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The Thirty Years War decimated the Holy Roman Empire. What began as a local conflict soon progressed into total war between all the major powers of Europe. Germany lost a quarter of its population, some principalities, like Brandenburg, lost nearly half to the plague and the bloodshed. Gottfried Wilhelm Leibniz was born in Leipzig on July 1, 1646, two years before the Peace of Westphalia.
As one of his first biographers wrote: “antiquity made only one person from several Hercules; we will make several savants from a single Leibniz.” In 1923 scholars began compiling his public and private manuscripts. So far twenty large volumes have been filled and it is expected the task will take another two centuries to complete. He worked as a diplomat, jurist, mathematician, philosopher, geologist, engineer, linguist, computational pioneer, inventor, logician and theologian. Often he pursued these endeavors all at once. Given his phenomenal output it is easy to believe him when he complained of sometimes spending the better portion of a week recording the thoughts of a single morning
The Encyclopedia Britannica describes him as “a man of medium height with a stoop, broad-shouldered but bandy-legged, as capable of thinking for several days sitting in the same chair as of travelling the roads of Europe summer and winter. He was an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilisation.”
Others, like George Ross, are not so plenary in their praise:
“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 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 faculty structure prevented the cross-fertilisation of ideas which he saw as essential to the advance of knowledge and of wisdom. The irony is that he was himself instrumental in bringing about an era of far greater intellectual and scientific specialism, as technical advances pushed more and more disciplines out of the reach of the intelligent layman and amateur.”
One of the most unyieldingly holistic thinkers in the Western canon, Leibniz wrote: “what Hippocrates said about the human body is true of the whole universe: namely that all things conspire and are sympathetic, i.e that nothing happens in one creature of which some exactly corresponding effect does not reach all others. Nor, again, are there any absolutely extrinsic denominations in things.”
His religious and philosophical opinions have been hotly debated since his death. He has been labeled, with varying amounts of evidence, as a Catholic, Lutheran, Kabbalist, Rosicrucian, atheist and Deist. Bertrand Russell believed he had two philosophies: a public one rooted in Christianity that Russell despised, and a private one rooted in logic. Leibniz’s father was a twice widowed professor of moral philosophy. Gottfried was the first son of Friedreich’s third wife. Like his son he was described by his colleagues as industrious and virtuous. One morning young Gottfried fell down from a table and rose up unscathed. His father interpreted this as a sign of God’s blessing and expected great things from his son. Friedreich passed away when Gottfried was six, but he bequeathed to him the greatest gift imaginable: a well-stocked library.
As Maria Antognazza notes, almost from the beginning Leibniz was accustomed to comparing multiple approaches and points of view. Far from distracting or confusing him, his forays into seemingly unrelated branches of knowledge set the tone and pace for the rest of his life. The connections he perceived laid the foundations for his loftiest aspirations. Leibniz admitted to initially understanding very little in his father’s books, partially because of the difficult subject matter, but also because many of them were written in Greek or Latin.
Leibniz enrolled in the Nikolaischule in 1653 and graduated in 1661. It was designed to prepare a select group of young men for the university, not to provide a well-rounded education to the general public. This is not so surprising when one takes into account its close ties to the university. Its head rector also served as a professor of dialectic and Greek.
Rather than focusing on the quadrivium, which included practical arts like geometry, astronomy and music, the school’s curriculum revolved around the Trivium: grammar, rhetoric and logic. Approximately 20 hours a week were dedicated to Latin compared to the 2 hours a week dedicated to arithmetic. Like many polymaths Leibniz was also a polyglot whose passion for language began early. He would go on to refute the idea that Hebrew was the first language and, as one of history’s most famous Sinophiles, was enchanted by Chinese language and culture.
While he was a toddler his father read histories to him and from that point onwards he was infatuated.When he picked up a book on the subject he would not put it down until he had finished. It seems the habit never left him. The school’s seemingly pedantic curriculum did him no disservice: he was able to write 300 Latin verses in one morning for an assembly. Today this may not be considered an especially desirable skill, but proficiency in the ancient tongue was once prized.
