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They are also not affected by time and are subject to only creation and annihilation. Using the principle of reasoning, Leibniz concluded that the first reason of all things is God.

The contingent world must have some necessary reason for its existence. Leibniz uses a geometry book as an example to explain his reasoning.

If this book was copied from an infinite chain of copies, there must be some reason for the content of the book.

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 pre-programmed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment.

By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.

The Theodicy [72] 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 an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist.

In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create the world that excluded those flaws.

Leibniz asserted that the truths of theology religion and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself.

The Theodicy is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity.

It was also shaped by Leibniz's belief in the perfectibility of human nature if humanity relied on correct philosophy and religion as a guide , and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity the natural laws identified by science.

Because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected.

Leibniz then approached one of the central criticisms of Christian theism: [74] if God is all good , all wise , and all powerful , then how did evil come into the world?

The answer according to Leibniz is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will power to act.

This predisposes humans to false beliefs, wrong decisions, and ineffective actions in the exercise of their free will. God does not arbitrarily inflict pain and suffering on humans; rather he permits both moral evil sin and physical evil pain and suffering as the necessary consequences of metaphysical evil imperfection , as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good.

Further, although human actions flow from prior causes that ultimately arise in God and therefore are known to God as metaphysical certainties, an individual's free will is exercised within natural laws, where choices are merely contingently necessary and to be decided in the event by a "wonderful spontaneity" that provides individuals with an escape from rigorous predestination.

For Leibniz, "God is an absolutely perfect being". He describes this perfection later in section VI as the simplest form of something with the most substantial outcome VI.

Along these lines, he declares that every type of perfection "pertains to him God in the highest degree" I. Even though his types of perfections are not specifically drawn out, Leibniz highlights the one thing that, to him, does certify imperfections and proves that God is perfect: "that one acts imperfectly if he acts with less perfection than he is capable of", and since God is a perfect being, he cannot act imperfectly III.

Because God cannot act imperfectly, the decisions he makes pertaining to the world must be perfect. Leibniz also comforts readers, stating that because he has done everything to the most perfect degree; those who love him cannot be injured.

However, to love God is a subject of difficulty as Leibniz believes that we are "not disposed to wish for that which God desires" because we have the ability to alter our disposition IV.

In accordance with this, many act as rebels, but Leibniz says that the only way we can truly love God is by being content "with all that comes to us according to his will" IV.

Because God is "an absolutely perfect being" I , Leibniz argues that God would be acting imperfectly if he acted with any less perfection than what he is able of III.

His syllogism then ends with the statement that God has made the world perfectly in all ways.

This also affects how we should view God and his will. In our view of God, Leibniz declares that we cannot admire the work solely because of the maker, lest we mar the glory and love God in doing so.

Instead, we must admire the maker for the work he has done II. Effectively, Leibniz states that if we say the earth is good because of the will of God, and not good according to some standards of goodness, then how can we praise God for what he has done if contrary actions are also praiseworthy by this definition II.

Leibniz then asserts that different principles and geometry cannot simply be from the will of God, but must follow from his understanding.

Leibniz wrote: " Why is there something rather than nothing? The sufficient reason 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.

Leibniz's calculus ratiocinator , which resembles symbolic logic , can be viewed as a way of making such calculations feasible.

Leibniz wrote memoranda [81] that can now be read as groping attempts to get symbolic logic—and thus his calculus —off the ground.

These writings remained unpublished until the appearance of a selection edited by C. Gerhardt Couturat published a selection in ; by this time the main developments of modern logic had been created by Charles Sanders Peirce and by Gottlob Frege.

Leibniz thought symbols were important for human understanding. He attached so much importance to the development of good notations that he attributed all his discoveries in mathematics to this.

His notation for calculus is an example of his skill in this regard. 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.

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.

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 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 , 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 calculus, may never be established.

Leibniz has been noted as one of the most important logicians between the times of Aristotle and Gottlob Frege. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:.

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

Russell's principal work on Leibniz found that many of Leibniz's most startling philosophical ideas and claims e. He regarded such relations as real qualities of things Leibniz admitted unary predicates only : For him, "Mary is the mother of John" describes separate qualities of Mary and of John.

This view contrasts with the relational logic of De Morgan , Peirce , Schröder and Russell himself, now standard in predicate logic.

Notably, Leibniz also declared space and time to be inherently relational. Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in and , to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa , ordinate , tangent , chord , and the perpendicular.

Leibniz also believed that the sum of an infinite number of zeros would equal to one half using the analogy of the creation of the world from nothing.

Leibniz's discoveries of Boolean algebra and of symbolic logic , also relevant to mathematics, are discussed in the preceding section.

The best overview of Leibniz's writings on calculus may be found in Bos Leibniz arranged the coefficients of a system of linear equations into an array, now called a matrix , in order to find a solution to the system if it existed.

Leibniz laid down the foundations and theory of determinants , although Seki Takakazu discovered determinants well before Leibniz.

Finding the determinant of a matrix using this method proves impractical with large n , requiring to calculate n! This method for solving systems of linear equations based on determinants was found in by Leibniz Cramer published his findings in Leibniz wrote that circles "can most simply be expressed by this series, that is, the aggregate of fractions alternately added and subtracted".

Leibniz is credited, along with Sir Isaac Newton , with the discovery of calculus differential and integral calculus. Leibniz did not publish anything about his calculus until The concept became more transparent as developed through Leibniz's formalism and new notation.

In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule.

Leibniz exploited infinitesimals in developing calculus, manipulating them in ways suggesting that they had paradoxical algebraic properties.

A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms.

From until his death, Leibniz was engaged in a dispute with John Keill, Newton and others, over whether Leibniz had invented calculus independently of Newton.

This subject is treated at length in the article Leibniz—Newton calculus controversy. The use of infinitesimals in mathematics was frowned upon by followers of Karl Weierstrass , [] [] but survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential.

Beginning in , Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory , in the context of a field of hyperreal numbers.

The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning.

Robinson's transfer principle is a mathematical implementation of Leibniz's heuristic law of continuity , while the standard part function implements the Leibnizian transcendental law of homogeneity.

Leibniz was the first to use the term analysis situs , [] 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, citing a 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 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. But Hideaki Hirano argues differently, quoting Mandelbrot : [].

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 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.

Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: Natura non facit saltus.

As for "packing", Leibniz told 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 with three even smaller circles by the same procedure.

This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.

Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge.

Much of his writing on physics is included in Gerhardt's Mathematical Writings. Leibniz contributed a fair amount to the statics and dynamics emerging around him, often disagreeing with Descartes and Newton.

He devised a new theory of motion dynamics based on kinetic energy and potential energy , which posited space as relative, whereas Newton was thoroughly convinced that space was absolute.

An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense.

For instance, he anticipated Albert Einstein by arguing, against Newton, that space , time and motion are relative, not absolute: "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions.

Leibniz held a relationist notion of space and time, against Newton's substantivalist views. Leibniz's relationism, in contrast, describes space and time as systems of relations that exist between objects.

