Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  What is the Leviathan? It is much more than an absolute ruler, or even an absolute state. It is the literal embodiment of all the members of the commonwealth in one man: the sovereign. In their desperation to escape the state of nature, men conclude that the only way out is for each of them to give up his own free will and invest it in the sovereign. The sovereign consequently absorbs the individual wills of all the members of the commonwealth, and his actions are therefore their actions. This is the key. Men do not simply submit to the will of an overlord, subjugating their own will to his. Rather, whatever the sovereign wills is also the will of every one of his subjects. Every person in the commonwealth, Hobbes argues, will “own and acknowledge himself to be the author” of whatever the sovereign chooses to do. Under the Leviathan there can be no civil war, because the Leviathan embodies the wills of his subjects, and no one would will civil war. The end result is a perfectly unified body politic: “[It] is more than consent, or concord: it is a real unity of them all, in one and the same person.”

  “The multitude so united is called a COMMONWEALTH,” Hobbes writes:

  This is the generation of that great LEVIATHAN, or rather (to speak more reverently) of that Mortal God to which we owe, under the Immortal God, our peace and defence. For by this authority, given him by every particular man in the commonwealth, he hath the use of so much power … that by terror thereof he is enabled to conform the wills of them all to peace …

  And that, to Hobbes, is the very essence of the Leviathan: “one person, of whose acts a great multitude … have made themselves every one the author” in order that peace will prevail.

  Hobbes’s theory of the state is breathtaking in its audacity. He has no interest in discussing the different forces operating in human society or evaluating the different forms of political organization. Instead, with no qualifications or equivocation, he plunges ahead in a take-no-prisoners philosophical style. The problem of human society, he claims, is clear, and it is the perpetual war that exists in the state of nature. The solution is just as clear: the creation of the absolutist “Leviathan” state. Hobbes drives through his argument by sheer intellectual force, moving step by logical step, and leaving no room for dissent or contradiction: human nature leads to the state of nature, which leads to civil war, which leads to surrender of personal will, which leads to the Leviathan. Consequently, the Leviathan is the only viable political order. QED.

  From the very beginning, many found Hobbes’s Leviathan state abhorrent. Where would Parliament fit in? Where would the Anglican Church—or any church, for that matter? But even those critics who were repelled by his conclusions were hard pressed to find flaws in his arguments. For where exactly was Hobbes’s error? His assumptions were sound, and each step seemed reasonable in itself: Yes, humans are acquisitive and self-interested. Yes, they compete with and fear one another. Yes, they are prone to attacking each other out of fear, and one attack leads inevitably to more. It all seems oh so reasonable, and few would be inclined to argue with each particular step. By the time a reader realizes where all this is leading, it is too late. Somehow, without ever taking a false or even dangerous-looking step, the reader unwittingly concedes that the only viable state is Hobbes’s “living God,” the Leviathan.

  To many of Hobbes’s contemporaries, this was a completely repellent conclusion, but such is the power of Leviathan’s reasoning that it proved extraordinarily difficult to point out where exactly it goes astray. Hobbes followed his deductions to their logical conclusions, whatever those might be, and carried his readers along for the ride. It was as if he were conducting a geometrical demonstration.

  The Leviathan, composed of innumerable individuals united in a single will, is, to be sure, a beautiful thing. But bold as it is, and beautiful as it is, the Leviathan as a political organization is bound to give one pause. It is not just a powerful and centralized state as existed in Hobbes’s time in France, where political opposition was difficult and state measures repressive. It is, rather, a state in which political opposition is literally impossible. Opposition to the sovereign by his subjects means that they willfully oppose their own will—a paradox and a logical impossibility. Indeed, in the Leviathan the subjects do not have the same relationship to the state as we understand it, because the Leviathan is not a political organization but a unified organic whole. It is a living being composed of the bodies of all its subjects, and a will entrusted to the sovereign alone. Hobbes says as much when he explains in the introduction that “the great LEVIATHAN, called a COMMONWEALTH … is but an artificial man, though of greater stature and strength than the natural.” In a human body, a hand, or a foot, or a follicle of hair cannot oppose a human’s will. In just the same way, the members of the commonwealth are simply components of the body of the commonwealth, and are incapable of opposing its will.

