A Strange Wilderness

by Amir D. Aczel

  The rivalry between Gottfried Leibniz and Isaac Newton over who had first discovered calculus lasted until Leibniz’s death in 1716. In this frontispiece to Voltaire’s 1738 Eléments de la philosophie de Newton, the light of Newton’s wisdom is reflected onto Voltaire from the heavens by his muse and lover Émilie du Châtelet, who translated Newton’s work for him.

  NINE

  THE GREATEST RIVALRY

  In 1676, twenty-six years after the death of Descartes in Stockholm, a budding German intellectual living in Paris was determined to read all the works of the late French mathematician—both those that had been published and those that were hitherto unknown to the general public. The young German man managed to find a French gentleman named Claude Clerselier, a relative by marriage of Descartes. Clerselier had also been Descartes’s publisher and now jealously guarded all the documents that the great philosopher and mathematician had left behind when he died. It is a miracle that these papers survived: after the French ambassador to Sweden shipped them to Paris following Descartes’s death, the boat carrying the documents capsized while sailing up the Seine. Amazingly, the box containing these precious writings—most of them unpublished—was found floating on the river, and Clerselier and his servants spent days drying the wet documents and reassembling them. Clerselier then refused to show them to anyone, but something compelled him to let the young German see them, if only very briefly—he imposed a strict time limit on how long he could look at them. As it turned out, this young man was Gottfried Wilhelm Leibniz (1646–1716), a brilliant mathematician who then managed to decipher and copy Descartes’s writings within the little time allotted him by Clerselier. Leibniz’s copied notes now reside in an archive in Hanover, Germany.

  GOTTFRIED LEIBNIZ

  Gottfried Leibniz was born on July 1, 1646, in Leipzig, in a Germany that had been devastated by the Thirty Years’ War. Until he was four years old, Swedish troops remained in a garrison, holding the city. His father, Friedrich Leibniz, taught moral philosophy at the University of Leipzig, and his mother, Catharina Schmuck, was the daughter of a law professor. In 1652 Leibniz’s father died, and the six-year-old was sent to the Nicolai School, where he learned Latin—a normal practice among the European elite.

  Young Leibniz retained access to the vast library left behind by his father, and it was here that he received his real education. He read voraciously the works of classical Greece and ancient Rome, and, according to the French historian Yvon Belaval, the budding child-scholar was able to use combinatorial reasoning to decipher the meaning of the words and sentences of the Latin language.1 This deep understanding of language may have helped him years later to derive the key ideas of his masterpiece, De Arte Combinatoria. Guessing, defining, and combining the words of an unknown language as he deciphered their meaning had led Leibniz to combinatorial analysis—today a major component in modern probability theory.

  Leibniz also developed a great appreciation for the life, customs, writings, and history of the ancients, which became evident throughout his adult life as he developed an interest in the historical continuity of ideas from antiquity to the present. By age twelve he was fluent in ancient Greek, having deciphered that language as well. He read Plato and Aristotle, which fueled his interest in logic and the foundations of pure reason. He also developed an interest in theology and its polemics. Thus we see in these adolescent investigations the roots of Leibniz’s future groundbreaking work in theology, philosophy, and mathematics.

  In 1661, at the age of fourteen, Leibniz enrolled at the University of Leipzig, where he studied the works of his contemporaries, including Thomas Hobbes, Francis Bacon, and Galileo. He also took courses on rhetoric, as well as Latin, Greek, and Hebrew.2 In 1663 he presented his thesis, De Principio Individui, which dealt with ideas of individuation and totality, and that summer he took courses in jurisprudence, medieval history, and mathematics. He chose to concentrate his studies on law. The following year, Leibniz lost his mother, which was a devastating blow from which he took a long time to recover. Slowly, he returned to his studies and pursued the idea that proof in legal matters should rest on mathematical or logical evidence and its analysis—an approach that continues today with the use of scientific and statistical reasoning in criminal cases whenever such evidence and its analysis are possible.