The rise of general relativity and subsequent work in the history of physics has put Leibniz's stance in a more favorable light.

One of Leibniz's projects was to recast Newton's theory as a vortex theory. The principle of sufficient reason has been invoked in recent cosmology , and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense.

Those who advocate digital philosophy , a recent direction in cosmology, claim Leibniz as a precursor. In addition to his theories about the nature of reality, Leibniz's contributions to the development of calculus have also had a major impact on physics.

Leibniz's vis viva Latin for "living force" is m v 2 , twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter.

His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibniz's idea.

In reality, both energy and momentum are conserved, so the two approaches are equally valid. By proposing that the earth has a molten core, he anticipated modern geology.

In embryology , he was a preformationist, but also proposed that organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers.

In the life sciences and paleontology , he revealed an amazing transformist intuition, fueled by his study of comparative anatomy and fossils.

One of his principal works on this subject, Protogaea , unpublished in his lifetime, has recently been published in English for the first time.

He worked out a primal organismic theory. Psychology had been a central interest of Leibniz. His discussions in the New Essays and Monadology often rely on everyday observations such as the behaviour of a dog or the noise of the sea, and he develops intuitive analogies the synchronous running of clocks or the balance spring of a clock.

Bodies act according to the laws of efficient causes, i. And these two realms, that of efficient causes and that of final causes, harmonize with one another.

Leibniz found his most important interpreter in Wilhelm Wundt , founder of psychology as a discipline.

Wundt used the "… nisi intellectu ipse" quotation on the title page of his Beiträge zur Theorie der Sinneswahrnehmung Contributions on the Theory of Sensory Perception and published a detailed and aspiring monograph on Leibniz [] Wundt shaped the term apperception , introduced by Leibniz, into an experimental psychologically based apperception psychology that included neuropsychological modelling — an excellent example of how a concept created by a great philosopher could stimulate a psychological research program.

He believed that by the principle that phenomena found in nature were continuous by default, it was likely that the transition between conscious and unconscious states had intermediary steps.

His theory regarding consciousness in relation to the principle of continuity can be seen as an early theory regarding the stages of sleep.

In this way, Leibniz's theory of perception can be viewed as one of many theories leading up to the idea of the unconscious.

In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine.

He worked to set up a coherent medical training program, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance program, and discussed the balance of trade.

He even proposed something akin to what much later emerged as game theory. In sociology he laid the ground for communication theory.

In , Garland published a volume of Leibniz's writings bearing on his many practical inventions and engineering work.

To date, few of these writings have been translated into English. Nevertheless, it is well understood that Leibniz was a serious inventor, engineer, and applied scientist, with great respect for practical life.

Following the motto theoria cum praxi , he urged that theory be combined with practical application, and thus has been claimed as the father of applied science.

He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc.

With Denis Papin , he created a steam engine. He even proposed a method for desalinating water.

From to , he struggled to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains , but did not succeed.

Leibniz may have been the first computer scientist and information theorist. Leibniz interpreted a diagram which showed yin and yang and corresponded it to a zero and one.

Leibniz may have plagiarized Juan Caramuel y Lobkowitz and Thomas Harriot , who independently developed the binary system, as he was familiar with their works on the binary system.

His calculus ratiocinator anticipated aspects of the universal Turing machine. In , Norbert Wiener suggested that Leibniz should be considered the patron saint of cybernetics.

In , Leibniz began to invent a machine that could execute all four arithmetic operations, gradually improving it over a number of years.

This " stepped reckoner " attracted fair attention and was the basis of his election to the Royal Society in A number of such machines were made during his years in Hanover by a craftsman working under his supervision.

They were not an unambiguous success because they did not fully mechanize the carry operation. Couturat reported finding an unpublished note by Leibniz, dated , describing a machine capable of performing some algebraic operations.

Leibniz was groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace. In , while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards.

At this library, Leibniz focused more on advancing the library than on the cataloging. For instance, within a month of taking the new position, he developed a comprehensive plan to expand the library.

After working at this library, by the end of Leibniz was appointed as privy-councilor and librarian of the Bibliotheca Augusta at Wolfenbuettel.

It was an extensive library with at least 25, printed volumes. He was not allowed to make complete changes to the existing closed catalog, but was allowed to improve upon it so he started on that task immediately.

He created an alphabetical author catalog and had also created other cataloging methods that were not implemented. While serving as librarian of the ducal libraries in Hanover and Wolfenbuettel , Leibniz effectively became one of the founders of library science.

He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University.

He also called on publishers to distribute abstracts of all new titles they produced each year, in a standard form that would facilitate indexing.

He hoped that this abstracting project would eventually include everything printed from his day back to Gutenberg. Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library.

He called for the creation of an empirical database as a way to further all sciences. His characteristica universalis , calculus ratiocinator , and a "community of minds"—intended, among other things, to bring political and religious unity to Europe—can be seen as distant unwitting anticipations of artificial languages e.

Leibniz emphasized that research was a collaborative endeavor. More specifically, in his correspondence and travels he urged the creation of such societies in Dresden, Saint Petersburg , Vienna, and Berlin.

Only one such project came to fruition; in , the Berlin Academy of Sciences was created. Leibniz drew up its first statutes, and served as its first President for the remainder of his life.

That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works.

With the possible exception of Marcus Aurelius , no philosopher has ever had as much experience with practical affairs of state as Leibniz.

Leibniz's writings on law, ethics, and politics [] were long overlooked by English-speaking scholars, but this has changed of late.

While Leibniz was no apologist for absolute monarchy like Hobbes , or for tyranny in any form, neither did he echo the political and constitutional views of his contemporary John Locke , views invoked in support of liberalism, in 18th-century America and later elsewhere.

The following excerpt from a letter to Baron J. Boyneburg's son Philipp is very revealing of Leibniz's political sentiments:.

As for I am, however, quite of the opinion of Grotius , that one ought to obey as a rule, the evil of revolution being greater beyond comparison than the evils causing it.

Yet I recognize that a prince can go to such excess, and place the well-being of the state in such danger, that the obligation to endure ceases.

This is most rare, however, and the theologian who authorizes violence under this pretext should take care against excess; excess being infinitely more dangerous than deficiency.

In , Leibniz called for a European confederation, governed by a council or senate, whose members would represent entire nations and would be free to vote their consciences; [] this is sometimes considered an anticipation of the European Union.

He believed that Europe would adopt a uniform religion. He reiterated these proposals in But at the same time, he arrived to propose an interreligious and multicultural project to create a universal system of justice, which required from him a broad interdisciplinary perspective.

In order to propose it, he combined linguistics especially sinology , moral and legal philosophy, management, economics, and politics.

Leibniz devoted considerable intellectual and diplomatic effort to what would now be called ecumenical endeavor, seeking to reconcile first the Roman Catholic and Lutheran churches, and later the Lutheran and Reformed churches.

In this respect, he followed the example of his early patrons, Baron von Boyneburg and the Duke John Frederick —both cradle Lutherans who converted to Catholicism as adults—who did what they could to encourage the reunion of the two faiths, and who warmly welcomed such endeavors by others.

