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Gottfried Wilhelm Leibniz was a German polymath who wrote mostly in French and Latin.
Educated in law and philosophy, and serving as factotum to two major German noble houses (one becoming the British royal family while he served it), Leibniz played a major role in the European politics and diplomacy of his day. He occupies an equally large 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. He also invented the binary system, foundation of virtually all modern computer architectures. In philosophy, he is most 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 great 17th century rationalists, but his philosophy also both 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, knowledge engineering, 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 His contributions to this vast array of subjects are scattered in journals and in tens of thousands of letters and unpublished manuscripts.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, and a complete account of his accomplishments is not yet possible.
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 (in Leibniz's view) contradicted Christian orthodoxy.
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 Hapsburgs. 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 favor 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 honor his passing. His grave went unmarked for more than 50 years. Thus the indifference of official Germany and England to the passing of the most accomplished European mind since Aristotle. 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
Leibniz wrote in three languages: scholastic Latin, French, and (least often) German. 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. Only one substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain. Only in 1895, when Bodemann completed his catalogs 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:
Gottfried Leibniz
“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 in (Mates III: 194))
The extant parts of the critical edition of Leibniz's writings 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.
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 unsuspected.
Much of Europe came to doubt that Leibniz had invented 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 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, they made Leibniz somewhat respectable among 20th century analytical and linguistic philosophers. 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 literature on Leibniz did not really blossom until after WWII. This is especially true of English speaking countries; in Gregory Brown's bibliography[4] 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.
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
It is very difficult to grasp Leibniz's philosophical thinking, 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 only wrote two philosophical treatises, and the only 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 Suarez, 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, 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, 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 converse of the second principle, known as the Indiscernibility of identicals, together with the Identity of Indiscernibles is often referred to as Leibniz's Law [[1]]. However, it is the Identity of Indiscernibles that has attracted the most controversy and criticism, especially from corpuscular philosophy and quantum mechanics.
Leibniz would on occasion give a rationale 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 his Monadologie. Monads are to the mental realm what atoms are to the physical. Monads are the ultimate elements of the universe, and are also entities of perception. 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 God's existence can be inferred from the harmony prevailing among all other monads; God wills the pre-established harmony.
Monads are purported to solve the problematic:
* Interaction between mind and matter arising in the system of Descartes;
* Lack of individuation inherent to the system of Spinoza, which represent individual creatures as merely accidental.
The monadology was thought arbitrary, even eccentric, in Leibniz's day and since. It now seems less so, in the light of key notions in contemporary physics such as field, and the action at a distance and entanglement characterizing quantum mechanics.
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. However, scientific developments in recent decades enable a more sweeping defense of optimism:
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 dimensions. 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.
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.
Symbolic thought
Leibniz had a remarkable faith that a great deal 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 very much brings symbolic logic to mind, can be viewed as a way of making calculations of this sort 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 after modern formal logic had emerged in Frege's Begriffsschrift and in various 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 very important for human understanding. He attached so much importance to the invention of good notations that he attributed to this alone the whole of his discoveries in mathematics. His notation for the infinitesimal calculus affords a splendid 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 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, however Loemker, 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)
More complex thoughts would be represented by combining in some way the 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 at the time 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 was to include 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 perhaps never be unambiguously 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).
The importance of the characteristica and calculus goes beyond their value for understanding Leibniz's legacy, and extends to mathematics, modernity, the European Enlightenment, and, more controversially, even to postmodern theory. The characteristica and calculus are also possible ways in which Leibniz's thinking can contribute to contemporary thinking in thermodynamics, biology, climate change, and resource policy, and consequently how ethics and metaphysics can meaningfully engage with such currently topical issues. Moreover, computer software employing networks of block diagrams and pictograms to generate the mathematics and kinetics of ecological, thermodynamic, and dynamic socioeconomic systems, all appear to aim at formal systems of the sort Leibniz dreamed about.
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 Louis Couturat found in the Nachlass and published in 1903. Selections from this volume have been translated into English, mainly by Parkinson (1966) and Loemker (1969). Our present understanding of Leibniz the logician emerges mainly from the work of Wolfgang Lenzen, beginning around 1980; for a summary, see Lenzen (2004).
Charles Peirce, Hugh MacColl, Frege, and Bertrand Russell all shared Leibniz's dream of combining symbolic logic, mathematics, and philosophy. The culmination of Leibniz's approach to logic is, arguably, the algebraic logic of Ernst Schröder and the modal logic founded by Clarence Irving Lewis. Related to modal logic, Leibniz thus stumbled upon the problem of future contingents concerning the famous "sea battle" example. For an example of how present-day work in logic and metaphysics can draw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000).