  In 1666, after receiving his master’s degree in law, Leibniz published his treatise De Arte Combinatoria, an in-depth study of permutations that identified ideas as the basic elements of a logical system. Leibniz’s point of view was that all concepts in the world are combinations of simple ideas—a notion that anticipated the monads, which would make Leibniz famous later on. Akin to atoms, monads are the eternal, indivisible building blocks of the metaphysical universe and make up everything complex in all creation. Leibniz felt that these primary elements should be few in number and as simple as possible so that everything in the universe could be reduced to the combination of these basic elements. Relations among ideas, Leibniz argued, could be deduced from uncovering the ways in which the simplest concepts are combined. For example, an interval of numbers is obtained through the combination of its elements—the numbers themselves—and 3-D objects can be seen as combinations of intervals. Similarly, all sentences can be seen as combinations of words, the words as combinations of sounds, and the sounds as combinations of the letters of the alphabet. Thus, at age twenty, Leibniz already possessed the basic concepts of his philosophy.

  Next, Leibniz moved to Altdorf, Germany, and earned a doctorate at the local university. Then the intellectual life beckoned him to nearby Nuremberg, where he is rumored to have encountered the Rosicrucians. He served for a while as the secretary of the alchemical society’s local chapter, although later he would call alchemy a “deception.” Whatever the case may be, this connection to the Rosicrucians unites him with Descartes, who had also been rumored to have been involved with the society—if only at the suggestion of his adversaries. Through his supposed involvement with the Rosicrucians and with alchemy and astrology, Leibniz made the acquaintance of Baron Johann Christian von Boineburg, the former councilor to the elector of Mainz, one of several German princes who together were charged with electing the Holy Roman Emperor.3 Boineburg was a celebrated European statesman, and the young Leibniz was attracted to the possibility of working with him.

  Recognizing the young man’s genius, Baron Boineburg invited Leibniz to Frankfurt, where he employed him as his personal librarian. Sometime later Boineburg took Leibniz with him to Mainz and introduced him to the princely court. Leibniz was a Protestant, but he was so attracted to the idea of working for a prince that he entertained the thought of converting to Catholicism, although he never did follow through with this notion. Through Boineburg and his royal contacts, Leibniz began his apprenticeship in the field of politics.

  At one point Leibniz wrote a paper about his idea of basing jurisprudence on logical principles and, through Boineburg, presented it to the elector of Mainz. The elector was duly impressed by the brilliance of the young lawyer, and this move helped seal Leibniz’s appointment as councilor to the chancellery of Mainz. In 1668 Leibniz wrote a treatise arguing for the existence of God and the immortality of the soul entitled Nature’s Testimony Against the Atheists, which hinted at his future grand scheme for unifying the religions of Europe.

  Over the next few years, Leibniz turned his attention away from politics, devoting himself to the study of mathematics, physics, and metaphysics. He wrote a number of papers and sent them abroad to the scientific academies in countries that had retained their supremacy in science and culture despite any religious wars they may have gone through—wars that had devastated his native Germany. In 1670 Leibniz sent a seminal paper on universal movement entitled Theoria Motus Abstracti to the French Academy of Sciences—one of the most important scientific bodies in Europe and, at the time, the arbiter of what science was and where it was heading. At the same time, he sent the Royal Society in London a paper entitled Theoria Mo
tus Concreti, which applied to concrete, rather than abstract, notions of movement in mechanics. Leibniz also commenced a correspondence with various intellectuals in France and Britain, hoping one day to turn his new ties into a permanent, paying position in either of these two countries.

  Leibniz read Descartes and quickly made up his mind that the French philosopher’s ideas were anathema to him. “I am nothing less than a Cartesian,” he wrote to the French philosopher Antoine Arnauld, a man who had written diatribes against Descartes.4 What Leibniz objected to in Descartes’s philosophy was the latter’s separation of body and soul. In his own writings, which were then attacked by the Cartesians, Leibniz sought to find the soul within the physicality of the body. These debates enhanced his reputation as a philosopher, and he extended his correspondence with European intellectuals, including the famous Dutch philosopher Spinoza.