Leibniz was fortunate to have been born into a wealthy city with a flourishing book trade, although Leipzig was hardly the intellectual capital of Europe. His earliest teachers discouraged independent study and “unseasonable” reading. Ignoring the admonishments against unorganized study, he wandered freely in his father’s library. Livy’s history of Rome was a particularly formative text for him. His recollection of the order in which he perused the materials should be taken with a grain of salt since it seems unlikely he would have had any structure in mind.
At 13 he was exposed to and immediately fascinated by Aristotelian logic. Shortly after this first encounter, however, he mercilessly questioned the validity of the ten categories and other aspects of Aristotle’s thought. This not only is an example of a ferociously independent mind at an age when most information is accepted without question, but also a mind living in an age in which authority still carried more weight than empirical evidence.
The Ramist tradition of education reinforced Leibniz’s love of logic and strengthened his belief in its powers to organize and simplify reality. Among his father’s book Leibniz discovered Georg Calixtus, a offbeat theologian his father had studied to refute. Irenicism, an attempt to unify different Christian sects with one another, was what Leibniz found in Calixtus. Being already so widely read and so obsessed with finding unity in diversity, irenicism must have immediately resonated with him.
At 14 he entered the university of Leipzig. During his studies Leibniz was introduced to the philosophers Francis Bacon and Descartes, the mathematicians Cardano and Campanella and the astronomers Kepler and Galileo. He completed his bachelor’s degree in nineteen months. Six months later he wrote and defended his dissertation on the principle of individuation, how to distinguish items within a group. The budding metaphysician attempted to solve the problem by claiming two objects with the same characteristics have the same numerical properties. Studious, but far from reclusive, Leibniz participated in the Collegium Conferentium, a society where students and professors met to discuss books and ideas. He served as the treasurer of this unique club founded to dissect the world’s many intellectual traditions. Thus, his horizons continued to broaden.
Nine days after his graduation he lost his mother. His inheritance was modest. While he composed his first judicial paper he also began to write On the Art of Combinations. Influenced by Descartes and the Catalan polymath Ramon Llull, Leibniz sought a universal alphabet of primitives. In other words, he was looking for a way to simply and accurately describe reality and human thought through combinations of basic elements. He believed with such an alphabet everything could be discovered from first principles.
This was not an entirely new idea. Scholastics like William of Ockham aspired to create a “mentalese.” Thomas Hobbes and several of his contemporaries also wished to free themselves from the constraints of natural language. Dr. Antognazza calls the task he set for himself analogous “to the reduction of numbers into primes.” With this fabulous calculus he hoped to resolve all disputes quickly and easily.
“But to go back to the expression of thoughts through characters, this is my opinion: it will hardly be possible to end controversies and impose silence on the sects, unless we recall complex arguments to simple calculations and terms of vague and uncertain significance to determinate characters…Once this has been done, when controversies will arise, there will be no more need of a disputation between two philosophers than between two accountants. It will in fact suffice to take pen in hand, to sit at the abacus, and – having summoned, if one wishes, a friend – to say to one another: ‘let us calculate.’”
Because of his age the University in Leipzig would not give him a teaching position. He submitted his thesis instead to Altdorf and received his doctorate shortly thereafter. The graduation speech he delivered, in Latin, was so eloquent the audience members believed he was reading from a prepared document. To his amusement and to their awe he could extemporaneously speak the language with the eloquence of Cato and Cicero. Afterwards he became secretary to a group of alchemists at Nuremberg by submitting an application he, by his own admission, did not fully understand. Somehow his jargon-laden letter gave the members the impression he was already an adept. During this period he was convinced he would find the philosopher’s stone in phosphorous, but later rebuked alchemy altogether. Leibniz did not care for the petty squabbles of academia. He happily forsook them for the petty squabbles of entire nations.
Johann Christian von Boineburg became a mentor to him. With his assistance Leibniz found a post in the court of the archbishop of Mainz, one of the most powerful princes in the Holy Roman Empire. He became increasingly interested in public policy and was prescient in his emphasis of proper medical attention and the role medical advancements would play in improving the human condition.
