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All the quotations are my own translations. I have not specified any particular edition for the Monadology , which is included in most collections, and is referred to here by paragraph in the normal way. The rest are from the following editions:
Traditionally, university courses on the history of modern philosophy have been structured round a pantheon of seven great philosophers: three ‘continental rationalists’: Descartes, Spinoza and Leibniz; three ‘British empiricists’: Locke, Berkeley and Hume; and Kant. The empiricists were supposed to have believed that all our knowledge was built up out of the data of sense, whereas the rationalists were supposed to have restricted genuine knowledge to what could be deduced from indubitable truths of reason. Kant, on the other hand, created a new synthesis out of what was right in both empiricism and rationalism. Needless to say, this way of viewing the history of philosophy was invented by Kant himself. It has, however, had a remarkably long run for its money.
Historians of philosophy have always found particular difficulty in forcing Leibniz into this Kantian mould, since his approach was markedly less rationalist than that of either Descartes or Spinoza. During the present century, there has been a growing tendency among professional philosophers to see logic as central to their discipline. Consequently the focus of interest in Leibniz's thought has shifted to his work on formal logic, and the extent to which his philosophy can be interpreted as derived from his logic. However, this approach leaves many questions unanswered, and commentators have been positively embarrassed by the many quirkier aspects of Leibniz's philosophical work. These obviously have nothing to do with logic.
In fact, Leibniz stood on the interface between the holistic and vitalist worldview of the Renaissance, and the atomistic
Another common distortion is to see Leibniz as primarily a philosopher, as if his role in life were the same as that of the twentieth-century professional philosopher. Not only was he never employed as a professor of philosophy, but the range of his interests was so wide that his philosophical work was no more than one activity among many. He was, as the Germans quite rightly call him, an Universalgenie – a ‘universal genius’. A balanced account of his achievement must place his philosophy in the context of everything else he did. Only then is it possible to appreciate how Leibniz, far from being the extreme ‘rationalist’ Kant made him out to be, was really himself aiming to create a new synthesis out of the apparently irreconcilable conflicts between earlier traditions in various spheres of intellectual activity.
Gottfried Wilhelm von Leibniz was born in Leipzig, on Sunday 1 July 1646. His father, Friedrich Leibnütz (1597-1652), was Professor of Moral Philosophy at Leipzig University. His mother, Catherina Schmuck (1621-64), was Friedrich's third wife. Leibniz had a half-brother Johann Friedrich (d. 1696), and a sister Anna Catherina (1648-72), whose son Friedrich Simon Löffler was eventually his sole heir. During his twenties, Leibniz changed the spelling of his name from ‘-ütz’ to ‘-iz’: he himself never used the ‘-itz’ form, which became the normal spelling during his own lifetime, and which has only recently gone out of fashion.
Little is known about Leibniz's early education, beyond what he tells us in incidental reminiscences, which probably exaggerate the extent to which he was self-taught. He learned to read from his father, well before going to school at the age of seven. He claims that he taught himself Latin from an illustrated Livy, and that by the age of eight he was avidly reading his late father's books. It is hard to imagine what he can have made of them at first, since he was not fluent in Latin until the age of twelve, and was only then beginning Greek. But they certainly formed the basis of his later massive erudition in the classics, the Church Fathers, and scholastic philosophy. On top of this, his school syllabus was itself very demanding, including German literature and history, Latin, Greek, theology and logic. This last was of particular interest to him, and at the age of thirteen he was already trying to improve on the Aristotelian theory of the categories, despite discouragement from his teachers.
Leibniz moved on to Leipzig University after Easter 1661, when he was still only fourteen (young, but not exceptionally so in those days). He followed the standard two-year arts course, which included philosophy, rhetoric, mathematics, Latin, Greek and Hebrew. After graduating, he could proceed only by studying for a doctorate in one of the ‘higher’ faculties of Theology, Law, or Medicine. He opted for Law, but before starting his course, he spent the short summer term at the nearby university of Jena. Here he came into contact with more unorthodox ideas, in particular Erhard Weigel's Neopythagoreanism, according to which Number is the fundamental reality of the universe.
Following his return to Leipzig, Leibniz spent the next three years working at a series of ‘disputations’ which he had to publish and defend in open debate at each stage (or ‘degree’) of his student career. In addition to bachelor's and master's disputations in law, in his final year he wrote a Dissertation on the Art of Combinations, by which he ‘habilitated’, that is, became qualified to lecture in philosophy. However, he did not exercise his right to teach philosophy, since such lectureships were purely honorary, and he was already heavily in debt to his relatives. Rather, he had his eyes on one of twelve established law tutorships, which went to doctors of law whenever vacancies occurred, in order of date of graduation. Unfortunately, there were too many doctoral candidates that year, and the younger ones, including Leibniz, were told to wait until a later degree day. He took this very badly, and suspected a conspiracy directed against him personally by the Dean's wife, for motives which he never explained. So he moved on to the little university of Altdorf, just outside Nuremberg, which was then a major centre of science and technology. Almost as soon as he had registered (4 October 1666), he submitted his already prepared doctoral thesis, and
Leibniz's first job was a stopgap, and he may already have had it while still officially a student at Altdorf. It was the secretaryship of a society of Nuremberg intellectuals interested in alchemy (not Rosicrucians, as has often been asserted). It is unclear what his duties were – on alchemical questions Leibniz consistently adhered to the tradition of secretiveness. In contrast with his contemporary Isaac Newton, it is unlikely that he ever did any actual laboratory work, but he certainly acquired a reputation as an adept with deep theoretical understanding of the art. To his dying day he retained a close interest in alchemy (he talked about it with his doctor on his death-bed), and he periodically arranged tests of the claims of various alchemists. His declared motives were scientific: if transmutation were a practical possibility, the process should yield valuable information about the structure of matter. But in fact he also hoped to make his fortune from it. Thus, in about 1676, he entered into a formal profit-sharing agreement with two practising alchemists (G. H. Schuller and J. D. Crafft), his side of the bargain being to provide capital and technical advice: he was always a soft touch for people wanting to borrow money for alchemical experiments. His main reservation about gold-making was that gold would lose its value if it could be made too cheaply.
Whatever his precise relationship with the Nuremberg alchemists, he did not stay with them for long. Sometime during the summer he was on the move again, intending to
Despite this appointment, he was still very much Boineburg's protégé, and for the next five years he spent as much time in Frankfurt as in Mainz. Leibniz's close relation ship with Boineburg was important, not merely for launching him on his career, but also at an intellectual and personal level. In particular, Boineburg and other members of his circle were converts from Lutheranism to Catholicism. Leibniz very nearly followed their example, and it says much for his sincerity that in later years, when offered the prestigious librarianships of the Vatican (in 1689), of Paris (in 1698), and perhaps also of Vienna, he turned them down only because he was not prepared to go through a formal conversion. How ever, despite his loyalty to Lutheranism, he moved easily in Catholic circles, and was ideally placed to further the reunification of the churches, which was one of his life's ambitions. With Boineburg's encouragement, he drafted a number of monographs on religious topics, mostly to do with points at issue between the churches, such as the doctrine of transubstantiation.
As we shall see in Chapter 4, one of the corner-stones of Leibniz's philosophy was his vision of a ‘universal encyclopaedia’, which would incorporate all knowledge into a
As it stood, German law comprised a chaotic mixture of the Roman code, traditional Germanic common law, and the statute and case law of the various states. Leibniz hoped to reduce it to order by defining all legal concepts in terms of a few basic ones, and deducing all specific laws from a small set of incontrovertible principles of natural justice. Among his papers there survive many draft attempts at such a system, and he published a number of short treatises on the topic during his Mainz period (for example, the New Method of Teaching and Learning Jurisprudence of 1667). Although the focus of his interests moved away from law as he got older, he kept on returning to this youthful project. However, not only was it a huge task to reduce the whole of natural law to a system, but he never began to solve the problem of extending it to civil legislation.
More generally, Leibniz's scheme for a universal encyclopaedia required a pooling of existing knowledge, of research in hand, and of future efforts. There had already been attempts to encompass all knowledge in a single work, for example J. H. Alsted's seven-volume Encyclopaedia of 1630, which Leibniz once thought of adapting to his own purposes. But the vast bulk of current knowledge was in books scattered throughout the libraries of Europe, and he soon saw that the most feasible way of centralising access to it would be to compile a master subject-catalogue. At the time the only useful subject-catalogue in existence was that of the Bodleian Library at Oxford, and Leibniz had no knowledge of it. In 1670 he produced as a model a catalogue of Boineburg's rich book collection; but despite repeated pleas, he was never
As for setting up a central register of new discoveries, Leibniz devised a scheme for a review of books, which he called a nucleus librarius, to include abstracts of all new serious publications. His long-term plan was to expand it to cover earlier publications, unpublished works, research in progress, and a cumulative subject-index. Twice he applied for the necessary imperial licence (in 1668 and 1669), but on each occasion he was turned down, presumably because of fears that it would harm the retail book trade. However, despite the failure of the more grandiose scheme, he continued to do what he could in the same general direction. Throughout his life he kept a card-index of all the important books he read (an immense number), he was a regular contributor to such review journals as existed (notably the Journal des Sçavans, founded in 1665, and the Acta Eruditorum, founded in 1682); and much later, in 1700, he started his own journal, the Monatliche Auszug, under the editorship of his assistant Johann Georg von Eckhart. This, however, folded after only two years.
The need to co-ordinate research naturally suggested the foundation of learned and scientific societies. Like many of his contemporaries, Leibniz dreamed up various Utopian schemes for communes of researchers, and he also proposed exhibitions and museums for popularising and funding science. Until he was influential enough for his plans to have any real chance of success, the only practical step was for him to join such societies as already existed. With this in mind, he composed a number of treatises on scientific topics, two of which he had printed with dedications to the Royal Society of London, and to the Paris Academy. He was elected to the
Although scientific societies and periodicals were gradually coming into existence during the seventeenth century, by far the most important medium of intellectual co-operation and dissemination of ideas was the exchange of letters. These were often widely distributed among the acquaintances of the correspondents, and it was also common for collections of such letters to be published in book form. For example, in 1697 Leibniz published a selection from his correspondence, mainly with Jesuit missionaries, about China, under the title Novissima Sinica (‘The Latest from China’). Boineburg was an avid letter-writer, and he helped Leibniz to build up his own circle of correspondents by putting him in touch with intellectuals from all over Europe. Within a few years, Leibniz was in correspondence with literally hundreds of people at a time on almost every subject under the sun – science, mathematics, law, politics, religion, philosophy, literature, history, linguistics, numismatics, anthropology. He was obsessive about preserving his letters, and over 15,000 still survive. It is on these, and on a comparable mass of private notes and drafts, that we rely for most of our knowledge of his work, especially in the areas of philosophy, logic and mathematics. As he once wrote, ‘Anyone who knows me only by my publications does not know me at all’ (D vi i 65). Most of the manuscript material is in Latin, which was still (though not for much longer) the lingua franca of the scholarly world. He often corresponded in French (even with fellow Germans), but hardly ever in German. When he did, he fre quently lapsed into Latin because of the lack of abstract technical terms in German. As a keen nationalist he much regretted the fact, and proposed a German Academy to enrich and promote the German language. He occasionally tried to
Although Leibniz's interests were clearly developing in a scientific direction, he still hankered after a literary career. All his life he prided himself on his poetry (mostly Latin), and boasted that he could recite the bulk of Virgil's Aeneid by heart. During his time with Boineburg he would have passed for a typical late Renaissance humanist. His Latin style was still elaborate and florid, he never missed any opportunity to parade his classical learning, and his principal publication was an edition of the Antibarbarus of the sixteenth-century Italian humanist Mario Nizolio. Leibniz was broadly sympathetic with Nizolio's theme, which was that a pure style in Latin was a surer route to knowledge and wisdom than the logic and linguistic barbarism of university philosophy. In 1673 Leibniz promised to do the Delphin Classics edition of Martianus Capella (the fifth-century author of a fantastic allegory on the seven liberal arts), but he never got round to it. In 1676 he translated Plato's Phaedo and Theaetetus into Latin, and was the first modern scholar to detect a sharp contrast between the philosophy of the historical Plato, and the mystical and superstitious ‘ Neoplatonism’ (or ‘ Pseudo Platonism’, as Leibniz called it) of Plato's later followers.
Leibniz also applied his mind to political questions. For example, soon after his arrival in Mainz, he published a short treatise using deductive arguments to solve the question of the Polish succession. A more long standing problem was the French threat to Germany, now seriously weakened by the Thirty Years War. Leibniz periodically came up with anti-French ideas, such as undercutting the brandy trade with cheap rum from West Indian sugar, and in 1684 he published an anonymous satire on Louis XIV's bellicosity, under the
On arriving in Paris in the spring of 1672, and while waiting for an opportunity to carry out his political objective, Leibniz set about getting himself known in intellectual circles. He soon had a wide range of acquaintances, including the philosophers Arnauld and Malebranche, and the mathematician Huygens. Through his philosophical contacts, he managed to get access to the unpublished writings of the two greatest French philosophers of the previous generation, Pascal and Descartes, and some of the latter survive only through the copies he himself made. As we shall see, his close but critical study of Descartes' work was one of the major influences on his mature philosophical system. But at this stage, his main interest was in mathematics. Because of the narrowness of his mathematical education in backward Germany, he had arrived in Paris with exaggerated ideas of his own achievements. After a number of embarrassing encounters with various leading mathematicians, mainly French and English , he was forced to realise that he still had a lot to learn. Far from being discouraged, he immersed himself in mathematical studies under the guidance of Huygens, and by the time he left Paris he had already made most of the discoveries that were to earn him his leading place in the history of the subject.
But Leibniz's official reason for being in Paris came to nothing: he never found an opportunity to present his Egyptian plan to the King. In November 1672, Boineburg sent his son Philipp Wilhelm to Paris to finish his education under Leibniz's charge. He arrived in the company of his brother-in-law, the Elector's nephew Melchior Friedrich von Schönborn, who was on a diplomatic mission. This meant that Leibniz had both continued financial support, and the status of a semi-official attaché. Most importantly, he took part in a trip to London, in January 1673, which enabled him to make personal contact with members of the Royal Society, in particular its secretary, his fellow German Henry Oldenburg. The Society had given a mixed reception to his treatise, The Theory of Concrete Motion, which he had sent them (see p. 8 above), but they were very intrigued by another of his projects which he had brought along to show them. This was the prototype of a mechanical calculator he had been working on while still in Germany.
He was very proud of his invention. He once thought of commemorating it with a medal bearing the motto SUPERIOR TO MAN, and much later he had a machine made for Peter the Great of Russia to send to the Emperor of China as an example of superior Western technology. Its immediate applications were obvious: it would save considerable labour and improve accuracy in accountancy, administration, surveying, scientific research, production of mathematical tables and so on. This was all more significant than we might now appreciate, since at the time even educated people rarely understood multiplication, let alone division (Pepys had to learn his multiplication tables when already a senior administrator). For the long term, he envisaged a larger version of his calculator being used to mechanise all reasoning processes, once all possible thoughts had been given a number
The calculator itself was a considerable advance on earlier adding machines, such as Wilhelm Schickard's of 1623, or Pascal's of 1642. Leibniz designed it specifically as a multiplier and divider, and invented a number of devices which became standard in later technology – in particular the stepped reckoner (or ‘Leibniz wheel’), which had cogs of varying lengths. However, despite spending a small fortune on the project right up to the end of his life, he never developed a version which could do carrying completely automatically. One of his models still survives, and is now in the Hanover State Library.
While in Paris, Leibniz was full of other technological ideas. The one he had most fully worked out was a watch with two symmetrical balance wheels working in tandem, of which he demonstrated a model to the Paris Academy in April 1675. Others were a device for calculating a ship's position without using a compass or observing the stars, a method for determining the distance of an object from a single observation point, a compressed-air engine for propelling vehicles or projectiles, a ship which could go under water to escape enemy detection (though he rejected space flight on the grounds that the air would be too thin), an aneroid barometer (subsequently reinvented by Vidi of Paris in 1843), and various improvements to the design of lenses.