  Nothing captures the true essence of the Hobbesian state better than the image that adorned the frontispiece of the early editions of Leviathan (and many later ones as well). Engraved by the French artist Abraham Bosse, it shows a peaceful land of hills and valleys, fields and villages, with a prosperous and orderly town in the foreground, where small and neatly arranged houses are dwarfed by a great Church. The eye, however, does not dwell on this peaceful scene, but is drawn to the figure that looms behind it: a giant king who towers over the land like Gulliver among the Lilliputians, his arms spread wide, as if to embrace his domain. On his head is a crown; in his left hand, a bishop’s crosier, or staff; in his right, a sword to rule the land and defend it from all enemies. He dominates the land, and there is no question that it was he who brought it peace, order, and prosperity.

  At a glance, the image seems like an advertisement for the virtues of a strong centralized monarchy on the French model. But there is something strange about this towering king. His body appears rugged, and he seems to be wearing some sort of scale armor. A closer inspection reveals the truth: they are not scales, but people. What appears to be the king’s human body is in fact composed of the men of the commonwealth. Each single man is powerless, nothing but a minuscule component of this enormous body. But together, and working with a single will, they are the all-powerful Leviathan.

  So overpowering and all-encompassing is the state depicted in the frontispiece that it leaves no room for any independent institutions. For Hobbes, any institutions that are not directly dependent on the sovereign are a threat to the unity of the Leviathan and the stability of the state, a threat that, if not checked, will breed disagreements and conflicts and lead, once again, to civil war. The worst offender in Hobbes’s book is the Catholic Church, which claimed ascendancy over all civic authorities and consequently earns itself an entire section in the Leviathan, entitled “The Kingdom of Darkness.” In England, the rebellious Parliament is, of course, anathema to Hobbes, but so are seemingly tame institutions such as the Anglican Church and the universities of Oxford and Cambridge. Anglicanism is preferable to most churches because it is at least nominally subject to the king, even if, in Hobbes’s opinion, Anglican clerics showed far too much independence. Other denominations, and especially Presbyterianism, are far worse, because they set up their own rule separate from the commonwealth, and Hobbes is not above blaming them directly for the onset of the civil war. The universities earn Hobbes’s ire partly because they traced their intellectual roots to medieval Scholasticism, and so are tainted by association with the “Kingdom of Darkness.” But more fundamentally, the universities seem to Hobbes dangerous breeding grounds for doctrines and ideas that might conflict with the will of the sovereign, leading to open-ended controversies. And controversy, the Earl of Newcastle, Hobbes’s protector, warned Charles II some years after he was restored to the throne, “is a Civill Warr with the Pen, which pulls out the sorde soon afterwards.”

  Deciding which opinions and doctrines should be taught and which should be banned, in the universities and elsewhere, is the prerogative of the sovereign alone, Hobbes insists. If clergymen are allowed to preach w
hat they want, and professors to teach what they want, then division, conflict, and civil war will soon follow. But Hobbes goes even further: the Leviathan decides not just which teachings are harmful to the state and which beneficial, but more fundamentally, what is right and what is wrong. In the state of nature, there is no right and wrong, according to Hobbes, since every man acts as best he can to secure his own interests. The notions of right and wrong arrive on the scene only with the Leviathan, and the standard is simple: “the law, which is the will and appetite of the state, is the measure,” and nothing else. Following the law laid down by the sovereign is right, breaking it is wrong, and that’s all there is to it. Anyone who appeals to other sources of authority, such as God, tradition, or ancient rights, is undermining the unity of the commonwealth, and likely plotting against it to benefit his own interests. Right and wrong, good and evil—all are in the hands of the sovereign.

  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.