  The theologian Antoine Arnauld (1612–94) carried on a lively correspondence with Leibniz regarding the philosophy of René Descartes.

  Leibniz was different from many other geniuses throughout history in that he achieved so much in so many disparate fields. While battling the Cartesians on philosophy, he also invented a machine that could perform addition, subtraction, multiplication, division, and calculate square and cubic roots. This device of immense genius and value was seen as superior to a similar mechanical calculator created by Blaise Pascal. Leibniz also invented lenses, air pumps, and a nautical navigation instrument—all while serving the princely court of Mainz and Baron von Boineburg. It has been said that Leibniz lived several lives; each of his occupations gave him the equivalent of a full lifetime of knowledge. Some lamented this fact, believing that he wasted his genius for mathematics by not concentrating more on that field.5

  Then Leibniz developed the strange idea of trying to manipulate the king of France. Sometimes even great minds can hatch senseless schemes. This was one such example.

  A HAREBRAINED CONSPIRACY

  Leibniz was a product of the Thirty Years’ War and, hence, sensitive to the existence of religious differences among the peoples of Europe. He viewed these differences as eternal causes of conflict, so he came to the idea that Protestantism and Catholicism should be united in some way. In this respect, he was especially worried about the very powerful king of France, a Catholic whom Leibniz saw as a threat to European peace because of his supposed desire to attack Protestants, which were numerous in German principalities as well as in Holland.

  Four decades earlier, in 1628, Louis XIII killed off many of his own Protestant citizens in the siege of La Rochelle, witnessed by Descartes. Understandably, Leibniz feared that Louis XIII’s successor, Louis XIV—a.k.a. the Sun King—posed a threat to Protestants outside his realm. Leibniz conceived a political scheme to divert the French king from what he thought was a design on Protestant nations and regions. Would it be possible, Leibniz asked himself, to tempt Louis XIV to launch an attack on Muslim Egypt as a way of countering the increasing influence of the Ottomans? If France were to use its might against Egypt, he reasoned, it would be unlikely to launch attacks on European Protestants.

  Boineburg had been aware of Leibniz’s idea that the king of France might be induced, if approached correctly, to change direction and turn his military might against the Muslim infidels of Egypt instead of against his Christian brothers in northern Europe. In 1672 the baron sent his librarian and budding diplomat to Paris with the charge of finding a way to approach Louis XIV and propose his plan. Leibniz’s other mission was to arrange for the education of Boineburg’s son in the French capital. Thus, with Boineburg’s son under his charge, Leibniz happily left for Paris, a city he had always wanted to see and hoped to live in.

  Hyacinthe Rigaud’s 1701 portrait of Louis XIV, whom Leibniz tried to coax into invading Egypt, appears against the background of a battle scene painted by Charles Parrocel.

  The French were enemies of the Germans—they occupied German cities, as did their allies, the Swedes—so the audacity of a German statesman sending a minor diplomat to France to convince the king to invade Egypt strikes us today as naive and childish. In fact, Leibniz’s complete plan was even more outlandish. According to his strategy, the French, after conquering Egypt, could also be induced to invade the rest of North Africa and the Levant, a region that included several nations on the east coast of the Mediterranean. Additionally, he dreamed, Sweden and Poland could invade and “civilize” Siberia, the Crimea, the Black Sea, and the Sea of Azov; England and Denmark could invade all of North America; Holland could take over the East Indies; and Spain could consolidate its control over all of South America, without Portugal—a nation Leibniz considered unimportant. Thus, according to Leibniz, the entire world could eventually be occupied by European nations, whose religions would be unified under the rubric of “Christianity,” without distinction between its denominations.

  The silly plan came to nothing before it even had a chance of being presented. Leibniz had barely arrived in Paris when, on May 6, 1672, Louis XIV declared war on Protestant Holland. Leibniz’s master, the Prince of Mainz, offered to serve as mediator between France and Holland, but Louis XIV rebuffed him, and French troops invaded Holland.

