Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz was born in Leipzig, in the Electorate of Saxony, on June 21, 1646, near the end of the Thirty Years' War. His father, Friedrich Leibniz, taught moral philosophy at the University of Leipzig and died when the boy was only six years old. The most valuable inheritance he left was a substantial scholarly library, and the young Leibniz was given free access to it around the age of eight ^[1].

That library did much of the work a formal education might otherwise have done. Any Gottfried Wilhelm Leibniz biography has to reckon with how quickly he absorbed material meant for adults: he taught himself Latin from the books at hand, moved on to Greek, and read widely in classical authors, church fathers, and scholastic philosophy before entering the University of Leipzig in 1661, at fourteen ^[2]. He completed a bachelor's degree in philosophy in 1663 with a thesis on the principle of individuation, then continued into law.

A telling episode closed his student years. Leipzig declined to grant him the doctorate in law in 1666, reportedly on grounds connected with his youth, so he simply left. The University of Altdorf, near Nuremberg, awarded him the degree within months and offered him a professorship on the strength of his dissertation. He turned the chair down ^[1]. That refusal set the pattern for his whole career: Leibniz would spend his life in courts, chanceries, and libraries rather than lecture halls.

In 1667 Leibniz entered the service of Johann Christian von Boyneburg, a former chief minister of the Elector of Mainz, and soon worked for the Elector himself on projects of legal reform. His early ambition was enormous: he hoped to recodify the entire body of Roman law on rational principles, an idea that stayed with him for decades even though it was never completed ^[2]. He also drafted diplomatic proposals, including a scheme to redirect the military attention of Louis XIV toward an expedition in Egypt rather than the German states.

The Egyptian plan gave him the excuse for the most consequential journey of his life. Leibniz arrived in Paris in 1672, then the scientific capital of Europe, and stayed for four years. There he met Christiaan Huygens, who recognized that the brilliant young jurist had large gaps in his mathematical training and set him a course of reading and problems ^[3]. Leibniz learned fast. During a short visit to London in 1673 he demonstrated a calculating machine of his own design to the Royal Society, which elected him a fellow. The machine, later refined as the stepped reckoner, could multiply and divide mechanically, something Blaise Pascal's earlier adding machine could not do ^[4].

The Paris years ended for a mundane reason: he needed a salary. In 1676 Leibniz accepted the post of counsellor and librarian to the Duke of Brunswick in Hanover. On the journey there he stopped in London and in Holland, where he held long conversations with the philosopher Baruch Spinoza. He would remain attached to the House of Brunswick, serving three successive rulers, for the remaining forty years of his life ^[1].

The list of Gottfried Wilhelm Leibniz achievements begins with the calculus. During his Paris period, in work concentrated around 1675, he arrived at the fundamental methods of the differential and integral calculus, including the insight that differentiation and integration are inverse operations. Isaac Newton had reached equivalent results earlier, in the mid 1660s, but published nothing at the time. Leibniz printed his differential calculus first, in a 1684 paper in the journal Acta Eruditorum, followed by an account of the integral calculus in 1686 ^[3].

What gave his version lasting power was the notation. The elongated S of the integral sign, first used in his manuscripts of 1675, and the d notation for differentials proved far easier to work with than Newton's fluxional dots, and continental mathematicians such as the Bernoulli brothers and later Leonhard Euler built the analysis of the eighteenth century on Leibniz's symbols ^[3]. The dy/dx form that appears in every modern textbook is his.

Calculus was only one thread. Leibniz worked out binary arithmetic, the representation of all numbers using only 0 and 1, and published a paper on it in 1703, noting a resemblance between binary patterns and the hexagrams of the ancient Chinese I Ching ^[4]. He did significant work on determinants, on the theory of series, and on what he called analysis situs, an early gesture toward topology. In physics he argued against Descartes that the conserved quantity in collisions was not sheer quantity of motion but vis viva, proportional to mass times the square of velocity, a concept ancestral to kinetic energy ^[5]. He also dreamed of a characteristica universalis, a universal symbolic language in which reasoning could be carried out by calculation, an idea later historians of logic have treated as a distant anticipation of formal symbolic logic ^[5].

Leibniz never wrote a single systematic treatise setting out his philosophy in full. It has to be assembled from short published pieces, from the book length Theodicy of 1710, from the New Essays on Human Understanding (a reply to John Locke, withheld from publication after Locke died), and from an immense correspondence ^[6]. The mature system centers on the monad: a simple, unextended substance without parts, of which the universe is ultimately composed. Monads do not interact causally; instead each unfolds its own internal states in a pre-established harmony arranged by God, like clocks set to agree from the start ^[6].

