Gottfried Leibniz

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Gottfried Leibniz

Western Philosophy 18th-century philosophy
Gottfried Wilhelm Leibniz
Full name	Gottfried Wilhelm Leibniz
Birth	July 1 ( June 21 Old Style) 1646, Leipzig, Electorate of Saxony
Death	November 14, 1716, Hanover, Electorate of Hanover
School/tradition	Rationalism
Main interests	Metaphysics, Mathematics, Theodicy
Notable ideas	Infinitesimal calculus, Calculus, Monadology, Theodicy, Optimism
Influenced by Plato, Aristotle, Aquinas, Suárez, Descartes, Spinoza, Ramon Llull
Influenced Many later mathematicians, Christian Wolff, Kant, Bertrand Russell, Martin Heidegger
Signature

Gottfried Wilhelm Leibniz (also Leibnitz or von Leibniz ( July 1, 1646 ( June 21 Old Style) – November 14, 1716) was a German polymath who wrote primarily in Latin and French.

He occupies an equally grand place in both the history of philosophy and the history of mathematics. He invented calculus independently of Newton, and his notation is the one in general use since then. He also discovered the binary system, foundation of virtually all modern computer architectures. In philosophy, he is mostly remembered for optimism, i.e. his conclusion that our universe is, in a restricted sense, the best possible one God could have made. He was, along with René Descartes and Baruch Spinoza, one of the three greatest 17th century rationalists, but his philosophy also looks back to the scholastic tradition and anticipates modern logic and analysis. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in biology, medicine, geology, probability theory, psychology, linguistics, and information science. He also wrote on politics, law, ethics, theology, history, and philology, even occasional verse. His contributions to this vast array of subjects are scattered in journals and in tens of thousands of letters and unpublished manuscripts. To date, there is no complete edition of Leibniz's writings.

Biography

Early life

Gottfried Leibniz was born on 1 July ( 21 June Old Style) 1646 in Leipzig to Friedrich Leibniz and Catherina Schmuck. The name Leibniz was originally Slavonic - Lubeniecz . His father had passed away when he was six, so he learned his religious and moral values from his mother. These would exert a profound influence on his philosophical thought in later life. As an adult, he often styled himself "von Leibniz", and many posthumous editions of his works gave his name on the title page as "Freiherr [Baron] G. W. von Leibniz." However, no document has been found confirming that he was ever granted a patent of nobility.

When Leibniz was six years old, his father, a Professor of Moral Philosophy at the University of Leipzig, died, leaving a personal library to which Leibniz was granted free access from age seven onwards. By 12, he had taught himself Latin, which he used freely all his life, and had begun studying Greek.

He entered his father's university at age 14, and completed university studies by 20, specializing in law and mastering the standard university courses in classics, logic, and scholastic philosophy. However, his education in mathematics was not up to the French and British standards. In 1666 (age 20), he published his first book, also his habilitation thesis in philosophy, On the Art of Combinations. When Leipzig declined to assure him a position teaching law upon graduation, Leibniz submitted the thesis he had intended to submit at Leipzig to the University of Altdorf instead, and obtained his doctorate in law in five months. He then declined an offer of academic appointment at Altdorf, and spent the rest of his life in the service of two major German noble families.

1666–74

Leibniz's first position was as a salaried alchemist in Nuremberg, even though he knew nothing about the subject. He soon met Johann Christian von Boineburg (1622–1672), the dismissed chief minister of the Elector of Mainz, Johann Philipp von Schönborn. Von Boineburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for his Electorate. In 1669, Leibniz was appointed Assessor in the Court of Appeal. Although von Boineburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674.

Von Boineburg did much to promote Leibniz's reputation, and the latter's memoranda and letters began to attract favorable notice. Leibniz's service to the Elector soon followed a diplomatic role. He published an essay, under the pseudonym of a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main European geopolitical reality during Leibniz's adult life was the ambition of Louis XIV of France, backed by French military and economic might. Meanwhile, the Thirty Years' War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows. France would be invited to take Egypt as a stepping stone towards an eventual conquest of the Dutch East Indies. In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained the Elector's cautious support. In 1672, the French government invited Leibniz to Paris for discussion, but the plan was soon overtaken by events and gone irrelevant. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting implementation of Leibniz's plan.