The House of Brunswick remained Lutheran, because the Duke's children did not follow their father.

He evidently thought that the thoroughgoing application of reason would suffice to heal the breach caused by the Reformation.

Leibniz the philologist was an avid student of languages, eagerly latching on to any information about vocabulary and grammar that came his way.

He refuted the belief, widely held by Christian scholars in his day, that Hebrew was the primeval language of the human race.

He also refuted the argument, advanced by Swedish scholars in his day, that a form of proto- Swedish was the ancestor of the Germanic languages.

He puzzled over the origins of the Slavic languages and was fascinated by classical Chinese. Leibniz was also an expert in the Sanskrit language.

He published the princeps editio first modern edition of the late medieval Chronicon Holtzatiae , a Latin chronicle of the County of Holstein.

Leibniz was perhaps the first major European intellectual to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other works by, European Christian missionaries posted in China.

He apparently read Confucius Sinarum Philosophus in the first year of its publication. He mulled over the possibility that the Chinese characters were an unwitting form of his universal characteristic.

He noted with fascination how the I Ching hexagrams correspond to the binary numbers from to , and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.

Leibniz's attraction to Chinese philosophy originates from his perception that Chinese philosophy was similar to his own. Hughes suggests that Leibniz's ideas of "simple substance" and "pre-established harmony" were directly influenced by Confucianism, pointing to the fact that they were conceived during the period when he was reading Confucius Sinarum Philosophus.

While making his grand tour of European archives to research the Brunswick family history that he never completed, Leibniz stopped in Vienna between May and February , where he did much legal and diplomatic work for the Brunswicks.

He visited mines, talked with mine engineers, and tried to negotiate export contracts for lead from the ducal mines in the Harz mountains.

His proposal that the streets of Vienna be lit with lamps burning rapeseed oil was implemented. During a formal audience with the Austrian Emperor and in subsequent memoranda, he advocated reorganizing the Austrian economy, reforming the coinage of much of central Europe, negotiating a Concordat between the Habsburgs and the Vatican , and creating an imperial research library, official archive, and public insurance fund.

He wrote and published an important paper on mechanics. Leibniz also wrote a short paper, Primae veritates , first published by Louis Couturat in pp.

The paper is undated; that he wrote it while in Vienna in was determined only in , when the ongoing critical edition finally published Leibniz's philosophical writings for the period — But after a meticulous study of all of Leibniz's philosophical writings up to —a study the additions to the critical edition made possible—Mercer begged to differ with Couturat's reading; the jury is still out.

When Leibniz died, his reputation was in decline. Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description.

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.

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 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 calculus and Leibniz's charge that Newton's theory of universal gravitation was incorrect.

Leibniz's long march to his present glory began with the publication of the Nouveaux Essais , which Kant read closely. In , Louis 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.

In , Bertrand Russell published a critical study of Leibniz's metaphysics. 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.

This is especially true of English speaking countries; in Gregory Brown's bibliography fewer than 30 of the English language entries were published before American Leibniz studies owe much to Leroy Loemker — through his translations and his interpretive essays in LeClerc Nicholas Jolley has surmised that Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive.

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.

In , the German government created the Leibniz Prize , offering an annual award of 1. It was the world's largest prize for scientific achievement prior to the Fundamental Physics Prize.

Leibniz mainly wrote in three languages: scholastic Latin , French and German. 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 , when Bodemann completed his catalogue of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's Nachlass become clear: about 15, letters to more than recipients plus more than 40, other items.

Moreover, quite a few of these letters are of essay length. Much of his vast correspondence, especially the letters dated after , remains unpublished, and much of what is published has appeared only in recent decades.

The amount, variety, and disorder of Leibniz's writings are a predictable result of a situation he described in a letter 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.

The extant parts of the critical edition [] of Leibniz's writings are organized as follows:. The systematic cataloguing of all of Leibniz's Nachlass began in It was hampered by two world wars and then by decades of German division into two states with the Cold War's "iron curtain" in between, separating scholars, and also scattering portions of his literary estates.

The ambitious project has had to deal with writings in seven languages, contained in some , written and printed pages.

In it was reorganized and included in a joint program of German federal and state Länder academies. Since then the branches in Potsdam , Münster , Hanover and Berlin have jointly published 57 volumes of the critical edition, with an average of pages, and prepared index and concordance works.

Six important collections of English translations are Wiener , Parkinson , Loemker , Ariew and Garber , Woolhouse and Francks , and Strickland The ongoing critical edition of all of Leibniz's writings is Sämtliche Schriften und Briefe.

Leibniz is receiving popular attention. The Google Doodle for July 1, celebrated Leibniz's nd birthday. One of the earliest popular but indirect expositions of Leibniz was Voltaire 's satire Candide , published in Leibniz was lampooned as Professor Pangloss, described as "the greatest philosopher of the Holy Roman Empire ".

Leibniz also appears as one of the main historical figures in Neal Stephenson 's series of novels The Baroque Cycle.

Stephenson credits readings and discussions concerning Leibniz for inspiring him to write the series. An updated bibliography of more than From Wikipedia, the free encyclopedia.

German mathematician and philosopher. For other uses, see Leibniz disambiguation. Portrait by Christoph Bernhard Francke.

Alte Nikolaischule — Leipzig University — B. This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources.

Unsourced material may be challenged and removed. Further information: Best of all possible worlds and Philosophical optimism.

Main article: Algebraic logic. Stanford Encyclopedia of Philosophy. Retrieved 19 August Schmaltz eds. Maurice and Walter Rauschenbusch PhD thesis.

Montreal: McGill University. Retrieved 6 February Mannheim: Bibliographisches Institut GmbH. History of Western Philosophy: Collectors Edition revised ed.

Extract of page Leibniz-Nachlass i. From Plato to Derrida. Although there is some reason to be skeptical of the details, the spirit of the self-portrait Leibniz paints to Nicolas Remond in can be a helpful guide for approaching his work.

He writes:. Again, there is some reason to doubt whether Leibniz was really fifteen when he made his philosophical perambulations and whether and to what extent he had actually read any of the moderns.

Nevertheless, this self-portrait does express something that one sees in Leibniz's writings: the weaving together of varying strands of ancient and modern philosophy in a remarkably creative and sophisticated manner.

The letter to Remond makes clear that Leibniz had reservations about certain aspects of the modern philosophy, qualms that arose from and led him back to this eclectic mix of Aristotle and Christian Platonism.

It is probably most helpful, then, to see Leibniz's philosophy as a reaction to two sets of modern opponents: on the one hand, Descartes and his followers; on the other hand, Hobbes and Spinoza.

Leibniz's critique of Descartes and his followers was focused principally on the Cartesian account of body or corporeal substance.

According to Descartes, the essence of body is extension; that is, a corporeal substance is simply a geometric object made concrete, an object that has size and shape and is in motion.

This view, indeed, is the cornerstone of the new mechanical philosophy to which Leibniz was originally attracted.