Leibniz's trip to London was cut short by the news of the sudden deaths of both his patrons: of Boineburg in December 1672, and of the Elector in February 1673. He arrived back in Paris in early March, but continued as the young Boineburg's
The administration at Hanover was typical of that of the hundred or so independent states under the titular leadership of the German Emperor in Vienna. The autocratic head of state, in this case Duke Johann Friedrich of Brunswick- Luneburg, acted through a council composed largely of lesser aristocrats and law graduates, some of whom were delegated more specialist functions. Leibniz managed to negotiate partial relief from normal council duties because of the burden of his particular responsibilities as librarian, political adviser, in ternational correspondent and, increasingly, as technological adviser.
His duties as librarian were onerous, but fairly mundane: general administration, purchase of new books and second-
Throughout this period of his career, Leibniz's principal efforts were directed towards technological innovation. He seems to have spent much time in conclave with the Duke (as also with his successor) discussing alchemical recipes and testing the claims of itinerant alchemists. Among the alchemists he got to know at this time were J. D. Crafft (see p. 5) and J. J. Becher. He remained a close friend of Crafft for many years, but he soon fell out with Becher. Leibniz had put a stop to one of Becher's more idiotic alchemical schemes, and in revenge Becher satirised him in his book Foolish Wisdom and Wise Folly, for claiming to have invented a coach which could travel from Amsterdam to Hanover in six hours (i.e.
It was through Crafft that Leibniz came into contact with another German alchemist, Heinrich Brand, the discoverer of phosphorus. It seems that Brand was working from an old alchemical text which hinted that the philosopher's stone was to be found in the dregs of the human body. He took this literally, tried distilling urine, and produced phosphorus. Quite apart from the possibility of its being a step towards the philosopher's stone, it had considerable commercial value as a curiosity for the then fashionable courtly scientific demonstrations, and potential as a weapon of war. In 1678, Leibniz managed to engage Brand to the services of Hanover with an exclusive contract. From Brand's point of view, he had the advantage of virtually unlimited supplies of urine from the workers' latrines in the Duchy's mines in the Harz mountains, about 100 kilometres south-east of Hanover.
At the same time, Leibniz became obsessed by the problem of draining water from the Harz mines. Early in 1679 he conceived the idea of using wind power, and he persuaded the Duke to let him try various experiments, with a handsome life pension if he succeeded. From then until the end of 1686 he
As far as we know, every single one of these projects ended in failure. Leibniz himself believed that this was because of deliberate obstruction by administrators and technicians, and the workers' fears that technological progress would cost them their jobs. He certainly did face repeated attacks in the Court Council – not entirely unjustified, since his over- ambitious and often half-baked schemes must have caused considerable expense and disruption of normal working. It says much for the patience of successive Dukes that Leibniz's interference was tolerated as long as it was. Quite apart from the Harz mines, he kept on submitting memoranda on all sorts of technical projects, such as canals, inland navigation, water supply, fountains for the Herrenhausen palace gardens, linen production, porcelain manufacture, exploitation of waste heat in chimneys; and also on socio-political issues such as monetary policy, tax reform, balance of trade, and a primitive national insurance scheme.
Duke Johann Friedrich had died in December 1679, his successor being his younger brother Ernst August. In order to further his dynastic ambitions by establishing and publicising the historical rights of the House of Brunswick, Ernst August had suggested to Leibniz that he carry out research for a book on its recent history. Nothing much came of this at the time,
He soon exhausted the archival material available locally, and was given permission for a long trip to Bavaria, Austria and Italy. He was away from November 1687 to June 1690. Apart from his archival work, he took the opportunity to get to know many more scholars and scientists, and was elected a member of the Physico-Mathematical Academy of Rome. He had many discussions on church unity, and in Vienna he made an impression on the Emperor, Leopold I, though not enough to get him a post as Imperial Councillor and Official Historian, or permission to set up a ‘universal library’. About the same time he completed his first successful diplomatic mission, which was to negotiate the marriage of Duke Johann Friedrich's daughter Charlotte Felicitas to the Duke of Modena.
On returning to Hanover, Leibniz set about improving his position by acquiring part-time appointments at other courts ruled by branches of the Brunswick family. It was then that he was given charge of the Bibliotheca Augusta at Wolfenbüttel by the co-Dukes Rudolf August and Ulrich of Brunswick-Wolfenbüttel. They also agreed to pay a third of the costs of publishing the Guelf history, much of the material for which was in their library. Early in 1691, Duke Georg Wilhelm of Celle, Ernst August's brother, also gave Leibniz
Despite repeated complaints from his various employers that he was doing everything but what he was being paid for, Leibniz did a considerable amount of work on the history of the Guelfs. As a preliminary to the history itself, he edited a vast mass of mostly unpublished archival material. He published six enormous volumes in 1698-1700, and a further three relating specifically to the Guelfs (the Scriptores rerum Brunsvicensium) in 1707-11. More saw the light of day after his death. He also completed two preliminary essays which show how literally he took the commission to start from earliest times. The first was called Protogaea, about the formation of minerals and fossils, and depending mainly on observations made in the Harz; the second was about European tribal migrations, as inferred from linguistic and place-name evidence. He collected an enormous amount of information about the origins of European languages, primarily in order to find evidence for a single archetypal language (Hebrew, perhaps), although his actual findings led him to conclude that different language groups had separate origins. More specifically, he disproved the claim of certain Swedish scholars that
Among the side-products of his archival work in Italy was a detailed refutation, not published in his lifetime, of the legend that there had been a female English Pope (Flowers Scattered on the Grave of Pope Joan), and an edition, in 1696, of Johann Burchard's scurrilous diary of life at the court of the Borgia Pope Alexander VI (the only one of Leibniz's works to get onto the Vatican's Index of Prohibited Books). But despite a permanently agonised conscience, he never started on the history itself. It is ironical that his employers really wanted only a readable but authoritative little book which would impress their rivals.
Leibniz's access to different courts, together with his genealogical expertise, made him diplomatically very useful to Hanover. Already in 1676 he had published a lengthy treatise, which rapidly went through five editions, defending the right of German states to be treated as sovereign, and not merely as vassals of the Emperor. For this he adopted the fanciful pseudonym Caesarinus Fürstenerius, or ‘Prince-as-Emperor’'. Later, he managed to persuade Anton Ulrich of Brunswick to drop his claim to the ninth Electorate of the German Empire, and in 1692 it was Ernst August who was duly elevated to the status of Elector of Hanover. Ernst August died in 1698, and under his successor, the Elector Georg Ludwig, intra-dynastic relations became strained almost to the point of war. The main bone of contention was whether France should be contained by a strong federation of northern states (the Hanoverian policy) or by appeasement. In 1702, just before the outbreak of the War of the Spanish Succession, Leibniz was summoned home from Brunswick
For many years Leibniz busied himself with the question of the English succession, especially in his role as confidant to Sophie, Ernst August's wife. The 1689 Bill of Rights, by excluding Catholics from the throne of England, made it almost inevitable that the succession would pass through Elizabeth of Bohemia (James I's daughter, and Sophie's mother), and hence to Sophie's eldest son, Georg Ludwig. The presumption was eventually enshrined in the Act of Succession of 1701; but both before and after the passing of the Act, there were delicate negotiations between London and Hanover. Leibniz later prided himself on the importance of his role in these discussions; but in fact he had no official status, and may even have endangered matters by his naïve intriguing with the Scottish spy, Ker of Kersland. Most of his published writings in his last few years were pamphlets about British politics (for example, his Anti-Jacobite of 1715).
Evidently, Leibniz saw himself as an authority on questions of succession, since he wrote memoranda and pamphlets on every important case (in particular the Spanish and Austrian successions after the death of Charles II of Spain in 1700, and the Tuscan succession in 1713 ). In 1700 he was invited by Georg Ludwig's sister, the Electress Sophie Charlotte of Brandenburg to help in negotiations to have her husband Friedrich elevated to the status of King of Prussia, but he received her letter just too late to take part. The following year he was twice in Berlin promoting Hanoverian policy against the pro-French faction. In 1708 he went on a secret but inconclusive mission to Vienna on behalf of the Duke of Brunswick, and against the interests of Hanover, in order to obtain part of the Bishopric of Hildesheim for the House of
Of all Leibniz's enthusiasms around the turn of the century, the dominant one was the promotion of scientific academies. He was involved in tentative schemes for Mainz, Hanover, Hamburg and Poland; but his main efforts were devoted to specific proposals for Berlin, Dresden, Vienna and St Petersburg. He had been lobbying for an academy in Berlin since 1695, using the good offices of the Electress Sophie Charlotte, with whom he was on very close terms. He first went to Berlin in person during 1698, and on a subse quent visit in 1700 the Elector Friedrich gave final approval to the project, and also to proposals, less fully worked out, for an observatory and a Book Commission. Leibniz himself was made Life President of the Brandenburg Society of the Sciences, and given a Brandenburg Court Councillorship, with an expense account of 600 thalers a year to cover the costs of an annual tour of duty in Berlin. This, incidentally, was the first occasion on which his name was given as ‘von’ Leibniz, although nothing is known about any official elevation to the barony. The society was not in itself particularly successful. Although it met regularly to discuss scientific papers, only one volume of its proceedings was ever published, and Leibniz was generally disappointed with the standard of the papers (apart from the many he wrote himself). However, it did add to the growing prestige of Prussia, and it formed the basis of the later Deutsche Akademie der Wissenschaften zu Berlin.
After his initial success in Prussia, Leibniz made consider able efforts during 1703-4 to persuade the Elector of Saxony to set up a similar institution in Dresden, but nothing came of the project. A few years later he was much more nearly suc cessful in Vienna, where early in 1712 the Emperor made him an Imperial Councillor and appointed him Director of the proposed academy. However, it did not finally come into existence until after Leibniz's death. Lastly, in the period between 1711 and 1716, he managed to obtain frequent audiences with Peter the Great of Russia on three different trips to Europe. He persuaded him of the value of a scientific academy for St Petersburg, but again his efforts did not bear fruit until after his death. However, he was once more (in 1712) rewarded with a salaried court appointment as adviser on mathematics and science. The Tsar was very keen to foster scientific co-operation between Russia and Europe, and promised to commission research on the position of magnetic north (which would assist a project of Leibniz's for a magnetic globe as a navigational aid); on whether there was a land bridge between Russia and America; and on the origins of the Slavs and their language, for which Leibniz wrote a preliminary essay.
Besides promoting scientific societies, Leibniz never lost any opportunity to advocate other pet schemes. One in particular was his idea that the German economy might be rejuvenated by introducing silk production. He himself experimented on it in his own garden, using mulberry trees grown from seeds imported from Italy. In 1703 he was granted production licences in both Berlin and Dresden , and in the former city the industry became quite important later in the century. Other projects he suggested for Berlin included a public health system, a fire service, a land drainage scheme, and steam-powered fountains; and he was also asked
After becoming involved with Berlin and Vienna, Leibniz spent less time at home than away. By 1712 he was in the pay of five different courts: Hanover, Brunswick-Lüneburg, Berlin, Vienna and St Petersburg (Celle had by now been amalgamated with Hanover). Not surprisingly, he was con stantly receiving complaints from all of them about his not giving value for money, and periodically had one or other salary stopped until he reappeared. The strongest complaints came from Hanover, where Georg Ludwig not unreasonably thought that he had first claim on Leibniz's services – and the history of the Guelf family had now been in preparation for over thirty years. In the autumn of 1712, Leibniz went back to Vienna and tried unsuccessfully to get a full-time post (open to non-Catholics) as Chancellor of the Siebenburgen. He stayed there for nearly )two years, ignoring repeated commands to return to Hanover. Nor was he moved by rumours that he had become a Catholic spy, the stopping of his Hanover salary, or the death of his patroness the Electress Sophie. He finally came back in September 1714, on hearing of the death of Queen Anne and of Georg Ludwig's accession to the English throne. On arrival, he discovered that Georg Ludwig had left for England three days earlier.
The usual picture of Leibniz's last years is one of miserable neglect in Hanover. It is true that he was rather miserable, but not because he particularly wanted to be in England. Still less had he suddenly acquired a yearning to emigrate with the
Leibniz's problem was that he was now in his late sixties and getting too infirm either to travel around as he used to, or to start a new life elsewhere. He did propose going to London, and in 1715 he made the extraordinary suggestion that he should be made the official historian of England. But he was much more tempted by Paris. Despite his anti-French politics, he had been invited there by Louis XIV, and would probably have gone if Louis had not died in 1715. At the same time he was actively considering a move to Vienna, and he even started negotiations to buy some property there. Other possibilities were Berlin, where he was still president of the Academy, and St Petersburg, where he held a councillorship.
He had the unwritten history of the Guelfs very much on his conscience, and worked extremely hard at it. He still hoped not merely that he would finish it before he died, but that having finished it he would then have time to get down to some serious philosophical writing. As it happens, some of his most important philosophical correspondence dates from this period (for example with the English theologian, Samuel Clarke, and with the French Jesuit, Bartholomew des Bosses). But the history and the philosophical magnum opus were not to be.
On 14 November 1716, after a week in bed with gout and colic, Leibniz died peacefully in the presence of his amanuensis and his coachman. He was seventy years old. The rump of the Council still in Hanover refused to attend his
Leibniz's life was dominated by an unachievable ambition to excel in every sphere of intellectual and political activity. The wonder is not that he failed so often, but that he achieved as much as he did. His successes were due to a rare combination of sheer hard work, a receptivity to the ideas of others, and supreme confidence in the fertility of his own mind. Whenever he tackled a new subject, he would read everything he could lay his hands on, but without submitting to orthodox concepts and assumptions. On the other hand, his desire to produce monuments to his genius, which would be both complete and all his own work, made it impossible for him to finish anything. Despite all his notes, letters and articles, he never wrote a systematic treatise on any of his special interests. His assistant Eckhart put it nicely when he said of the Guelf project that, as with numbers, Leibniz knew how to extend his historical journey to infinity.
His self-importance led him into spending too much time travelling from one court to another, and ingratiating himself with the aristocracy. When he was younger he had a reputation as an elegant courtier, savant, and wit (the Duchess of Orleans said of him: ‘It's so rare for intellectuals to be smartly dressed, and not to smell, and to understand jokes’). But in his later years he was an object of ridicule for his old-fashioned and over-ornate clothes, his enormous black wig and his halfbaked schemes. Georg Ludwig's younger brother once described him as an ‘archaeological find’, and suggested that Peter the Great must have taken him for the Duke of Wolfenbüttel's clown.
There were all sorts of reasons why he should have been
Leibniz's unpopularity at home was the inevitable price he had to pay for his universalist vision in politics and religion. He was indeed prepared to sacrifice the narrow interests of his own state and sect for a wider unity, and this set him against the nationalism and sectarianism of his age. It is sad that he spent his declining years among people who regarded him as a pampered traitor; but he would probably have fared no better as a resident alien in Paris, Vienna, Berlin or London.
Throughout the seventeenth century, the majority of university mathematicians continued in the restricted tradition of scholasticism, and the main impetus for mathematical advance came from the Renaissance humanist reaction against the universities. The most fruitful and original research was carried out by gifted amateurs, who were some times called virtuosi, as being endowed with a special, individual genius (or virtù, in Italian). This tendency to single people out as intellectual heroes fostered a spirit of competitive individualism, rather than of co-operative research – an attitude which probably encouraged the development of new ideas, but which tended to recede as mathematics became more and more technical.
The competitive spirit gave rise to considerable jealousies as to priority over the discovery of new theorems and methods. One manifestation of this was the custom of setting challenge problems. Often the challenger had already solved the problem himself, and wanted to publicise his individual achievement. Leibniz was involved in many such challenges, and they stimulated him to a number of useful discoveries, such as equations for the curves known as the catenary, the brachistochrone and the isochronous curve.
The emphasis on inventive genius encouraged greater interest in ideas themselves than in their detailed elaboration. This reinforced Leibniz's natural reluctance to follow his ideas through, and to censor the wilder ones. The resultant mass of partly-formed thoughts meant that some of his best
Most of Leibniz's mathematical discoveries came to him during his stay in Paris. Potentially one of the most important was that of binary arithmetic – though he was not actually the first to discover it, since it had already been thought of by Thomas Hariot early in the century, and again by Juan Caramuel y Lobkowitz in 1670. Leibniz himself later came to believe that the Chinese must have known about it, on the grounds that it was implicit in the theory of i ching.