Thus Leibniz began several years in Paris, during which he greatly expanded his knowledge of mathematics and physics, and began contributing to both. He met Malebranche and Antoine Arnauld, the leading French philosophers of the day, and studied the writings of Descartes and Pascal, unpublished as well as published. He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus; they corresponded for the rest of their lives. Leibniz was extraordinarily fateful when came into acquaintance of the Dutch physicist and mathematician Christiaan Huygens, who was active in Paris then. Soon after arriving in Paris, Leibniz received a rude awakening; his knowledge of mathematics and physics was spotty. Finding Huygens as mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including inventing his version of the differential and integral calculus.

When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the English government in London, early in 1673. There Leibniz came into acquaintance of Henry Oldenburg and John Collins. After demonstrating a calculating machine to the Royal Society he had been designing and building since 1670, the first such machine that could execute all four basic arithmetical operations, the Society made him an external member. The mission ended abruptly when news reached it of the Elector's death, whereupon Leibniz promptly returned to Paris and not, as had been planned, to Mainz.

The sudden deaths of Leibniz's two patrons in the same winter meant that Leibniz had to find a new basis for his career. In this regard, a 1669 invitation from the Duke of Brunswick to visit Hanover proved fateful. Leibniz declined the invitation, but began corresponding with the Duke in 1671. In 1673, the Duke offered him the post of Counsellor which Leibniz very reluctantly accepted two years later, only after it became clear that no employment in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court was forthcoming.

House of Hanover 1676–1716

Leibniz managed to delay his arrival in Hanover until the end of 1676, after making one more short journey to London, where he possibly was shown some of Newton's unpublished work on the calculus. This fact was deemed evidence supporting the accusation, made decades later, that he had stolen the calculus from Newton. On the journey from London to Hanover, Leibniz stopped in The Hague where he met Leeuwenhoek, the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza, who had just completed his masterwork, the Ethics. Leibniz respected Spinoza's powerful intellect, but was dismayed by his conclusions that contradicted both Christian and Jewish orthodoxy.

In 1677, he was promoted, at his request, to Privy Counselor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for the period.

Among the few people in north Germany to warm to Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and her avowed disciple, and Caroline of Ansbach, the consort of her grandson, the future George II. To each of these women he was correspondent, adviser, and friend. In turn, they all warmed to him more than did their spouses and the future king George I of Great Britain.

The population of Hanover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick was quite an honour, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the Holy Roman Empire. The British Act of Settlement 1701 designated the Electress Sophia and her descent as the royal family of the United Kingdom, once both King William III and his sister-in-law and successor, Queen Anne, were dead. Leibniz played a role in the initiatives and negotiations leading up to that Act, but not always an effective one. For example, something he published anonymously in England, thinking to promote the Brunswick cause, was formally censured by the British Parliament.

The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting the calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on the calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum. That journal played a key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy.

The Elector Ernst August commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes.

In 1711, John Keill, writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarized Newton's calculus. Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of the calculus.

In 1711, while traveling in northern Europe, the Russian Tsar Peter the Great stopped in Hanover and met Leibniz, who then took some interest in matters Russian over the rest of his life. In 1712, Leibniz began a two year residence in Vienna, where he was appointed Imperial Court Councillor to the Habsburgs. On the death of Queen Anne in 1714, Elector Georg Ludwig became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the dowager Electress Sophia, died in 1714.

Leibniz died in Hanover in 1716: at the time, he was so out of favour that neither George I (who happened to be near Hanover at the time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Berlin Academy of Sciences, neither organization saw fit to honour his passing. His grave went unmarked for more than 50 years. Leibniz was eulogized by Fontenelle, before the Academie des Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans, a niece of the Electress Sophia.