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 arrays, now called matrices, which can be manipulated to find the solution of the system, if any. This method was later called Cramer's Rule. 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 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 Cauchy, Riemann, 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. However, 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 employ the term analysis situs (LL §27), later employed 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 Freudenthal, argues as follows:
"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 Konigsberg 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."
Hirano (1997) argues differently, quoting Mandelbrot (1977: 419) as follows:
"...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 Mandelbrot's well-known fractal geometry 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 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.
Scientist and engineer
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. His writings on other scientific and technical subjects are mostly scattered and relatively little known, because the Academy edition has yet to publish any volume in its Series Scientific, Medical, and Technical Writings .
Physics
Leibniz contributed a fair amount to the statics and dynamics emerging about him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic and potential energy, which posited space as relative, whereas Newton felt strongly space was absolute. While he may have been Newton's peer as co-discoverer of calculus, he was not in Newton's league as a physicist and may even deserve to be ranked below his mentor Huygens. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695. (AG 117, LL §46, W II.5) On Leibniz and physics, see the chapter by Garber in Jolley (1995) and Wilson (1989).
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 Einstein by arguing, against Newton, that space, time and motion are relative, not absolute. Leibniz's rule in interacting theories plays a role in supersymmetry and in the lattices of quantum mechanics. His 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.
The vis viva
Leibniz 's vis viva (Latin for living force) is an invariant mathematical characteristic of certain mechanical systems (see AG 155-86, LL §§53-55, W II.6-7a). It can be seen as a special case of the conservation of energy. Here too his thinking gave rise to another regrettable nationalistic dispute. 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. Engineers eventually found "vis viva" useful when making certain calculations, so that the two approaches eventually were seen as complementary.
Other natural science
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. He worked out a primal organismic theory. On Leibniz and biology, see Loemker (1969a: VIII). In medicine, he exhorted the physicians of his time -- with some results -- to ground their theories in detailed comparative observations and verified experiments, and to distinguish firmly scientific and metaphysical points of view.
Social science
In psychology he anticipated the distinction between conscious and unconscious states. On Leibniz and psychology, see Loemker (1969a: IX). 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 programme, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance scheme, 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.
Technology
In 1906, Gerland 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 praxis, 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 invented a steam engine. He even proposed a method for desalinating water. He struggled, 1680-85, to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains, but his efforts were not crowned with success. (Aiton 1985: 107-114, 136)
Information technology
Leibniz may have been the first computer scientist and information theorist. Early in life, he discovered the binary number system (base 2), the one subsequently employed on most computers, then revisited that system throughout his career. On Leibniz and binary numbers, see Couturat (1901: 473-78). Leibniz anticipated Lagrangian interpolation and algorithmic information theory. His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1934, Norbert Wiener claimed to have found in Leibniz's writings a mention of the concept of feedback, central to Wiener's later cybernetic theory.
In 1671, Leibniz began to invent a machine that could execute all four arithmetical 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 1673. A number of such machines were made during his years in Hanover, by a craftsman working under Leibniz's supervision. It was not an unambiguous success because it did not fully mechanize the operation of carrying. Couturat (1901: 115) reported finding an unpublished note by Leibniz, dated 1674, 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, 1830-45. In 1679, 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.[2] Modern electronic digital computers replace Leibniz's marbles moving by gravity with shift registers, voltage gradients, and pulses of electrons, but otherwise they run roughly as Leibniz envisioned in 1679. Davis (2000) discusses Leibniz's prophetic role in the emergence of calculating machines and of formal languages.
The librarian
In his capacity as librarian of the ducal libraries in Hanover and Wolfenbuettel, Leibniz effectively became one of the founders of library science.[3] The latter library was enormous for its day, as it contained more than 100,000 volumes, and Leibniz helped design a new building for it, believed to be the first building explicitly designed to be a library. 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 means of furthering all the 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.g., Esperanto and its rivals), symbolic logic, even the World Wide Web.
Advocate of scientific societies
Leibniz emphasized that research was a collaborative endeavor. Hence he warmly advocated the formation of national scientific societies along the lines of the British Royal Society and the French Academie Royale des Sciences. 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 1700, 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. On Leibniz’s projects for scientific societies, see Couturat (1901: App. IV).