The binary system is the simplest possible notation for numerals. Our ordinary decimal system has a choice of ten characters for each place (units, tens, hundreds, etc.). In the binary system there are only two characters: one to designate an empty place, the other to mark that it is filled. If the places can be defined independently, for example by a grid, all that is needed is an arbitrary mark or signal wherever a place is filled. Using the convention of 0 for empty, and 1 for filled, the system runs as follows:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | etc. |
| 0 | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 | etc. |
Although Leibniz was tremendously proud of his discovery, he did very little with it. Unlike Lobkowitz and modern mathematicians, he failed to generalise it into a theory of modular arithmetic with its own special theorems. Nor, apart from one very tentative draft, did he try to design a calculating machine which used it. It may seem odd to us, in the age of the computer, that someone who invented both a calculator and binary arithmetic should not have put the two together, and come up with something closer in principle to the modern computer. But in the context of the technology of the time, a binary machine would only have increased Leibniz's difficulties. There would have been more wheels, more friction, more carrying, and there would have had to be an extra mechanism for translating between binary and decimal, in order to make the calculator usable by ordinary people. The binary system came into its own only with the advent of electronics. As far as Leibniz was concerned, the greatest significance of his discovery was metaphysical, or indeed mystical, as showing how the whole universe could be seen as constructed out of Number. We shall come back to this in Chapter 6.
By far the most important of Leibniz's mathematical achievements was his discovery of the infinitesimal calculus, which he made at the end of his Paris period, in 1675. As before, he was not the first – Newton beat him to it by nine years – though Leibniz was the first to publish the discovery, in 1684.
The significance of the calculus is so great that it is
The infinitesimal calculus has two parts: the differential and the integral calculus. The differential calculus provides a general method for determining the rate of change at any instant of a quantity which is continuously changing in relation to another quantity of which it is a ‘function’. Leibniz was, incidentally, the first person to use the word ‘function’ in this modern sense of a variable, the value of which is uniquely determined by some other variable (as, for example, the distance covered by a falling body is a ‘function’ of the time it has been falling). Applications of the differential calculus include the calculation of planetary orbits; describing the motion of pendulums, waves or vibrating strings; Finding values of otherwise insoluble equations; establishing highest and lowest values of functions; calculating the bending of loaded beams – and so on.
Integration is the reverse of differentiation, and consists in reconstructing a whole from a given value at an instant – in other words, in going up one dimension. From a rate of change at a point you can reconstruct a whole line, from a line you can reconstruct an area it defines, and from an area you
Leibniz
was much exercised by the problem of the logical
respectability of infinitesimal quantities. At the time, mathematicians
generally
insisted that the objects of mathematics
should be real, in the sense of being representable
geometrically. They were
deeply suspicious of ‘imaginary’ quantities which could not be
constructed
by
ruler and compass – for
instance,
. Infinitesimals, such
as rates of
change at an
instant, clearly fell into this class. For example, speed is change of
distance
divided by time; but at an instant, no time
elapses and no distance is travelled – so the notion of distance
divided by time is strictly meaningless. As Leibniz wrote to
Bartholomew des Bosses in 1706:
Infinitesimals are mental fictions, though they have their place in calculations like imaginary roots in algebra. (G ii 305)
Although Leibniz went along with the prejudice that infinitesimals needed a geometrical foundation, the route by which he arrived at the calculus was algebraic rather than geometrical. His discovery arose from the concept of an infinite series converging on a limit: the differential calculus was a technique for determining the limit of such a series, and the integral calculus for finding its sum.
Leibniz's earlier researches in Paris had been directed in particular towards the age-old problem of squaring the circle. Traditionally, the problem was that of finding a method for constructing a square of exactly the same area as a given circle. In algebraic terms, it meant giving an exact numerical value to pi, that is, to the ratio between the circumference and the diameter of a circle. Closer and closer approximations had been obtained by taking regular polygons with more and more sides fitting just outside and just inside a circle, and calculating the difference between the lengths of their circumferences. But these results were necessarily only approximate.
Leibniz
believed that he had made an important step towards
the goal of a precise value for pi, when he discovered for
himself an infinite series of which the sum would be
exactly 
. This series went as
follows: 
...
All that was needed was a method for calculating the sum
of the series. It was eventually recognised that this was an
impossibility. Given the diameter of a circle, any numerical
expression for the length of its circumference would run to
infinite decimal places. But, spurred on by his partial success,
Leibniz worked intensively at the
theory of infinite series and
their summation. In the course of his labours, he produced a
number of results of considerable importance in their own
right. To give just one example, he discovered what has since
become known as ‘Leibniz's
test’ for whether an infinite
series
will converge on a single limit.
Of course he never found his crock of gold of squaring the circle, but his researches did lead up to the far greater prize of the infinitesimal calculus. What he discovered was that the gradient of a curve at a point (corresponding to a rate of
Leibniz's work on the calculus was, if anything, more successful even than he realised. If he had fulfilled his ambition of squaring the circle, his discovery would have been self-contained, whereas the discovery of the calculus ushered in a whole new era of mathematics. As for his failure to give a sound geometrical basis to the calculus, it was precisely the fact that he approached the issue algebraically that made his system superior to Newton's.
While Leibniz was developing and publicising his calculus during the last decades of the century, he remained genuinely unaware that Newton had discovered it before him. During his Paris period, he learned from a brief correspondence with Newton that the latter had made certain advances in the direction of the calculus, but Newton's letters revealed nothing about his general method. Leibniz's impression must have been confirmed much later, in 1689, on his first reading of Newton's Principles of Mathematical Philosophy, since the work was presented almost entirely in terms of traditional mathematics. This was despite the fact that the calculus was indispensable for some of the proofs, and would have made most of the rest far less cumbersome and obscure.
Leibniz knew that the calculus was of tremendous importance, and he could not believe that his rival would have deliberately suppressed it just where it would have been most useful. In this he was adopting a characteristically futurist perspective, seeing his own work and that of his contemporaries in the light of future advances yet unmade. We too tend to see the Principles historically, as inaugurating the ‘Age of Newton’, and so we can easily share Leibniz's puzzlement at the omission of the calculus. By contrast, Newton himself believed he was fulfulling the prophetic work of his ancient and modern predecessors, rather than creating something entirely new. He was therefore more motivated to stress his continuity with the giants on whose shoulders he was standing. Quite apart from this, there was the purely practical consideration that he could expect a much wider readership for a work that presupposed only conventional mathematics.
Even when
Newton's ‘method of fluxions’ was eventually
unveiled in print, it was by no means obvious that it was
essentially the same as Leibniz's
infinitesimal calculus. His
approach was basically geometrical; his terminology was suspiciously
reminiscent of the scholastic jargon of the ‘flowing’
of points and lines; and his notation, which involved the
addition or subtraction of dots over letters, was clumsy and
difficult to work with. Leibniz's
approach, on the other hand,
was algebraical; his language was fresh and appropriate,
incorporating such terms as differential, integral,
coordinate
and function, and his notation, which we still use
today, was
clear and elegant. It was based on the letter d
for ‘difference’
(as in 
, the symbol for a
differential), and
the contemporary long
s 
for ‘
sum’, or
integral.
If matters had rested with Newton and Leibniz, there
It is beyond reasonable doubt that Leibniz's discovery was in fact independent, but the nationalistic fervour aroused by the dispute, and the incontrovertible evidence in favour of Newton's priority, had disastrous consequences for English mathematics. While the Continental mathematicians of the eighteenth century made great strides in the theory of the calculus, and in its applications to Newtonian physics, the English stuck loyally to Newton's own much less suitable method of fluxions, and remained in a backwater for over a century.
Leibniz's major contribution to science lay in his clarification of certain key concepts in dynamics. Theorists were still struggling to turn the vague notions of everyday life into precise scientific concepts. The first step, due primarily to Galileo, had been to abandon the common-sense assumption that things stop moving unless they are kept going by a force, and to postulate instead the principle that things conserve their motion unless acted upon by a force. Descartes tried to systematise the laws of motion Galileo had established, but he ran into difficulties because of ambiguities in expressions like 'quantity of motion' and ‘force’. Leibniz's earliest success, acting on suggestions he owed to Huygens in Paris, was to give a clear diagnosis of the deficiencies in Descartes' conceptual apparatus.
Descartes based his system on a law of the conservation of motion. His idea was that God, having created the world of matter, then set it in motion, and although motions could be exchanged between one thing and another, and the direction in which things moved could change, the total quantity of motion had to remain constant. Leibniz showed that Descartes' formulation was seriously confused in at least two respects.
The first confusion was that Descartes talked of motion, without making an adequate distinction between speed and velocity. Speed is a measure of distance covered in a particular time, whereas velocity is what is known as a vector quantity, and is a measure of distance covered in a particular direction in
Descartes himself was well aware of this loophole in his mechanics, and actually exploited it in order to account for the influence of mind over matter. He believed that human behaviour was controlled by the direction of motion of tiny particles in the cavities of the brain, which made them enter this nerve rather than that. The principal seat of the soul was the pineal gland, situated in the main cavity at the centre of the brain. The soul could use the gland to deflect the direction of motion of particles without itself adding any new motion, rather as the rider of a horse can change its direction without actually pushing it round himself. But, as Leibniz pointed out, the soul would have to exert some sort of physical force in order to move the pineal gland so as to change the direction of motion of particles in the brain – and how an immaterial soul could influence matter was precisely what needed explaining in the first place.
The second problem with Descartes' concept of quantity of motion was its inability to relate a fast motion of a small body to a slow motion of a large one. Here again metaphysical considerations were involved. Descartes wanted to base his account of reality on the simplest possible metaphysical categories, and opted for the directly measurable, spatio-temporal properties of size, shape and motion, as constituting the essence of the material world. That is, he saw the physical universe as consisting of nothing but sizes, shapes and relative positions changing through time. In Descartes' terminology, the essence of matter was geometrical extension. However, Leibniz showed that the dynamically relevant measure of an object's size was not its geometrical dimensions, but its mass.
So instead of talking loosely of ‘size’ times ‘motion’ as being conserved, Descartes should have defined ‘quantity of motion’ as mass times velocity – in other words, as momentum.
As it happens, momentum is conserved
in certain
systems – for example, hard balls colliding on a plane surface
(as in snooker). But Leibniz showed
that in the case of falling
bodies, it is not momentum, but a different measure of energy
that is conserved. He explained the distinction by means of
the following thought-experiment. A certain weight falling
through a certain distance will obviously release the same
amount of energy as something four times the weight falling
through a quarter of the distance. But the velocity of a falling
body is proportional only to the square root of the distance
travelled, so that the momentum of the smaller weight will
only be a multiple of 
, whereas
that of the
larger weight will be a multiple of
.
Leibniz
concluded
that, to compensate for the effect of the square root, there had
to be a quantity proportional to the mass (m) times the
square
of the velocity (v), which would be the true measure of
the
energy conserved in all interactions. He called this quantity
vis viva (‘live force/energy/power’
– the concepts were not
yet distinguished), and he assumed that it was simply 
. In
this he came very close to the modern concept of kinetic
energy, or 
, which a
body has
by virtue of its motion.
There are, however, considerable problems over the extent to which kinetic energy is conserved. For a start, there needs to be a complementary potential energy, which a body has by virtue of its position. Thus, if the energy of a swinging pendulum is to be constant, it will all be in the form of kinetic energy at the bottom of the swing, and of potential energy at the top. Leibniz did postulate what he called ‘dead force’ (vis mortua), but he made this a much more general concept, covering all
A more serious problem is that our everyday experience includes innumerable instances of kinetic energy being generated out of completely different forms of energy, and the other way round – for example, an animal waking up, gunpowder exploding, steam pushing out of a boiling kettle, brakes stopping a moving vehicle. Leibniz's explanation of the dissipation of kinetic energy through friction is plausible enough: the energy is taken up by faster motions of the particles of the bodies concerned – they get hotter. In the other cases, however, there is no independent evidence that the energy pre-exists in the form of motions among the particles of the muscles, gunpowder, or unburnt fuel. Instances such as these show the need for a non-mechanical concept of energy, and it is here that Leibniz's explanations become metaphysical rather than scientific. We shall look at the metaphysical dimension of Leibniz's theory of energy in greater detail in Chapter 5.
As significant as his critique of Descartes' mechanics was Leibniz's attack on Newton's account of force. In the Principles, Newton limited himself to describing interactions between bodies in terms of general mathematical laws. This limitation was both a strength and a weakness. Newton succeeded in making the complexities of nature amenable to mathematical description only by simplifying the phenomena: by treating material particles as if they were infinitely hard yet infinitely elastic, concentrated at points, capable of exchanging any amount of force all at once, connected by forces operating instantaneously at a distance, and so on. Leibniz complained that this made Newton's system an
A more damaging criticism which Leibniz brought against Newton was that he gave pseudo-explanations in terms of magical ‘occult virtues’. Just as Molière had joked about the scholastic explanation that opium sends one to sleep because of its ‘dormitive virtue’, so Leibniz laughed at Newton for explaining the gravity of things as due to a gravitational force. The trouble was that Newton's forces were defined in terms of directly measurable masses and changes in velocity. This meant that these masses and velocities themselves were the primary realities. The forces he postulated added nothing new to reality, and therefore explained nothing.
Leibniz held that it was not enough to formulate mechanical laws to describe and predict the behaviour of physical systems. A genuine science also had to explain the phenomena by postulating underlying mechanisms and powers of which perceptible motions were the results. Motions must be derived from powers, not powers from motions. In other words, what was needed was a dynamics, or science of powers, not just a kinematics, or science of motions. This requirement dominated Leibniz's approach to two scientific questions which were to have a significant influence on his metaphysics: gravitation, and the transfer of forces between colliding bodies.
In the New Physical Hypothesis, which comprised the two treatises dedicated to the Royal Society and the Paris Academy
Much later, in his Specimen of Dynamics ( 1695), Leibniz tried to give an account of the mechanism which mediated exchanges of force between colliding bodies. In real collisions (unlike Newton's idealisations), there had to be a finite period during which one body slowed down and the other picked up speed. This implied that bodies had a certain size, and were not absolutely hard or elastic, since the only conceivable mechanism for transfer of force was that bodies were first squashed together, and then gradually sprang back from each other once all the kinetic energy had been taken up. However, as soon as it is accepted that transfer of force between every day objects must be mediated by a mechanism, there is no point at which you stop needing smaller and smaller sub-mechanisms. At no level can you suddenly say that force is transferred directly.
Elasticity is itself a phenomenon requiring explanation in terms of pushings of particles. At the first instant of impact, the outermost particles of each colliding body push against their neighbours, and these in turn push against their neighbours, and so on right through each body. But then each of these pushings needs to be explained by the compression of sub-particles, and so on to infinity. The conclusion Leibniz drew was that, ultimately, forces were not really transferred at all. All action was, as he put it, spontaneous. The energy required for a body's motion on the occasion of an impact, had to be drawn from its own resources, since it could not actually take up any energy from bodies impinging on it.
Leibniz's theory of the spontaneity of all motion is not as silly as it might seem. It is a commonplace that every force has an equal and opposite reaction. In the case of colliding bodies, the reaction is the force holding each body together. If either of the bodies has less cohesive force than the kinetic energy involved in the collision, it will shatter instead of moving as predicted by the laws of mechanics. So Leibniz was right to say that bodies can take up only as much energy as they have the capacity to absorb, even though it does not follow that they cannot absorb energy from each other at all.
An even more significant aspect of the theory was its abandonment of the traditional notion that matter was essentially inert. Leibniz saw that if the only function of matter was as a passive carrier of forces, then it had no role to play in scientific explanation. Its only role would be the metaphysical one of satisfying the prejudice that forces must inhere in something more substantial than themselves. He maintained that matter was nothing other than the receptive capacity of things, or their ‘passive power’, as he called it. Matter just was the capacity to slow other things down, and to be accelerated rather than penetrated (capacities which ghosts and shadows
However, although Leibniz was ahead of his time in aiming at a genuine dynamics, it was this very ambition that prevented him from matching the achievement of his rival Newton. Newton succeeded in producing a comprehensive theory of kinematics precisely because he avoided ‘inventing hypotheses’ about dynamics, or the powers and mechanisms underlying the kinematics. It was only by simplifying the issues in this way, that Newton succeeded in reducing them to manageable proportions.
Leibniz's perception of his own scientific and philosophical position was, to an important extent, defined by reference to his interpretation of Newton. It is therefore surprising that there was no direct confrontation between the two men in later life. The nearest was an exchange of letters between Leibniz and Newton's friend, the theologian Samuel Clarke , who wrote in close consultation with Newton himself. In a letter of November 1715 to Caroline, Princess of Wales , Leibniz had criticised certain theological implications of Newton's physics, and she had invited Clarke to reply. In accordance with custom, both sides wrote with a view to eventual publication, and the letters were in fact printed in 1717, within a year of Leibniz's death. The two most important topics of the correspondence were entropy, and whether space was absolute or relative.