Leibniz never married. He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had, by and large, paid him well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which cannot be excused or defended and which put him in a bad light during the calculus controversy. On the other hand, he was charming and well-mannered, with many friends and admirers all over Europe.

Writings and edition

Leibniz mainly wrote in three languages: scholastic Latin (ca. 40%), French (ca. 35%), and German (less than 25%). During his lifetime, he published many pamphlets and scholarly articles, but only two "philosophical" books, the Combinatorial Art and the Théodicée. (He published numerous pamphlets, often anonymous, on behalf of the House of Brunswick-Lüneburg, most notably the "De jure suprematum" a major consideration of the nature of sovereignty.) One substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain, which Leibniz had withheld from publication after the death of John Locke. Only in 1895, when Bodemann completed his catalogues of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's Nachlass become clear: about 15,000 letters to more than 1000 recipients plus more than 40,000 other items. Moreover, quite a few of these letters are of essay length. Much of his vast correspondence, especially the letters dated after 1685, remains unpublished, and much of what is published has been so only in recent decades. The amount, variety, and disorder of Leibniz's writings are a predictable result of a situation he described as follows:

"I cannot tell you how extraordinarily distracted and spread out I am. I am trying to find various things in the archives; I look at old papers and hunt up unpublished documents. From these I hope to shed some light on the history of the [House of] Brunswick. I receive and answer a huge number of letters. At the same time, I have so many mathematical results, philosophical thoughts, and other literary innovations that should not be allowed to vanish that I often do not know where to begin". (1695 letter to Vincent Placcius in Gerhardt)

The extant parts of the critical edition of Leibniz's writings (see photograph there) are organized as follows:

• Series 1. Political, Historical, and General Correspondence. 21 vols., 1666–1701.
• Series 2. Philosophical Correspondence. 1 vol., 1663–85.
• Series 3. Mathematical, Scientific, and Technical Correspondence. 6 vols., 1672–96.
• Series 4. Political Writings. 6 vols., 1667–98.
• Series 5. Historical and Linguistic Writings. Inactive.
• Series 6. Philosophical Writings. 7 vols., 1663–90, and Nouveaux essais sur l'entendement humain.
• Series 7. Mathematical Writings. 3 vols., 1672–76.
• Series 8. Scientific, Medical, and Technical Writings. In preparation.

The systematic cataloguing of all of Leibniz's Nachlass was begun in 1901. Two World wars, the NS dictatorship (with Jewish genocide, including an employee of the project, and other personal consequences), and decades of German division (two states with the cold war's "iron curtain" in between, separating scholars and also scattered portions of his literary estates), greatly hampered the ambitious edition project which had and has to deal with seven languages used on ca. 200 000 pages of written and printed paper. In 1985 it was reorganized and included in a joint program of German federal and state ("Länder") academies. Since then the branches in Potsdam, Münster, Hannover and Berlin have jointly published 25 volumes of the critical edition (until 2006) with an average of 870 pages (compared to only 19 volumes since 1923), plus preparing index and concordance works (so, had that "speed" of work been possible from the beginning, the project would already be completed).

Posthumous reputation

When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée, whose supposed central argument Voltaire lampooned in his Candide. Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description (this misapprehension may still be the case among certain lay people). Thus Voltaire and his Candide bear some of the blame for the lingering failure to appreciate and understand Leibniz's ideas. Leibniz had an ardent disciple, Christian Wolff, whose dogmatic and facile outlook did Leibniz's reputation much harm. In any event, philosophical fashion was moving away from the rationalism and system building of the 17th century, of which Leibniz had been such an ardent exponent. His work on law, diplomacy, and history was seen as of ephemeral interest. The vastness and richness of his correspondence went unrecognized.