Nevertheless, Leibniz came to see two distinct problems with this view. First, in claiming that the essence of body is extension, Descartes is endorsing the view that matter is infinitely divisible.

But if matter is infinitely divisible, then one can never arrive at the simple unities that must exist at some ontological ground level.

Second, if matter is simply extension, then there is in its nature no source of activity. If this is so, Leibniz thought, then the bodily objects of the world cannot count as substances.

Hobbes and Spinoza, despite their own differences, advanced, or were read as advancing, a number of objectionable and deeply troubling theses which Leibniz and most of his contemporaries saw as an enormous threat: materialism, atheism, and necessitarianism.

It is Leibniz's response to Hobbesian and Spinozistic necessitarianism that is perhaps of greatest interest, for he sought to develop an account of action and contingency that would preserve divine and human freedom.

As will be shown, central to Leibniz's philosophy was the view that God freely chose the best world from an infinite number of possible worlds and that a person could be said to act freely when the contrary of that action does not imply a contradiction.

This topic will be addressed principally in the article on Leibniz's Modal Metaphysics. To these two great principles could be added four more: the Principle of the Best , the Predicate-in-Notion Principle , the Principle of the Identity of Indiscernibles , and the Principle of Continuity.

The relation among these principles is more complicated than one might expect. And while the Principle of the Identity of Indiscernibles is often presented in contemporary discussions in analytic metaphysics as a stand-alone axiom, Leibniz tells us that it follows from the two great principles.

Finally, the Principle or Law of Continuity is actually a principle that Leibniz takes from his work in mathematics and applies to the infinite hierarchy of monads in the world and to the quality of their perceptions; it appears to derive only tenuous support from the Principle of Sufficient Reason.

Leibniz presented a number of arguments for the existence of God, which represent great contributions to philosophical theology and which will be discussed below.

But one of the most basic principles of his system is that God always acts for the best. See Adams And Leibniz sometimes implicitly, sometimes explicitly, appeals to this principle in his metaphysics, most notably when he is also employing the Principle of Sufficient Reason.

Leibniz has a very distinctive notion of truth, one which underlies much of his metaphysics. But this notion of truth goes back to Aristotle's Organon cf.

Posterior Analytics I. As he tells us in the Primary Truths and the Discourse on Metaphysics , many things follow from the Predicate-in-Notion Principle PIN , including what he believes to be the correct analysis of necessity and contingency.

Leibniz also follows Aristotle cf. Metaphysics IV. Furthermore, the combination of PC and PIN will mean that, since in any true proposition the predicate is contained explicitly or implicitly within the subject, this is so for all affirmative truths, whether they be universal or particular, necessary or contingent.

Leibniz will use this seemingly innocuous principle to draw profoundly strong metaphysical conclusions about the nature of substance and modality.

The Principle of Sufficient Reason PSR in its classic form is simply that nothing is without a reason nihil est sine ratione or there is no effect without a cause.

In the Principles of Nature and Grace , Leibniz suggests that the claim that nothing takes place without a sufficient reason means that nothing happens in such a way that it is impossible for someone with enough information to give a reason why it is so and not otherwise.

While the idea that every event must have a cause and that there is a reason why everything is so and not otherwise again might not seem novel, it is the connection that Leibniz sees between this principle and his other metaphysical principles that is noteworthy.

According to Leibniz, PSR must actually follow from PIN, for if there were a truth that had no reason, then there would be a proposition whose subject did not contain the predicate, which is a violation of Leibniz's conception of truth.

See also the entry on identity of indiscernibles. What is particularly important to note, however, is that Leibniz is adamant that certain kinds of properties are excluded from the list of properties that could count as difference-making properties, chief among these spatio-temporal properties.

This is what Leibniz means in part when he asserts that there can be no purely extrinsic i. Therefore, it is not the case that there could be two chunks of matter that are qualitatively identical but existing in different locations.

In Leibniz's view, any such extrinsic difference must be founded on an intrinsic difference. As he puts it in the New Essays ,.

Sometimes, unfortunately, only the Principle of the Indiscernibility of Identicals is so called. Briefly, one way to sketch the argument is this:.

Now, it was said above that Leibniz excludes purely extrinsic denominations or relational properties from the kinds of properties that are constitutive of an individual.

To allow purely extrinsic denominations would be to accept the possibility that that two things could be discernible in terms of their relational properties while being identical in terms of their intrinsic properties, for their relational properties would not follow from their intrinsic properties.

Again, if relational properties were allowed to factor into the nature of an individual, then PII would be relatively weak.

Of course two things that exist in different spatio-temporal locations are distinct, and that is what Leibniz admits in the passage from the New Essays above.

But if we follow Leibniz in excluding such relational properties as difference-making properties and reflect on the above argument, then we see that worlds are distinguished in terms of intrinsic properties of individuals and that this difference has a bearing on the relative greatness or perfection of a world.

We could not. There must be a reason why a is here and b is there , and this reason has to do with the intrinsic properties of a and b.

In other words, even the relational properties must be somehow derivative of the intrinsic properties of substances. As we shall see, Leibniz employs this principle in a range of arguments: against the mind as a tabula rasa , against atomism, against Newtonian absolute space, and so on.

For more on this subject, consult the entry on identity of indiscernibles. The first concerns human freedom, the latter the composition of the continuum.

Leibniz, however, thought that he had found the way out of each labyrinth, and his solution to the problem of the continuum is related ultimately to a maxim or law that he employs not only in his mathematical writings but also in his metaphysics.

More exactly, Leibniz believes that this law or principle implies that any change passes through some intermediate change and that there is an actual infinity in things.

For Leibniz, the fundamental questions of metaphysics were reducible to questions of ontology: What is there? What are the most basic components of reality?

What grounds what? In a certain sense, his answer remained constant throughout his life: everything is composed of or reducible to simple substances ; everything is grounded in simple substances.

There he claims that the Aristotelian idea that a substance is that which is the subject of predication and which cannot be predicated of something else is insufficient for a true analysis of the nature of substance.

He next appeals to the PC and the PIN: in every true predication, the concept of the predicate is contained in the concept of the subject.

In other words, x is a substance if and only if x has a complete individual concept CIC , that is, a concept that contains within it all predicates of x past, present, and future.

The CIC, then, serves to individuate substances; it is able to pick out its bearer from an infinity of other finite created substances.

Leibniz gives as an example Alexander the Great. The concept of Alexander contains being a King, being a student of Aristotle, conquering Darius and Porus, and so on.

Leibniz's invocation of the Scotist notion of a haecceity is intriguing. What Leibniz is telling us is that Alexander's thisness is determined by the sum of his qualitative properties.

Moreover, we can see a metaphysical aspect to this logical conception of substance: the complete individual concept of a substance is the notion or essence of the substance as it known by the divine understanding.

The doctrine of marks and traces, therefore, claims that, because the CIC contains all predicates true of a substance past, present, and future, the entire history of the universe can be read if only by God in the essence of any individual substance.

The consequences that Leibniz draws from the logical conception of substance and the doctrine of marks and traces are remarkable. Unfortunately, Leibniz's reasons for drawing these consequences are not in all cases obvious.