At the very end of the Optics, Newton had suggested that God might eventually have to intervene in order to restore the orderly motions of the planets. Leibniz read this as implying that the clockwork of nature would eventually run down unless God wound it up from time to time. In other words, God would prevent a state of entropy, in which all energy would have become evenly distributed, and therefore incapable of doing any work. At one level, the argument was overtly theological: Leibniz claimed that it was blasphemy to suppose that God would need to correct what he had created; Clarke replied that it was tantamount to atheism to suppose that creation could function without God. But at another level, they were really having a scientific argument about the conservation of energy.
Leibniz assumed as a basic principle of the mechanistic world-view, that the total amount of energy in the universe remains constant. That there can be no increase in energy, is equivalent to saying that there can be no interference from outside – no Cartesian deflections of particles in the brain by the immaterial soul, no Newtonian adjustments to planetary orbits by God. God might miraculously suspend the laws of nature as part of his divine plan, but it makes no sense to suppose that the ordinary workings of nature might depend on miraculous interventions.
That there can be no decrease in the amount of energy, Leibniz took as implying that the universe as a whole must be a perpetual motion. Although he strenuously denied that there could be perpetual motion machines within the universe, he was wrong about the universe as a whole, since he failed to take account of the fact that work is only achieved in a system in which energy passes from a part with a higher level of energy to one with a lower level. So, the more work that is done in the universe, the more evenly energy is spread, and
Part of the reason for Leibniz's mistake was the limitation of his concept of energy to kinetic energy, which cannot exist without motion. He believed that whenever a mechanism ran down through friction, the energy would always be taken up by motions of particles, with the consequence that motion in the universe could never cease. But even with this conception of energy, Leibniz failed to see that the law of the conservation of energy would still permit the universe to degenerate into a gas of randomly moving particles with the same total kinetic energy as at present.
In fact, what is lost as the universe approaches a state of entropy is not energy, but what is now known as information, or the degree to which it is non-random: variety and order give way to uniformity and chaos. So what Leibniz really needed in order to deny that the universe would naturally run down, was a ‘principle of the conservation of information’. As it happens, he did believe in such a principle, but his reasons for it were metaphysical rather than scientific.
The other main point at issue in the correspondence with Clarke was the question of whether space was absolute or relative. Newton's system was premised on the assumption that there was an absolute difference between a body's being at rest, in motion, or under acceleration. The distinction required the concept of a fixed frame of reference, such as the 'fixed stars', to define the absolute space relative to which bodies moved or accelerated. As for what space was, over and above the bodies in it, Newton proposed in the Optics an analogy with the ‘sensorium’, or subjective perceptual space. We mortals perceived things by means of perceptual images
Leibniz opened with objections to Newton's theology, but he soon progressed to a fundamental critique of the very notion of absolute space. His main point was that, if space was distinct from everything in it, then it must itself be completely uniform and homogeneous. But in that case, it could not conceivably fulfil its function as an absolute frame of reference, since it would have no markers to which one could refer, in order to tell whether anything moved relative to it. We could imagine lines, like the grid lines on a map, relative to which we might suppose things to be moving. But there were no such lines marked out in real space, and even if there were, we could have no grounds for saying they were stationary. Consequently, it made no sense to suppose some privileged frame of reference as absolutely at rest, whether the fixed stars, or real space itself.
Leibniz's conclusion was that space was unreal. He thought it was a mere superstition to suppose that there was some imperceptible container which the whole of the material universe was ‘in’. Ultimately, only things existed. We could make true statements about their ‘order of coexistence’ – statements like ‘Mercury is nearer the Sun than Venus’ – but space itself was an abstraction. The only basis for truths about spatial relations was how they appeared to different observers, especially to God as the only unbiased and perspective-free observer.
Given Leibniz's relativism about space, it is natural to ask how far he anticipated Einstein's theory of relativity. This is not an easy question to answer, since Leibniz, unlike Einstein, did not produce any mathematical theory of his own to rival Newton's. But the very fact that he failed to attempt an alter
Leibniz was exceptional among his contemporaries for his belief in the importance of logic. This is undoubtedly a major reason for the recent revival of interest in Leibniz's philosophy, now that logic has regained a central position on the philosophical stage, for the first time since the heyday of scholasticism.
Ever since the ancient Greeks first tried to systematise the principles of good reasoning, logic was in constant rivalry with the art of rhetoric. Rhetoricians specialised in the refine ment of concepts through the proper use of language. Logicians, on the other hand, aimed to abstract completely from the subject matter or content of reasoning, and deal exclusively with purely formal relationships among concepts, and among propositions composed out of concepts. Just as mathematicians used letters such as x and y to stand for any number, so logicians learned to do the same for concepts. Until Leibniz, however, little was done to avoid the imprecisions of ordinary language by symbolical representation of the logical relationships themselves.
Throughout later antiquity and the middle ages, and even into the eighteenth century in places, the standard curriculum at schools and universities included both logic and rhetoric. With the rise of scholasticism in the thirteenth and fourteenth centuries, the emphasis became heavily biased towards logic. As part of their reaction against university education in general, the humanist intellectuals of the Renaissance tended to espouse the cause of rhetoric as the only true art of reasoning. Nizolio (see p. 10 ) was typical of this approach.
At the end of the sixteenth century, logic was faced with a more serious challenge from the new scientific movement. It became increasingly evident that the new scientific methods far outclassed those of the old scholastic theorisers, and that they succeeded not merely through the amassing and organising of experimental data (as advocated by Bacon in particular), but through abstract problem-solving techniques which owed nothing either to traditional logic or to rhetoric.
The most popular alternative model for effective reasoning was the axiomatic method of Euclidean geometry, which systematically derived sets of theorems from a minimum of axioms and definitions. But the applicability of the method outside mathematics was highly problematic. In particular, it could not show how to draw general conclusions about the world from experience, nor how to discover or invent any thing. Nevertheless, there was a brief fashion for presenting work ‘in the geometrical style’ – for example Newton's Principles, Spinoza's Ethics and Leibniz's own work on jurisprudence.
Most of the major philosophers of the period wrote extensively on the problem of the best method for reasoning out new truths, especially Descartes in France, and Bacon and Hobbes in England. Leibniz's approach was to try and reconcile the logical, rhetorical and geometrical traditions by blend ing their three distinct emphases (on formalism, on linguistic propriety and on mathematicisation) into the single vision of a formal language notated mathematically.
One of the more conventional aspects of Leibniz's logic was his acceptance of the traditional system of definition by genus and differentia (the ‘method of division’ ). According to this
Whole trees of branching classes could be defined in this way. One of the simplest and best-known examples was the so-called ‘Tree of Porphyry’', named after the third-century commentator on Aristotle's Categories. This started by dividing things into the material (bodies) and the immaterial; bodies into the animate (living things) and the inanimate; living things into those that had sensation (animals) and those that did not (vegetables); and animals into the rational (man) and the non-rational (brutes). The Tree of Porphyry served as a model for most subsequent systems of taxonomy, such as the modern classification of the animal kingdom based on the work of the English naturalist, John Ray ( 1627-1705), and the botanical classification devised by the Swedish taxonomist, Linnaeus (Carl von Linné, 1707-78 ).
In common with many of his contemporaries, Leibniz believed that, in principle, all concepts could be defined in terms of their position in one single hierarchy of this sort. And however unrealistic the ideal, it had important repercussions on his philosophy as a whole.
The method of division encouraged a belief that, at any stage, the concept defined must be more complex than the concepts used to define it. So, if the concept ‘man’ was a combination of the concepts ‘rational’ and ‘animal’, these components had to be simpler than the concept compounded from them. It seemed to follow that, ultimately, there must be certain absolutely simple, atomic concepts out of which all others were constructed, otherwise there would be an infinite
At the other end of the scale, there was the question of where to stop the process of subdivision into species – in traditional terms, the problem of the ‘lowest’ species. In the example of the Tree of Prophyry, a biologist would say that Man was a lowest species, on the grounds that its members were inter-fertile. But a geneticist or ethnologist would recognise subspecies such as races, tribes and even families. From a purely logical point of view, it is possible to go on subdividing as long as there remain characteristics which at least one member of a species has, and others lack. Consequently, all the members of a logically lowest species will be absolutely identical with each other.
Aristotle, whose interests were primarily biological, saw an absolute difference between species-defining characteristics, and the mass of characteristics peculiar to individuals. He explained the difference in terms of an analogy with carpentry. Some of the characteristics of a carpenter's finished work are due to the form he imposes on his material, and these will be common to all pieces of furniture of the same type. Any remaining differences are due to the individual character of the wood of which each piece is made ( Aristotle actually used ?λη, the Greek word for 'wood', to mean material or matter, as contrasted with form or essence). Similarly, God the Creator imposed specific forms on different portions of matter when creating plants and animals of various species.
Medieval philosophers were much exercised by the question of whether there was an absolute distinction between formal and material characteristics, and whether individual things were individuals in virtue of having a unique set of qualities, or a unique piece of matter. This was the issue discussed by Leibniz in his undergraduate dissertation On the Principle of Individuation. There he sided with the thirteenth- Century Irish or Northern British philosopher, John Duns Scotus (the original ‘Dunce’), who held that each individual had its own haecceity, or 'thisness', which was distinct both from its qualities and its matter. However, Leibniz soon came to the view that the process of subdivision into species ended only with a complete description of an individual, so that to be an individual was to be a lowest species. He was therefore committed to what he called the ‘Principle of the Identity of Indiscernibles’ (now often known as ‘Leibniz's Law’'), to the effect that a lowest species could not have more than one member: if two things were distinct individuals, there had to be something that was true of the one but not of the other, thereby making them of different species.
Leibniz believed that ultimately the only realities were individaul substances and their properties. Precisely what he meant by ‘individual substances’ will be one of the main topics of the next chapter. But as a first approximation, we can take him as meaning things like trees, sheep or people, in contrast with items that have only a secondary or dependent existence, such as collections of things, or relations. These secondary beings are dependent in at least two different ways.
Firstly, other things can exist without them, but they can not exist without other things. For example, you can have sheep without having a flock, but you cannot have a flock
Secondly, collections and relations are as they are only in virtue of some sort of mental attitude. A flock of fifty sheep is not any arbitrary collection of fifty sheep, but fifty particular sheep considered by intelligent beings as forming a unit in relation to some specific function. For instance, they may have the same owner or be in the charge of the same shepherd, despite being scattered over the hillside or mingled with another flock. Again, relationships presuppose a particular set of criteria or point of view in respect of which they hold: a large thing is large only considered in relation to something smaller.
Leibniz called things like collections and relations 'semi-mental': semi-mental because they were mind- dependent, but only semi-mental, because they were based in reality, and true assertions could be made about them. This then invites the question of how such assertions can be true, if what they are true of is not fully real.
The most natural account of truth is to say that it consists in a correspondence between reality itself and a linguistic item, whether spoken, written or silently thought. For the correspondence to be perfect, it must apply at the structural level as well as at the level of content. So, given that the primary constituents of reality are substances and their properties, the basic linguistic structure will be the one that most closely mirrors the having of properties by substances. The appropriate structure is the one known as the subject-predicate form. In sentences of this form, the subject identifies a substance, and the predicate attributes a certain property to it. So, in the sentence ‘Socrates is wise’ the subject-term ‘ Socrates’
When we make true assertions about dependent beings, it ought to be possible to paraphrase what we say into sentences of the subject-predicate form, in which the subjects refer to genuine substances. For example, in the sentence ‘The flock is grazing’ subject refers not to a substance, but to a collection of substances. But it could in principle be rewritten as a series of sentences about individual sheep.
However, relational statements are more problematic. Leibniz made various attempts at analysing them in terms ofconjunctions of subject-predicate sentences. One example he discussed (C 287) was the sentence ‘ Paris is the lover of Helen’, which asserts a relationship between two subjects, Paris and Helen. Leibniz said that it was equivalent to the two subject-predicate sentences ‘ Paris loves’ and ‘ Helen is loved’, together with the proviso that each was true only because the other was true. This type of paraphrase removed relations from within sentences and turned them into causal relations between sentences. So we should not expect relations to correspond to distinct components of reality, but to facts about why things are as they are. Ultimately, God made Paris a lover because he was making Helen loved, and Helen loved because he was making Paris a lover.
However, Leibniz was not particularly interested in tinkering with ordinary language as it was. He was mainly concerned with the possibility of an uncompromisingly ideal language in which every truth could be expressed without recourse to relational properties at all. If you could describe all the properties of substances, you could deduce any relational truths you liked, but they would give no new information about reality. For instance, if you knew the weights of two objects, the assertion that one was heavier than the other
Leibniz's thesis that relations were not primary constituents of reality gave a new twist to his Principle of the Identity of Indiscernibles (see p. 53 above). It is a truism to say that if there are two distinct things, there must be something that is true of the one but not of the other, otherwise there could be no grounds for saying that there were two things at all. In everyday situations, the difference would normally be a relational one, such as spatial position. For example, I might hold two otherwise indiscernible objects in different hands, and it would be true of only one of them that it was in my right hand. But if, as Leibniz maintained, relational truths are reducible to truths about the intrinsic properties of substances, then it must follow that everything is intrinsically different from everything else.
This is a much more radical thesis than the original truism, and it gave Leibniz a further defence against the Newtonian theory of absolute space (see pp. 46 ff). Clarke claimed that God had created innumerable atoms differing only in their spatial location, and that therefore space was the prior reality. Leibniz replied that since space was only relational, it had to depend on the diversity of related things, not the other way round. Clarke's argument showed not that space was absolute, but that there could not be two absolutely identical atoms.
Leibniz's thesis that only substances and their properties exist gave rise to another problem, namely ‘the problem of existence’. As we have seen, a complete set of properties defines an individual substance. But not all possible substances are actualised. So what is the difference between a substance that exists, and one that does not? It cannot be any
One solution which Leibniz studiously avoided was that of Locke. For Locke, the essential difference between a mere abstraction and an actual substance was that the concept of the latter included, as a distinct ingredient, the ‘idea of substance in general’. Locke had to admit that the idea was the particularly obscure one of ‘an I know not what’, underlying and supporting its properties. Leibniz could no more define exist ence than Locke could, but at least he did not add to the difficulty by attributing it to an occult base, which was, as it were, garnished with properties.
In Leibniz's view, substances were nothing other than groups of existent properties. For a property to belong to, or 'inhere in' a substance was simply for it to be a member of a particular group. When people join together to form a club, there is no extra entity over and above the membership (some 'club-in-itself') to which every member directly belongs. Similarly with existent things, there is no extra entity (an underlying substance, or ‘thing-in-itself’ ) in which all the detectible properties ‘inhere’, like pins in a pincushion.
In any account of the nature of truth, it is essential to distinguish between truths about abstractions, and truths about individual substances. If we take an abstraction, such as triangularity, the only predicates that can be truly ascribed to triangularity as such, are those that belong to its definition, or are deducible from it. So, if we define a triangle as a plane figure with three angles and three sides, we can say: ‘A triangle is a plane figure’, or: ‘A triangle has three angles’, or even: ‘A
Truths about abstractions are described as analytic, since it can be discovered by a process of analysis whether or not the predicate ascribed to the subject is a component of the definition of the subject. They are also necessary (necessarily true), since they are entirely self-contained: they risk no claims about what might or might not be the case in the real world, and so they cannot be falsified. Contingent truths, on the other hand, refer to concrete individuals, and are contingent upon what happens to be the case.