Much of Europe came to doubt that Leibniz had discovered the calculus independently of Newton, and hence his whole work in mathematics and physics was neglected. Voltaire, an admirer of Newton, also wrote Candide at least in part to discredit Leibniz's claim to having discovered the calculus and Leibniz's charge that Newton's theory of universal gravitation was incorrect. The rise of relativity and subsequent work in the history of mathematics has put Leibniz's stance in a more favorable light.

Leibniz's long march to his present glory began with the 1765 publication of the Nouveaux Essais, which Kant read closely. In 1768, Dutens edited the first multi-volume edition of Leibniz's writings, followed in the 19th century by a number of editions, including those edited by Erdmann, Foucher de Careil, Gerhardt, Gerland, Klopp, and Mollat. Publication of Leibniz's correspondence with notables such as Antoine Arnauld, Samuel Clarke, Sophia of Hanover, and her daughter Sophia Charlotte of Hanover, began.

In 1900, Bertrand Russell published a critical study of Leibniz's metaphysics. Shortly thereafter, Louis Couturat published an important study of Leibniz, and edited a volume of Leibniz's heretofore unpublished writings, mainly on logic. While their conclusions, especially Russell's, were subsequently challenged and often dismissed, they made Leibniz somewhat respectable among 20th century analytical and linguistic philosophers in the English speaking world (Leibniz had already been of great influence to many Germans such as Bernhard Riemann). For example, Leibniz's phrase salva veritate, meaning interchangeability without loss of or compromising the truth, recurs in Willard Quine's writings. Nevertheless, the secondary English-language literature on Leibniz did not really blossom until after World War II. This is especially true of English speaking countries; in Gregory Brown's bibliography fewer than 30 of the English language entries were published before 1946. American Leibniz studies owe much to Leroy Loemker (1904–85) through his translations (Loemker) and his interpretive essays in (LeClerc).

Nicholas Jolley (Jolley 217–19) has surmised that Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive because:

• Work in the history of 17th and 18th century ideas has revealed more clearly the 17th century "Intellectual Revolution" that preceded the better known Industrial and commercial revolutions of the 18th and 19th centuries.
• The doctrinaire contempt for metaphysics, characteristic of analytic and linguistic philosophy, has faded;
• Analytic and contemporary philosophy continue to invoke his notions of identity, individuation, and possible worlds;
• The 17th and 18th century belief that natural science, especially physics, differs from philosophy mainly in degree and not in kind, is no longer dismissed out of hand. That modern science includes a " scholastic" as well as a "radical empiricist" element is more accepted now than in the early 20th century;
• He is now seen as a major prolongation of the mighty endeavor begun by Plato and Aristotle: the universe and man's place in it are amenable to human reason.

The University of Hannover (German spelling) is named after him.

In 1985, the German government created the Leibniz Prize, annual awards of 1.55 million Euros for experimental results, and 770,000 Euros for theoretical ones. It is the world's largest prize for scientific achievement.

Philosopher

Leibniz's philosophical thinking appears fragmented, because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and many letters to many correspondents. He wrote only two philosophical treatises, and the one he published in his lifetime, the Théodicée of 1710, is as much theological as philosophical.

Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, which he composed in 1686 as a commentary on a running dispute between Malebranche and Antoine Arnauld. This led to an extensive and valuable correspondence with Arnauld (Ariew & Garber 69, Loemker §§36,38); it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with a journal article titled "New System of the Nature and Communication of Substances" (Ariew & Garber 138, Loemker §47, Wiener II.4). Over 1695–1705, he composed his New Essays on Human Understanding, a lengthy commentary on John Locke's 1690 An Essay Concerning Human Understanding, but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie, composed in 1714 and published posthumously, consists of 90 aphorisms.

Leibniz met Spinoza in 1676, read some of his unpublished writings, and has since been suspected of appropriating some of Spinoza's ideas. While Leibniz admired Spinoza's powerful intellect, he was also forthrightly dismayed by Spinoza's conclusions, (Ariew & Garber 272–84, Loemker §§14,20,21, Wiener III.8) especially when these were inconsistent with Christian orthodoxy.