“Moral and medical matters: these are the things which ought to be valued above all. For this reason I value microscopy above telescopy; and if someone were to find a certain and tested cure of any disease whatsoever, he would in my judgement have accomplished something greater than if he had discovered the quadrature of the circle.”
He went on to say in a later essay, “indeed, I believe that this aspect of public policy will become almost the chief concern of those who govern, second only to the concern for virtue; and that one of the greatest results of sound morality and sound politics will be our promoting an improved medical science.” He was also in favor of governments actively fostering collaboration between physicians for better therapies and for the control of epidemics.
“Human life is a sacred thing,” therefore it “should never be subject to the marketplace.”
In 1672 Leibniz went to Paris to convince the French king to undertake a crusade against Egypt. Of this time he said, he “thought neither of jurisprude nor of literature nor of controversies, things which were my concern in Germany. Instead I began a completely new study to deepen my knowledge of mathematics.”Although he had strived to achieve a knowledge “a bit above average”, by the end of his stay he had devised the infinitesimal calculus. His tremendous intellect was molded by the Dutch astronomer Christiaan Huygens.
The two were a scholarly odd couple. Whereas Huygens was cautious, oriented towards experimentation and concerned with minutiae, his eager pupil was always concerned with a bigger picture, with how all of his assorted undertakings tied together. Oldenberg, secretary of the royal society, arranged a meeting between Morland and Leibniz to compare their arithmetic machines. Gottfried’s machine could perform multiplication and division as well as addition and subtraction. Although not terribly reliable, it was an improvement on earlier attempts. It was dismissed by Robert Hooke, a man never lauded for his amicability, but admired by others in London.
By a unanimous decision he was admitted into the Royal Society at the age of twenty six. During his stay Leibniz met with Robert Boyle. John Pell, a former prodigy who had not quite lived up to the promise he showed as a youth, was also present. When Leibniz proudly announced one of his mathematical discoveries Pell implicitly accused him of plagiarism by telling him one Mr. Mouton of Lyon had made it already. This was incredibly embarrassing. Hurriedly he borrowed Mouton’s book and found Pell was telling the truth. Though marred by his clashes with Pell and Hooke, his visit was a resounding success. He was now familiar with the intellectual life of the island and had made a few key contacts and, just as important, had learned prudence and caution in announcing discoveries. The lesson may have made too deep a mark on the still impressionable Saxon.
He returned to Paris in March to tutor Phillip Wilhelm von Boineburg. His financial position was not particularly good. This period may explain why later in life he always made sure to have multiple patrons and sources of income. Befitting the model for Professor Pangloss he proclaimed: “one who has less must work more and the more he works the more accomplished he becomes.”
During his stay he received a report on the state of British mathematics by John Collins. It mentioned a means of finding the “quadrature of all Curvilinear figures” being developed by a rising star named Isaac Newton. Given the brevity and broadness of the report, as well as Collins’s less than stellar grasp of the subject matter, it is unlikely it did anything but plant a seed in Gottfried’s mind—if it did anything at all. Much ink was spilled while the two titans lived, and much more after they were both long dead. One could say Bertrand Russell summarized the scandal best when he called it “discreditable to all parties.”
In 1675 Leibniz found the area under the graph of a function with his integral calculus. Afterwards he devised a number of notations that are still in use, including the integral sign (from the Latin word summa) and the d for differentials.
In 1676 he traveled back to Germany. In Holland he met with the microbiologists van Leeuwenhoek and the pantheist philosopher Benedict Spinoza. When he reached Hannover he found in Duke Johann Friedrich a kindred spirit. To the Duke Leibniz proposed a number of reforms, including the reorganization of the legal system, the introduction of publicly funded schools and vocational institutes, public health care, programs to assist the poor, a luxury tax to encourage thrift, labor laws and an early form of social security. He also wished to reorganize the bureaucracy and, quite predictably, wished to expand upon the royal library. As its caretaker he designed one of the first cataloguing systems.