Most philosophers were in a position to distinguish between necessary and contingent truths. Leibniz, however, had spoilt the contrast between abstractions and concrete individuals, through his doctrine that an individual was a lowest species. At higher levels, he could still contrast the abstract concept with the individuals it covered. But the concept of a lowest species was a complete specification of all the properties of an individual – it was a ‘complete concept’, or ‘individual notion’. Consequently Leibniz could not differentiate between what was necessarily true of an individual in virtue of its concept, and what was contingently true of it as a concrete individual. All truth had to be analytic, since to say that a predicate belonged to an individual subject was to say that it was part of the complete concept of that subject. As he wrote to Arnauld:
In every true) affirmative proposition, whether necessary or contingent, universal or singular, the notion of the predicate is somehow included in that of the subject – otherwise I do not know what truth is. (G ii 56)
It has often been held against Leibniz that, if all truth is
Leibniz defined a ‘necessary’ truth as one which could not have been otherwise, in that its opposite would imply a contradiction. So, it is necessary that a triangle has three sides, since the idea of a non-three-sided, three-sided figure is self-contradictory. By a ‘contingent’ truth, he meant one that could have been otherwise, in that its opposite would be noncontradictory, or logically possible. To use his own example, it is a contingent fact that Caesar crossed the Rubicon, since denying that he did so would not contradict anything else in his complete concept.
The difference between Leibniz and his critics lies in the question of what might or might not be contradicted by the idea of Caesar's turning back. In so far as the actual Caesar was the realisation of a complete concept including the predicate ‘crossed the Rubicon’, it is indeed contradictory to deny that that Caesar crossed the Rubicon. But this presupposes God's decision to actualise a Rubicon-crossing, rather than a non- Rubicon-crossing Caesar. And it is a trivial truth that, given God's final decision, any alternative would contradict it.
However, nothing has been said about the internal relations between the predicates constituting the subject. These cannot be analytically connected, since they are logically distinct. The infinitely many possible Caesars are nothing other than different combinations of separate predicates. If we imagine two Caesars with identical lists of predicates up to the point of
In fact, corresponding to every contingently true proposition, there will be a false one which is analytically true of another possible, but unactualised, subject, which differs only in that respect. Consequently, contingent truths are not true because they are analytic, but because the subjects of which they are analytically true are actual. And the difference between the actual and the merely possible involves the notion of existence, which, as we have seen, is not part of the concept of any created being.
The fact that God actualised this world rather than some other, cannot itself be an analytic truth. But given this one fundamental and infinitely complex contingent fact, all else is indeed analytic as part of it. Since God knows the concepts he has chosen in every detail, he himself has no use for knowledge in the form of propositions at all – he merely thinks the subject-concepts. And the more we mortals learn about individual subjects, the more we approach God's state in this respect.
In defining truth as analytic, Leibniz was trying to get away from the sirnplistic view that truth consisted in a correspondence between language and reality. But instead of abolishing the question of correspondence altogether, he shifted it over to the relation between our concepts, and concepts in the mind of God. And since he held that reality was nothing other than the realisation of a privileged set of divine concepts, Leibniz's thesis was much less radical than it might at first seem.
In common with most logicians of his time, Leibniz made a
Leibniz was dominated by the vision of a foolproof logic of discovery. His central idea was that the logic of discovery and that of judgement should be perfectly complementary. Since judgements of truth were always analytic, symmetry suggested that the process of discovery should be synthetic, or combinatorial, to use Leibniz's preferred term. In effect, it would be an extension of his student work on the art of combinations. However, he could not altogether eliminate the need for judgement and intuition, since the combinatory art was only a method for generating all the possible combinations of a set of concepts. It could not tell you how to analyse complex concepts into simples, nor could it tell you which combinations to prefer:
The human mind is analogous to a sieve: the process of thinking consists in shaking it until all the subtlest items pass through. Meanwhile, as they pass through, Reason acts as an inspector snatching out whatever seems useful. (C 170)
Moreover, the further mankind progressed towards a perfect language, the less scope there would remain either for the
With the passage of time, certain operations which were once combinatorial will become analytic, after everyone has become familiar with my method of combination, which is within the grasp of even the dullest. This is why, with the gradual progress of the human species, it can come about, perhaps after many centuries, that no one will any more be praised for accuracy of judgement; for the analytic art (which is still virtually confined to mathematics in its correct and general use) will have become universal and applied to every type of matter through the introduction of a scientific notation or ‘philosophical character’ such as I am working on. Once this has been accepted, correct reasoning, given time for thought, will be no more praiseworthy than calculating large numbers without any error. Furthermore, if there is also a trustworthy catalogue of facts [a ‘Universal Encyclopaedia’] . . . written in the same notation, together with the more important theorems . . . derived from the notation either alone or with observational data, it will come about that the art of combination will lose all its glory. (C 168)
We must now consider what grounds Leibniz had for believing that both analytic and synthetic reasoning could be reduced to purely mechanical operations. The key lies in what he called the Principle of Identity. This is the principle that a proposition is proved to be necessarily true if it either is itself an identical proposition, or can be reduced to one. An identical proposition is one in which the predicate is explicitly identical with or included in the subject. The simplest cases are of the form a is a, or ab is a. For example, ‘Green is green’, or ‘Green grass is green’. Conversely, a proposition is necessarily false, or contradictory, if it is reducible to the form a is not-a.
The only legitimate procedure for reducing a proposition to an identical one is that of replacing a term by definitionally
Now the predicate is explicitly included in the subject, and the proposition is proved.
In principle it would be possible to have a purely mechanical procedure for checking whether a proposition was identical, contradictory or as yet unproved. You merely had to see if the symbol on the right could be found on the left, and if so, whether either of them was preceded by a negation sign:
The only proposition of which the contrary implies a contradiction without one's being able to demonstrate it, is one of formal identity. The identity is formulated explicitly in the proposition, so it cannot be demonstrated – demonstrated, that is, made evident by reason and inferences. Here the identity can be made visible to the eye, so in this case it cannot be demonstrated. The senses make it evident that a is a is a proposition of which the opposite, a is not-a, formally implies a contradiction. But that which the senses make evident is indemonstrable. So the real, indemonstrable axioms are identical propositions. (C 186)
Although the checking process could in principle be done by a machine, human reason was still necessary for substituting definitions. To bypass human reason altogether would require a means of symbolising all the components of all complex concepts. In general terms, Leibniz's vision of a 'Universal Characteristic' to fulfil this function was very much in the tradition of Aristotle's tentative classification of concepts in the Categories; of the far more elaborate ‘Great Art’ of the medieval Spanish missionary Ramón Lull; of contemporary ‘universal
A perfect notation must, of course, reflect the logical structure of the concepts themselves. It is tempting to assume that the complete concepts of individuals will be complexes of elementary components that are in themselves positive. So the individual concept of a particular physical object would be the sum of all its actual, positive properties – a particular shade of colour, its precise shape, size and weight and so on. Its complete concept would therefore be a sort of super-recipe, specifying absolutely every ingredient, and with infinite precision; and like other recipes, it would specify only what went in, and not what was left out. This was the approach that Locke adopted. However, Leibniz had three reasons for rejecting the assumption that all the components of individual complete concepts were positive simple predicates. The first arose from his belief that the created universe was differentiated from God only by the inclusion of an essential negative element (we shall come back to this in Chaper 6).
The second reason was that, if all the components of complex concepts were positive, there would be no way in which anything could be incompatible with anything else. Leibniz held that two concepts were incompatible only if there was some element in the one which was negated in the other, so that a contradiction of the form a and not-a could be derived from them. Since many concepts are mutually incompatible, for example those of the various possible Caesars, the constituents of complex concepts had somehow to include negation.
The third reason for rejecting a purely positive notation, was that it could represent only one aspect of an individual's complexity. As a concrete substance it might be nothing but the sum of its actual, positive characteristics. But as a lowest species, each individual also belonged to the realm of abstraction. As such, the individual was essentially the product of limitation or negation, and was what it was by virtue of its position within the hierarchy of abstract concepts. It was an act of faith on Leibniz's part that definitions of individuals arrived at by compounding concrete, simple predicates (a particular shape, size, colour and so on) would coincide with definitions reached by the repeated subdivision of abstract concepts (as in the Tree of Porphyry). But any notation capable of doing justice to the latter aspect had both to represent limitation and also to register the pedigree of each individual concept.
Leibniz's two-number notation was geared to fulfilling both requirements, even if rather clumsily. Limitation was symbolised by the negative number, and the pedigree would be traceable through the prime factors of the numbers, of which every genuinely distinct complex concept would have a unique combination.
The overwhelming disadvantage of the system was that numbers would rapidly become unmanageably large. It was possible to produce small-scale models of how it might work,
An alternative approach which Leibniz envisaged, but never actually experimented with, was to give concepts a single number, but in binary notation. This way, even the most complex number might run to only a few millions, and the problem of incorporating the negative element was neatly overcome by interpreting all the 0s in any number as negative, and the 1s as positive. The main snag was that there would be no obvious way of registering how a large number could have been generated from a unique combination of smaller ones.
To sum up so far, Leibniz's long-term goal for logic involved a merging of the logic of discovery, the logic of judgement, and the rhetorical ideal of a perfect language. Given a binary notation and a set of combinatory rules, all possible thoughts could be generated and validated by purely mechanical means. In the mean time, we would have to content ourselves with improving the logic we already had.
For most people, at least until well into the ineteenth century, logic was the theory of the syllogism. The theory is based on four general types of subject-predicate proposition, which can be interpreted as involving different permutations of ‘all’ and ‘not’. They are as follows (with S representing the subject-term, and P the predicate-term):
| 1 | Universal Affirmative | All S is P. | |
| 2 | Universal Negative | No S is P. | (All S is not-P.) |
| 3 | Particular Affirmative | Some S is P. | (Not all S is not-P.) |
| 4 | Particular Negative | Some S is not-P. | (Not all S is P.) |
A syllogism consists of three propositions, two premises and a conclusion, such that the premises have one term in common (the ‘middle term’, = M), and the terms of the conclusion are the remaining two terms of the premises. This yields four possible combinations of positions of the terms (reversing the order of the premises makes no difference). These four combinations are known as the syllogistic figures, and are as follows:
| Figure | 1 | 2 | 3 | 4 |
| Major Premise | MP | PM | MP | PM |
| Minor Premise | SM | SM | MS | MS |
| Conclusion | SP | SP | SP | SP |
In every figure, each proposition can be of any of the four types, giving a grand total of 256 moods, or possible combinations of different types of proposition.
One of the tasks of the logician was to say which of the moods constituted valid arguments, and why. But the rules evolved for validating syllogisms were clumsy and arbitrary, and logicians could not even agree on a list of valid moods. The main problem was the ambiguity of the word ‘some’ in ordinary language. Take, for example, the following controversial syllogism:
The first reason is that normally, when we say some individuals are something, we imply that others are not. Since all
The other reason for rejecting the syllogism would be that 'Some unicorns are vertebrates' can be interpreted as meaning 'There exists at least one vertebrate unicorn.' But this is false, since unicorns do not exist. The syllogism is valid only if the propositions beginning with ‘all’ and with ‘some’ have the same existential import – that is, if either both or neither of them imply that there actually exist members of the class in question.
Leibniz tackled the chaos of syllogistic theory as early as in his student thesis On the Art of Combinations. In this he tried to eliminate the arbitrariness of the rules for syllogistic validity, by structuring the theory round the permutations and combinations of ‘all’, ‘not’, and the subject, predicate and middle terms. As it stood, his project was over-ambitious and somewhat naive in conception, but it was a promising start in the direction of making logic amenable to mathematical treatment. As far as conventional logic was concerned, he produced a symmetrical list of valid syllogisms by allowing six in each Figure (including the one about unicorns).
His next step was to see that logic was essentially concerned with concepts or classes of things, and how they did or did not overlap. The distinction between subject and predicate, however important in grammar, had no ultimate significance for logic, since the two could always be swapped around. Instead of ‘All men are mortal’, you could say ‘Mortality is an essential part of the human condition’, or ‘Mortals include all men among their number’. Besides, as we have already seen, Leibniz held that subjects, as lowest species, were really nothing other than highly complex predicates. So subject predicate propositions merely asserted a certain relationship
But, as Leibniz discovered, it makes a crucial difference to the relationship whether you are talking about concepts or about classes. For example, if the universal form ‘All S is P’ is interpreted in terms of concepts, it means that the concept S includes the concept P. Thus, in the proposition ‘All men are animals’, the concept ‘man’ includes the concept ‘animal’, in that man is by definition a rational animal. But if the proposition is about classes, it means that the class P includes the class S – to use the same example, men are a subclass of animals. The position with propositions containing ‘some’ is analogous, only more complicated because of the ambiguities already mentioned. The two interpretations are known as the intensional and the extensional. The conceptual interpretation is intensional, because it has to do with what you mean or intend by the concept; the class interpretation is extensional, because it has to do with the scope or extension of the class defined by the concept.
The next stage in Leibniz's thought was the one which made him the first true symbolic logician, and brought him to the very edge of abandoning traditional syllogistic logic altogether. We have already seen how he was searching for a way of notating concepts which would make their inner structure evident to the eye, and mechanically calculable. Given that logic was concerned with relations between concepts, he now hoped to make these relationships equally evident to the eye, and calculable.
The visual approach consisted in representing the extensions of classes geometrically. The simplest system Leibniz devised involved circles which either overlapped, or were entirely separate, or of which one included the other. They are now usually known as ‘Venn diagrams’, though John Venn himself (1834-1923) called them ‘ Eulerian’, after Leonhard
The second approach constituted an even more significant step towards modern logic, as also towards Leibniz's ambition of an arithmeticised universal characteristic. It consisted in representing the relationships between the intensions of concepts by a purely arithmetical notation, and in establishing the axioms of a formal calculus of deductive logic. He experimented with various different schemes, of which the most advanced are to be found in some private notes written during the 1690s.
In one of these papers, Leibniz used the symbol ‘ = ’ to mean 'is the same as; ‘ + ’ for the combining of concepts; and ‘ - ’ for the subtraction of a simpler concept from a more complex one. So, if A is ‘man’. B is ‘rational’ and C is ‘animal’, then ‘A = B + C’ means ‘The concept of a man is the same as the concept of a rational animal’; and ‘A - B = C’ means ‘If you subtract the qualification "rational" from the concept of a man, you are left with the concept of an animal.’ He was well aware of the difference between his ‘-’ and negation: the concept of a man with his rationality negated was a contradiction in terms (a non-rational rational animal); whereas the concept of a man with rationality subtracted merely reverted to that of an animal (Gvii232n.).
In a later paper, Leibniz dropped the subtraction sign as
unnecessary, and altered the ‘+’ to ‘
’, to
show that it was different from the ‘+’ of arithmetic. For
example,
his calculus had as an axiom ‘A + A = A’, meaning that to
add the
same qualification a second time made no further difference to a
concept. But in arithmetic, A + A = 2A.
Leibniz did not in fact
get very far with his logical calculus.
Clearly, what he envisaged was something like the system finally produced by George Boole (1815-64). Boole's calculus was a purely formal deductive system, which could do everything that syllogistic logic could do, while avoiding its ambiguities and apparent arbitrariness. But although Leibniz's failure to arrive at Boolean algebra itself was a near miss, he had an even nearer miss from the twentieth-century concept of truth-tables.
When
Leibniz saw that his was not the ‘+’ of
arithmetic, he failed to consider the possibility of its being some other
mathematical operator. This was despite the fact that he himself
had previously notated ‘A
B’ as ‘AB’, which is an
alternative for ‘A x B’ in algebra. Now, if we symbolise
being-a-member-of-a-class by ‘1’, and not-being-a-member by
‘0’, we
can then draw up a simple table defining membership of the
class A B in terms
of membership of the classes A
and B. The only circumstance in which something is a member of A
B is when it is a
member of A and is a member of B:
If we now interpret the 0 and 1 as binary
numbers, it
is clear that
is not
the ‘+’ of arithmetic, which would have given:
0, 1, 1, 10 in the third
column, since in binary, 0 + 1 = 1,
1 + 0 = 1 and 1 + 1 =
10. In fact it is equivalent to multiplication:
0 x 0 = 0, 0 x 1 =
0, 1 x 0 = 0, 1 x 1
= 1. This is why
logicians call the class defined by the combination of two concepts their
‘logical product’.
If we now ask what the mathematical ‘ + ’ is equivalent to in
| A | B | A+B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Translated into logical terms, we have the operator ‘or’, in the everyday, exclusive sense of ‘one or other, but not both’ (in this sense it would be wrong to say, ‘ Leeds is either in England or in Yorkshire’, since it happens to be in both).