Why should PII follow from the complete individual concept account of substance? If we consider the CIC as that which allows us to pick out and individuate any individual substance from an infinity of substances, then we realize that, if the individual concepts of two substances, a and b , do not allow us or God to distinguish the one from the other, then their individual concepts are not complete.

That is, there must always be a reason, found within the complete individual concept of substances and issuing from the free decree of God, that a is discernible from b.

And this fact points to another important fact about the interpretation suggested above: it is not only the case that each substance has a complete individual concept—the essence of the substance as it exists in the divine mind—but for every essence or complete individual concept there is one and only one substance in a world.

The argument here is essentially that which was given above in the section describing the relation between PSR and PII; namely, what reason could God have had for instantiating two substances with identical CICs?

Further, why should it be the case that substances can only arise naturally in God's creation of the world and end in his annihilation?

If one takes quite literally Leibniz's claim that the CIC contains within it all predicates true of the substance past, present, and future, then one might be able to say that this must include truths extending back to the creation and forward either infinitely or to the end of time.

This argument might be somewhat weak in itself, but it certainly would seem to follow from Leibniz's logical notion of substance and one of the other consequences, namely, that each substance is a mirror of the entire universe.

If this is the case, then a substance, insofar as it is a mirror of the entire universe, must have within its complete individual concept predicates that extend back to creation and forward in time.

At first glance, it is also not readily apparent merely from the CIC and doctrine of marks and traces why a substance cannot be constructed from two substances or be divided into two new substances.

Let substance x have within its complete individual concept predicates g, h, i… which are true of it past, present and future. One might imagine that both new substances would have all of x 's pre-division predicates in common and unique predicates thereafter.

But the relevant part of Leibniz's logical notion of substance is that the CIC is sufficiently rich to allow us or God to deduce from it all predicates past, present and future.

Leibniz's implicit suggestion is that the pre-division predicates would not allow the logical deduction of branching or divided substances.

A similar argument works against the possibility of the fusion of two substances. Since substances can only naturally arise during God's creation of the world and since substances cannot undergo fusion or fission, it is obvious that the number of substances must remain constant.

Finally, if it is the case that it is of the nature of a substance to have a notion so complete that one can deduce from it all its predicates past, present, and future and if substances exist from the creation of the world, then it would seem relatively natural to conclude that each substance contains within it a kind of story of the entire universe from its own particular perspective.

While more will be said below, what Leibniz is suggesting here is a set of doctrines that he will develop in greater detail: the worlds apart doctrine, the mirroring or expression thesis , and the doctrine of universal harmony.

Another notable consequence of the logical conception of substance is the denial of the causal interaction of finite substances.

Not only is it the case, Leibniz claims, that genuine physical influx — the transfer of some property within one substance to a second substance — is inexplicable, but more important the logical conception of substance shows us that the reasons for any property that a substance may have are already contained within its CIC.

In other words, every state of a substance is explained, grounded, or caused by its own notion or CIC. Of course, the ground or reason for the existence or actuality of any particular substance is to be found in God and his free choice of worlds.

A more detailed account of Leibniz's views on causation is available in the entry Leibniz on Causation. As we shall see below, the denial of the causal interaction of substances forms an essential premise of Leibniz's argument for pre-established harmony.

That is, what kind of thing could have such a CIC or such a nature? Leibniz's answer to this question brings to the fore another paradigm of substancehood: unity.

While it is the nature of an individual substance to have a CIC, only a genuine unity can qualify as a substance. In later years, the Scholastic way of speaking fades away, but the fundamental idea remains the same: there must be something that guarantees or makes possible the unity of a substance, and this is the substantial form or the soul.

The point Leibniz wants to make is that only a soul or a substantial form is the kind of thing that can be said to have or underlie a complete individual concept, for only a soul or substantial form is by its nature an imperishable unity.

Thus, unity is the hallmark of a genuine substance, but equally important is Leibniz's paradigm case of a substance: the self.

This thought underlies much of Leibniz's reflections on the nature of substance and has important consequences.

Material atoms, as advocated by Democritus in the classical period and by Gassendi and others in the seventeenth century, are excluded, Leibniz thinks, because they violate PII; that is, two purely material atoms would seem to be qualitatively identical and yet distinct, which is impossible if one accepts PC, PSR, and the derivation of PII.

Leibniz is not as clear as one would like him to be, for at this point in his career it is possible to read him as seeing that something is a substance so long as it has a soul or a substantial form, whereas later in his career it seems more clearly to be the case that the only substances are souls or soul-like entities, the monads.

In other words, Leibniz can be interpreted as advocating, at least in this period, a kind of Aristotelian hylomorphism, in which substances are composites of matter and form.

This has been the subject of debate in the field, but this entry cannot adjudicate the matter. For more on this dispute, see Look Nevertheless, in declaring that a substance is necessarily indivisible, Leibniz renders it impossible for a body , or matter alone, to be a substance.

Thus, Cartesian corporeal substance , the essence of which is simply extension, cannot exist as substance.

Put differently, Leibniz's argument is that nothing that is divisible is a substance; a Cartesian chunk of matter is divisible; therefore, a Cartesian chunk of matter is not a substance.

This points to the first part of Leibniz's critique of the Cartesianism mentioned above: namely, that according to Leibniz, Cartesian matter fails to have the unity required of a genuine substance.

Indeed, in the Correspondence with Arnauld, Leibniz considers the case of a human body deprived of a soul and says the body, or cadaver, would not be a substance at all but merely an aggregate of substances.

Aggregates of simple substances, therefore, have a different ontological status from simple substances. The distinction between simple substances and aggregates becomes an important one in Leibniz's philosophy.

If this is the case, then aggregates of simple substances are merely phenomena and fail to have the reality of the underlying simples.

Further, the bodies of natural philosophy, the bodies of the world we observe around us, would seem to be in some sense mere phenomena.

While some scholars of Leibniz's thought have suggested this, it does not get at the full story of Leibniz's metaphysical system.

The distinction that Leibniz draws is one between a real unity and a phenomenal unity, or as he also puts it, between a unum per se and a unum per aggregationem.

Leibniz's favorite comparison in the case of the latter is to a rainbow: bodies, for example, fail to have intrinsic unity, but we do represent them as being single and unified objects much as we represent a rainbow as being one thing when it is in fact merely the result of the refraction of light through innumerable water droplets.

But just as the rainbow results from the presence of genuine unities, the water droplets to continue the metaphor, even if this is not true when speaking with Leibniz in metaphysical rigor , so do the bodies of the natural world result from the genuine simple substances.

Put differently, the simple substances ground the phenomena of bodies in the world. But insofar as the bodies of the natural world are well-founded phenomena — that is, insofar as they are grounded in the simple substances — they are not simply phenomena as in Berkeley's philosophy.

This view is also not uncontroversial. To compare Leibniz with Berkeley, see the entry on Berkeley. The second part of Leibniz's critique of the Cartesian doctrine of corporeal substance relates to the notion of activity.