Two principles anchor his reasoning. The principle of contradiction governs necessary truths, and the principle of sufficient reason holds that nothing occurs without a reason why it is so rather than otherwise. From these he argued that God, choosing among all possible worlds, must have created the best of them. That phrase, the best of all possible worlds, became famous and then notorious: Voltaire mocked it mercilessly in Candide in 1759, embodying it in the absurdly cheerful tutor Dr. Pangloss ^[1].

His theological energies were practical as well as speculative. For years Leibniz corresponded with the French bishop Jacques-Bénigne Bossuet in an effort to find terms for reunifying the Catholic and Protestant churches, and he later worked on schemes to unite the Lutheran and Reformed confessions. None of these projects succeeded, but they reflect a lifelong conviction that apparent conflicts, whether between churches or between philosophical schools, usually concealed a deeper reconcilable truth ^[2].

Leibniz never married and had no children. Contemporaries described a man of restless sociability, moderate habits, and almost unbelievable industry, who often worked through the night in his chair and whose rooms filled with unsorted heaps of notes ^[1]. His real social world was epistolary: he exchanged letters with roughly 1,100 correspondents across Europe, from Jesuit missionaries in China to fellow mathematicians, and the surviving correspondence and papers run to an estimated 200,000 sheets, still being edited today in a long running academy project ^[2].

His closest intellectual friendships were with women of the Hanoverian court. Sophia, Electress of Hanover, and her daughter Sophia Charlotte, who became the first Queen in Prussia, treated him as a philosophical companion rather than a servant, and the Theodicy grew out of conversations with Sophia Charlotte ^[6]. At her urging Leibniz helped found the Berlin Society of Sciences in 1700, the ancestor of the Prussian Academy, and served as its first president ^[4].

He was also, by profession, a historian and archivist. Commissioned to write a history of the Guelf dynasty, he traveled through Germany, Austria, and Italy between 1687 and 1690 hunting documents. The research pulled him sideways into geology: his Protogaea, drafted in the early 1690s though printed only after his death, argued from fossils and rock strata that the Earth had a long developmental history, an unusual position for the time ^[5].

The final decades brought both honors and grief. In 1711 Leibniz met Tsar Peter the Great and advised him on plans for promoting science in Russia, ideas that fed into the later founding of the St. Petersburg Academy. The Emperor in Vienna made him an imperial court councillor in 1712 ^[2]. Yet in Hanover his standing sank. Elector Georg Ludwig valued the still unfinished Guelf history far more than metaphysics, and when the Elector departed for London in 1714 to reign as King George I of Great Britain, Leibniz was pointedly ordered to stay behind and finish the book ^[1].

The exclusion had another cause. The dispute over priority in the invention of the calculus, simmering since the 1690s, had turned poisonous. Accusations that Leibniz had plagiarized Newton were pressed by Newton's partisans, and in 1712 a Royal Society committee, whose report Newton himself largely drafted, ruled against Leibniz ^[3]. Modern historians of mathematics conclude that the two men made their discoveries independently, Newton first in time, Leibniz first in print, but during Leibniz's lifetime the verdict in England was unforgiving, and a courtier under such a cloud could hardly be welcomed in London.

Leibniz kept working almost to the end, conducting a final philosophical exchange with the Newtonian Samuel Clarke on the nature of space, time, and God. He died in Hanover on November 14, 1716, at the age of seventy. The funeral was strikingly modest for a man of his fame: no court dignitary attended, and only his secretary followed the coffin ^[1].

Recognition arrived slowly, then overwhelmingly. Eighteenth century continental mathematics ran on Leibnizian notation, and through Christian Wolff his philosophy dominated German universities until Kant, who framed much of his own critical project as a response to the Leibnizian tradition ^[6]. Bertrand Russell's 1900 study of Leibniz's logic helped revive philosophical interest, and twentieth century logicians looked back on the characteristica universalis and the calculus ratiocinator as prophetic sketches of symbolic logic and mechanical reasoning ^[5].

The computing connection is the fact most often cited among Gottfried Wilhelm Leibniz facts today, and it is genuine on two counts: he built a working four function calculating machine, and he developed the binary arithmetic on which digital computers now run ^[4]. Norbert Wiener, the founder of cybernetics, called him a patron saint of the field. The Gottfried Wilhelm Leibniz Prize, Germany's most generous research award, carries his name, as does the university in Hanover and the Leibniz Association of research institutes.

His papers, preserved in the Gottfried Wilhelm Leibniz Library in Hanover, were inscribed in UNESCO's Memory of the World register: the correspondence in 2007, with further manuscripts added later ^[2]. Three centuries on, the edition of his complete writings remains unfinished, which is perhaps the most fitting monument to a man whose projects always outran any single lifetime.