In 1679 Leibniz penned his famous political satire of the French king called the Most Christian War-God.
“The majority of men having the habit of regarding their particular interest sooner than the public good, and the present sooner than the future, I am not surprised to see that there are men who see clearly that the salvation of the Church depends solely on the greatness of France, but who [yet] have more regard for the interest of their princes or of their own nation, than for the general good of Christendom, on the pretext of conserving the liberty of their country – which, however, they will not preserve against Ottoman arms, if France does not secure them against slavery. One could, however, pardon to some degree the indiscreet zeal which they show for their country, if they did not allow themselves to speak indignantly of the good intentions of the King”
Of more enduring interest is a letter he wrote to Christian Huygens in which he claimed “it will be possible to represent figures and even machines and motions in characters, as algebra represents numbers or sizes.” He aspired to express geometrical concepts and manipulations, such as congruence, with symbols instead of figures. Leibniz is sometimes credited with anticipating topology because of his “in situ” analysis. Leonhard Euler later used the term in his paper on the bridges of Konigsberg, however the two were using it different ways. While it can be said fractal geometry draws upon Leibniz’s concept of self-similarity, it is hard to say to what extent he foresaw the development of non-euclidean geometries.
Around this time he also began to think about a binary arithmetic. Binary could easily be dismissed as an arcane plaything of the numerically inclined. Yet, working in the 17th century, he could see its immense potential. His work in determinants had more immediate applications. Determinants are used to solve systems of linear equations. What Leibniz discovered is now better known as Cramer’s Rule. It does not carry his name because, for reasons unknown, he did not publish his results.
In his physics he elucidated upon, in contrast to Newton, a relativistic interpretations of space.
“If motion is nothing but the change of contact or of immediate vicinity, it follows that we can never define which thing is moved. For just as the same phenomena may be interpreted by different hypotheses in astronomy, so it will always be possible to attribute the real motion to either one or the other of the two bodies which change their mutual vicinity or position.”
In another essays he explains it in this way:
“No eye, wherever in matter it might be placed, has a sure criterion for telling from the phenomena where there is motion, how much motion there is and of what sort it is, or even whether God moves everything around it, or whether he moves that very eye itself.”
Between 1670 and 1690 he continued to contribute to propositional and modal logic while working on his universal calculus. Like much of his best work, it was published posthumously. His first serious undertaking in the field was the axiomatization of a syllogism. By systematizing, he made proving (or disproving) syllogisms simpler. His “linear diagrams”, later dubbed “Euler-circles” dealt with simple statements like “some men are wise” and “no man is a stone.”
His contribution to modal logic came partially from his philosophical writings, wherein he anticipated what is now called “possible-world-semantics.” It is a concept that deals with true, false, possible, contingent, impossible and necessary statements. Leibniz makes extensive use of all these analytical tools in his metaphysics. One of his ideas that came from his musings on this topic, namely that this is the best of all possible worlds God could have created, was mercilessly mocked by those who disagreed with him. Most famously by Voltaire in his novel Candide. Poor Professor Pangloss, no matter befell him, could not let go of his belief in God’s goodness and wisdom.
Much of his energy during this period was consumed by a comparatively mundane assignment: draining the ducal mines in the Harz, mountains from which the Guelf dynasty drew much of its income. Although silver production had been reduced by the ravages of the Thirty Years War. In retrospect students of philosophy and mathematics have bemoaned the waste of Leibniz’s intellect on such an impure pursuit, but it must be remembered he was not by any means a rich man. Nor was he one to brood over it. He saw it as an opportunity to serve his patron while applying his knowledge of physics and mechanics. Since the 10th century miners had utilized different means of draining the water. They eventually settled on the use of mountain streams to power hydraulic pumps.
In dry years productivity plummeted. Never content to do the bare minimum, he also worked to improve the way horse mills were used to lift ore out of the mines by solving the problem of entangling chains. After some testing he decided upon a conical spiral winding drum. A number of his sketches have survived, but due to the restrictions of his time, could not be implemented. His endless cable is still used today, but he did not succeed in his primary objective.