Leibniz came very close to defining this function when he introduced the notion of ‘incommunicating’ classes. They were such that a member of a pair of incommunicating classes was a member of one or the other, but not of both. However, although the function was essential for his calculus, he never got as far as representing it symbolically. If he had done so, he might have arrived at a general logic of ‘either/or’, which could have served equally for his own ‘either a member of a class or not’; for the ‘either true or false’ of truth-tables; or for the ‘ either carrying an electric current or not’, which is the basis of the modern computer,
Leibniz regularly referred to his metaphysical theory as a system – in particular, the System of Pre-Established Harmony. This raises the question of whether he intended to stress the systematic character of his metaphysics, or whether he merely meant ‘system’ in the sense of ‘theory’, as when we talk of ‘the Copernican system’, for instance. Many commentators have assumed that Leibniz's metaphysics was systematic in the former sense, despite the fact that his actual writings are a complete jumble in comparison with the systematic treatises of contemporaries such as Spinoza, Hobbes or Malebranche. The question of the systematic character of Leibniz's philosophy is an important one, and to settle it we must distinguish between his ambition and his achievement.
It will be obvious from the previous chapter that Leibniz's ideal must have been a fully deductive system. Indeed, when he was about forty, he seems to have found his philosophical ideas suddenly falling into place, and he went as far as to claim that everything followed from the Principle of Identity (see p. 62 above) and the Principle of Sufficient Reason (the principle that there must be some reason why anything is as it is rather than otherwise). But he never showed how the rest of his system followed from those two principles, and the nearest he ever came to producing a systematic exposition of his philosophy as a whole was in a number of short summaries wrttten at various times during his life – in particular, the Discourse on Metaphysics (1686), the New System (1695) and the so-called Monadology (1714).
Although these tracts are by no means devoid of reasoned argument, they read more like dogmatic creeds or manifestos, and they certainly fall far short of showing how his various philosophical tenets are supposed to come together as a deductive system. Those who stress the systematic character of Leibniz's philosophy tend to appeal to the structure that they, as commentators, can detect in his writings, especially in his private notes. For example, in his Critical Exposition of the Philosophy of Leibniz (1900), Lord Russell dismissed most of Leibniz's correspondence and publications as obsequious panderings to aristocratic patrons. Russell used Leibniz's unpublished jottings as the basis for ‘a reconstruction of the system which Leibniz should have written’ (my italics). But it is surely an extreme measure to ignore precisely those writings to which Leibniz gave his seal of approval, and to base one's interpretation on his tentative speculations. In fact much of what Leibniz published seemed strange even to his contemporaries, and it was hardly calculated to ingratiate him with anyone. It is clear that Leibniz aspired to a system, but the system remained half formed in his mind.
The actual Leibniz was nearly always in dialogue – with real live opponents in the course of his voluminous correspondence; with imaginary interlocutors when writing in explicit dialogue form (for example, in his New Essays, a lengthy dialogue between a Leibnizian and a Lockean); and with himself in his private notes. It may be that, in the light of his hankering after a system, the open-endedness of dialogue seemed to him a second best. On the other hand, his great philosophical hero was Plato. and Plato never used anything but the dialogue form. Both believed that truth ultimately rested in the logical relationships between hierarchically structured objective concepts, and both believed that the most appropriate method for us to discover the truth was the dialectical method. Once the system of
Leibniz was perfectly willing to admit that he did not have a complete system. As he wrote to the Paris Academician, Gilles des Billettes in l696:
My system ... is not a complete body of philosophy, and I do not claim to have a reason for everything that other people have thought they can explain. Progress must be gradual to be assured. (G vii 451)
As we shall see, it was one of the main theses of his philosophy that objective truth is the summation of the different viewpoints of all individuals. Quite properly, he applied the thesis to the philosophical disputes in which he himself was involved:
Most philosophical schools are largely right in what they assert, but not so much in what they deny. (G iii 607)
Consideration of this system [of mine] makes it evident that when one comes down to the basics, one finds that most philosophical schools have more of the truth than one would have believed ... They come together as at a centre of perspective, from which an object (confused if looked at from any other position) displays its regularity and the appropriateness of its parts. The commonest failing is the sectarian spirit in which people diminish themselves by rejecting others. (G iv 523-4)
Leibniz's use of dialogue, far from obscuring some supposed underlying system, in fact provides the key to the genuine structural unity of his writings, both public and private. He believed that in most disputes each side had part of the truth and was wrong only in failing to recognise what was true in the opposing position. Truth was therefore best served, not by
During the period when Leibniz was developing his philosophy, it was comparatively rare for the professional, scholastic philosophers to pay much attention to the ‘modern’ philosophy of amateurs such as Bacon, Descartes or Hobbes. With the passage of time, the traditionalists felt themselves increasingly under threat, but they tended to retaliate with violent abuse or attempts at censorship, rather than by any sort of compromise. This is hardly surprising, since for their part the modern philosophers were unanimous in their wholesale rejection of scholasticism. Their various schemes for starting completely afresh from indubitable axioms, crystal-clear definitions and the evidence of the senses, and their rejection of traditional terminology as meaningless obfuscation, almost wholly precluded the possibility of finding any common ground between the two approaches. With the benefit of hindsight, we can now see that the moderns were not wholly successful in escaping from the traditional modes of thought which dominated their education and culture. But it was their conscious attitude that mattered: they all saw themselves as at war with the scholastics.
From an early age, Leibniz felt himself torn between the two camps. He loved the subtle complexity, the precision and the
Leibniz believed that the scholastics were at fault, not in their respect for tradition as such, but in allowing their traditionalism to become so ossified and inward-looking that they were incapable of absorbing new techniques and discoveries. Conversely, the moderns threw the baby out with the bathwater in their wholesale rejection of scholasticism. It was easy to poke fun at the logic-chopping and clumsy neologisms of university metaphysics, but its technical terms and distinctions provided a valuable means for conceptualising certain unavoidable metaphysical issues. Wilful renunciation of scholastic terminology had lulled the moderns into glossing over problems at the very roots of their philosophies.
Leibniz's usual formula for mediating in any dispute thrown up by the modern philosophy was to show that it arose from neglect of traditional metaphysical concepts. He believed that scholastic terminology could generally be used to express a deeper theory in terms of which both sides could be seen as right in what they asserted, and wrong only in feeling themselves threatened by the insights of their opponents. Obviously this technique was most directly applicable within
Within the non-scholastic philosophy of the mid-seventeenth century, the two main rival accounts of the nature of material substance in general were those of the Cartesians and of the atomists (especially Gassendi). Cartesianism was hardly known in Germany while Leibniz was a student, so it made all the more impression on him during his stay in Paris . For a while he was almost totally absorbed in Descartes' writings, and although he did not emerge as a Cartesian, his sympathetic critique of crucial aspects of Cartesianism played a significant role in the development of his own philosophical system.
The main difference between the two theories of matter was this: For the Cartesians, matter was essentially a continuous, homogeneous quantity, and its division into apparently distinct physical objects was something requiring explanation. For the atomists, matter consisted essentially in discrete bits separated by empty space, and it was its cohesion into apparently homogeneous physical objects that required
Descartes, true to his profound anti-scholasticism, wanted to escape as far as possible from what he saw as mere quibbling about elementary scholastic terms, such as substance, accident, essence, matter and form. His radically simplified categorisation of reality divided it into two things: thought (or consciousness) and extension in space. Every item of the world as we knew it was a mode, or particular manifestation, either of thought or of extension. Our beliefs, intentions, emotional attitudes and conscious perceptions were modes of thought; and the qualities genuinely belonging to physical objects themselves, and not just to our perceptions of them, were modes of extension.
But since the only real properties of matter were the spatial or geometrical ones of shape, size and motion (or change of relative position), Descartes could not accommodate any absolute distinction between matter itself and the space occupied by matter. The notion of empty space was a contradiction in terms, since matter was not distinguished from it by anything such as substantiality, mass or solidity. For Descartes, the material universe was essentially nothing other than an infinite sea of homogeneous, extended matter.
However, in order to ensure some connection between his metaphysics and the world of ordinary experience, Descartes had to explain how his essentially homogeneous matter could give rise to the variety of things like stones, water, air, plants and animals. His immediate solution was to say that matter was divided into corpuscles of three different orders of size: the largest made up tangible solids and liquids; the medium
But this only moved the problem one stage back. Since matter was strictly homogeneous, meaning had to be given to the notion of a particular mathematical point belonging to one corpuscle rather than to another; or of a particular volume of space constituting a single large corpuscle, rather than a densely packed collection of smaller ones. Descartes' answer was in terms of the only other available intrinsic property of matter, namely motion. His criterion for saying that a given volume constituted an individual corpuscle distinct from the surrounding matter, was that the whole volume moved together. In effect, he pictured the universe as like an infinite tank of water, containing frozen lumps of various shapes and sizes: the ice was of the same nature as the water, and differed from it only in the mutual cohesion of its parts.
Leibniz's objection was that it made no sense to say that one part of space was in motion relative to another, unless they were already distinguished from each other. In the ice example, there are criteria for a piece of ice having moved in water, since ice is intrinsically different from water in many ways: it is solid, and colder and refracts light differently. But if the only difference were the cohesion of the ice when it moved, then there would be no specifiable difference between the state of affairs in the tank before and after the supposed motion. In accordance with the Principle of the Identity of Indiscernibles, there would have been no change, and therefore no motion. The very concept of motion presupposes intrinsically distinct things relative to which motion can take place. Consequently, intrinsic difference cannot, without circularity, depend on motion. Descartes' theory of matter collapses for lack of an inherent basis of variegation (in scholastic
The atomists avoided Descartes' problem over variegation by conceiving solid atoms as intrinsically different from the surrounding empty space. Given atoms of different shapes and sizes, it was not difficult in principle to imagine how their various combinations might give rise to all the variety of the world as we know it. Their problem, in Leibniz's view, was to explain cohesion. Suppose we have a collection of atoms: the problem is to give an account of the difference between their constituting a single, solid object, and their being merely a heap or aggregate, like a pile of sand. If atoms were to be the sole constituents of material reality, the atomists could not appeal to any occult ‘cohesive forces’, or to quasi-material ‘gluons’ sticking them together, as in some modern physical theories. One solution was to say that the atoms of solid objects held together by interlocking hooks and eyes, like Velcro, or three-dimensional puzzles.
Leibniz's objection was that, far from explaining cohesion, this account presupposed it, in the form of the internal cohesion of the atoms themselves. If the atomists said that atoms held together because they were composed of smaller atoms with hooks and eyes, the process would go on to infinity. So they had to say that atoms were intrinsically indivisible (as implied by the very name α-τομον, meaning ‘uncuttable’). But this made cohesion an ultimate, and hence inexplicable, property of nature.
Not only would cohesion be inexplicable, but, in Leibniz's view, the point at.which it was appealed to would be entirely arbitrary. He held (wrongly, it would now seem) that size is purely relative. Consequently, if it is a puzzle how, say, the moon holds together, or how a cannon-ball holds together, then it must equally be a puzzle how a material atom holds
Leibniz claimed that both the Cartesians and the atomists had gone astray in what he called ‘the labyrinth of the composition of the continuum’ (G vi 65). The Cartesians took continuity as their point of departure, but could not succeed in crystallising discrete objects out of it. The atomists postulated discrete atoms, but could not explain their composition into continuous wholes. Leibniz believed that each were right in what they asserted, and that the material world was both a continuum, and composed of atomic units.
As he saw it, the problem of the composition of the continuum was this: if its atomic units were small enough to be genuine units (that is, logically indivisible, unlike material atoms, which were indivisible only by arbitrary assumption), then they would have to be mathematical points. But since mathematical points had no dimensions, they could not contain any substance, so as to be real in their own right. Besides, even if there could be such things as real mathematical points, they certainly could not generate matter by being laid next to each other, since any number of points together collapse into a single point. In short, anything small enough to be part of a continuum would be too small to be an ultimate building-block of matter.
In his early days, Leibniz thought he could get round the problem by appealing to the scholastic concept of the flowing of a point. His idea was that, if a point was in continuous motion, it would at any instant be moving away from where it was, and would therefore be occupying fractionally more than the zero
So Leibniz's first solution was to say that the essence of matter was not Descartes' extension, or the atomists' solidity, but motion: every part of matter was composed of infinitely many continuously moving points. However, this position was inherently unsatisfactory, since it suffered from the very same mistake that Leibniz had accused Descartes of: motion presupposes that which moves, and therefore cannot constitute its essence. Motion had to be grounded in something else; and whatever that something was, would have to be capable of existing at a point.
As we saw in Chapter 3, Leibniz criticised Descartes for failing to see that motion had to be grounded in energy. Applied to his own theory, this led him to the conclusion that it was energy that existed at a point and constituted the essence of matter. Other thinkers saw motion or energy as something extra, added to the world after it had already been created (like a clock wound up by its maker). For Leibniz, the world consisted of nothing but point-particles of energy permanently expressed in motion. This energy was the source not only of the activities of physical objects (in particular, kinetic energy), but also of their passive aspect, or matter itself, which just was the energy to resist penetration or acceleration, and to react to applied forces (see pp. 43 f.).
In short, Leibniz's way out of the labyrinth of the composition of the continuum was to see the world of continuously extended matter as secondary and derivative. He realised that he could not explain matter and space without circularity,
At its sharpest, the dispute between mechanists and vitalists was over the choice of a single model for understanding the whole of nature. Extreme mechanists, such as Hobbes, interpreted everything, including biological and psychological phenomena, as the products of underlying mechanisms operating in accordance with deterministic laws. Extreme vitalists, such as Leibniz's close friend Francis Mercury van Helmont, tended to interpret everything, including the functioning of everyday machinery, as due to the operations of invisible, goal-directed organic principles. For the former, the universe was a huge machine, of which the parts were also machines; for the latter it was a huge animal, of which the parts were also animals.
Descartes was strongly inclined towards extreme mechanism, but he baulked at including reason and consciousness within the mechanist model. His compromise was to allow the laws of mechanics complete sway over the realm of matter, but to take the human soul right out of nature. As we saw (see pp. 38 f.), Leibniz criticised him for wanting to eat his cake and have it. If he was really serious about exempting the soul from the laws of nature, he would be committed to denying that the soul could act upon the body at all, even though it might still be indirectly influenced by the body through consciousness of what was happening to it.
Leibniz was as convinced as Descartes of the universality of the laws of mechanics, and of the futility of vitalist explanations
Leibniz's solution was not to restrict the scope of mechanism in any way, but to make all mechanical explanation ultimately dependent on a metaphysical version of vitalism. As he wrote to Arnauld:
Nature must always be explained mathematically and mechanically, provided it is remembered that the principles themselves, or laws of mechanics or force, do not depend on mathematical extension alone, but on certain metaphysical reasons. (G ii 58)
He had a number of grounds for denying the sufficiency of mechanical explanations. One relatively trivial reason was that the laws of mechanics could not be used to explain themselves. If there were to be any explanation of why precisely this set of laws held rather than some other possible set, it would have to be found outside mechanics itself, in metaphysics or theology for example, in that the actual laws were chosen by God as the best. But many philosophers would have agreed with Leibniz on this, and it does nothing to account for the radical consequences which he alone drew from the insufficiency of mechanics.
Of much deeper significance for his philosophy was his analysis of the nature of mechanical interaction itself. We have already looked at his scientific grounds for denying real interaction, and for concluding that all action must be spontaneous
It is a serious misconception to think of causation simply in terms of a chain of influences passing on from one individual object to another. Even in the case of balls ricocheting off each other on a snooker table, we must never forget that their motions are permanently dependent on an infinity of other considerations, such as the characteristics of the table, the lie of the felt, air currents, all the gravitational forces acting on them and so on. In fact, the resultant motion of a particular ball is more like the expression of the solution to an infinitely complex equation, than like receiving the baton in a relay race. Leibniz was quite right to interpret the laws of mechanics, not as laws governing the amount of force transferred from one colliding body to another, but as elegant mathematical formulae governing the evolution of whole complex systems from their state at one time to their state at the next.
At the everyday level, we quite properly see such systems as involving interactions between different bodies. But if, strictly speaking, all action is really spontaneous, then the law-guided evolution of the system as a whole must ultimately derive from the internal evolution of each individual substance. In other words, there are two different dimensions to the changes in an individual's states: its spontaneous development from within its own nature, and the adaptation of that development to the harmony of the total system of which it is a part. Of the two dimensions, the former is more fundamental, given that the part is prior to the whole. But since God selected individuals for
Typically, Leibniz chose to express his position in the terminology of the Aristotelian-scholastic tradition. The two dimensions of things were essentially active and passive – things were active in so far as their development was spontaneous; passive in so far as it was determined by the requirements of the surrounding system. Leibniz identified the two dimensions with the traditional form (active) and matter (passive). From a scholastic point of view, it was then utterly orthodox to say that the two aspects were complementary, and that every earthly substance had to have both.