  Leibniz noted that Louis XIV was universally hated in Europe, and he saw the king as an aggressive leader consumed with the desire to attack his neighbors, while “a large part of his people eats only once a day.”6 Although he had never met the king, the young German diplomat was able to make connections with many French statesmen, scientists, and intellectuals. Among them were the Duke of Chevreuse and the philosopher Antoine Arnauld, with whom he already had an established correspondence. He also met the Prince of Condé, who expressed great interest in Leibniz’s idea of a religious unification of the continent.

  Not only did Louis XIV invade Holland—obviating Leibniz’s mission in Paris—but in December 1672 Leibniz’s patron Boineburg died, and a few months later, in early 1673, so did Prince Johann Philip, the elector of Mainz. Leibniz continued to tutor Boineburg’s son until sometime in 1674, but he refused to return to Germany, where he had no employment prospects left.

  After failing to get an audience with Louis XIV, Leibniz settled in Paris for four years. Now free to do whatever he wanted, he chose to focus on mathematics and the physical description of the universe, expanding upon the physics of Galileo and his contemporaries. Leibniz also drew upon the ideas of Eudoxus, Archimedes, Descartes, and Fermat, while developing the key mathematical theory we know as calculus.

  In 1673 Leibniz took a trip to Britain, remaining there for three months. In England he met the physicist Robert Boyle, as well as the mathematicians Christopher Wren, John Pell (of the famous Pell equation), and the German intellectual Henry Oldenburg, who served as secretary of the Royal Society in London. That same year, Leibniz published the philosophical treatise Confessio Philosophi (A Philosopher’s Creed).

  Leibniz is depicted in this ca. 1700 portrait by Johann Friedrich Wentzel.

  Leibniz returned to Paris invigorated by the mathematical and physical ideas he had discussed with British mathematicians, but he soon realized that without employment his days in the lovely French capital were numbered. He needed a new patron, and he soon found one in the person of Duke Hans Friedrich of Hanover. The duke agreed to bankroll Leibniz’s stay in Paris. Leibniz was now free to reside in the city and turn his full attention to mathematics. Here, under the care and guidance of Christiaan Huygens, a brilliant mathematician, astronomer, and physicist seventeen years his senior, Leibniz was to become a first-rate mathematician. As Leibniz wrote in his memoir:

  It was thus that Huygens who, I think, saw in me more than there was, brought me with gentility a recently published copy of his book on the pendulum [Horologium Oscillarium, 1673]. This was for me the beginning of the opportunity for a deeper study of mathematics. While doing so, Huygens realized that I had not an understanding of the concept of center of mass and he gave me a brief description, noting that [Blaise Pascal] had treated it re
markably well.7

  Leibniz went back to the ideas of his younger days and tried to apply his combinatorial analysis to a wider range of problems in mathematics. In 1674 he was studying a figure in a work of Pascal when he had a brilliant insight into functions and rates of change. A year later, this insight led him to propose a coherent and complete system of computing derivatives, which are instantaneous rates of change of functions; and a general way of finding areas under the curves of functions—i.e., integrals. This system comprised what later became known as infinitesimal calculus.8 Leibniz even introduced the universal notation we use in calculus today: the differential symbols dx and dy, as well as the integral sign, , an elongated S, from the Latin word for sum: summa. The integral operation is, in fact, a continuous form of summation. Leibniz introduced this sign for the first time in a paper he wrote in November of 1675.

  THE CALCULUS CONTROVERSY

  Once Leibniz published his calculus, he was fiercely attacked by British mathematicians, who accused him of stealing from the works of Isaac Newton. Some of them knew that Newton had been working on the same calculus methods for years; others even claimed that all Leibniz had done was to “steal the ideas of Descartes.” Leibniz became convinced that the only way to defend himself was to read everything Descartes had written. It was at this point in time that he found Claude Clerselier, the keeper of all Descartes’s works. It was also at this time that, by reading Descartes anew, he became a convert to the Cartesian approach to understanding the universe.