The traditional narrative is Leibniz’s genius was not recognized by the mining officials and his failure to make wind power into a reality came from their lack of support. He was well-liked by some of the employees and intensely disliked by others. Ultimately his ambitious wind power project proved too costly. He needed thicker pipes to handle the compressed air and special sails to regulate the speed of the windmills. By 1683 costs had risen to 7 times his initial estimate. Needless to say, many prominent people were upset with him at this point. He showed a bit of haughtiness when dealing with the miners and their overseers. When one of the officials suggested others should review his plans, he replied, “experience is in my opinion a better judge than those gentlemen.”
As an inventor Leibniz was not a dilettante. Besides his calculating machine he devised a system of wheels for watches, an aneroid barometer, a submarine, and a spacecraft (though he tossed aside this plan due to the thinness of air at higher altitudes). While thinking about binary arithmetic he planned out a machine that could be controlled by marbles representing 1’s or 0’s. He had designed a computer.
Shortly after being contracted by the Guelf family to write a 2,000 year history he embarked on a trip to Italy to find links between their lines and the Este family. Although now in his forties his passionate curiosity had not diminished. His stay was pleasant and he was warmly received by Italian scholars. In November he returned to Germany.
At weddings Leibniz would offer the brides “useful maxims” in lieu of more traditional, and more expensive, gifts. He never married. It seems he had no interest in sex since he never expressed an interest in men or women, although he had a number of female confidants.
“All my difficulties derive from the fact I am not in a great city like Paris or London, which have a plethora of learned men from whom one can obtain instruction and assistance. For there is much that one cannot do by oneself. Here one finds hardly anyone with whom to talk; indeed, around here one is not regarded as a proper courtier if one speaks of learned matters, and without the Duchess one would discuss such things even less.”
Psychology was yet another field his thoughts impacted. Because he believed things in nature generally existed on a continuum, Leibniz postulated there are stages of consciousness that lead to deep sleep. He rejected Cartesian dualism in favor psychophysical parallelism. The brain and body act separately from one another, but in harmony. Moreover, he believed the brain had an active role in shaping perceptions and that many perceptions are below conscious experience, thus paving the way for the concept of a mind operating underneath ordinary awareness. His work influenced both William Wundt and Ernst Platner. The former was among the first to scientifically study sense perception and the latter coined the term Unconscious.
In 1690 he sent a letter to Christian Huygens:
“If I had the age and leisure which I had in Paris I would hope it would serve to make the progress in physics which your first gift helped me make in geometry. But that mental vigour is quite diminished and in addition I am distracted by completely different studies which seem to require my complete attention. From time to time I manage to escape from this prison where I find myself, and the trip as far as Italy which I have just undertaken has cheered me up a bit; but now I must return more than ever to my ordinary duties, and in particular to a far-reaching historical work laden with facts, which requires great precision.”
He was suffering from a midlife crisis. He had become a hypochondriac and feared he would die before completing his manifold and multiplying missions. To a friend he confided:
“If death wants to give me all the time I need to finish the projects which I have already conceived, I will promise it in exchange not to start any new ones and to work with great diligence on those which I already have, though even with this deal I shall be seriously late. But death cares nothing for our plans or the advancement of science”
Yet somehow these stressors did not deter him from creating work for himself. His Guelf History did not commence with the beginning of the dynasty, nor the Holy Roman Empire or the Battle of Gaugamela. No, no, no. Instead he chose to start with the origins of earth.This resulted in a geological investigation in Lower Saxony and an attempt to understand how and when people arrived to the region. Yet this seemingly mad pursuit resulted in more insights far ahead of their time. He concluded the earth began as a molten mass that cooled and solidified. Underneath the solid crust is a hot liquid core. The evidence for this is volcanic eruptions. Breaking from a literal interpretation of the Old Testament, Leibniz contended there was not a single massive flood, but many small ones.
He also again threw himself into a project near and dear to his heart: public health. He envisioned a council of experts who would come together to discuss pressing matters. He also advocated the keeping of accurate records in order to better treat diseases and handle epidemics. He also, with Ramazzini’s assistance, tried to formalize medical statistics and make annual reports a legal requirement.