According to Leibniz, substances are not only essentially unities, but also active. But Cartesian corporeal substance, insofar as its essence is extension, cannot be itself a source of activity.

First, Leibniz holds that this is so because he adheres to the classical and Scholastic idea that actions pertain to supposita ; that is, only something that can be the subject of predication can be active, and only true unities can be genuine subjects of predication and not mere phenomena.

Put differently, Cartesian extended stuff cannot, insofar as it is infinitely divisible, constitute a suppositum , or subject of predication.

But, second, Leibniz believes that something is active if and only if the source of its activity can arise within itself, that is, if and only if its activity arises spontaneously from within itself.

This is another reason, then, that individual substances will be understood as mind-like, for Leibniz believes that only minds or mind-like things can originate and alter their modifications.

In saying that substances are essentially active, Leibniz means that they are endowed with forces. The idea here again sounds Aristotelian: a substance has a certain essentially active component, the soul or substantial form or first entelechy, and a passive component, primary matter.

Since simple substances are minds, their modifications are representations or perceptions, and the activity of the simple substance will relate to the change or succession of its perceptions.

One way to think of this is that each substance has a unique series of perceptions programmed by God to play in harmony with all other substances, and the internal tendency of a substance to move from perception to perception is its active force, or what Leibniz also calls appetite or appetition.

While separate entries detail Leibniz's account of causation and his account of the mind, it will still be useful to provide a short exegesis of Leibniz's celebrated solution to the mind-body problem which Leibniz had inherited from Descartes and his followers.

The problem, briefly, is this: if mind is essentially thought and nothing else , and body is essentially extension, then how can mind and body interact or form a unity as we know from experience they must?

Or how do thinking substance and extended substance unite in the substance of a human being? Leibniz answers this question by, first, denying the possibility of the causal interaction of finite substances.

In this way, Leibniz undermines Cartesian dualism because it takes as a premise the idea that mind-body interaction is to be explained by the influence of the one on the other via the pineal gland.

In one of Leibniz's best-known metaphors, he asks his readers to imagine the mind and body as two pendula hanging from a beam.

Whence comes their agreement? One could imagine that the motion of the one is communicated through the wooden beam to the other, thus causing them eventually to swing harmoniously the theory of influx.

Or one could imagine that God intervenes and moves the pendula, guaranteeing their synchronicity the theory of occasionalism.

Or, Leibniz says, one could imagine that God, the supreme artificer, created the world and the pendula so perfectly that, by their own natures, they would swing in perfect harmony.

Naturally, it is this last thesis that Leibniz endorses and asks his readers to endorse as well. More precisely, Leibniz argues that God created the world so perfectly that each substance acts according to its own law of unfolding and is at the same time in perfect harmony with all others substances; further, that the mind has a distinct point of view of the world by virtue of its being the center of some mass body , and that the law of unfolding of the mind is in accord with the laws of the corporeal machine.

He puts this most succinctly in his essay, A New System of Nature , in which he effectively presents a five-step argument for pre-established harmony :.

Now, when Leibniz speaks in metaphysical rigor, he denies the underlying premise of Cartesian dualism: body is not a substance; so there can be no question of how it qua substance interacts with or is related to the mind, or thinking substance.

Nevertheless, Leibniz was able to express his view for the vulgar — that is, for those expecting a Cartesian metaphysics — by saying that the mind and body can be said to form a union and interact insofar as the mind follows its laws, the body follows its laws, and they are in perfect harmony.

The body and soul are not united to each other in the sense that Descartes had suggested, but the perceptions and appetitions of the soul will arise spontaneously from its own stores and will correspond to the actions of the body as well as to the events of the world.

In other words, while the perceptions and appetitions of the mind or soul will be independent of the body, they will nevertheless correspond precisely to the actions of the particular body to which it is attached and be in perfect conformity with all the other substances of the world.

On Leibniz's view, to individual substances there belong only perceptions and appetitions, and these perceptions and appetitions can be understood to form a series within the individual substance.

In other words, every individual substance can be thought to have a set of perceptions and appetitions such that one could say that, at any given time, a particular substance was experiencing such-and-such a perception and such-and-such an appetition.

In fact, the position is more complex; for, as will be shown in a subsequent section, the mind has at any moment an infinity of petites perceptions within it, perceptions of everything that is occurring in the universe, but the human mind at least will be truly aware of one thing at a time.

For example, the reader of this article could be said to have a temporally-ordered series of perceptions — with t 1 corresponding to the first sentence, t 2 the second sentence, etc.

Moreover, the series of perceptions and appetitions are generated from within the individual substance itself. That is, Leibniz speaks as if perceptions and appetitions follow naturally from prior perceptions and appetitions — and it is in this respect, after all, that a finite individual substance is causally independent from all other finite created substances.

The crucial idea is that the body will follow its own laws, the mind its own laws, and there will be no true influence between the two.

The mind and body thus seem to constitute, as it were, worlds apart, as indeed Leibniz claims later when he explains the world in terms of monads, and these worlds apart are, according to Leibniz, unified solely by virtue of the correspondence of their actions and perceptions.

Further, to these separate realms there will apply two distinct means of explaining the events of the world: we may explain things according to the final causes of the mind or according to the efficient causes of the body or of bodies in general.

Thus, not only do the mind and body each seem to follow a different set of laws, but the world, according to Leibniz, can be described in terms of either set of laws.

Leibniz's account of the pre-established harmony of mind and body is part of a more general position in his metaphysics: the existence of parallel modes of explanation.

As we saw above, Leibniz believes that the mind will act according to its laws and the body according to its laws and the two will be in harmony.

But Leibniz also believes that the mind or soul operates for particular ends and that therefore its actions are explicable in terms of final causes , whereas the actions of the body, purely instances of matter in motion according to the claims of the mechanical philosophy, are to be explained in terms of efficient causes.

According to this system, bodies act as if there were no souls though this is impossible ; and souls act as if there were no bodies; and both act as if each influenced the other.

Though Leibniz speaks here of the kingdoms of power and wisdom, the two-tiered explanatory approach — the phenomena of the natural world explained through efficient causes and the actions of the mind explained through final causes — leads to the distinction between what he more commonly calls the kingdom of nature and the kingdom of grace.

Thus far we have seen that Leibniz rejected the Cartesian account of matter, according to which matter, the essence of which is extension, could be considered a substance.

Leibniz held instead that only beings endowed with true unity and capable of action can count as substances.

The ultimate expression of Leibniz's view comes in his celebrated theory of monads, in which the only beings that will count as genuine substances and hence be considered real are mind-like simple substances endowed with perception and appetite.

What was said above concerning the unity and activity of simple substance should suffice to explain Leibniz's reasons for holding such a position.

Now a fuller version of Leibniz's idealism must be presented. According to Leibniz, if the only genuinely real beings are mind-like simple substances, then bodies, motion, and everything else must result from or be derivative of those simple substances and their perceptual states.

Yet, this position, denying the reality of bodies and asserting that monads are the grounds of all corporeal phenomena, as well as its metaphysical corollaries has shocked many.

