Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (34 page)

In his own work, Hobbes believed he fully followed the geometrical example: his philosophy (once all its parts were published) begins with simple definitions in
De corpore
, just as Euclid’s
Elements
begins with definitions and postulates. And just as
The Elements
moves from the simple and self-evident to the complex and surprising, so Hobbes’s opus proceeds through its three sections:
De corpore
(“On Matter”),
De homine
(“On Man”), and
De cive
(“On the Citizen”). From a discussion of definitions (which he calls “names”), he proceeds to the nature of space, matter, magnitudes, motion, physics, astronomy, and so on. Finally, at the end of this long chain of reasoning, he reaches the most complex and the most urgent topic of all, the one that justifies the entire enterprise: the theory of the commonwealth. Certainly there were those who disputed whether he had succeeded in living up to the geometrical standard, but Hobbes paid them no mind. His systematic and careful reasoning from first definitions, he was convinced, ensured that his conclusions about the proper organization of the state were absolutely certain. As certain, in fact, as Euclid’s Pythagorean theorem.

THE GEOMETRICAL STATE

If Hobbes’s entire philosophy was structured as a grand geometrical edifice, this was particularly true of his political theory. This was because the commonwealth shared a fundamental feature with geometry: both were created entirely by humans, and therefore were fully and completely known to humans. “Geometry is … demonstrable, for the lines and figures from which we reason are drawn and described by ourselves; and civil philosophy is demonstrable, because we make the commonwealth ourselves.” Our knowledge of how to create the ideal state is perfect, Hobbes claims, just like our knowledge of geometrical truth. In
Leviathan
, Hobbes put this principle into practice, creating what he believed was a perfectly logical political theory whose conclusions were in all respects as certain as geometrical theorems.

It wasn’t just the broad principles of the commonwealth that possessed the certainty of geometrical demonstrations. The actual laws established by the Leviathan to govern the state also had the inescapable logical force of a geometrical theorem, and were as indisputably correct. As Hobbes puts it, “the skill of making and maintaining commonwealths consisteth in certain rules, as does arithmetic and geometry.” This is because the laws themselves define what is right and true, and what is wrong and false. Before the commonwealth, in the state of nature, the terms
right
and
wrong
or
true
and
false
were empty words that referred to nothing. There was no justice or injustice, right or wrong, in the state of nature. But once men gave up their personal will to the great Leviathan, he laid down the law and gave the terms meaning:
right
is following the law;
wrong
is breaking it. Anyone charging that the decrees of the sovereign are “wrong” and should be changed is speaking nonsense, since what is “wrong” is defined by the decrees themselves. Opposing a law is as absurd as denying a geometrical definition.

Hobbes, of course, was far from the only early modern intellectual to idealize geometry. Only a few decades earlier, Clavius, too, had extolled the virtues of geometry, promising his fellow Jesuits that it would be a powerful weapon in the struggle against Protestantism. But apart from their admiration of geometry, Clavius and Hobbes had almost nothing in common. Clavius was a Jesuit scholar trained in Aristotelian philosophy and the methods of Scholastic disputation, which were embraced by his order and perfected at the Collegio Romano. He was also a Counter-Reformation warrior who fought to spread the word of God and bring about a Catholic spiritual awakening, and he abhorred Protestants, materialists, and heretics of all kinds. His life’s ambition was to establish the Kingdom of God on Earth, which to him meant setting the Pope above all secular rulers, and the Church above all civic institutions. Hobbes, in contrast, had nothing but scorn for Scholastic disputation, believed that
spirit
was a meaningless term, and that only matter and motion existed in the world. The sole purpose of terms such as
spirit
and
immortal soul
was to allow unscrupulous and corrupt clergymen to frighten men and subject them to their will. Finally, the notion that the Pope would rule over kings was intolerable to Hobbes. Any infringement on the absolute power of the civil sovereign would lead to disagreements, divisions, and, inevitably, civil war.

Clavius died in 1612, long before Hobbes published his tracts and almost certainly without ever having heard the Englishman’s name. But if he had had a chance to read any of Hobbes’s works—
De cive
,
De corpore
,
Leviathan, De homine—
there is no question what his reaction would have been. To a devout Jesuit such as Clavius, Hobbes was a godless materialist and a heretic, an enemy of the Catholic Church whose books should be banned. If Hobbes had ever been so unfortunate as to fall into the hands of Clavius and his brethren, he would have been lucky to escape the stake. Meanwhile, Hobbes’s verdict on the Jesuits was no less harsh: their goal, he argued, was to scare men and “fright them from obeying the laws of their country.” This, as far as Hobbes was concerned, was true of all clergymen, but he reserved special scorn for the Catholic Church. The Jesuit dream of a universal and all-powerful Church, ruled by the Pope, was to Hobbes the darkest of nightmares.

Only on the role of geometry were these two natural enemies in perfect agreement. Euclidean geometry, Clavius believed, was a model of correct logical reasoning, which would ensure the triumph of the Roman Church and the establishment of a universal Christian kingdom on earth, with the Pope at its apex. Hobbes’s Leviathan state was, in many ways, the precise opposite of the Jesuits’ Christian kingdom: it was ruled by a civic magistrate who embodied the will of the people, not by the Pope, who derived his authority from God; its laws were derived from the Leviathan’s will, not from divine or scriptural injunctions, and the Leviathan would never tolerate any clerical infringement on his absolute powers. But in their deep structure, the Jesuit papal kingdom and the Hobbesian commonwealth are strikingly similar. Both are hierarchical, absolutist states where the will of the ruler, whether Pope or Leviathan, is the law. Both deny the legitimacy, or even the possibility, of dissent, and each assigns to each person a fixed and unalterable place in the order of the state. Finally, both rely on the same intellectual scaffolding to guarantee their fixed hierarchy and eternal stability: Euclidean geometry.