Lawyer, moralist
No philosopher has ever had as much experience with practical affairs of state as Leibniz, Marcus Aurelius possibly excepted. Leibniz's writings on law, ethics, and politics (e.g., AG 19, 94, 111, 193; Riley 1988; LL §§2, 7, 20, 29, 44, 59, 62, 65; W I.1, IV.1-3) were long overlooked by English speaking scholars but this has changed of late; see (in order of difficulty) Jolley (2005: chpt. 7), Gregory Brown's chapter in Jolley (1995), Hostler (1975), and Riley (1996).
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 democracy, first in 18th century America and subsequently elsewhere. The following excerpt from a 1695 letter to Baron J. C. Boineburg's son Philipp is very revealing of Leibniz's political sentiments:
"As for.. the great question of the power of sovereigns and the obedience their peoples owe them, I usually say that it would be good for princes to be persuaded that their people have the right to resist them, and for the people, on the other hand, to be persuaded to obey them passively. 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." (LL: 59, fn 16. Translation revised.)
Leibniz foresaw the European Union. In 1677, he (LL: 58, fn 9) 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. Europe would adopt a uniform religion. He reiterated these proposals in 1715.
Ecumenism
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, later the Lutheran and Reformed churches. In this respect, he followed the example of his early patrons, Baron von Boineburg 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.) These efforts included corresponding with the French bishop Bossuet, and involved Leibniz in a fair bit of theological controversy. He evidently thought that the thoroughgoing application of reason would suffice to heal the breach caused by the Reformation.
Philologist
Leibniz 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 some sort of proto-Swedish was the ancestor of the Germanic languages. He puzzled over the origins of the Slavic languages,[citation needed] was aware of the existence of Sanskrit, and was fascinated by classical Chinese. Scholarly appreciation of Leibniz the philologist is hampered by the fact that no volume of the planned Academy edition series "Historical and Linguistic Writings" has appeared.
Sinophile
Leibniz was perhaps the first major European intellect to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other work by, European Christian missionaries posted in China. He concluded that Europeans could learn much from the Confucian ethical tradition. 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 0 to 111111, and mistakenly concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.
On Leibniz, the I Ching, and binary numbers, see Aiton (1985: 245-48). Leibniz's writings on Chinese civilization are collected and translated in Cook and Rosemont (1994), and discussed in Perkins (2004).
Leibniz as polymath
The following episode from the life of Leibniz illustrates the breadth of his genius, and the difficulties awaiting those who try to come to terms with it. While making his grand tour of European archives to research the Brunswick family history he never completed, Leibniz stopped in Vienna, May 1688 – February 1689, 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, first published by Louis Couturat in 1903, later translated as LL 267 and WF 30, summarizing his views on metaphysics. The paper is undated; that he wrote it while in Vienna was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677-90. Couturat's reading of this paper was the launching point for much 20th century thinking about Leibniz, especially among analytic philosophers. But after a meticulous study of all of Leibniz's philosophical writings up to 1688 -- a study the 1999 additions to the critical edition made possible -- Mercer (2001) begged to differ with Couturat's reading; the jury is still out.
La Monadologie (Monadology) (1714) is a highly condensed outline of Leibniz's metaphsics. Complete individual substances, or monads, are dimensionless points which contain all of their properties—past, present, and future—and, indeed, the entire world. The true propositions that express their natures follow inexorably from the principles of contradiction and sufficient reason. Leibniz
The same themes are presented more popularly in the Discours de Metaphysique (Discourse on Metaphysics) (1686). There Leibniz emphasized the role of a benevolent deity in creating this, the best of all possible worlds, where everything exists in a perfect, pre-established harmony with everything else. Since space and time are merely relations, all of science is a study of phenomenal objects. According to Leibniz, human knowledge involves the discovery within our own minds of all that is a part of our world, and although we cannot make it otherwise, we ought to be grateful for our own inclusion in it.
True Propositions
The basis for Leibniz's philosophy is pure logical analysis. Every proposition, he believed, can be expressed in subject-predicate form. What is more, every true proposition is a statement of identity whose predicate is wholly contained in its subject, like "2 + 3 = 5." In this sense, all propositions are analytic for Leibniz. But since the required analysis may be difficult, he distinguished two kinds of true propositions: (Monadology 33)
Truths of Reason are explicit statements of identity, or reducible to explicit identities by a substitution of the definitions of their terms. Since a finite analysis always reveals the identity-structure of such truths, they cannot be denied without contradiction and are perfectly necessary.
Truths of Fact, on the other hand, are implicit statements of identity, the grounds for whose truth may not be evident to us. These truths are merely contingent and may be subject to dispute, since only an infinite analysis could show them to be identities.