But how so? When Leibniz argues that bodies are the results of monads and that matter itself is a phenomenon, he has something very specific in mind.

First, in Leibniz's system there is a special kind of order in the natural world corresponding to a hierarchy of monads. But, ultimately, the picture is even more complex than this, for each of the subordinate monads can be considered as having an organic machine attached to it, and this relation continues on to the infinitely small.

In other words, each monad will have an organic body which is in turn composed of other monads, each of which likewise has an organic body.

Similarly, any seemingly inanimate chunk of matter — a stone or, yes, a drop of urine — will be the result of an infinity of monads and their organic bodies, which are nothing more than more monads and their organic bodies.

This view is associated with a panorganicist strand of Leibniz's thought. Second, there is what can best be described as a genuinely idealist strand of Leibniz's thought.

That is, if idealism is the thesis that the only things that truly exist are minds and their ideas, then Leibniz clearly espouses this doctrine.

Here the operative idea is that bodies, and in particular the bodies associated with particular minds, are intentional objects — though they result from or are grounded in monads.

Thus, the only real things are simple substances; the bodies that we perceive in motion around us are phenomena and not themselves substances, though they are grounded ultimately in simple substances or monads.

Furthermore, the bodies of the natural world ought be considered intentional objects in that they are objects about which we have certain beliefs.

This is what Leibniz means in saying that they have reality insofar as there is a harmony between perceivers or between the same perceivers' beliefs or perceptions at different times.

In other words, one's body or even a stone is real because it is an object of perception that fits into an account of the world that is both coherent from the point of view of the single perceiver and in harmony with the perceptions of other minds.

Leibniz's point, however, is that, while monads are not extended, they do have a situation insofar as they bear an ordered relation to other bodies through the body in which they are present or through the body to which they represent themselves as being attached.

Leibniz's conception of such a perspectival universe has, however, a distinctively Platonist origin. Again, each mind-like simple substance represents itself as having a body and a position relative to other bodies, but in doing so each simple substance offers a perspective on the world for the divine mind.

This is a striking passage. Leibniz is telling us that each finite substance is the result of a different perspective that God can take of the universe and that each created substance is an emanation of God.

The argument here can be expressed in several different ways. First, since God could occupy any and all points of view of the universe, there must be a simple substance to represent the world from that perspective.

And since the simple substance must have representations of its unique perspective, it must be a mind-like substance, a monad, capable of having perceptions.

Second, and stronger, God's omniscience entails knowledge of the world from every perspective simultaneously, and the infinite perspectives of the world originating from God's nature simply are monads.

If the only things that truly exist are mind-like entities, monads, then the differences between them must be explicable in terms of mental features.

Now, it was stated above that a central feature of Leibniz's account of substance was his claim that substances are endowed with active and passive forces.

In his mature metaphysics, Leibniz expresses this view somewhat differently by saying that a substance is active insofar as it has distinct perceptions and passive insofar as it has confused perceptions.

The fundamental idea here is two-fold: first, activity and passivity are features of the relative clarity and distinctness of the representations of the monad, and, second, insofar as the organic bodies of a particular monad are themselves constituted by monads, they — the monads of the organic body — will have confused perceptions.

This chain goes down to the infinitely small, with monads having only very confused and inexact perceptions of the world. Since there is a hierarchy among monads within any animal, from the soul of a person down to the infinitely small monad, the relation of domination and subordination among monads is a crucial feature of both Leibniz's idealism and his panorganicism.

But the hierarchy of substances is not simply one of containment, in which one monad has an organic body which is the result of other monads, each of which has an organic body, and so on.

What is it then that explains the relation of dominant and subordinate monads? As Leibniz tells Des Bosses, domination and subordination consists of degrees of perfection.

Since monads are to be differentiated in terms of their perceptions, one natural reading would simply be that suggested in the paragraph above: monad x is dominant over monad y when x has clearer perceptions than y.

Monad x is dominant over monad y when x contains within it reasons for the actions of y. This is why the mind of an animal can be said to direct the actions of its body, and why, for example, there will be a hierarchy of functionality within any one animal.

Thus, one's mind has clearer perceptions than those contained in the monads of its organic body, but it contains the reasons for everything that happens in one's body; one's liver contains the reasons for what happens in its cells; a cell contains the reasons for what happens in its mitochondria; and, according to Leibniz, this relation continues infinitely on down.

Leibniz's reflections on epistemological matters do not rival his reflections on logic, metaphysics, divine justice, and natural philosophy in terms of quantity.

Nevertheless, he did think deeply about the possibility and nature of human knowledge, and his main doctrines will be presented here.

In , Leibniz published a short treatise with the above title. It was his first mature publication and one to which he often referred in the course of his philosophical career.

In it, Leibniz sets out a series of distinctions for human knowledge or cognition cognitio : knowledge is either obscure or clear; clear knowledge is either confused or distinct; distinct knowledge is either inadequate or adequate; and adequate knowledge is either symbolic or intuitive.

Now, according to Leibniz, clear knowledge means being able to recognize something that is represented to us, for example, a rose; and knowledge is both clear and distinct when one can enumerate marks sufficient to distinguish a rose from other things.

When one can give such an enumeration, one possesses a distinct notion or concept and is thus able to give a nominal definition of the thing.

Further, if all the marks that form part of a distinct notion are themselves distinctly known, then the cognition is adequate.

And, finally, if a notion is complex and we are able to consider all its component notions simultaneously, then our knowledge of it is intuitive.

Ultimately, Leibniz holds that human beings have intuitive knowledge only of primary notions and propositions, whereas God, of course, has intuitive knowledge of all things.

Leibniz believes his distinctions also serve to show the difference between true and false ideas. Now, possibility can be established a priori and a posteriori.

On the one hand, we can know a priori that something is possible if we can resolve it into its component notions which are themselves possible and if we know that there is no incompatibility among those component notions.

On the other hand, we know a posteriori that something is possible merely through experience, for the actual existence of a thing is proof of its possibility.

While Leibniz's Principle of Contradiction and Principle of Sufficient Reason were discussed above, it was not mentioned that these two principles are employed in the service of Leibniz's distinction between truths of reasoning and truths of fact , that is, between necessary truths and contingent truths.

Leibniz's account of modality is treated elsewhere, but a short account of this distinction is here required.

Ultimately, all truths of reasoning will be resolvable into primitives or identities, and the Principle of Contradiction is thereby operative.

In the case of a truth of fact, on the other hand, its reason cannot be discovered through a finite process of analysis or resolution of notions.

However, there must be a reason that some particular fact is so and not otherwise PSR , and, according to Leibniz, this reason is found outside the series of contingent things.

See below. Leibniz is often put in the camp of rationalists and opposed to the empiricists for example, Locke, Berkeley, and Hume.

While there are good grounds to be unhappy with this standard textbook distinction, Leibniz does fit the bill in two important respects: he is a rationalist insofar as he holds to the Principle of Sufficient Reason, and he is a rationalist insofar as he accepts innate ideas and denies that the mind is at birth a tabula rasa or blank slate.