Unlike Descartes and Spinoza, Leibniz had a thorough university education in philosophy. His lifelong scholastic and Aristotelian turn of mind betrayed the strong influence of one of his Leipzig professors, Jakob Thomasius, who also supervised his BA thesis in philosophy. Leibniz also eagerly read Francisco Suárez, a Spanish Jesuit respected even in Lutheran universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle, but viewed their work through a lens heavily tinted by scholastic notions. Yet it remains the case that Leibniz's methods and concerns often anticipate the logic, and analytic and linguistic philosophy of the 20th century.

The Principles

Leibniz variously invoked one or another of seven fundamental philosophical Principles (Mates 1986: chpts. 7.3, 9):

• Identity/ contradiction. If a proposition is true, then its negation is false and vice versa.
• Identity of indiscernibles. Two things are identical if and only if they share the same properties. Frequently invoked in modern logic and philosophy.
• Sufficient reason. "There must be a sufficient reason [often known only to God] for anything to exist, for any event to occur, for any truth to obtain." (LL 717).
• Pre-established harmony. See Jolley (1995: 129–31), Woolhouse and Francks (1998), and Mercer (2001). "[T]he appropriate nature of each substance brings it about that what happens to one corresponds to what happens to all the others, without, however, their acting upon one another directly." (Discourse on Metaphysics, XIV) A dropped glass shatters because it "knows" it has hit the ground, and not because the impact with the ground "compels" the glass to split.
• Continuity. Natura non saltum facit. A mathematical analog to this principle would go as follows. If a function describes a transformation of something to which continuity applies, then its domain and range are both dense sets.
• Optimism. "God assuredly always chooses the best." (LL 311).
• Plenitude. "Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in Théodicée that this best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature's perfection." (From Plenitude.)

The second principle here is often referred to as Leibniz's Law . The Identity of Indiscernibles has attracted the most controversy and criticism, especially from corpuscular philosophy and quantum mechanics.

Leibniz would on occasion give a speech for a specific principle, but more often took them for granted. For a precis of what Leibniz meant by these and other Principles, see Mercer (2001: 473–84). For a classic discussion of Sufficient Reason and Plenitude, see Lovejoy (1957).

The monads

Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in Monadologie. Monads are to the metaphysical realm what atoms are to the physical/phenomenal. Monads are the ultimate elements of the universe. The monads are "substantial forms of being" with the following properties: they are eternal, indecomposable, individual, subject to their own laws, un-interacting, and each reflecting the entire universe in a pre-established harmony (a historically important example of panpsychism). Monads are centers of force; substance is force, while space, matter, and motion are merely phenomenal.

The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads possess no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, by virtue of the principle of pre-established harmony, each monad follows a preprogrammed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. (These "instructions" may be seen as analogs of the scientific laws governing subatomic particles.) By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, in which case free will is problematic. God, too, is a monad, and the existence of God can be inferred from the harmony prevailing among all other monads; God wills the pre-established harmony.

Monads are purported to having gotten rid of the problematic:

• Interaction between mind and matter arising in the system of Descartes;
• Lack of individuation inherent to the system of Spinoza, which represents individual creatures as merely accidental.

The monadology was thought arbitrary, even eccentric, in Leibniz's day and since.

Theodicy and optimism

The Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. It must be the best possible and most balanced world, because it was created by a perfect God. Rutherford (1998) is a detailed scholarly study of Leibniz's theodicy.

The statement that "we live in the best of all possible worlds" drew scorn, most notably from Voltaire, who lampooned it in his comic novel Candide by having the character Dr. Pangloss (a parody of Leibniz) repeat it like a mantra. Thus the adjective "panglossian", describing one so naive as to believe that the world about us is the best possible one.

The mathematician Paul du Bois-Reymond, in his "Leibnizian Thoughts in Modern Science," wrote that Leibniz thought of God as a mathematician.