In the case of living organisms, the scholastics had identified the form with the soul: as the essence of the living thing, the form must be its principle of life. In the case of man, his form was his rationality, or rational soul; in the case of animals it was the capacity for sensation, or the ‘sensitive’ soul; in the case of plants it was their principle of organic life, or ‘vegetative’ soul. Leibniz's crucial step was to say that there was no difference in kind between the forms of living substances and those of everything else: everything had a form as well as matter, and thereby had a vital principle as the ultimate source of its spontaneous activity.
This, in general terms, was how Leibniz thought he could reconcile mechanism and vitalism. Absolutely everything was subject to the deterministic laws of mechanics in virtue of interaction with other substances. But the capacity to interact depended ultimately on spontaneous change, which arose from forms or vital principles. However, in order to understand more precisely how the two aspects were supposed to be connected, we must look now at Leibniz's resolution of yet another metaphysical conflict.
A phenomenalist is one who believes that nothing exists apart from perceivers and their perceptions (or ‘phenomena’). A realist, in this context, means someone who believes that there also exists a real world underlying our perceptions. Leibniz had an ingenious idea for reconciling these two apparently contradictory metaphysical theories. What he did was to start from an unambiguously phenomenalist position, and then to reconstruct a realist picture of the universe on the basis of his conception of forms as vital principles capable of something analogous to perception.
Unlike later phenomenalists, Leibniz did not rest his case on the unknowability of some supposedly more real world lying behind the one we are immediately aware of in experience. He preferred to argue directly that the material world of experience had two essential features which consigned it to the realm of appearance rather than of reality.
The first feature was the same as had led Plato to reject the reality of matter, namely that it was essentially in a state of becoming, not of being:
The objects of perception, and in general all compounds or what one might call artificial substances, are in a flux, and in a state of becoming rather than of being. (E 445)
As we have seen, Leibniz held that the essence of matter was energy, and that everything was always moving, since energy, in the form of kinetic energy, could be actualised only through motion. Since things were essentially in motion, they had to differ from things at rest, at every instant of their existence. The difference was that, like a ‘flowing point’ (see p.82), a moving object would always be already entering its next position – it both did and did not occupy a precisely defined space, unlike a stationary object, which occupied a space
The other feature of matter which made it merely pheno menal was the fact that it was irreducibly composite. Leibniz held that compounds were not real in themselves, but only in virtue of their components (see p. 53 f.), and as he had argued against the atomists (see p. 81), the parts of spatially extended, material compounds had also to be extended, and therefore themselves capable of further subdivision, and so on to infin ity. The only way out of the infinite regress was to postulate genuine unities, or ‘monads’ (from the Greek μ?νας, meaning ‘unit’), which would not be parts of matter, but on which matter would depend in some other way. In Leibniz's terminology, monads were not parts, but ‘requisites’ of matter. Granted that matter consisted of appearances, its exist ence would have to depend on perceivers to which it appeared. Leibniz's theory was that monads, as principles of energy and life, were the perceivers on which material bodies ultimately depended:
Extension, mass and motion are no more things than images in mirrors, or rainbows in clouds . . . Anything in nature apart from perceivers and their perceptions is invented by us, and we struggle with chimeras created by our own minds, as if with ghosts. (G ii 281)
We normally understand the world as consisting of objects of perception separate from and common to different perceivers. Leibniz held that such objects are only mental constructs. For example, if we imagine a number of people looking at a solid cube, they will all have different perspectives on it. None will have a perceptual image answering to the geometrical description of a cube as a solid with six
The fiction is, however, a useful one, since the specification of the cube is encapsulated in a neat mathematical formula from which the details of every possible perspective can be straightforwardly derived. But all that matters is the math ematical formula itself – a purely mental thing. Once we have that, there is no need for some extra-mental cube itself in which the formula has a physical embodiment. The reason why our perceptions are coherent and orderly is not that they depend on supernumerary, unperceiving substances, but because they were chosen by God with a view to maximising their mutual harmony in accordance with mathematical principles.
The only channel of communication between different perceivers is through their perceived bodies. If one person is in communication with another, it is by means of speaking, gesturing and touching, and by means of hearing, watching and feeling the other person. It is precisely this importance of the perceived body that leads us to conceive of the body as the most immediate reality, and the conscious soul as somehow secondary, and hidden within the body. For instance, Descartes' theory of the pineal gland as the seat of the soul suggested that the soul was in a particular part of the body, though in a way which put it magically just beyond the reach of the neurosurgeon's scalpel. But for Leibniz, the unreality of the body meant that the soul or monad could not be literally
The next question is: Which bodies are animated? Since Leibniz identified monads or souls with forms, a first answer is that all parts of matter that exhibit form are animated. Unlike the scholastics, however, Leibniz was not prepared to attribute forms to purely material objects. As principles of unity, he restricted them to genuinely unitary wholes, or organisms.
A material object, even one which has deliberately been given a certain form by a human craftsman, is not a genuine whole, since it is no more than the sum of its parts. If you dismantle a table, you have the parts of a table, and can put them back together again. But if you dismantle a person, you end up with parts of a corpse, and reassembling the parts will give you only a complete corpse, and not a person. What distinguishes a person from a corpse is his being a living organism. And this distinction does not depend on the presence of some extra ingredient, for example an immaterial soul inserted by God at conception, or the electrical energy that Frankenstein added to his monster. Such an addition would be just another part, and the whole would still be no more than the sum of its parts.
What makes an organism an organic whole is the way in which its parts are interconnected. For Leibniz, this was not simply a question of co-ordinated activity and mutual respon siveness, since these could also be found in sophisticated machinery. In his view, the defining characteristic of an organism was that each part depended for its very identity on its relation to that particular whole. Unlike human artefacts,
So every organic body of a living being is a sort of God-made machine, or natural robot, which infinitely excels all man-made robots. For human skill cannot make machines of which all their parts are machines. For example) the tooth of a brass cog-wheel has parts or segments which are not themselves man-made and no longer have any mechanical role in the functioning of the wheel. But machines of nature, that is to say living bodies, are still machines in their smallest parts, right down to infinity. It is that which makes the difference between nature and art, or rather between God's art and ours.(section 64)
But if there are to be organic bodies within organic bodies, there must also be perceptions within perceptions, since bodies are ultimately nothing but perceptions. However, when we perceive bodies, we do not consciously perceive the microscopic organisms of which they are composed – so we must perceive them unconsciously. This consideration led Leibniz to formulate a distinction, remarkable for his time, between the conscious and the unconscious. Consciousness he termed ‘apperception’, and unconscious perceptions he described as little perceptions'. His idea was that little per ceptions are components of our conscious perceptions, but ones that are not separately (or ‘distinctly’) perceived in them selves. For example, we cannot hear the distant roar of the sea
From a metaphysical point of view, the doctrine of organ isms within organisms gives a means of filling out the reality of phenomenal bodies. Although we can talk as if bodies contained souls, it makes no sense to say where the soul is within the body. It is not even like a pea rattling around inside an otherwise empty pod, since the relation between body and soul is logical rather than spatial. But if bodies are in fact colonies of smaller bodies, then there must be as many souls 'inside' the higher-level bodies as there are smaller bodies, and the scope for the indeterminacy of the position of each soul is confined to the volume of these smaller bodies. If the smaller bodies are' in their turn colonies of yet smaller bodies, and so on to infinity, then it is impossible to define a portion of a body so small that it will not include an infinity of souls. Literally, every specifiable part of the phenomenal body is backed up by infinitely many real substances. If, as realism maintains, the reality of phenomena is to consist in their correspondence with underlying real substances, then Leibniz's phenomena are as real as any realist could demand.
At a purely empirical level, Leibniz found dramatic confirmation of his approach in what had been discovered through the newly invented microscope. Examination of the human body revealed all sorts of hitherto invisible components which
So far, Leibniz's theory accounts for the reality of the organic bodies in our experience. However, it would be absurd to picture a world in which all our perceptions of living beings were grounded in reality, but in which all the surrounding inanimate objects were mere appearances in the mind. So Leibniz was more or less forced into maintaining that they too were collections of living beings, though not such as to form organic wholes. Material objects were mere ‘aggregates’ of organisms, like banks of living animal tissues in a laboratory. Leibniz thought he had empirical confirmation of this too, from microscopic examination of chalk, which showed it to be a mass of shells and skeletons of tiny sea creatures. So, although matter was in itself only a phenomenon, it was a 'well-founded' phenomenon, in that it was constructed from bodies of organisms, analysable in their turn as perceptions of real, living beings.
The idea of a hierarchy of organisms within organisms established more than just their reality. Since organisms were 'machines of nature', it explained how, at every level, there were always sub-mechanisms available for mediating interactions. In addition, the infinite complexity of monads made it possible for Leibniz to claim that each one registered the influence of, or ‘mirrored’, not only its own body, but the whole of the universe. Its perceptual state was the whole phenomenal world from a particular point of view, so it was in principle possible to read off everything else from it. Finally, given
Had Leibniz lived a little later, he might have expressed his theory in the more neutral terms of field dynamics. But in his day, the vitalist and psychological model was the only one available for conceptualising reality as something other than material, atomic and inert. Terms like ‘life’, ‘soul’ and ‘ perception’ were all he had for describing coordinated energy and activity without consciousness. So we should not read too much into the more fanciful expositions of his philosophy, such as the following almost poetic passage from the Monadology:
‘There is a world of creatures, living beings, animals, substantial forms, souls in the very smallest part of matter. Each bit of matter can be thought of as a garden full of plants or as a pond full of fish – except that every branch of a plant, every part of an animal's body, every drop of the liquids they contain is in its turn another such garden or pond. And although the earth and the air occupying the spaces between the plants in the garden, or the water occupying the space between the fishes in the pond, is not itself a plant or a fish, yet they contain still more of them, only mostly too small to be visible. Thus there is nothing uncultivated, sterile or dead in the universe – no chaos or confusion, except in appearance. It is rather as a pond appears from a distance, when you can see a confused motion and milling around, so to speak, of the fishes in the pond, but without being able to make out the individual fishes themselves. One sees from this how every living body has a dominant substantial form which is the soul in the animal; but the members of this living body are full of other living bodies, plants and animals, each one of which also has its own substantial form, or dominant monad. (¶¶ 66-70)’
It was not enough for Leibniz merely to assert that there was a correspondence between monads and the material world of phenomena. He had also to explain how they mapped off against each other. His answer can best be understood through his resolution of the dispute as to whether or not nature was purposeful – in traditional terms, whether or not there were final as well as mechanical or efficient causes.
Leibniz's compromise between those who rigidly excluded final causes from nature, and those who made them central to it, consisted in revamping the traditional view that all events have both efficient and final causes. In terms of his philosophy, they have efficient causes when considered as events in the material world, and final causes when considered as changes in the perceptions of monads. The best example is that of human behaviour, since it is the only case in which we are directly aware of the spiritual as well as of the bodily dimension. By introspection we know our own purposes and intentions, and can make sense of the development of our personalities in terms of a rational progression of final causes. Parallel to this, we could in theory discover a purely material causal network fully determining our behaviour – genes, environment, brain-states and so on. Far from being incompatible with each other, Leibniz believed that the two types of explanation were mutually indispensable. However, before we see how they interlocked, we must first distinguish two different levels at which Leibniz believed final causes to operate.
At the level of the individual monad, finality is essential to its internal principle of activity. In higher monads, it finds expression in conscious desires and purposes; in lower ones, it is simply an unconscious motivation (or ‘appetition’, as Leibniz called it) towards a better state. An alternative way of seeing it is as the driving force behind the progressive
The other level at which finality operates is that of the universe as a whole. Given that the universe was created by a per fect and omnipotent God, it must be the best possible, at least taken over its whole history. If it is still imperfect, it must be advancing towards better and better states. This thesis is generally known as the Principle of the Best . But the ‘best’ is not just the morally best. Leibniz was thinking primarily of functional and aesthetic criteria, according to which the universe was the perfect product of the divine craftsman. From the functional point of view, this meant that nothing was redundant or without a sufficient reason. Aesthetically, it meant that the widest possible variety of phenomena was included underthe simplest and most elegant mathematical formulae – precisely the criterion adopted by many modern philosophers of science for justifying choices between competing scientific theories that are equally compatible with the known facts. So we can now understand Leibniz's claim that mechanics depends on metaphysics: the mechanical laws which actually prevail in the universe are those that result in the best possible compromise between the conflicting requirements of variety and simplicity.
The obvious question now is: How could the independent and selfish purposes of infinitely many individual monads conspire to produce the most perfect possible whole? This was
He illustrated his idea by an analogy of two clocks keeping perfect time with each other. There was no need to suppose that there must be some hidden connexion between their workings, nor that they were periodically adjusted. It could simply be that they were perfectly made. Likewise God, with his infinite skill, had created an infinity of monads so well that they would.keep in perfect harmony to eternity.
Modern technology offers more striking examples – space-ships programmed to dock in orbit, synchronised robots in factories and so on. A better model of Leibniz's system would be an elaboration of the example of the cube (see p. 89 f.). Computer graphics can be used to create animated film sequences representing the changing shapes and positions of imaginary objects from particular perspectives. We can imagine an infinity of such films, each from infinitesimally different viewpoints, all being run simultaneously. Even though the objects and their interactions are entirely Fictional, it will be as if there had been infinitely many cameras filming one and the same scene from different points of view. The simplest way of describing what they portrayed would be by adopting that fiction, even though its only reality would be as a formula in a computer program. But although this formula would not be real in the sense of having a physical embodiment outside the computer, it would be objective. It would be the only representation not biased towards one or other perspective, and all the others could be derived from it.
This is very much how Leibniz pictured his universe of
Finally, we must now return to the question of how Leibniz managed to preserve a distinction between active, purposeful perceivers, and passive, mechanically determined matter, given that they both ultimately consisted in perceptions. For this, he borrowed Descartes' terminology of ‘distinct’ as opposed to ‘confused’ ideas: a monad was active, purposeful and spiritual in so far as its perceptions were distinct; and pas sive, mechanically determined and material in so far as they were confused. The language is highly metaphorical. What he actually meant was that, even though both sorts of cause always operate, the final cause is primary when an agent's state follows most naturally from the previous states of its own body; and the efficient cause is primary when it follows most naturally from the previous states of surrounding bodies.
For example, if I steal up on someone and hit him on the head, his perception of the event will be one of shock and confusion, whereas I will have had a distinct and orderly pro gression of perceptions culminating in the intended action. Anyone who wanted to explain what had happened to the other person would do so more naturally by reference to what was going on in me immediately beforehand, even though it would in theory be possible to unravel the relevant information from the victim's confused ‘little perceptions’. As Leibniz himself wrote in the Monadology:
‘Created things are said to be active in proportion to their degree of
In short, there exist only monads, and monads are nothing other than actualised sets of perceptions defined by a particular point of view. Every perception is both spontaneous (arising from the essence of the individual monad) and harmonious (adapted to the rich pattern of the whole universe). Form and matter represent these two complementary aspects. A monad is a form or spirit in so far as it is spontaneous, active and purposeful; it belongs to the realm of material bodies in so far as it is accommodated to the actions of other substances through the laws of mechanics. For all created beings, the bodily dimension is inescapable. Without it, they would be wholly active and perfect, which is a privilege reserved for God alone.
Leibniz's metaphysical account of the difference between God and the world had both mystical and moral repercussions. We have just seen how he held that the created universe was distinct from God in virtue of its passive, material and mechanistic aspect. But if matter is unreal, this means that the materiality of the world consists in an admixture of unreality, or not-being. God is pure being: matter is a compound of being and nothingness (S ii 411).