Anything that human beings can believe or know, Leibniz held, must be expressed in one or the other of these two basic forms. The central insight of Leibniz's system is that all existential propositions are truths of fact, not truths of reason. This simple doctrine has many significant consequences.
Complete Individual Substances
Consider next how this logic of propositions applies to the structure of reality itself for Leibniz. The subject of any proposition signifies a complete individual substance, a simple, indivisible, dimensionless being or monad, while the predicate signifies some quality, property, or power. Thus, each true proposition represents the fact that some feature is actually contained in this substance.
Each monad is a complete individual substance in the sense that it contains all of its features—past, present, and future. Because statements of identity are timeless, the facts they express perpetually obtain. (Thus, for example, I am the person whose daughter was born in 1982 and the person who now develops this web site and the person who will vacation in Manitoba next summer; since each of these predicates can be truly affirmed of me, each of these features is contained in me.) Everything that was, is, or will ever be true of any substance is already contained in it. (Monadology 22)
Moreover, each monad is a complete individual substance in the sense that its being is utterly independent of everything else. Because statements of identity are self-contained, any apparent relation between substances must actually be a matching pair of features that each possesses alone. (Thus, for example, I happen to have the property of being Aaron's father, and Aaron happens to have the property of being my son, but these are two facts, not one.) Hence, on Leibniz's view, there can be no interaction between substances, each of which is purely active. Monads are "windowless." (Monadology 7)
Where Spinoza saw the world as a single comprehensive substance like Descartes's extended matter, then, Leibniz supposed that the world is composed of many discrete particles, each of which is simple, active, and independent of every other, like Descartes's minds or souls. The rationalists' common reliance upon mathematical models of reasoning led to startlingly different conceptions of the universe. Yet the rationality, consistency, and necessity within each system is clear.
Logical Principles
Another way of summing up the structure of the universe on Leibniz's view is by reviewing the great logical principles from which all truths are said to flow:
The Principle of Contradiction generates the truths of reason, each of which states the connection between an individual substance and one of its finite number of essential features. (Monadology 31) It would be a contradiction to deny any of these propositions, since the substance would not be what it is unless it had all of these features. Truths of reason, then, are not influenced by any contingent fact about the world; they are true "in all possible worlds." Thus, for example, "Garth Kemerling is a human being" would be necessarily true even if my parents had been childless.
The Principle of Sufficient Reason generates the truths of fact, each of which states the connection between an existing individual substance and one of its infinitely many accidental features or relations. (Monadology 32) The sufficient reason for the truth of each of these propositions is that this substance does exist as a member of the consistent set of monads which constitutes the actual world. Truths of fact, then, depend upon the reciprocal mirroring of each existing substance by every other. Thus, for example, "Garth Kemerling is an oldest child" is contingently true only because my parents had no children before I was born.
The Principle of the Identity of Indiscernibles establishes the fact that, within the set of monads that constitutes any possible world, no two can be exactly alike. (Monadology 9) If, on the contrary, there were two distinct but perfectly identical substances, Leibniz argued, then there could be no sufficient reason for each to occupy its own location rather than that of the other. More positively, since each monad mirrors the entire structure of the world, each must reflect a unique set of relations to every other.
Finally, the Principle of the Plenum (or principle of plenitude) affirms that the actual world, considered as a set of monads, is as full as it can possibly be. Since there is no genuine interaction among distinct substances, there would be no sufficient reason for the non-existence of any monad that would be consistent with the others within a possible world. Hence, anything that can happen will; every possibility within this world must be actualized. The world in which we live, then, is but one among the infinitely many possible worlds that might have existed. What makes this one special?
Space and Time
Since we experience the actual world as full of physical objects, Leibniz provided a detailed account of the nature of bodies. As Descartes had correctly noted, the essence of matter is that it is spatially extended. But since every extended thing, no matter how small, is in principle divisible into even smaller parts, it is apparent that all material objects are compound beings made up of simple elements. But from this Leibniz concluded that the ultimate constitutents of the world must be simple, indivisible, and therefore unextended, particles—dimensionless mathematical points. So the entire world of extended matter is in reality constructed from simple immaterial substances, monads, or entelechies.
In fact, Leibniz held that neither space nor time is a fundamental feature of reality. Of course individual substances stand in spatial relation to each other, but relations of this sort are reducible in logic to the non-relational features of windowless monads. In exactly the same way, temporal relations can be logically analyzed as the timeless properties of individual monads. Space and time are unreal, but references to spatial location and temporal duration provide a convenient short-hand for keeping track of the relations among the consistent set of monads which is the actual world.
