroiled Hobbes in a personal war that lasted for the rest of his life. For, even before the book saw light, a man determined to be his enemy was secretly obtaining the unpublished text from Hobbes’s printer and preparing a devastating response. This man was not a rival philosopher waiting to challenge Hobbes on some technical definition of matter or motion. His name was John Wallis, and he was a mathematician.

7

Thomas Hobbes, Geometer

IN LOVE WITH GEOMETRY

Hobbes’s introduction to mathematics is the stuff of legend. Having studied no mathematics at Oxford, he encountered it quite accidentally at age forty, while visiting Geneva with one of his young charges. “Being in a gentleman’s library,” his biographer Aubrey reports, “Euclid’s
Elements
lay open, and ’twas the 47 El. Libri I” (i.e., theorem number 47 in the first book of
The Elements
). This, as anyone educated in classical mathematics knew, was the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two legs. Hobbes read the proposition. “‘By God’ sayd he, ‘this is impossible!’ So he reads the demonstration of it, which referred him back to such a proposition,” which in turn referred him to an earlier one, and so on, “that at last he was demonstratively convinced of that trueth.” This, according to Aubrey, “made him in love with geometry.”

In the years that followed, Hobbes worked hard to make up for his late start in the field. By the 1640s he was in regular contact with some of the leading mathematicians of the day, including Descartes, Roberval, and Fermat, and when the English geometer John Pell fell into dispute with his Danish colleague Longomontanus, he considered Hobbes enough of an authority to seek his public support. When Descartes died some years later, the French courtier Samuel Sorbière hailed him as one of the world’s greatest mathematicians, along with “Roberval, Bonnel, Hobbes, and Fermat.” Sorbière, it must be said, was Hobbes’s good friend, and his high opinion of the Englishman’s mathematical talents may not have been shared by all. His evaluation does nevertheless show that when Hobbes was made mathematical tutor to the exiled Prince of Wales, he was one of the most respected English mathematicians of the day.

Why was Hobbes so taken with mathematics? Aubrey’s tale provides a crucial clue here: in geometry, each result is built on another, simpler one, so one can proceed, step upon logical step, starting with self-evident truths and moving to ever-more-complex ones. By the time a reader reaches some truly unexpected results, such as the Pythagorean theorem, he is “demonstratively convinced of that trueth.” This, to Hobbes, was an amazing accomplishment: here was a science that could actually prove its results, leaving not a shadow of a doubt about their veracity. Consequently, he considered it “the only science it hath pleased God hitherto to bestow on Mankind,” and the proper model for all other sciences. All sciences, he believed, should proceed like geometry, since “there can be no certainty of the last conclusion, without a certainty of all those affirmations and negations on which it was grounded and inferred.” No field but geometry, Hobbes noted, has as yet achieved the required level of systematic certainty, but this was about to change: now that Hobbes had arrived upon the scene, he was ready to provide the true philosophy, which would be structured as systematically as geometry, and whose results would be just as certain.

Over the more than four centuries since his birth, Hobbes has been accused of many things: during his lifetime he was accused (probably falsely) of living a dissolute, immoral life and (probably correctly) of being an atheist and promoting irreligion. In later days he has been accused of an unjustifiably grim view of human nature, and of serving as inspiration to some of the most oppressive regimes in human history. But no one, in all this time, has ever accused Hobbes of excessive modesty. Indeed, when Hobbes came to present his new geometrically inspired philosophical system, there was not a trace of this lamentable trait. To the contrary, his texts simmer with brashness and provocation.

For most of its history, Hobbes explains in the dedication of
De corpore
to his Cavendish patron, the Earl of Devonshire, the world had known almost no philosophy. True, the ancients made great advances in geometry, and more recently there had been some important steps forward in natural philosophy, thanks to the work of Copernicus, Galileo, Kepler, and several others. As for the rest of philosophy, from Plato and Aristotle to the present, it was worse than useless. “There walked in old Greece a certain phantasm,” he wrote, “… full within of fraud and filth, a little like philosophy,” which some people mistook for the real thing. Instead of teaching the truth, this pseudo-philosophy taught people to disagree and dispute, and all just so that the supposed “philosophers” would be lavishly compensated. The worst of it was “school divinity,” the medieval Aristotelianism taught at the universities. This, Hobbes charges, was a “pernicious philosophy that hath raised an infinite number of controversies … and from those controversies, wars.” Hobbes calls this abomination “Empusa,” the Greek monster with one leg of bronze, the other of an ass, which was a harbinger of ill fortune.

Hobbes was about to change all that. Natural philosophy, he explains, may be young, dating no further back than Copernicus, but civil philosophy was even younger, according to Hobbes, being “no older than my own book …
de Cive
.” In that book, for the first time, he had used unchallengeable reasoning to prove that all authority in the state, whether religious or civic, must derive from the sovereign alone. Now, in
De corpore
, he would complete the job: he would set down the true philosophy that would replace the fake and pernicious ones, and finally conquer the monster Empusa. For Hobbes, his philosophy was not a contribution to an ongoing conversation that had lasted thousands of years: it was, rather, a philosophy to end all philosophies, the one and true doctrine that would put an end to all discussion and debate. His book, he wrote, may be short, but it is nothing short of great, “if men count well as great.” His critics, of whom there were many, might dispute this, but Hobbes cared little for them. They were simply envious of the work, and he, after all, was not “striving to appease,” he noted with perfect candor. The brilliance of
De corpore
would vanquish them: I will “revenge myself of envy by encreasing it,” he announced, without a hint of irony.

How would Hobbes’s new philosophy conquer Empusa? The answer was simple: through geometry. The reason that the so-called philosophers of the past had failed was that they relied on flawed and inconclusive methods of reasoning. They taught dispute instead of wisdom, Hobbes charges, and they “determine[d] every question according to their own fancies.” As a result, instead of bringing peace and unanimity, they fostered strife and civil war. Geometry, in contrast, compelled agreement: “For who is so stupid as both to make a mistake in geometry and also to persist in it, when another detects his error to him?” Consequently, geometry produces peace rather than discord, and Hobbes’s philosophy would follow its lead.
De corpore
, he explains in the dedication, was written for “the attentive readers versed in the demonstrations of mathematicians,” and some parts of it were written “to geometricians alone.” But the implications of the geometric method extended to all fields: “Physics, ethics, and politics, if they are well demonstrated, are no less certain than the pronouncements of mathematics.” If one but follows the clear and indisputable method of geometrical reasoning, he will without trouble “fright and drive away this metaphysical Empusa.”