In terms of Leibniz's classical allegiances, it is interesting to see that in the realm of metaphysics, he often couched his philosophy in Aristotelian and Scholastic terms but that in the realm of epistemology, he was a fairly open Platonist — at least in terms of the existence of innate ideas.

Indeed, in the opening passages of his New Essays on Human Understanding , his book-length commentary on Locke's Essay Concerning Human Understanding , Leibniz explicitly aligns himself with Plato on the fundamental question of the origin of ideas.

Leibniz has several straightforwardly metaphysical reasons for denying that the mind could be a tabula rasa. First, and most obvious, since there can be no genuine causal interaction among substances, then there could be no way that all our ideas could come from experience; indeed, no ideas could, strictly speaking, come from experience.

Leibniz will, however, adopt a more liberal understanding of sense experience, so that this is not mooted tout court. But, second, and rarely remarked upon, Leibniz believes that the view that our minds are blank slates at birth violates the Principle of the Identity of Indiscernibles.

In short, PII works against qualitatively identical physical atoms and against qualitatively identical because blank souls.

Further, in one telling passage, he shows us the metaphysical underpinnings of the empiricist view that he finds so objectionable.

But how could experience and the senses provide the ideas? Does the soul have windows? Is it similar to writing-tablets, or like wax?

Throughout his career, Leibniz expresses no doubt that the human mind or soul is essentially immaterial, and Locke's skepticism about the nature of substance is fundamentally at odds with Leibniz's most deeply held philosophical commitments.

But, of course, the consequence of this is that Leibniz seeks to undermine Locke's position with respect to the origin and nature of ideas.

That the mind, according to Leibniz, must be essentially immaterial has been shown above in the section on metaphysics. But Leibniz does have a particular argument for the mind's immateriality or against its mechanism that concerns the nature of thought and ideas.

This is his famous metaphor of a mill, which comes forth both in the New Essays and the Monadology. According to Leibniz, perceptions cannot be explained in mechanical or materialistic terms.

Even if one were to create a machine to which one attributes thought and the presence of perceptions, inspection of the interior of this machine would not show the experience of thoughts or perceptions, only the motions of various parts.

But even when Leibniz accepts the common way of speaking — that is, as if the senses are causally responsible for some ideas — he has arguments against the empiricist claim that the senses are the origin of all ideas.

According to Leibniz, while the empiricist position can explain the source of contingent truths, it cannot adequately explain the origin and character of necessary truths.

For the senses could never arrive at the universality of any necessary truth; they can, at best, provide us with the means of making a relatively strong induction.

Rather, it is the understanding itself, Leibniz claims, which is the source of such truths and which guarantees their very necessity.

While we are not aware of all our ideas at any time — a fact demonstrated by the function and role of memory — certain ideas or truths are in our minds as dispositions or tendencies.

This is what is meant by an innate idea or an innate truth. On this subject, Leibniz uses a distinctive metaphor: a piece of marble has veins that indicate or are disposed to indicate shapes that a skillful sculptor can discover and exploit.

The hierarchy of monads mentioned above has a corollary in Leibniz's epistemology. Monads are more or less perfect depending upon the clarity of their perceptions, and a monad is dominant over another when the one contains reasons for what happens in the other.

But some monads can also rise to the level of souls when, for example, they experience sensations , that is, when their perceptions are very distinct and accompanied by memory.

This is a position occupied by animals. Furthermore, some souls are sometimes also in a position to engage in apperception , that is, to reflect on their inner states or perceptions.

The point that Leibniz wants to make is clearly an anti-Cartesian one: it is not the case that animals lack souls and are mere machines.

There is a continuum here from God, angels, and human beings through animals to stones and the dull monads which underlie the muck and grime of the world; and this continuum is not solely to be understood in terms of the comparative clarity of the mind's perceptions but also in terms of the kinds of mental activity possible for a particular being.

Indeed, according to Leibniz, animals operate not as mere automata as they do in the Cartesian philosophy, but rather have fairly sophisticated mental faculties.

At the same time, Leibniz is quick to add that the mental activity of the dog is the same as the mental activity of human beings in three fourths of their actions, for most of us most of the time are not actually reasoning from causes to effects.

And yet we are different from the beasts, Leibniz believes. Thus, what makes human beings and higher minds special is the capacity, via apperception, to formulate a conception of the self.

Indeed, as we see in this passage, Leibniz suggests that rationality itself follows from the capacity for reflection: we begin with a conception of the self; we move from this point to thinking of being, of substance, of God; and we become aware as well of eternal and necessary truths.

In other words, animals and most human beings most of the time are purely empiricists; a rational person, however, is one who can engage in genuine a priori reasoning, moving from knowledge of a true cause via deduction to necessary effects.

One of the fundamental theses of Leibniz's philosophy is that each substance expresses the entire universe. In other words, everything that takes place in the universe really is expressed by each finite mind, but the infinite perceptions present in the mind — from the butterfly's flight in the Amazonian jungle to the penguin's waddling in Antarctica — are usually too minute or too indistinct to outweigh, for example, the appearance of this computer screen or the feeling of hunger.

Indeed, this infinity of perceptions is likened by Leibniz to the roar of the sea. The infinity of petites perceptions is, then, simply epistemological white noise.

For Leibniz, the simplicity and unity of the mind still allows for the multiplicity of perceptions and appetitions. The multiplicity, however, should not only be interpreted as diachronous but also synchronous ; that is, the mind despite its simplicity and unity has within it at any time an infinity of different petites perceptions.

A human being, in a waking state, is conscious of particular perceptions, but never all. And here we see that Leibniz's doctrine is important, insofar as it offers a contrast to the Cartesian theory of the mind.

According to Leibniz, the mind is always active, for there are always perceptions present to it, even if those perceptions are minute and do not rise to such a level that we are cognizant of them.

Thus, even in a deep and dreamless sleep, the mind is active, and perceptions are in the mind. Moreover, if Descartes really did advocate the perfect transparency of the mind, then it should be clear that Leibniz allows for a subtler picture of mental contents: there are many things in the mind that are confused and minute and to which we do not always have complete access.

Leibniz, however, does not simply disagree with Locke about the nature of the mind and the possibility of innate ideas. It is also Leibniz's contention that human beings are capable of knowledge in a way that Locke had clearly denied.

As shown above, Leibniz is convinced that our knowledge of necessary truths has a completely different foundation from that for which Locke argues.

Similarly, Leibniz holds that we can have genuine knowledge of the real essences of things, something called into question by Locke.

Leibniz, however, holds that we can know certain things not only about individuals but also about their species and genera.

It would seem, then, that Leibniz has something like the following in mind: experience informs us of a certain consistent set of sensible properties in, for example, gold; that is, a certain set of properties is compossible.

And, more important, we ought to be able to assert with certainty that if some object has the greatest ductility, then it also has the greatest weight.

Like most of his great contemporaries Descartes, Spinoza, Malebranche , Leibniz developed a number of arguments for the existence of God.

But they have long histories in Leibniz's thought.

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