What is at work here again is Leibniz's notion of complete individual substances, each of which mirrors every other. A monad not only contains all of its own past, present, and future features but also, by virtue of a complex web of spatio-temporal references, some representation of every other monad, each of which in turn contains . . . . In a universe of windowless mirrors, each reflects any other, along with its reflections of every other, and so on ad infinitum. It is for this reason that an infinite analysis would be required to reveal the otherwise implicit identity at the heart of every truth of fact. In order fully to understand the simple fact that my eyes are brown, one would have to consider the eye-color of all of my ancestors, the anatomical structure of the iris, my personal opthalmological history, the culturally-defined concept of color, the poetical associations of dark eyes, etc., etc., etc.; the slightest difference in any one of these things would undermine the truth of this matter of fact. Existential assertions presuppose the reality of just this one among all possible worlds as the actual world.
The Best of All Possible Worlds
Both in the Monadology and at the more popular level of presentation that characterizes the Discourse on Metaphysics, Leibniz (like Descartes) resolved some of the most thorny philosophical problems by reference to god. God (alone) exists necessarily, and everything else flows from the divine nature. Limited only by contradiction, god first conceives of every possible world—the world with just one monad; the worlds with exactly two monads; those with three, with seventeen, with five billion, etc. Then god simply chooses which of them to create.
Of course even god must have a sufficient reason for actualizing this world rather than any other. The most direct advantage of this world is that (as the plenum principle requires) it is the fullest. That is, more things exist and/or more events actually take place in this world than in any other consistent set of interrelated monads. In a more lofty tone, Leibniz declared that a benevolent god would choose to create whatever possible world contained the smallest amount of evil; hence (in a phrase that would later be mocked by Voltaire) this is "the best of all possible worlds," according to Leibniz. Nothing about it could be changed without making things worse rather than better on the whole.
Similarly, the existence of a benevolent god can be used to account for the smooth operation of a universe that consists of indefinitely many distinct individual substances, none of which have any causal influence over any other. (Monadology 51) A crucial element of god's creative activity, Leibniz held, is the establishment of a "pre-established harmony" among all existing things. Like well-made clocks that have been synchronized, wound, and set in motion together, the monads that make up our world are independent, self-contained, purely active beings whose features coincide without any genuine interaction among them.
One special case of this pre-established harmony, of course, accounts for the apparent interaction of mind and body in a human being as nothing more than the perfect parallelism of thier functions. In fact, the human mind is just the dominant member of a local cluster of monads which collectively constitute the associated human body. (Monadology 63) Neither has any real effect on the other, but these monads are most clearly reflected in each others' foreground. Thus, in both sensation and volition, the divinely-ordained coincidence of bodily movements and mental thoughts creates an illusion of genuine causal influence.
Knowledge and Freedom
The possibility human knowledge emerges more clearly from a slightly more technical account of Leibniz's position. All monads have the capacity for perception of the external world in the sense that, as complete individual substances, each of them contains as properties unconscious images of its spatio-temporal relations to everything else. (Monadology 19) These innate ideas constitute the unique point of view from which any monad may be said to represent the world as a whole.
But Leibniz held that some monads—namely, the souls of animals and human beings—also have conscious apperception in the sense that they are capable of employing sensory ideas as representations of physical things outside themselves. And a very few monads—namely, spirits such as ourselves and god—possess the even greater capacity of self-consciousness, of which genuine knowledge is the finest example. Although Leibniz himself did not draw the inference directly, notice that if a cluster of dimensionless monads can make up an extended body, it might be equally possible for a cluster of unconscious monads to constitute a thinking thing.
What Leibniz did claim is that we have the free will required for moral responsibility even though all of our future actions are already contained in us (along with the future of the entire actual world). Any awareness of those contingent future actions would follow from the principle of sufficient reason only upon an infinite analysis of my nature. Hence, since I lack knowledge of what I will do tomorrow, it will seem to me as if I act freely when I do it. Like space and time, freedom is a benevolent illusion that adequately provides for life in an uncertain world.
Concluding note on the Rationalists
Descartes, Spinoza, and Leibniz illustrate well the range of diverse outcomes that may result from an effort to understand the world through a priori knowledge. If their systems of thought seem implausibly remote from the world of ordinary experience, it may help to remember that modern science leads to a similar result. Once we grant that the reality of things may be quite different from the way they appear to us, only the internal coherence of the scheme of thought makes much difference. Next we'll look at modern philosophers who were more determined to make sense out of the materials provided in everyday life.