Leibniz elevated this into what he called a ‘mystical theology’, by taking up two of the principal ideas of Pythagoras , and adding one of his own. Pythagoras believed both that numbers were the ultimate realities, and that the universe as a whole was harmonious, in that it manifested simple math ematical ratios, like those of the basic intervals in music (the 'harmony of the spheres'). Leibniz accepted both these positions. His novel contribution was to make the numbers binary. Just as the whole of arithmetic could be derived from 1 and 0, so the whole universe was generated out of pure being (God) and nothingness. God's creative act was therefore at one and the same time a voluntary dilution of his own essence, and a mathematical computation of the most perfect number derivable from combinations of 1 and 0. Binary arithmetic was not merely a convenient notation for the hierarchy of all possible concepts, but it was the most faithful possible way of representing their very essence, with 1 and 0 themselves func tioning as the only absolutely simple concepts. As Leibniz himself wrote, probably as early as his Paris period:
‘Perhaps only one thing is conceived independently, namely God himself– and also nothing, or absence of being. This can be made clear by a superb analogy . . . [He then outlines the binary system, and continues:] I shall not here go into the immense usefulness of this system; it would be enough to note how wonderfully all numbers are thus expressed by means of Unity and Nothing. But although there is no hope in this life of people being able to arrive at the secret ordering of things which would make it evident how everything arises from pure being and nothingness, yet it is enough for the analysis of ideas to be continued as far as is necessary for the demonstration oftruths.’ (C 430-1)
Leibniz was so proud of this idea, that he planned to commemorate it with a medal bearing the legends: THE MODEL OF CREATION DISCOVERED BY G.W.L., and ONE IS ENOUGH FOR DERIVING EVERYTHING FROM NOTHING. His design emphasised his debt to Pythagoras and Plato , in depicting the sun, or 1, radiating its light on formless earth, or 0. The theme of sun and light also occurs elsewhere. For instance, On the True Mystical Theology (S ii 410-13) is centred round a dualism of the worlds of light and of shadows; and in the Monadology he writes:
‘So God alone is the primitive unity, or original simple substance, of which all created or derivative monads are products. They are, so to speak, born from moment to moment through continual flashes of divine light, up to the capacity of the created substance, which is of its very nature limited.’ (¶47)
The ethical counterpart of the doctrine that the world is differentiated from God by the inclusion of not-being, is that the element of not-being explains why the world must be morally less perfect than its creator. It provided Leibniz with a solution to the age-old Problem of Evil, namely the problem of
A primitive solution to the problem had been to ascribe evil to a rival power, but from the first few centuries of the Chris tian era this had been outlawed as the heresy of Manicheism (after the third-century Eastern heretic, Manichaeus, who held that good and evil were equal principles). St Augustine, the great extirpator of Manicheism, had tried to get round the difficulty by appealing to the Platonic idea that evil is only a privation. If it was an absence of goodness rather than any thing positive in itself, then it would require no special apology. Leibniz agreed that evil was not some real force opposed to God's goodness. On the other hand, he saw that even if evil were nothing other than an absence of goodness, it still seemed to be incompatible with God's perfection.
Leibniz's solution had two parts. The first was to admit that the universe was indeed imperfect, but to point out that its imperfection was logically necessary in order to preserve its distinctness from God, the only perfect being. God could not be blamed for failing to contravene the laws of logic. The other part of his answer was to say that, although the universe was not perfect, it was the best possible – it was as perfect as it could be without collapsing back into God himself. Consequently, to blame God for creating this universe as he did would be tantamount to saying that he should not have created anything at all.
The trouble with Leibniz's solution to the problem was that, although it may have been theologically sound, it seemed to fly in the face of common-sense experience of natural disasters, misery, disease, cruelty, poverty and so on. Indeed, many people would regard Leibniz's optimism as not merely false in point of fact. but outrageously and wickedly complacent. One person to take this attitude was Voltaire, who bitterly satirised Leibniz as Dr Pangloss (who ‘glossed over everything’), in his novel Candide.
Leibniz himself was well aware of the objection, and tried to forestall it by focusing on a separate logical limitation to the possibility of a perfect world. This was not that the world could not be better without becoming God, but that it could not be better without becoming worse. That is, the elimination of what might seem a fault from one perspective, would constitute a greater evil from other points of view.
To give a few examples: At the moral level, our personalities would be diminished if there were no possibility of sin, or no temptations to overcome. At the level of nature, the absence of disasters and discomforts would require such irregularities in causal laws as to preclude the possibility of science and engineering. At the aesthetic level, we should not judge a whole from a tiny portion. Looked at too closely, a part of a painting will seem to be an ugly and meaningless jumble of pigment. Similarly, a chord in a piece of music may be a cacophonous discord in itself, but crucial for the harmony of the whole.
So, whatever might be said about how imperfect the world in fact is, Leibniz can always produce some account of how this is compatible with its being the best possible. But he is thereby vulnerable to a different objection, namely that his thesis is meaningless, in that nothing could count as evidence against it. Leibniz himself had made just the same point against Descartes. Descartes had maintained that it would
At the purely moral level, this is probably fair comment. Leibniz does not propose any set of moral values by which we can judge the actual world. So far his position is, like Descartes', the basically Stoic one, that whatever actually happens must be for the best, as ordained by Providence. But at the aesthetic level, Leibniz does have perfectly clear criteria for what the perfection of the world consists in, and how it could have been worse. For instance, it would have been worse if it had not been amenable to description in terms of elegant causal laws – and in this case, any local irregularities could not be explained away as contributing to the greater harmony of the whole.
Leibniz had objected to Descartes' taking the human soul right out of nature. But he himself was prone to the opposite danger of understating the differences between man and everything else. He certainly agreed with earlier philosophers that man (or rather, man and other superior spirits) was distinguished from animals by possessing both consciousness and
‘There is also no fear of dissolution, and there is no conceivable way for a simple substance to die in the natural course of events – nor for one to come into being, since it could not be formed by a process of construction. So you can say that monads can only come in and out of being abruptly, that is by creation or annihilation; as contrasted with compounds, which come in or out of being bit by bit.’ (§§ 4-6)
Leibniz concluded that birth was only a ‘growth and development’, and death only an ‘envelopment and diminution’ (Monadology, § 73), so that there was both pre-existence and survival on this earth after death. He was well aware of his closeness to Pythagoras' belief in the transmigration of human and animal souls:
‘I have the highest opinion of Pythagoras, and I almost believe that he was superior to all other ancient philosophers, since he virtually founded not only mathematics, but also the science of incorporeals, having formulated that famous doctrine, worthy of a whole hecatomb, that all souls are immortal.’ (G vii 497)
Whether or not it would have been appropriate to celebrate this particular discovery with the slaughter of a hundred oxen, it is important not to interpret Leibniz as being more Pythagorean than he really was. There are two major respects
The first difference is that Pythagoras held that souls migrated from one body to another, whereas Leibniz maintained that a soul, as the form of its body, could not become attached to any body other than its original one. It is true that he did picture the soul as animating a minute animalcule pre-existing in the seed of the parent, and as remaining alive, once more as an animalcule, after the death of the animal it had grown into – rather as a king might survive the dissolution of the state he once ruled. But Leibniz's essential point was not far removed from modern biology. If a monad was a concentration of information about its organic body, and the organism itself was a larger or smaller embodiment of that information, then he was saying that animal bodies included organic components which embodied the information required for generating future unborn individuals – in other words, genetic codes. In a manner of speaking, monads were gonads. But the more radical thesis that genetic information could never be lost (an aspect of his ‘principle of the conservation of information’) was much wider of the mark.
The other difference between Leibniz and Pythagoras is that the Pythagorean theory was utterly heretical for a Christian. So Leibniz claimed that our special status as human beings exempted us from the cycle of rebirth. As mere biological entities, our organic bodies with their corresponding souls existed before and after this life. But it was an entirely separate question whether or not the souls always had human status.
At one level, human status consisted simply in the capacity for self-consciousness and reason. Leibniz maintained that human souls were miraculously elevated to this status on conception, but he refused to speculate on what might happen
At another level, Leibniz identified our humanity, not with our inner nature or capacities, but with a special moral status. Adapting Augustine's idea of a ‘City of God’, he distinguished between the realms of Nature and of Grace. All subhuman creatures belong only to the Realm of Nature, but we have the privilege of dual nationality, and are also citizens of the Realm of Grace. As such, we enter into a special set of relationships with God. We are not merely natural machines created by the Author of Nature, but we are the subjects of the Heavenly King. As such we have all sorts of privileges and duties that mere animals lack, and it is these that ultimately constitute our human status.
Leibniz's account side-steps many of the philosophical difficulties with other theories of what it is to be human. However, it only makes sense if he can explain how our status as moral agents is compatible with our being at the same time subject to the deterministic laws of nature.
Along with the problem of the composition of the continuum, that of freedom and determinism was one of the ‘two great labyrinths in which our reason very often gets lost’ (G vi 29). As usual, Leibniz aimed for a compromise between two extreme positions, each of which had part of the truth. The extremes were fatalism and indeterminism. He held that fatalists were wrong to deny the practical and moral significance of human action, and that indeterminists were wrong to deny that human behaviour was causally determined and in principle
He identified two main forms of fatalism. The first was what he called Muhammadan fatalism. The Muhammadan form consisted in arguing that there was no point in trying to achieve or avoid anything, since God would in any case bring about what he had willed, whatever we might do. Leibniz rejected the argument on the grounds that it separated men's actions from God's plan for the universe. He saw human behaviour as wholly embedded within the deterministic scheme of things, and hence as playing an essential role in the fulfilment of the divine plan. The plan was not carried out despite human volitions and actions, but through them. In effect, the Muhammadan position erred in being indeterminist as well as fatalist, since it assumed that God might miraculously interfere with the normal course of nature in order to play cat and mouse with us.
A more purely philosophical version of fatalism denied that our actions could influence the future, on the grounds that everything that happened in the world was logically necessitated. Descartes, Hobbes and Spinoza all tended towards this position. Leibniz diagnosed their error as arising from a failure to distinguish between physical and logical necessity. This was partly a consequence of their seeing physics as nothing other than geometry with an added fourth dimension of time. The history of the universe consisted in the unfolding through time of all possible geometrical configurations of matter, so that every possibility would be actualised at some time or other. There was therefore no room for the notion of a genuine possibility which could be either brought about or prevented by human action. As we saw earlier ( p. 59), Leibniz defined the logically
All the same, there is still a certain inevitability about the operations of the laws of nature. Leibniz described this as ‘moral’, or ‘physical’ necessity, as opposed to ‘logical’, ‘mathematical’ or ‘metaphysical’ necessity. Sometimes, in order to soften the impact of his determinism, he adapted the astrological tag ‘The stars incline without necessitating’, and said that we are only ‘inclined’, and not necessitated by the laws of nature. But this was no more than a rhetorical device, since we are no more capable of performing physically impossible actions than we are of logically impossible ones. Merely to have shown that we are not logically fated does not of itself explain how our freedom is compatible with determinism.
As for indeterminism, Leibniz thought it was wrong on two counts. The first was that it was incompatible with the Principle of Sufficient Reason to suppose that God could have stayed his hand from determining every detail of the universe during the act of creation. If he had left anything to the discretion of his creatures, the resultant history of the universe could only have been worse than the best possible.
His second argument was that what he called a ‘freedom of indifference’ would in any case be of no benefit. If a free action was one that could not be attributed to your character, ambitions, appreciation of the environment and so on, then that would make it an absurd and irrational whim – quite the
If Leibniz had been an atheist, he could have left it at that, and said that it makes no sense to want more freedom than we already have. We are not imposed upon by external forces, such as past history or the influence of our environment, since we are the most immediate of those forces. There is no ‘us’ over and above the sum of our previous experiences to be determined by those experiences. But Leibniz did have the problem of explaining how we could escape being mere puppets of God.
He attempted to solve the problem by means of his doctrine that a person's essence was his complete concept in the mind of God. He argued that complete concepts were not determined by God, since God merely reviewed all the logically possible permutations and combinations of predicates. In creating a person, he selected one of many possible complete concepts for actualisation, and the process of actualisation changed nothing in the concept. In particular, it could not add any element of necessity or divine determination linking its predicates together. The succession of predicates was as free and random in reality as it was in the original coming together of the concept.
Taken at face value, Leibniz's solution is completely inadequate. From our point of view as actually existent
However, it is likely that his remarks were intended only as a piece of philosophical diplomacy. For the benefit of anyone worried by the idea of necessitation, he stresses that we each have an infinite variety of logically possible futures. But the fact that we are no more free than animals or material objects, means that we have no special privilege of freedom granted by God. Leibniz is therefore committed to the Stoic position that all is predestined by Providence: rather than undergo the stress and frustration of struggling against the inevitable, we should apply our Reason to the task of aligning our perspectives on the world with the optimal perspective that God has. We have not been endowed with a miraculous freedom, but God has at least granted us the gift of Philosophy, which enables us to understand how all is ultimately for the best in the best of all possible worlds.
Leibniz's most immediate influence was as a mathematician. In particular, Continental mathematicians adopted his version of the infinitesimal calculus in preference to Newton's – a fact which opened up a divide between British and Continental science which was to take over a century to heal. His philosophical influence was rather less direct, but such as it was, it helped to widen the gulf still further. His principal disciple was Johann Christian Wolff (1679-1754), whose philosophy was dominant in Germany for most of the eighteenth century, and had a considerable influence on Kant. Leibniz's tendencies towards speculation and system-building found an exaggerated expression in Wolff's work, and still constitute a major point of difference between the styles of Continental and English-language philosophy, which has always preferred the more empiricist and ad hoc approach deriving from Bacon, Hobbes and Locke.
It is ironical that one so devoted to the cause of mutual understanding should have succeeded only in adding to intellectual chauvinism and dogmatism. There is a similar irony in the fact that he was one of the last great polymaths – not in the frivolous sense of having a wide general knowledge, but in the deeper sense of one who is a citizen of the whole world of intellectual inquiry. He deliberately ignored boundaries between different disciplines, and lack of qualifications never deterred him from contributing fresh insights to established specialisms. Indeed, one of the reasons why he was so hostile to universities as institutions was because their rigid faculty structure prevented the cross-fertilisation of ideas which he
It is difficult not to be impressed by the number of ways in which Leibniz's ideas were far ahead of his time. But his being out of his time made him all the less influential. Generally, it has only been after the independent rediscovery of his ideas that his priority has been noticed. For example, the mathematician and logician, George Boole (1815-64), had first to reinvent the idea of mathematical logic for the chief architects of modern logic. Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970) to appreciate that Leibniz was a fellow spirit. On the other hand, re-evaluation can also go too far. Thus followers of the current fashion for ‘'possible worlds logic'’ (according to which necessary truths are necessary because they are true in all possible worlds) have tried to father their approach on Leibniz, even though his concept of a possible world was radically different from theirs.
Leibniz's greatness as a philosopher did not consist just in his ability to back winners. If that had been so, the task of the scholar would merely be to search among his quaint and outmoded ideas, in order to ‘salvage’ those that anticipate modern beliefs. But this would be an arrogant and patronising attitude to adopt to one of the best minds in the history of philosophy. If we are to treat Leibniz as the master he was, we must be prepared to follow his lead.
But what example was Leibniz setting? As we have seen, in content his philosophy was largely an updating of the Pythagorean and Platonic traditions, using the concepts of Aristotelian scholasticism. In style and spirit, however, he
The principal sources for Leibniz's biography are: G. E. Guhrauer, Gottfried Wilhelm, Freiherr von Leibniz: Eine Biographie, 2 vols (Breslau, 1842), and K. Müller and G. Krönert, Leben und Werk von G. W. Leibniz: Eine Chronik ( Frankfurt am Main, 1969). The only biography in English is E. J. Aiton's Leibniz: A Biography (Bristol, 1985). This work also includes clear and detailed expositions of Leibniz's principal ideas.
Only a small proportion of Leibniz's writings is available in English. By far the most comprehensive selection is Philosophical Papers and Letters, ed. Leroy E. Loemker, 2nd ed. (Dordrecht, 1969). Useful smaller selections are: Discourse on Metaphysics, Correspondence with Arnauld, and Monadology, ed. G. R. W. Montgomery (Chicago, 1902, reprinted 1980), Philosophical Writings, ed. G. H. R. Parkinson (London, 1973), Leibniz Selections, ed. P. P. Wiener (New York, 1951), and Logical Papers: A Selection, trans. G. H. R. Parkinson (Oxford, 1966). Manchester University Press have pub lished very helpful editions of The Discourse on Metaphysics (1953), The Leibniz-Arnauld Correspondence ( 1967), and The Leibniz-Clarke Correspondence (1956). There is also a translation of the Theodicy by E. M. Huggard (London, 1952), and of the New Essays by P. Rem nant and J. Bennett (Cambridge, 1981; abridged, 1982).
In comparison with other major philosophers, there are relatively few commentaries on Leibniz in English. Two of the most accessible
marked up by Katherine Fenton July 2000