Leibniz was fortunate to find a friend in Sophia of Hanover who, as electress of Brandenburg and, later, queen of Prussia, served as his patroness. To Frederick I he suggested taxing tobacco, which he rightly believed was harmful and addictive. He spent the final two years of his life in Hanover. He was not particularly happy. He was expected to finally finish the long awaited Guelf History. and his emoluments from the Berlin Academy of Sciences were cut. Burdened by worsening health, a dismal financial situation and too many other obligations, he continued to work at reconciling Protestants with Catholics and laying the foundations for lasting peace in Europe. Though he wrote extensively on the subject, he summarized his mature theology in a single sentence: “it is possible to be saved in every religion, provided one truly loves God above all things.”
His quarrels with Newton and the Newtonians continued. His correspondence with Samuel Clarke, a proud Newtonian and Anglican clergyman, goes deeper than quibbles about space and matter. Newton and Leibniz were both deeply religious men, though the former was more fanatically and rigidly devout than the latter. Thomas Aquinas “considered the essence of God to be identical with his infinite intellect. The logical consistency of his properties was primary; from these followed his power to act. Intellect gave rise to the will and from this proceeded God’s love. Gottfried’s sympathies rested with Aquinas. On the other hand Augustine “emphasized the will, power and love of God in his active creation and intervention in the world.” In the Augustinian view God can create everything “immediately, spontaneously and directly out of nothing.” He placed the Divine Will before the Divine Intellect.
An intervening God made no sense to Leibniz. It appeared to him to be a limitation of his “wisdom and foresight.” For Clarke and his colleagues, the world could have turned out in many ways since it depended on the whims of the Creator. This did not mesh well with the Saxon’s Panglossian doctrine. The two also sharply disagreed on matter: Newton viewed matter as lifeless, mechanistic and unchanging. Anything seemingly vital had its source in God. There was nothing intrinsic to matter that resulted in its myriad of properties. Leibniz viewed all things as part of an organic whole. Nothing is static or fixed. Everything is in constant dynamic change. This interpretation has been confirmed by research into how atoms and subatomic particles behave. Newton resisted this view with good reason: for him and for his followers it seemed this would lead to Deism or atheism since it did not require the active participation of a supernatural force.
In these letters Leibniz formulated an early version of the First Law of Thermodynamics when he insisted “the same force and vigour” was always present in the universe, “passing from one particle to the next.” His interlocutor was not so prescient: he and his master believed God could add energy to the system when needed. For them this was necessary to support their religious convictions, for Leibniz the design of the universe itself was proof of God’s existence.
Though this was an enlightening exchange, there were also less civil parts of the conflict. The two senior statesmen of science ignored one lady’s plea to find a more “profitable” use of their minds. Finally, in 1716, Conti announced to Newton, “Mr. Leibniz is dead and the dispute is finished.” His passing brought another debate to an end. The majority of Catholic scholars believed Chinese customs were incompatible with Christianity. Leibniz, one who longed for international peace and cooperation, took the opposing position. There was nothing in Chinese culture, for which he had deep admiration, that could not be reconciled with Christianity. Furthermore, to convert the country their rituals needed to be accommodated as much as possible. Ironically, given his broadmindedness, his reputation in England had been sullied as much by unbridled nationalism as it had been by the unfounded accusations of plagiarism.
To his own countrymen he had become little more than a relic, a dinosaur in his dress and appearance. The attendance of his funeral was small, much smaller than one would expect for a man so widely known and so accomplished. Yet he never lost his faith in humanity or God’s plan:
“One can conclude that the human race will not always remain in this state, since it is not consonant with the divine harmony to fiddle away on the same string forever. And one should believe instead that, due to natural reasons of congruence, things are bound to progress for the better, whether gradually or sometimes by leaps and bounds. Although at times things do seem to change for the worse, this should be regarded as similar to the way in which we sometimes retrace our footsteps in order to leap forward with greater vigour.”
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