"As is well known, the theory of the maxima and minima of functions was indebted to him for the greatest progress through the discovery of the method of tangents. Well, he conceives God in the creation of the world like a mathematician who is solving a minimum problem, or rather, in our modern phraseology, a problem in the calculus of variations — the question being to determine among an infinite number of possible worlds, that for which the sum of necessary evil is a minimum."

A cautious defense of Leibnizian optimism would invoke certain scientific principles that emerged in the two centuries since his death and that are now thoroughly established: the principle of least action, the conservation of mass, and the conservation of energy. In addition, the modern observations that lead to the Fine-tuned Universe arguments seem to support his view:

1. The 3+1 dimensional structure of spacetime may be ideal. In order to sustain complexity such as life, a universe probably requires three spatial and one temporal dimension. Most universes deviating from 3+1 either violate some fundamental physical laws, or are impossible. The mathematically richest number of spatial dimensions is also 3 (in the sense of topological nontriviality).
2. The universe, solar system, and Earth are the "best possible" in that they enable intelligent life to exist. Such life has evolved on Earth only because the Earth, solar system, and Milky Way possess a number of unusual characteristics; see Ward & Brownlee (2000), Morris (2003: chpts. 5,6).
3. The most sweeping form of optimism derives from the Anthropic Principle (Barrow and Tipler 1986). Physical reality can be seen as grounded in the numerical values of a handful of dimensionless constants, the best known of which are the fine structure constant and the ratio of the rest mass of the proton to the electron. Were the numerical values of these constants to differ by a few percent from their observed values, it is unlikely that the resulting universe would contain complex structures.

Our physical laws, universe, solar system, and home planet are all "best" in the sense that they enable complex structures such as galaxies, stars, and, ultimately, intelligent life. On the other hand, it is also reasonable to believe that life might be more intelligent given some other set of circumstances.

Symbolic thought

Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion:

"The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right." (The Art of Discovery 1685, W 51)

Leibniz's calculus ratiocinator, which resembles symbolic logic, can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda (many of which are translated in Parkinson 1966) that can now be read as groping attempts to get symbolic logic—and thus his calculus—off the ground. But Gerhard and Couturat did not publish these writings until modern formal logic had emerged in Frege's Begriffsschrift and in writings by Charles Peirce and his students in the 1880s, and hence well after Boole and De Morgan began that logic in 1847.

Leibniz thought symbols were important for human understanding. He attached so much importance to the invention of good notations that he attributed all his discoveries in mathematics to this. His notation for the infinitesimal calculus is an example of his skill in this regard. Charles Peirce, a 19th century pioneer of semiotics, shared Leibniz's passion for symbols and notation, and his belief that these are essential to a well-running logic and mathematics.

But Leibniz took his speculations much further. Defining a character as any written sign, he then defined a "real" character as one that represents an idea directly and not simply as the word embodying the idea. Some real characters, such as the notation of logic, serve only to facilitate reasoning. Many characters well-known in his day, including Egyptian hieroglyphics, Chinese characters, and the symbols of astronomy and chemistry, he deemed not real. (Loemker, however, who translated some of Leibniz's works into English, said that the symbols of chemistry were real characters so there is disagreement among Leibniz scholars on this point.) Instead, he proposed the creation of a characteristica universalis or "universal characteristic," built on an alphabet of human thought in which each fundamental concept would be represented by a unique "real" character.

"It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus." (Preface to the General Science, 1677. Revision of Rutherford's translation in Jolley 1995: 234. Also W I.4)

Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic, a striking anticipation of Gödel numbering. Granted, there is no intuitive or mnemonic way to number any set of elementary concepts using the prime numbers.

Because Leibniz was a mathematical novice when he first wrote about the characteristic, at first he did not conceive it as an algebra but rather as a universal language or script. Only in 1676 did he conceive of a kind of "algebra of thought," modeled on and including conventional algebra and its notation. The resulting characteristic included a logical calculus, some combinatorics, algebra, his analysis situs (geometry of situation) discussed in 3.2, a universal concept language, and more.