Today, Euclidean geometry is just one narrowly defined area of mathematics, albeit one with an extraordinarily long and impressive pedigree. Not only is it just one among many mathematical fields, it is also, since the nineteenth century, just one among an infinite number of geometries. It is taught to high school students today partly because of tradition and partly because it is thought to impart the powerful method of rigorous deductive reasoning. Beyond that, it is of little interest to practicing mathematicians. But things were very different in the early modern world, when Euclidean geometry was viewed by many as one of the towering achievements of humanity, the unassailable bastion of reason itself. To Clavius, Hobbes, and their contemporaries, it seemed natural that geometry would have implications far broader than for objects such as triangles and circles. As the science of reason, it should apply to any field in which chaos threatened to eclipse order: religion, politics, and society, all of which were in a state of profound disarray in this period. All that was needed was to use its methods in the afflicted fields, and peace and order would replace chaos and strife.

Euclidean geometry thus came to be associated with a particular form of social and political organization, which both Hobbes and the Jesuits strived for: rigid, unchanging, hierarchical, and encompassing all aspects of life. To us, who can look back on the rise and fall of bloody totalitarian regimes in recent centuries, it is a chilling, repellent vision. But at the dawn of the modern age, with the old medieval world in shambles and nothing to replace it, perspectives were different. To Clavius, to Hobbes, and to many others, it seemed that the answer to uncertainty and chaos was absolute certainty and eternal order. And the key to both, they believed, was geometry.

THE PROBLEM THAT WOULD NOT BE SOLVED

Beautiful and powerful as it was, Euclidean geometry was not free from flaws, as Hobbes discovered to his dismay when he began studying the subject in depth in the years after his encounter with the Pythagorean theorem. The difficulty was that certain classical problems of mathematics, known since ancient times, still defied solution: the squaring of the circle, the trisection of an angle, the doubling of the cube. Despite the efforts of the greatest mathematicians over a span of nearly two millennia, these classical conundrums still defeated every effort to solve them.

This was very bad news for Hobbes’s science of politics. If geometry is fully known, as he declared it must be, then it should have no unsolved, not to mention insoluble, problems. The fact that it does suggests that it possesses dark corners where the light of reason does not shine. And if geometry, which deals with simple points and lines, is not fully understood, how can one expect the theory of the commonwealth, which deals with the thoughts and passions of men, to be perfectly known? If geometry has its blind spots, then the science of politics may well have some, too, and they are likely to be far greater and more significant than those of geometry. For Hobbes, as long as the classic geometrical problems remained unsolved, his entire philosophical edifice remained insecure, and the Leviathan state a political house built on sand.

In order to secure the foundations of his political theory, Hobbes set out to solve the three classical outstanding problems of geometry. Initially he seems to have believed that this should not be too difficult. Surely, he thought, just as he had corrected the errors of all past philosophers, he could also correct the errors of all past geometers. And he can perhaps be excused for his unwarranted optimism, because part of the reason the problems had attracted the attention of the greatest mathematicians over centuries was that they were easily stated, and appeared deceptively simple. “The quadrature of the circle” means constructing a square equal in area to a given circle; “the trisection of the angle” means dividing any given angle into three equal parts; and “the doubling of the cube” means constructing a cube of double the volume of a given cube. How hard could it be to solve such questions? As it turns out, very hard. In fact, impossible.

Figure 7.1. The Quadrature of the Circle 1. The Case of the Inscribed Polygon.

To understand why this is so, consider the problem that most interested Hobbes and to which he devoted an entire chapter in
De corpore
: the quadrature of the circle. Already in the second century BCE, the polymath Archimedes of Syracuse proved that the area enclosed in a circle is equal to that of a right triangle whose two perpendicular legs are equal to the radius and to the circumference of a circle—that is,
PQ
and
QR
in
Figure 7.1.
Archimedes proved this result by looking at polygons inscribing and circumscribing a circle. The greater the number of sides of a polygon, the closer its enclosed area is to that of the circle. Now, Archimedes reasoned, let us consider the circumscribed octagon
AHDGCFBE
. Its area is equal to that of a right triangle whose two perpendicular sides are equal to the apothem and the sum of the octagon’s sides (the “apothem” being a vertical line from the center of the polygon to its side). This is obvious if we think of the area of the octagon to be the sum of eight triangles with bases on the sides of the octagon and the apex at the center. The area of each triangle is half its base times the apothem, and consequently the area of the entire octagon is half all the bases together times the apothem—that is, the area of the right triangle in question.

Let us look, Archimedes continued, at the area of the circle, which we call
C
, and the area of the right triangle whose perpendicular sides are its radius and circumference, which we call
T
, and the area of an inscribed polygon with
n
sides, which we call
I
n
. Let us assume for the moment that the circle is greater than the triangle—that is,
C
>
T
. Archimedes had already shown that as we increase the number of sides of the inscribed polygon, it approaches the area of the circle as closely as we please. Consequently, there is a number of sides
n
for which the area of the polygon falls between the area of the circle and the triangle that is greater than the triangle but (being inscribed) still smaller than the circle. In modern notation:
I
n
>
T
.