What Leibniz actually intended by his characteristica universalis and calculus ratiocinator, and the extent to which modern formal logic does justice to the calculus, may never be established. A good introductory discussion of the "characteristic" is Jolley (1995: 226–40). An early, yet still classic, discussion of the "characteristic" and "calculus" is Couturat (1901: chpts. 3,4).

Formal logic

Leibniz is the most important logician between Aristotle and 1847, when George Boole and Augustus De Morgan each published books that began modern formal logic. Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:

1. All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought.
2. Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, analogous to arithmetical multiplication.

With regard to (1), the number of simple ideas is much greater than Leibniz thought. As for (2), logic can indeed be grounded in a symmetrical combining operation, but that operation is analogous to either of addition or multiplication. The formal logic that emerged early in the 20th century also requires, at minimum, unary negation and quantified variables ranging over some universe of discourse.

Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts.

In his book History of Western Philosophy, Bertrand Russell went as far as claiming that Leibniz had developed logic in his unpublished writings to a level which was reached only 200 years later.

Mathematician

Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular (Struik 1969: 367). In the 18th century, "function" lost these geometrical associations.

Leibniz was the first to see that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination. Leibniz's discoveries of Boolean algebra and of symbolic logic, also relevant to mathematics, are discussed in the preceding section.

A comprehensive scholarly treatment of Leibniz's mathematical writings has yet to be written, perhaps because Series 7 of the Academy edition is very far from complete.

Calculus

Leibniz is credited, along with Isaac Newton, with the discovery of infinitesimal calculus. According to Leibniz's notebooks, a critical breakthrough occurred on November 11, 1675, when he employed integral calculus for the first time to find the area under the function y = x. He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia. This ingenious and suggestive notation for the calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684. For an English translation of this paper, see Struik (1969: 271–84), who also translates parts of two other key papers by Leibniz on the calculus. The product rule of differential calculus is still called "Leibniz's law." In addition, the theorem that tells how and when to differentiate under the integral sign is called Leibniz integral rule.

Leibniz's approach to the calculus fell well short of later standards of rigor (the same can be said of Newton's). We now see a Leibniz "proof" as being in truth mostly a heuristic hodgepodge mainly grounded in geometric intuition. Leibniz also freely invoked mathematical entities he called infinitesimals, manipulating them in ways suggesting that they had paradoxical algebraic properties. George Berkeley, in a tract called The Analyst and elsewhere, ridiculed this and other aspects of the early calculus, pointing out that natural science grounded in the calculus required just as big of a leap of faith as theology grounded in Christian revelation.

From 1711 until his death, Leibniz's life was envenomed by a long dispute with John Keill, Newton, and others, over whether Leibniz had invented the calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamentally Newton's. Hall (1980) gives a thorough scholarly discussion of the calculus priority dispute.

Modern, rigorous calculus emerged in the 19th century, thanks to the efforts of Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, and others, who based their work on the definition of a limit and on a precise understanding of real numbers. Their work discredited the use of infinitesimals to justify calculus. Yet, infinitesimals survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential. Beginning in 1960, Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory. The resulting nonstandard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning.

Topology

Leibniz was the first to use the term analysis situs (LL §27), later used in the 19th century to refer to what is now known as topology. There are two takes on this situation. On the one hand, Mates (1986: 240), citing a 1954 paper in German by Jacob Freudenthal, argues:

"Although for [Leibniz] the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer Euler, in the famous 1736 paper solving the Königsberg Bridge Problem and its generalizations, used the term geometria situs in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ...it is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics."

But Hirano (1997) argues differently, quoting Mandelbrot (1977: 419):

"...To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in 'packing,'... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In "Euclidis Prota"..., which is an attempt to tighten Euclid's axioms, he states,...: 'I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets.' This claim can be proved today."

Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: natura non facit saltus. We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole..." he was anticipating topology by more than two centuries. As for "packing," Leibniz told to his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled