Gottfried leibniz

Gottfried Wilhelm Leibniz, sometimes Gottfried Wilhelm von Leibniz (Leipzig, July 1, 1646-Hannover, November 14, 1716), was a polymath, German philosopher, mathematician, logician, theologian, jurist, librarian and politician.

He was one of the great thinkers of the 17th and 18th centuries, and he is recognized as the "last universal genius", that is, the last person who could be sufficiently trained in all fields of knowledge; then there were only specialists. He made profound and important contributions in the areas of metaphysics, epistemology, logic, philosophy of religion, as well as mathematics, physics, geology, jurisprudence, and history. Even Denis Diderot, the French deist philosopher of the 18th century, whose views could not be more in opposition to those of Leibniz, did not he could help but be awed by his achievements, writing in the Encyclopédie: "Perhaps never has there been a man who has read so much, studied so much, meditated so much and written more than Leibniz... What he has elaborated on the world, about God, nature and the soul is of the most sublime eloquence. If his ideas had been expressed with the nose of Plato, the philosopher of Leipzig would not yield anything to the philosopher of Athens ».

Indeed, Diderot's tone is almost hopeless in another remark, equally true: "When one compares one's talents with those of Leibniz, one is tempted to throw away all one's books and go and die quietly in the darkness of some forgotten corner. Diderot's reverence contrasts with the attacks that another important philosopher, Voltaire, would launch against Leibniz's philosophical thought, a consequence of his appreciation for Newton and his contempt for the optimism that led to his philosophical system. Despite acknowledging the vastness of his work, Voltaire maintained that in all of it there was nothing useful that was original, nor anything original that was not absurd and laughable.

It occupies an equally important place in both the history of philosophy and mathematics. Independently of the work of Newton (who had developed it 10 years earlier but had not published it due to his trauma from the criticism that Hooke once made of him) he developed the infinitesimal calculus and its notation, which has been used ever since. He also invented the binary system, the virtual foundation of all current computer architectures. He was one of the first European intellectuals to recognize the value and importance of Chinese thought and of China as a power from all points of view.

René Descartes, Baruch Spinoza and Leibniz make up the shortlist of the three great rationalists of the 17th century. His philosophy is also linked to the scholastic tradition and anticipates modern logic and analytical philosophy. Leibniz also made contributions to technology and anticipated notions that appeared much later in biology, medicine, geology, probability theory, psychology, engineering, and computer science. His contributions to this vast list of topics are recorded in journals and in tens of thousands of letters and unpublished manuscripts. Until now, a complete edition of his writings has not been made, and therefore it is not yet possible to make a comprehensive account of his achievements.

Biography

Early Years

Gottfried Leibniz was born on July 1, 1646 in Leipzig, two years before the end of the Thirty Years' War, the son of Frederick Leibniz, a jurist and professor of moral philosophy at the University of Leipzig, and Catherina Schmuck, daughter of a law professor. As an adult, he frequently signed himself "von Leibniz" and numerous posthumous editions of his works name him as "Freiherr [baron] G. W. von Leibniz"; however, no document has been found to confirm that he was granted a noble title.

His father died when he was six years old, so his education was left to his mother and uncle, and in his own words, himself. On his father's death, he left behind a personal library that Leibniz was free to use from the age of seven, and proceeded to benefit from its contents, particularly the volumes on ancient history and the Church Fathers.

By the time he was twelve he had taught himself Latin, which he used for the rest of his life, and had begun to study Greek. In 1661, at the age of fourteen, he enrolled in the University of Leipzig and completed his studies at the age of twenty, specializing in law and showing mastery of the classics, logic, and scholastic philosophy. However, his education in mathematics was not up to the standards of the French or British.

In 1666 he published his first book and also his habilitation thesis, Dissertation on Combinatory Art. When the university declined to secure a law teaching position after his graduation, Leibniz opted to submit his thesis to the University of Altdorf and obtained his doctorate in five months. He later declined the offer of an academic post at Altdorf and devoted the remainder of his life in the service of two prominent families of the German nobility.

House of Schönborn (1666-1674)

Consultant in Mainz

Leibniz's first position was as a salaried alchemist in Nuremberg, although he had no knowledge of the subject. He came into contact with Johann Christian von Boineburg (1622–1672), former chief minister of the Elector of Mainz, Johann Philippe von Schönborn, who hired him as an assistant and shortly afterward introduced him to the Elector, after reconciling with him. Leibniz dedicated an essay to the voter in the hope of getting a job. The strategy worked, as the elector asked him to help redraft the legal code of his electorate, and in 1669 he was appointed assessor to the Court of Appeals. Although von Boineburg died in 1672, he remained in the service of his widow until 1674.

Von Boineburg did much to further his reputation, and his service to the electorate soon took on a more diplomatic role. He published an essay under the pseudonym of a Polish nobleman, in which he argued (unsuccessfully) in favor of the German candidate for the Polish Crown. The main factor in European geopolitics during his adult life was the ambitions of Louis XIV of France, backed by his military and his economic might. The Thirty Years' War had left German-speaking Europe exhausted, as well as fragmented and economically backward. Leibniz proposed protecting it by diverting Louis XIV as follows: France would be invited to take Egypt as a first step towards an eventual conquest of the Dutch East Indies. In return, France would agree not to disturb Germany or the Netherlands. The plan received cautious support from the voter. In 1672 the French government invited Leibniz to Paris for discussion, but the plan was soon overtaken by events and rendered irrelevant.

Stays in Paris and London

In this way, Leibniz began a stay of several years in Paris, during which he considerably increased his knowledge of mathematics and physics and began to make contributions in both disciplines. He met Malebranche and Antoine Arnauld, the leading French philosopher of the day, studied the writings of Descartes and Pascal, both published and unpublished, and became friends with the German mathematician Ehrenfried Walther von Tschirnhaus, with whom he corresponded until the end of his life. It was especially opportune to meet the Dutch physicist and mathematician Christiaan Huygens, who was also in Paris at the time. Arriving in Paris, Leibniz received a rude awakening, as his knowledge of physics and mathematics was fragmentary. With Huygens as a mentor, he began a self-taught program that soon resulted in great contributions to both fields, including the discovery of his version of differential calculus and his work on infinite series.

La Stepped Reckoner

In early 1673, when it became clear that France would not carry out its part of Leibniz's plan for Egypt, the Elector sent his own nephew, accompanied by Leibniz, on a diplomatic mission to the British government. In London Leibniz met Henry Oldenburg and John Collins. After showing the Royal Society a machine capable of performing arithmetic calculations known as the Stepped Reckoner, which he had been designing and building since 1670, the first machine of its kind that could perform the four "basic arithmetic operations," the Society made him an external member. The mission ended abruptly upon receiving the news of the elector's death. Leibniz immediately returned to Paris and not to Mainz, as he had planned.

The sudden death of both of Leibniz's patrons in the same winter meant that he had to seek a new direction for his career. In this connection, an invitation from the Duke of Brunswick in 1669 to visit Hanover was timely. There he declined the invitation, but began corresponding with the duke in 1671. In 1673 the duke offered him a counselorship, which he reluctantly accepted two years later, only after it was clear that he would get no employment in Paris (whose encouragement intellectual cherished) or in the imperial court of the Habsburgs.

House of Hanover (1676-1716)

Second trip to London

He managed to delay his arrival in Hanover until the end of 1676, after another brief trip to London, where he was possibly shown some of Isaac Newton's unpublished works (this is merely conjecture, of course, given Newton's well-known reluctance to show their writings), although most historians of mathematics now claim that Newton and Leibniz developed their ideas independently: Newton developed the ideas first and Leibniz was the first to publish them.

On the way from London to Hannover, he stopped in The Hague, where he met Leeuwenhoek, who improved the microscope and discovered microorganisms. He likewise engaged several days in intense discussion with Spinoza, who had recently completed his masterpiece, Ethics. Leibniz had respect for Spinoza's powerful intellect, but he was dismayed by his conclusions, which contradicted Christian orthodoxy.

Political adviser

In 1677 he was promoted, at his own request, to Privy Councilor of Justice, a position he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as a historian, political adviser, and as librarian of the Ducal Library. From then on he employed his pen on the various political, historical, and theological matters involving the House of Brunswick; the resulting documents constitute a valuable part of the historical records of the period.

Among the few people who welcomed Leibniz in northern Germany were the Elector, her daughter Sophia Charlotte of Hannover (1630–1714), Queen of Prussia and her avowed disciple, and Caroline of Brandenburg-Ansbach, the consort of his grandson, the future George II. For each of these women, Leibniz was a correspondent, adviser and friend. Each of them welcomed him more warmly than did their respective husbands and the future King George I of Great Britain.

Hanover then had only about 10,000 inhabitants and its parochialism displeased Leibniz. However, to be an important courtier in the House of Brunswick was a great honor, especially in view of the meteoric rise in that House's prestige during Leibniz's relationship with it. In 1692, the Duke of Brunswick became hereditary Elector of the Holy Roman Empire. The Settlement Act of 1701 designated Electress Sophia and her offspring as the royal family of the United Kingdom, once both King William III and her sister-in-law and successor Queen Anne had died. Leibniz participated in the initiatives and negotiations that led to the Law, but not always effectively. For example, something he published in England, thinking it would further the Brunswick cause, was formally censored by the British Parliament.

Intellectual works

Portrait of Gottfried Wilhelm Leibniz at the Hanover Public Library (Baja Saxony)

The Brunswicks tolerated the enormous efforts that Leibniz devoted to his intellectual projects unrelated to his courtly duties, projects such as perfecting calculus, his writings on mathematics, logic, physics, and philosophy, and maintaining a vast correspondence. He began working on calculus in 1674, and by 1677 he had a coherent system on his hands, but he did not publish it until 1684. His most important mathematical papers appeared between 1682 and 1692, usually in a journal that he and Otto Mencke had founded in 1682, the Acta Eruditorum. This magazine played a key role in the advancement of his scientific and mathematical reputation, which in turn increased his eminence in diplomacy, history, theology, and philosophy.

The Elector Ernest Augustus commissioned Leibniz a task of enormous importance, the history of the House of Brunswick, going back to the time of Charlemagne or earlier, in the hope that the resulting book would aid his dynastic ambitions. Between 1687 and 1690 Leibniz traveled extensively in Germany, Austria, and Italy in search of archival materials relevant to this project. Decades passed and the book did not arrive, so the next constituent was quite upset at the apparent lack of progress. Leibniz never completed the project, partly because of his enormous output elsewhere, but also because of his insistence on writing a meticulously researched and scholarly book based on archival sources. His patrons would have been quite satisfied with a short popular book, a book that was perhaps little more than an annotated genealogy, to be completed in three years or less. They never knew that he had, in fact, carried out a good part of the assigned task: when Leibniz's writings were published in the 19th century , the result was three volumes.

Last years

Leibniz residence in Hanover (first floor of the central building), from 1698 to his death Photochrome made around 1900.

In 1711 John Keill, writing in the journal of the Royal Society and, with Newton's supposed blessing, accused Leibniz of having plagiarized Newton's calculus, thus starting the the dispute over the paternity of calculus. He began a formal investigation by the Royal Society (in which Newton was a recognized participant) in response to Leibniz's request for retraction, thus endorsing Keill's accusations.

In the same year, during a trip through northern Europe, the Russian Tsar Peter the Great stopped in Hannover and met with Leibniz, who later took an interest in Russian affairs for the rest of his life. In 1712 Leibniz began a two-year stay in Vienna, where he was appointed counselor to the Imperial Court of the Habsburgs.

After Queen Anne's death in 1714, Elector George Louis became King George I of Great Britain under the terms of the 1711 Act of Settlement. Although Leibniz had done quite a bit to further that cause, he would not have to be his hour of glory. Despite the intervention of Princess Caroline of Brandenburg-Ansbach of Wales, George I forbade Leibniz to meet him in London until he had completed at least one volume of the Brunswick family history commissioned by his father nearly 30 years ago. back. Furthermore, the inclusion of Leibniz in his London court would have been insulting to Newton, who was seen as the winner of the dispute over the priority of calculus and whose position in British official circles could not have been better. Finally, his dear friend and defender, the Elector Sophia of Wittelsbach, died in 1714.

Death

Leibniz Tomb in Hanover on the 300th anniversary of his death

Leibniz died in Hanover in 1716: by then, he was so out of favor at court that neither George I (who happened to be near Hanover at the time) nor any other courtiers, other than his personal secretary, attended the funeral. Even though Leibniz was a life member of the Royal Society and the Prussian Academy of Sciences, neither entity considered it appropriate to honor his memory.

His tomb remained anonymous until Leibniz was exalted by Fontenelle before the French Academy of Sciences, which had admitted him as a foreign member in 1700. The exaltation was drawn up at the request of the Duchess of Orleans, granddaughter of the Electress Sofia.

Chronological summary

Work

Leibniz wrote primarily in three languages: Scholastic Latin (ca. 40%), French (ca. 35%), and German (less than 25%). During his lifetime he published many pamphlets and scholarly articles, but only two philosophical books, Dissertation Concerning the Combinatory Art and the Théodicée .

Manuscript of Leibniz preserved in the National Library of Poland

A letter from Leibniz to Kiel in 1716 concerning a publication (Biblioteca Gottfried Wilhelm Leibniz, Hamburg)

He published numerous, often anonymous, pamphlets on behalf of the House of Brunswick, including De jure suprematum, an important consideration on the nature of sovereignty. Another substantial book appeared posthumously: his Nouveaux essais sur l'entendement humain ( New Essays on Human Understanding ), which he had avoided publishing after the death of John Locke.

It was not until 1895, when Bodemann completed his catalog of Leibniz's manuscripts and correspondence, that the enormous extent of his legacy became clear: approximately 15,000 letters to more than 1,000 recipients, plus 40,000 additional items, not counting that many of these letters are essay length. Much of his vast correspondence, particularly letters dated after 1685, remain unpublished, and much of what has been published has been published only in recent decades. The quantity, variety, and disorder of Leibniz's writings are the predictable result of a situation that he described as follows:

I can't finish telling you how extraordinarily distracted and scattered I am. I'm trying to find several things in these files; I'm looking for old papers and I'm going behind unpublished documents. With this I hope to shed some light on the history of the Brunswick House. I receive and respond to an immense amount of letters. At the same time I have so many mathematical results, philosophical thoughts and other literary innovations, which should not be allowed to vanish, that I often do not know where to begin.

Letter from Leibniz to Vincent Placcius in Gerhardt, 1695.

The extant parts of Leibniz's writings in critical edition are organized as follows:

• Series 1. Political, Historical and General Correspondence. 25 vols. 1666-1701.
• Series 2. Philosophical Correspondence. 1 vol. 1663-1685.
• Series 3. Mathematical, Scientific and Technical Correspondence. 8 vols. 1672-1696.
• Series 4. Political writings. 7 vols. 1667-1699.
• Series 5. Historical and linguistic writings. Inactive.
• Series 6. Philosophical writings. 5 vols. 1663-1690 and Nouveaux essais sur l'entendement humain.
• Series 7. Mathematical writings. 6 vols. 1672-1676.
• Series 8. Scientific, medical and technical writings. 1 vol. 1668-1676.

The cataloging of the entire Leibniz legacy began in 1901. Two world wars (with the Jewish Holocaust involved, including a project employee and other personal consequences) and decades of German division (two states divided by an iron curtain, which separated the academics and also scattered parts of his literary legacy) greatly hampered the ambitious publishing project that must deal with the use of seven languages in about 200,000 pages of printed material.

In 1985 it was reorganized and included in a joint program of German federal and state academies. Since then the branches in Potsdam, Münster, Hannover and Berlin have collectively published 25 volumes of the critical edition (up to 2006), averaging 870 pages per volume (compared to 19 volumes since 1923), plus indexing. and the work of concordance.

Posthumous Celebrity

Monument to Leibniz at the University of Leipzig. Hähnel Work (1883)

At the time of Leibniz's death, his reputation was in decline; he was remembered only for one book, the Théodicée , whose supposed central plot was caricatured by Voltaire in his Candide . Voltaire's description of Leibniz's ideas was so influential that many took it to be an accurate description (this misinterpretation can still occur among certain laymen). So Voltaire bears some responsibility for the fact that many of Leibniz's ideas remain misunderstood. In addition, Leibniz had an ardent disciple, the philosopher Christian Wolff, whose dogmatic and superficial appearance did much to damage Leibniz's reputation. In any case, the philosophical movement was moving away from the rationalism and systems-building of the 17th century, from which Leibniz had been a great exponent. His work in law, diplomacy and history was perceived as ephemeral in his interest, and the vastness and richness of his correspondence was overlooked.

Much of Europe came to doubt that he had discovered calculus independently of Newton, and his entire work in mathematics and physics was therefore scorned. Voltaire, who admired Newton, also wrote his Candide , at least in part, to discredit Leibniz's claim to his discovery of calculus and his view that Newton's theory of universal gravitation it was incorrect. The rise of relativity and subsequent work in the history of mathematics placed Leibniz's position in a more favorable light.

Leibniz's long journey to his present glory began with the publication in 1765 of his Nouveaux Essais, which were rigorously read by Kant. In 1768 Dutens published the first multi-volume edition of Leibniz's work, followed in the 19th century by several more, including that of Erdmann, Foucher de Careil, Gerhardt, Gerland, Klopp and Mollat, as well as the publication of his correspondence with notable people such as Antoine Arnauld, Samuel Clarke, Sofía de Hannover and her daughter, Sofía Carlota de Hannover.

In 1900 Bertrand Russell published a critical study of Leibniz's metaphysics, and shortly thereafter Louis Couturat published a major study of Leibniz and edited a volume of previously undisclosed writings, mainly on logic. Although such conclusions, especially Russell's, were doubted and often dismissed, they gave Leibniz a little more respectability among analytic and linguistic philosophers of the XX of the English-speaking world (Leibniz had already been a great influence on several Germans, such as Bernhard Riemann). However, the English-language secondary literature on Leibniz did not really flourish until after World War II, in Brown's bibliography. Fewer than thirty of the English-language entries were published before 1946.

Nicholas Jolley has said that Leibniz's reputation as a philosopher is now perhaps higher than it was at any time since Leibniz's time, for the following reasons:

• Work in the history of the ideas of the centuries XVII and XVIII has revealed more clearly the "Intellectual Revolution" that preceded the most well-known industrial and commercial revolution of the centuries XVIII and XIX.
• The contempt of metaphysics, characteristic of analytical and linguistic philosophy, has diminished.
• Contemporary analytical philosophy continues to use Leibniz's diverse ideas about identity, individuation, possible worlds.
• It is now seen as an important extension of the powerful effort initiated by Plato and Aristotle: the universe and the place of man in it is attributable to human reason.

In 1985 the German government instituted the Gottfried Wilhelm Leibniz Prize, which is awarded annually. The economic amount of the prize in 2018, for each of the eleven winners, amounted to 2.5 million euros for nine of them and 1.25 million euros for two other winners. It is the highest prize awarded in Germany for scientific contributions.

In 1970 the International Astronomical Union decided to name an impact crater located in the southern hemisphere of the far side of the Moon after him "Leibniz".

In 2006, the University of Hannover was renamed "Gottfried Wilhelm Leibniz" in his honour.

Philosophy

Leibniz Portrait by Johann Friedrich Wentzel, about 1700

Leibniz's philosophical thought appears fragmented, as his philosophical writings consist mainly of a multitude of short texts: journal articles, manuscripts published long after his death, and large numbers of letters to multiple people. He wrote only two treatises on philosophy, and the one published during his lifetime, the Théodicée of 1710, is both theological and philosophical.

Leibniz himself dates his start as a philosopher with his Metaphysics Discourse, which he produced in 1686 as a commentary on a dispute between Malebranche and Antoine Arnauld. This led to a lengthy and valuable dispute with Arnauld; the commentary and the Discourse were not published until the XIX.

In 1695 Leibniz made his public entry into European philosophy with an article entitled New system of the nature and communication of substances. In the period 1695-1705 he elaborated his New Essays Concerning Human Understanding, a lengthy commentary on John Locke's Essay Concerning Human Understanding (1690), but upon learning of Locke's death in 1704 he lost the desire to publish it, so the New Essays were not published until 1765. The Monadology, another of his important works, composed in 1714 and published posthumously, consists of ninety aphorisms; the influence of Giordano Bruno, whose work he knew, has been seen in it, and for his composition the files that the author made during his last stage in Hannover were used.

Leibniz met Spinoza in 1676 and read some of his unpublished writings, and it has been suspected ever since that he appropriated some of his ideas. Unlike Descartes, Leibniz and Spinoza had a rigorous philosophical education. The scholastic and Aristotelian disposition of his mind reveal the strong influence of one of his professors in Leipzig, Jakob Thomasius, who also supervised his graduate thesis. Leibniz also voraciously read Francisco Suárez, the Spanish Jesuit respected even in Lutheran universities. He had a deep interest in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle, but viewed his work from a perspective heavily influenced by scholastic notions. However, it remains remarkable that his methods and concerns often anticipate the linguistic and analytical logic and philosophy of the 20th century .

He was one of the first European intellectuals to recognize the value and importance of Chinese thought.

The principles

Leibniz freely resorted to one or another of nine fundamental principles:

• Identity/contradiction. If a proposition is true, then his denial is false, and vice versa.
• Substance. The substance is that which in a preacher corresponds to the subject, and that individualizes the world. It is the basic individual unit of the world, which has capacity for perception and appetite and whose attributes can only be caused by itself (self-caused, since it is substance).
• Identity of indiscerns. Two things are identical if and only if they share the same properties. This principle is often called Leibniz Law. This principle has been the subject of major controversies, in particular corpuscular philosophy and quantum mechanics.
• Principle of reason enough. "There must be a sufficient reason (often only by God known) for anything to exist, for any event to occur, so that any truth can be obtained." (LL 717)
• Pre-established harmony. "The appropriate nature of each substance makes what happens to one that corresponds to what happens to others, however, without them acting directly among them." (Speech of metaphysics, XIV). "A glass that falls is shattered because it "knows" that has touched the ground, and not because the impact with the soil compels it from."
• Continuity. Natura non facit saltum. A similar concept in mathematics at this beginning would be the following: If a function describes a transformation or something to which continuity applies, then its domain and its rank will be both dense sets.
• Optimism. "Undoubtedly God always chooses the best." (LL 311).
• Plenitude. "The best of the possible worlds would update every genuine possibility, and the best of the possible worlds will contain all the possibilities, with our finite experience of eternity that does not provide reasons to dispute the perfection of nature."
• Principle of convenience: or "the choice of the best", which, unlike the logic that comes from the principle of necessity, is based on contingency (Monastery46).

Principle of Sufficient Reason

The principle of sufficient reason, enunciated in its most finished form by Gottfried Leibniz in his Theodicy, affirms that no fact occurs without there being a sufficient reason for it to be so and not otherwise mode. In this way, he maintains that the events considered random or contingent appear such because we do not have complete knowledge of the causes that motivated them.

Now we must go back to the metaphysics, serving the great principle for the common unused, which affirms that nothing is done without sufficient reason, that is to say that nothing happens without it being impossible for anyone who knew things enough, to give a reason that is sufficient to determine why this is so and not otherwise. Initiated the principle, the first question that is entitled to be raised will be: why there is something rather than nothing. Well, nothing is simpler and easier than something. In addition, the assumption that there must be things, it must be given reason why they must exist in that way and not in another way.

Gottfried Leibniz. Principles of Nature7.

The principle of sufficient reason is complementary to the principle of non-contradiction, and its preferred field of application are statements of fact; the traditional example is the statement "Caesar passed the Rubicon", from which it is affirmed that, if such a thing happened, something must have motivated it.

According to the rationalist conception, the principle of sufficient reason is the foundation of all truth, because it allows us to establish what is the condition —that is, the reason— of the truth of a proposition. For Leibniz, without a sufficient reason it is not possible to affirm when a proposition is true. And since everything that happens for something, that is, if everything that happens always responds to a determining reason, knowing that reason could know what will happen in the future. This is the foundation of experimental science.

However, given the limits of the human intellect, we have to limit ourselves to accepting that nothing happens without a reason, despite the fact that such reasons very often cannot be known to us.

One of the general consequences for physics of the principle of sufficient reason was condensed by Leibniz in the form of an aphorism: «In the best of all possible worlds nature does not jump and nothing happens suddenly», which links said principle with the problem of the continuum and the infinite divisibility of matter.

Monads

First page of the manuscript Monastery.

Leibniz's most important contribution to metaphysics is his theory of monads, as expounded in the Monadology. Monads are to the metaphysical realm, what atoms are to the physical/phenomenal realm; monads are the ultimate elements of the universe. They are "substantial forms of being" with the consequent properties: they are eternal, they cannot be decomposed, they are individual, they are subject to their own laws, they are not interactive and each one is a reflection of the whole universe in a pre-established harmony (an example historically important of pampsichism).

Monads, without going into great mystery, are simple substances. Furthermore, they have no extension, the first accident of matter, each monad is a spiritual substance, each monad has an appetite, and each monad, as has been said, develops according to its inner law.

Monads are centers of force; substance is force, while space, matter, and motion are merely phenomenal. Space is phenomenal and not absolute, but relative, and consists of the perception of the spatial relationships between some monads and others (or a set of them). Thus, spatiality occurs when I perceive that a chair is in front of a table, the table in the center of the room's walls, the window in one of them, etc. It cannot be absolute because there is not a sufficient reason to consider that the universe is located in one area and not in another. As for the materiality or extension of the monads, it does not exist because then we would have to accept that an object, when divided in two by something external, is being modified by a cause alien to itself, which would contradict the inherent self-causation of monads. the substance. This is solved, as far as the phenomenal world is concerned (that is, the world of natural sciences), with the principle of pre-established harmony, in which everything happens according to a simultaneous and coherent order of "reflections".

The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads do not have a material or spatial character. They also differ from atoms in their complete independence from each other, so that interactions between monads are only apparent. Rather, by virtue of the principle of pre-established harmony, each monad obeys a particular set of pre-programmed "instructions", so that a monad "knows" what to do at each moment (These "instructions" can be understood as analogous to the laws scientific rules governing subatomic particles). By virtue of these intrinsic instructions, each monad is like a small mirror of the universe. Monads are necessarily "small"; p. For example, each human being constitutes a monad, in which case free will becomes problematic. Likewise, God is a monad, and his existence can be inferred from the prevailing harmony among the remaining monads; God desires the pre-established harmony.

Monads are supposed to have gotten rid of the troublesome:

• the interaction between the mind and the body (see the mind-body problem that arises in the Descartes system);
• of the lack of individuation inherent to the Spinoza system, which presents individual creatures as merely accidental.

Monadology was seen as arbitrary, even eccentric, in Leibniz's time and ever since.

Existence of God

Plate of Gottfried Wilhelm Leibniz in Leibniz-Schule, Berlin.

The God of Leibniz is not the immobile Mover of Aristotle, the Natura naturans of Spinoza, nor the Great Being of Newton or the Universal Spirit in Hegel; but "a living and personal God who reveals himself to both the heart and reason", thus trying to rationally substantiate the Christian God with his classical attributes. Within Leibniz's philosophy you can find four types of arguments regarding the existence of God:

1. The ontological and/or modal argument;
2. the cosmological argument;
3. the argument of eternal truths;
4. the argument of pre-established harmony (or physiological argument according to Kant).

Leibniz held that the concept of God is possible and wrote various formulations of St. Anselm's ontological argument in his works and letters. In his Monadology he wrote:

(41) “From where it follows that God is absolutely perfect, not being perfection but the magnitude of positive reality, taken precisely, leaving aside the limits or boundaries in the things that have them. And where there is no limit, that is, in God, perfection is absolutely infinite.”

(44) “If there is any reality in the Essences or possibilities or in the eternal truths, it is necessary that such reality is founded on something existing and actual, and therefore in the Existence of the necessary Being, in which the Essence encloses the Existence, or in which it is possible to be actual.

(45) Thus, only God (or the necessary Being) enjoys the following privilege: it is necessary to exist, if possible. And as nothing can prevent the possibility of what has no limit, no denial, and therefore no contradiction, this is only enough to know the Existence of God a priori....”

Monastery § 41, 44, 45 (1714)

In addition, Leibniz formulated a cosmological argument from contingency for the existence of God with his principle of sufficient reason in his Monadology. "No fact can be found to be true or existent, nor any proposition true," he wrote, "without there being a sufficient reason why it should be so and not otherwise, though we cannot know these reasons in most cases." cases”. He formulated the cosmological argument succinctly: "Why is there something rather than nothing? The sufficient reason [...] is found in a substance that [...] is a necessary being that carries the reason for its existence within itself." Martin Heidegger called this question "the fundamental question". from metaphysics". This argument is one of the most popular cosmological arguments in philosophy of religion and has been restated by Alexander Pruss and William Lane Craig. Philosophers such as Kant and Bertrand Russell criticized both arguments respectively.

The argument of eternal truths is also based on the principle of sufficient reason: "eternal truths do not have in themselves the reason for their existence and, therefore, this must be sought in the Supreme Being. [...] The sufficient reason for the eternal truths is God himself, since the set of all of them is nothing other than divine understanding itself". The argument of pre-established harmony is based on the harmony of the monads: "according to Leibniz, the world and each of the creatures that compose it develop with their own forces, but the latter were created and chosen by God in a necessary way to pre-establish the best organization in the world".

Theodicy and optimism

Title page Théodicée in a 1734 version.

The term «optimism» is used here in the sense of «optimal», and not in the more common sense of the word, that is, «state of mind», contrary to pessimism.

The Theodicy tries to justify the evident imperfections of the world, affirming that it is the best of all possible worlds. It has to be the best and most balanced of all possible worlds, since it was created by a perfect God. See Rutherford (1998) for a detailed scholarly study of Leibniz's Theodicy.

The conception of «the best of all possible worlds» is justified by the existence of a God with ordering capacity, not morally but mathematically. For Leibniz, this is the best of all possible worlds, not understanding "better" in a morally good way, but mathematically good, since God, of the infinite possibilities of worlds, has found the most stable between variety and homogeneity. It is the most mathematically and physically perfect world, since its combinations (whether morally good or bad, it doesn't matter) are the best possible. At the end of this book, Leibniz rewrites a fable that comes to symbolize this very thing: the mathematical perfection of this real world compared to all possible worlds, which are always found in the imperfection and imbalance of heterogeneity and homogeneity, with hell being the most homogeneous. (sins are repeated eternally) and paradise the maximum heterogeneous.

The statement that "we live in the best of all possible worlds" drew Leibniz much ridicule, especially from Voltaire, who caricatured him in his comic novel Candide, introducing the character of Dr. Pangloss (a parody of Leibniz) who repeats the phrase like a mantra every time misfortune fell on his companions. This is where the adjective "Panglossian" comes from, to describe someone so naive as to believe that our world is the best of all possible worlds.

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

As is known, the theory of maximum and minimum functions is in debt to him for progress, thanks to the discovery of the method of tangents. Well, conceive God in the creation of the world as a mathematician solving a problem of minimums, or rather, in our modern phraseology, a problem in the calculation of variations — being the question to determine, among an infinite number of possible worlds, that in which the sum of the necessary evil is minimized.

A cautious defense of Leibniz's optimism would draw on certain scientific principles that emerged in the two centuries since his death and are now established: the principle of least action, the law of conservation of mass, and the conservation of energy.

Theory of Knowledge

Monads have perceptions. They can be light or dark. Things have perceptions without consciousness. When perceptions have clarity and awareness and are at the same time accompanied by memory, they are apperception, proper to souls. Humans can know universal and necessary truths. Thus, the soul is spirit. At the top of the scale of monads is the divine. A good source to deepen the latter is found in Monadology.

Leibniz distinguishes between truths of reason and truths of fact. The first are necessary. The latter are not justified a priori, without more. "Two and two are four" is a truth of reason. "Columbus discovered America" is a truth in fact, because it could have been otherwise, that is, "Columbus did not discover America." But Columbus discovered America because it was in his individual being, Columbus (monad). The truths of fact are included in the essence of the monad. But only God knows all the truths of fact, because in his omniscience and omnipotence there can be no distinctions of truths of reason and fact of each monad. Only God can understand the truths of fact, since this presupposes an infinite analysis.

Leibniz, in the order of knowledge, will affirm a type of nativism. All ideas without exclusion proceed from the internal activity that is proper to each monad. Ideas, therefore, are innate. Leibniz will oppose Locke and all English empiricism.

Symbolic thinking

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

The only way to rectify our reasoning is to make them as tangible as those of the Mathematics, so that we can find our error at a glance, and when there are disputes between people, we can simply say: Let us calculate [calculemus], no more delay, see who's right.

Leibniz's calculus ratiocinator, which resembles symbolic logic, can be seen as a way of making such calculations feasible. Leibniz wrote memoranda that can now be read as attempts to raise symbolic logic and thus its calculus. These writings remained unpublished until the appearance of a selection edited by Carl Immanuel Gerhardt (1859). Louis Couturat published a selection in 1901; at that time the main developments of modern logic had been created by Charles Sanders Peirce and by Gottlob Frege.

Leibniz thought symbols were important to human understanding. He attached such importance to the development of good notation that he attributed all his discoveries in mathematics to it. His notation for calculus is an example of his skill in this regard. Leibniz's passion for symbols and notation, as well as his belief that these are essential for the proper functioning of logic and mathematics, made him a forerunner of semiotics.

But Leibniz took his speculation much further. Defining a character as any typed sign, he then defined a 'real' character. as one that represents an idea directly and not simply as the word that embodies the idea. Some real characters, such as logic notation, serve only to facilitate reasoning. Many well-known characters in his day, including Egyptian hieroglyphics, Chinese characters, and symbols of astronomy and chemistry, he considered unreal. Instead, he proposed creating a characteristica universalis or "universal characteristic", built on an alphabet of human thought (alphabetum cogitationum humanarum) in which each fundamental concept would be represented by a single "real" character:

It is obvious that if we could find appropriate characters or signs to express all our thoughts with as much clarity and accuracy as the arithmetic expresses the numbers or geometry expresses the lines, we could do in all matters as soon as they are subject to reasoning everything we can do in arithmetic and geometry. Because all the research that depends on reasoning would be carried out by transposition of these characters and by a kind of calculation.

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

Because Leibniz was a novice in mathematics when he first wrote about the characteristic, he initially conceived of it not as an algebra but as a universal language or script. It was not until 1676 that he conceived a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting feature included logical calculus, some combinatorics, algebra, its analysis situs (geometry of the situation), a universal conceptual language, and more. What Leibniz actually intended with his characteristica universalis and calculus ratiocinator, and to what extent modern formal logic does justice to calculus, may never be established. Leibniz's idea of reasoning through a universal language of symbols and calculations remarkably heralds the great developments of the 20th century in formal systems, such as Turing completeness, where computation was used to define equivalent universal languages (see: Turing degree).

Formal Logic

Leibniz has been noted as one of the most important logicians between the times of Aristotle and Gottlob Frege. Leibniz enunciated the main properties of what we now call conjunction, disjunction, negation, identity, subset, and the empty set. The principles of Leibniz's logic, and possibly of all his philosophy, can be reduced to two:

1. All our ideas are made up of a very small number of simple ideas, which form the alphabet of human thought.
2. Complex ideas come from these simple ideas through a uniform and symmetrical combination, analogous to arithmetic multiplication.

The formal logic that emerged in the early XX century also requires, at a minimum, a unary negation and quantified variables that encompass some universe of discourse.

Leibniz published nothing on formal logic during his lifetime; most of what he wrote on the subject consists of working drafts. In his History of Western Philosophy, Bertrand Russell went so far as to claim that Leibniz had developed logic in his unpublished writings to a level he reached only 200 years later.

Russell's major work on Leibniz found that many of Leibniz's most striking philosophical ideas and claims (for example, that each of the fundamental monads reflects the entire universe) follow logically from Leibniz's conscious choice to reject the relations between things as unreal. He considered such relationships as (real) qualities of things (Leibniz admitted only unary predicates): for him, "Mary is the mother of John" it describes separate qualities of Mary and John. This view contrasts with the relational logic of De Morgan, Peirce, Schröder, and Russell himself, now standard in predicate logic. Notably, Leibniz also stated that space and time are inherently relational.

Leibniz's discovery in 1690 of his algebra of concepts (deductively equivalent to Boolean algebra) and the associated metaphysics, are of interest in current computational metaphysics.

Politics and law

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

While Leibniz was not an apologist for absolute monarchy like Thomas Hobbes, or for tyranny in any form, neither did he echo the political and constitutional views of his contemporary John Locke, views invoked in support of liberalism, in South America. 18th century and later elsewhere. The following excerpt from a 1695 letter to Baron JC Boyneburg's son Philipp is very revealing of Leibniz's political sentiments:

As for... the great question of the power of the sovereigns and the obedience that their peoples owe them, I say it would be good for the princes to be convinced that their people have the right to resist them, and that the people, on the other hand, be persuaded to obey them passively. I am, however, quite of the opinion of Grotius, that one must obey as a rule, being the evil of the greatest revolution without comparison to the evils that cause it. However, I recognize that a prince may come to such excess and put in such danger the welfare of the state, which ceases the obligation to endure. This is very rare, however, and the theologian who authorizes violence under this pretext must take care of excesses; excess is infinitely more dangerous than deficiency.

In 1677, Leibniz called for a European confederation, governed by a council or senate, whose members would represent entire nations and be free to vote conscientiously; this is sometimes seen as an anticipation of the European Union. He believed that Europe would adopt a uniform religion. He reiterated these proposals in 1715.

But, at the same time, he came to propose an interreligious and multicultural project to create a universal system of justice, which required of him a broad interdisciplinary perspective. To propose it he combined linguistics (especially sinology), moral and legal philosophy, management, economics and politics.

Law

Leibniz was trained as a legal scholar, but under the tutelage of the Cartesian sympathizer Erhard Weigel we already see an attempt to solve legal problems by rationalist mathematical methods (Weigel's influence is most explicit in the Specimen Quaestionum Philosophicarum ex Jure collectarum (An Essay of Collected Philosophical Problems of Law.) For example, the Inaugural Disputation on Puzzling Cases uses early combinatorics to resolve some legal disputes, while the 1666 Dissertation on the Combinatorical Art includes simple legal problems by way of illustration.

The use of combinatorial methods to solve legal and moral problems appears, through Athanasius Kircher and Daniel Schwenter, to be Lullian-inspired: Ramon Llull tried to resolve ecumenical disputes by resorting to a mode of combinatorial reasoning that he considered universal (a mathesis universalis)..

In the late 1660s, the enlightened Prince-Bishop of Mainz, Johann Philipp von Schönborn, announced a review of the legal system and made a position available to support his current legal commissioner. Leibniz left Franconia for Mainz before even getting the role. Arriving in Frankfurt am Main, Leibniz wrote The New Method of Teaching and Learning the Law, by way of application. The text proposed a reform of legal education and is characteristically syncretic, integrating aspects of Thomism, Hobbesianism, Cartesianism and traditional jurisprudence. Leibniz's argument that the function of legal education was not to imprint rules as one might train a dog, but to help the student discover his own public reason, evidently impressed von Schönborn when he got the job.

Leibniz's next great attempt to find a universal rational core for law and thus found a legal “science of law” came when Leibniz worked in Mainz between 1667 and 1672. Initially starting from the mechanistic doctrine of the power of Hobbes, Leibniz returned to combinatorial-logical methods in an attempt to define justice. As Leibniz's so-called Elementa Juris Naturalis advanced, he incorporated modal notions of right (possibility) and obligation (necessity) in which we see perhaps the earliest elaboration of his doctrine of possible worlds within a deontic framework. eventually the Elementa remained unpublished, Leibniz continuing to work on his drafts and promote his ideas to correspondents until his death.

Scientific Activities

Logic

German Seal of Leibniz of 1927

In the field of logic, Gottfried Wilhelm Leibniz developed the doctrine of analysis and synthesis. He understood logic as the science of all possible worlds. Leibniz belongs to the first in the history of the formulation of the law of sufficient reason; he is also the author of the expression law of identity adopted in modern logic. He considered the law of identity to be the supreme principle of logic. "The nature of truth in general consists in the fact that it is something identical."

The law of identity formulated by Leibniz is currently used in most modern logical-mathematical calculations. The principle of substitution is equivalent to the law of identity: “If A is B and B is A, then A and B are called 'the same' '“. Or: A and B are equal if they can be substituted for one instead of the other ".

For Leibniz, the principles of identity, equivalent substitution and contradiction are the main means of any deductive proof; Relying on them, Leibniz attempted to prove some of the so-called axioms. He believed that axioms are unverifiable sentences, that they are identities, but in mathematics not all positions given as axioms are identities and therefore, from Leibniz's point of view, it is necessary to prove it. The criterion of identification and distinction of names introduced by Leibniz corresponds to a certain extent to the modern distinction between the meaning and meaning of names and expressions, for example the well-known example with the equivalence of the expressions "Sir Walter Scott" and "the author of Waverley", going back to Russell, literally repeats this thought.

Leibniz did not develop a unified system of designations, he developed the negative plus sign calculus. Leibniz's successful presentation of the correct modes of syllogism was the presentation of judgments by means of parallel segments or circles ("Experience of Syllogistic Based in evidence" in the book Opuscules et fragments inédits de Leibniz). Leibniz's important place was occupied by the protection of the object and the method of formal logic. He wrote to G. Wagner the following:

... although Mr. Antoine Arnauld (son), in his art of thinking, argued that people seldom make mistakes in form, but almost in essence, in fact, the situation is completely different and already Huygens, along with me, noticed that usually mathematical errors, called paralogism, are caused by form disorder. And, of course, Aristotle did not derive in any strict laws for these forms and, therefore, he was the first to write mathematically out of mathematics.

Leibniz made the most complete classification of definitions for his time, in addition, he developed a theory of genetic definitions. In his work "The Art of Combinatorics", written in 1666, Leibniz anticipated some aspects of mathematical logic. Combinatorics called Leibniz developed by him under the influence of R. Lully the idea of "great art" of discovery, which, based on the "first truths" obvious, would logically allow the entire system of knowledge to be derived from them. This theme has become one of the key themes of all life and he developed the principles of "universal science", on which, according to him, "the well-being of humanity depends above all." of". Gottfried Wilhelm Leibniz wrote the idea of using mathematical symbols in logic and the construction of logical calculations. He advanced the task of corroborating mathematical truths about general logical principles, and also proposed using a binary number system, that is, binary, for the purposes of computational mathematics. Leibniz justified the importance of rational symbolism for logic and for heuristic conclusions; He argued that knowledge is reduced to proof of assertions, but to find proof it is necessary by a certain method.

According to Leibniz, the mathematical method itself is not enough to discover everything we are looking for, but it protects against errors. The latter is explained by the fact that in mathematics statements are formulated with the help of certain signs and act according to certain rules, and checking, which is possible at each stage, requires "only paper and ink". Leibniz also first expressed the idea of the possibility of machine modeling of human functions, he also has the term "model". Leibniz made a great contribution to the development of the concept of "necessity". He understood the need as something that must be. According to Leibniz, the first necessity is metaphysical, absolute, as well as the logical and geometric necessity. It is based on the laws of identity and contradiction, therefore it admits the only possibility of events. Leibniz also noted other characteristics of necessity. He contrasted the necessity of chance, understanding it not as a subjective appearance, but as an objective connection of phenomena, which depends on free decisions and the course of processes in the Universe. He understood it as a relative accident, objective in nature and arising at the intersection of certain necessary processes. In "New experiences" (Book 4), Leibniz made a deductive analysis of traditional logic, showing that figures 2 and 3 of the syllogism can be obtained as a consequence of the "Barbara" using the law of contradiction, and the 4th figure. - use the treatment law; here he gave a new classification of the modes of syllogism. Leibniz's original logical ideas, the most valued today, only became known in the 20th century. Leibniz's results had to be rediscovered, since his own work was buried in piles of manuscripts in the royal library in Hanover.

Math

Before Leibniz several techniques were created to solve tangent problems, find extrema and calculate quadrature, but in the works of his predecessors there was no study limited mainly by complete algebraic functions to any fractional and irrational and especially to transcendental functions. In these works, the basic concepts of analysis were not clearly distinguished in any way, and their interrelationships were not established, there was no developed and uniform symbolism. Gottfried Leibniz brought together private and disparate techniques into a single system of interrelated analysis concepts, expressed in notation, allowing to perform actions with infinitely small according to the rules of a certain algorithm.

• 1675: Leibniz created the differential and integral calculation and later published the main results of his discovery, ahead of Newton, who had reached similar results before Leibniz, but did not publish them at that time, although Leibniz had some of them known in private order.

• 1684: Leibniz published the first important work of the world on differential calculation: "The new method of maximums and minimums", work in which the name of Newton is not even mentioned, and in the second merit of Newton described is not entirely clear. Then Newton didn't pay any attention. His analysis works began to be published only with 1704. Later, on this topic, a long dispute arose between Newton and Leibniz about the priority of the discovery of differential calculus.

Leibniz's document establishes the basic concepts of differential calculation, the rules of differentiation of expressions. Using the geometric interpretation of the relationship danddx{displaystyle {frac {dy}{dx}}}}, briefly explains the signs of increase and decrease, maximum and minimum, convexity and concavity (so, sufficient extreme conditions and for the simplest case), as well as inflection points. On the way, the "differentials of differentials" (multiplos of differentials), denoted by "ddv{displaystyle ddv}", they are introduced without any explanation. Leibniz wrote: "What a person versed in this calculation can solve in three lines, other learned men were forced to seek following uncompromising complexes."

• 1686 For the first time, he introduced the symbol ∫ ∫ {displaystyle int } for integration, and indicated that this operation is inverse to differentiation.

• 1692: The general concept about a family of curves of a parameter is introduced, its equation is derived. The theory of the curved family wraps was developed by Leibniz simultaneously with X. Huygens in 1692 - 1694.

• 1693: Leibniz addressed the problem of solvency of linear systems; its results introduced the concept of determining. But this discovery did not awaken interest then, and the linear algebra arose only half a century later.

• 1695: Leibniz introduced the exponential function in the most general form: uv{displaystyle u^{v}}. Later, in 1697 and, Johann Bernoulli studied the calculation of the exponential function.

• 1702: Together with Johann Bernoulli, Leibniz discovered the method of decomposition in partial fractions. This solved many problems of integrating rational fractions.

In Leibniz's approach to mathematical analysis there were some features. Leibniz conceived of the higher analysis not kinematically, but algebraically, unlike Newton. In his early papers, he seemed to understand infinitesimals as real objects comparable to each other only if they are of the same order. He perhaps hoped to establish his connection with his concept of monads. At the end of his life, he spoke quite a bit in favor of potentially infinite variables, although he didn't explain what he meant by that. In general philosophical terms, he considered the infinitesimal as the support of continuity in nature. Leibniz's attempts to carry out a rigorous analysis of analysis were unsuccessful, he wavered between various interpretations of infinitely small, sometimes trying to resort to unspecified ideas of limit and continuity. Leibniz's views on the nature of the infinitely small and on the reason for the operations on them caused criticism even during his lifetime, and the reason for analysis satisfying modern scientific requirements could only be given in [19th century].

Leibniz binary numbers system. Page Explication de l’Arithmétique Binaire

Gottfried Wilhelm Leibniz demonstrated the robustness of his general methods by solving several difficult problems. For example, in 1691 he established that a heavy and uniform thread hanging at two ends had the shape of a catenary and, together with Isaac Newton, Jacob and Johann Bernoulli, and also L'Hôpital, in 1696, he solved the problem of the brachistochrone curve.

An important role in spreading Leibniz's ideas was played by his extensive correspondence. Leibniz stated some discoveries with only letters: the beginnings of the theory of determinants in 1693, and, a generalization of the concept of a differential to negative and fractional indicators in 1695, and, a sign of convergence of a series of alternate signs (attribute Leibniz, 1682), methods for solving quadratures of various types of ordinary differential equations.

Leibniz introduced the following terms: " differential", "differential calculus", "differential equation", " function", " variable", "constant", "coordinates", "abscissa", "algebraic and transcendental curves", "algorithm&# 34;(in a sense close to modern). Although the mathematical concept of a function was implicit in trigonometry and in the logarithmic tables that existed at the time, Leibniz was the first to use it explicitly to refer to any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, rope and normal.

Leibniz formulated the concept of differential as an infinitely small difference between two infinitely close values of a variable and integral as the sum of an infinite number of differentials and gave the simplest rules for the differentiation and integration already in his manuscript notes of Paris concerning October and November 1675; here in Leibniz for the first time there are modern signs of the differential "d{displaystyle d}"and the whole. Leibniz gave the definition and the differential sign in 1684, in the first memory on differential calculation, "A new method of maximums and minimums". In the same work, the rules to differentiate the sum, difference, product, partial, any constant degree, function of the function (invariance of the first differential), as well as the rules to find and distinguish (using the second differential) maximum and minimum and find inflection points. The differential of a function was defined as the relation of the orderly to the sub-tangent, multiplied by the differential of the argument, whose value can be taken arbitrarily; At the same time, Leibniz indicated that the differentials are proportional to infinitesimal increments of magnitudes and that, based on this, it is easy to obtain a test of its rules.

The 1684 essay was followed by a series of other essays by Leibniz, covering in their entirety all the basic divisions of differential and integral calculus. In these works, Gottfried Wilhelm Leibniz defined and the integral sign (1686), emphasizing the reciprocal nature of the two main analytical operations, indicated the rules for differentiating the exponential function and multiple differentiation of a work (Leibniz formula, [1695]), and also initiated the integration of rational fractions (1702 - 1703). In addition, Leibniz gave fundamental importance to the use of infinite power series for the study of functions and the solution of differential equations (1693).

Due not only to earlier publications, but also to significantly more convenient and transparent designations of Leibniz's work on the differential and integral calculus, they had a much greater influence on contemporaries than Newton's theory. Even Newton's compatriots, who for a long time preferred the fluxion method, gradually learned the more convenient Leibniz notation. Leibniz also described binary systems with the numbers 0 and 1. The modern binary system was fully described by him in the work Explanation de l’Arithmétique Binair. As a person interested in Chinese culture, Leibniz was introduced to the Book of Changes and noted that the Hexagrams correspond to binary numbers from 0 to 111111. He admired the fact that this mapping is evidence of important Chinese achievements in philosophical mathematics at the time. Leibniz may have been the first programmer and information theorist. He found that if you write certain groups of binary numbers one below the other, then the zeros and zeros in the vertical columns will repeat regularly, and this discovery led him to believe that there are entirely new laws of mathematics. Leibniz realized that the binary code is optimal for the system of mechanics, which can work on the basis of active, passive, and intermittent passive cycles. He tried to apply binary code in mechanics and even made a drawing of a computer that worked on the basis of his new mathematics, but soon realized that the technological capabilities of his time did not allow him to create such a machine.. The project of the computer operating on the binary system, in which the prototype punched card was used, was described by Leibniz in a paper written in 1679 and (before he described binary arithmetic in detail in 1703 a Explanation de l'Arithmétique Binaire ). The units and zeros in an imaginary machine were respectively represented by open or closed holes in a moving jar, through which balls were supposed to pass, falling into the slots below it. Leibniz also wrote about the possibility of machine modeling the functions of the human brain.

Natural Sciences

By proposing that the Earth has a molten core, he anticipated modern geology. In embryology, he was a preformist, but he also proposed that organisms are the result of a combination of an infinite number of possible microstructures and their powers. In the life sciences and paleontology, he revealed an astonishing quick-change intuition, fueled by his study of comparative anatomy and fossils. One of his major works on this subject, Protogaea, previously unpublished in his lifetime, has recently been published in English for the first time. He elaborated a primary organismic theory.In medicine, he urged physicians of his time, with some results, to base his theories on detailed comparative observations and verified experiments, and to distinguish firmly between scientific and metaphysical points of view.

Psychology

Psychology had been a central interest of Leibniz's. He appears to be an 'underappreciated pioneer of psychology'. He wrote on topics now considered fields of psychology: attention, consciousness, memory, learning (association), motivation (the act of "effort"), emergent individuality, general developmental dynamics (developmental psychology).^{[citation needed]} His discussions in Nuevos Essays and Monadology often draw on everyday observations, such as the behavior of a dog or the noise of the sea, and develop intuitive analogies (the synchronous operation of clocks or the spring of balance of a clock). He also devised postulates and principles that apply to psychology: the continuum from unnoticed small perceptions to distinct self-conscious apperception, and psychophysical parallelism from the point of view of causality and purpose: &# 34;Souls act according to the laws of the end". causes, through aspirations, ends and means. Bodies act according to the laws of efficient causes, that is, the laws of motion. And these two kingdoms, that of efficient causes and that of final causes, harmonize with each other". This idea refers to the mind-body problem, stating that the mind and the brain do not act on each other, but they act side by side separately but in harmony. Leibniz, however, did not use the term psychology. Leibniz's epistemological position, against John Locke and English empiricism (sensualism), became clear: "Nihil est in intellectu quod non fuerit in sensu, nisi intellectu ipse". – "There is nothing in the intellect that was not first in the senses, except the intellect itself". Principles that are not present in sensory impressions can be recognized in human perception and consciousness: logical inferences, categories of thought, the principle of causality and the principle of purpose (teleology).

Leibniz found his most important interpreter in Wilhelm Wundt, founder of psychology as a discipline. Wundt used the quote "... nisi intellectu ipse" of 1862 on the title page of his Beiträge zur Theorie der Sinneswahrnehmung (Contributions on the Theory of Sense Perception) and published a detailed and aspiring monograph on Leibniz. Wundt shaped the term apperception, introduced by Leibniz, in a psychology of apperception based on experimental psychology that included neuropsychological modeling, an excellent example of how a concept created by a great philosopher could stimulate a program of psychological research. A principle in Leibniz's thought played a fundamental role: "the principle of equality of separate but corresponding points of view". Wundt characterized this style of thought (perspectivism) in a way that also applied to him: views that "complement each other, at the same time that they can appear as opposites that are only resolved when considered more deeply" 34;. Much of Leibniz's work had a major impact on the field of psychology. Leibniz thought that there are many small perceptions that we perceive but are not aware of. He believed that because of the principle that phenomena found in nature are continuous by default, the transition between conscious and unconscious states was likely to have intermediate steps. For this to be true, there must also be a part of the mind that we are unaware of at any given time.His theory of consciousness in relation to the principle of continuity can be seen as an early theory of sleep stages.. In this way, Leibniz's theory of perception can be seen as one of many theories leading to the idea of the unconscious. Leibniz was a direct influence on Ernst Platner, who is credited with originally coining the term Unbewußtseyn (unconscious). Furthermore, the idea of subliminal stimuli can be traced back to his theory of small perceptions. Leibniz's ideas on the Music and tonal perception influenced Wilhelm Wundt's laboratory studies.

Math

Although the mathematical notion of function was implicit in trigonometry and logarithmic tables, which already existed in his time, Leibniz was the first, in 1692 and 1694, to use them explicitly to denote any of the various geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord and perpendicular. Leibniz was the first to propose the use of the point as a multiplier in mathematical notation instead of the letter x (x) used in England for it. The letter x (x) has been used ever since as a variable name, especially for the three-dimensional calculation XYZ. transform:lowercase">XVIII, the concept of "function" lost these merely geometric associations.

Leibniz was the first to see that the coefficients of a system of linear equations could be arranged in an array, now known as a matrix, which could be manipulated to find the solution of the system, if any. This method was later known as "Gaussian elimination." Leibniz also made contributions in the field of Boolean algebra and symbolic logic.

Infinitesimal calculus

Statues of Isaac Newton and Leibniz at the Museum of Natural History of the University of Oxford

The invention of the calculus is attributed to Leibniz and Newton. According to Leibniz's notebooks, a pivotal event took place on November 11, 1675. That day he used integral calculus for the first time to find the area under the curve of a function y = f (x) .

Leibniz introduced several notations used today, such as, for example, the «integral» sign ∫, which represents an elongated S, derived from the Latin summa, and the letter «d» for refer to "differentials", from the Latin differentia. This ingenious and suggestive notation for calculus is probably his most enduring mathematical legacy. Currently, the calculus notation created by Leibniz is used, not Newton's.

Leibniz did not publish anything about his calculus until 1684. The product rule of differential calculus is still called "Leibniz's rule for the derivation of a product". Furthermore, the theorem that tells when and how to differentiate under the integral symbol is called the "Leibniz rule for the derivation of an integral".

From 1711 until his death, Leibniz's life was poisoned by a long dispute with John Keill, Newton, and others over whether he had invented calculus independently of Newton, or merely invented another notation for Newton's ideas. Leibniz then spent the rest of his life trying to prove that he had not plagiarized Newton's ideas.

Geometry

Leibniz's formula for π/4 states that:

1− − 13+15− − 17+19− − =π π 4{displaystyle 1,-,{frac {1}{3}{3},+,{frac {1}{1}{5}},-,{frac {1}{7}}}}}{,{frac {1}{1}{9}}}}{,-cdots ,=,{,{frac {, {,{c {c {c}{, {c {c}{, {, {c {c}{pos(p}{c}{, {, {c {c}{pos(p}{,}{pos(p}{,}{c {c}{,}{,}{,}{,}{,}{,}{1}{,}{,}{1}{,}{,}{,}{1}{,}{,}{,}{

Leibniz wrote that circles "can be expressed in the simplest way by this series, that is, the addition of fractions alternately added and subtracted". However, this The formula is only accurate with a large number of terms, using 10,000,000 terms to get the correct value of π/4 to 8 decimal places. Leibniz attempted to create a definition for a straight line by trying to prove the Parallel Postulate. If Although most mathematicians defined a straight line as the shortest line between two points, Leibniz believed that this was simply a property of a straight line rather than the definition.

Topology

Leibniz also published the idea of the science now called Topology, which is concerned with the properties of space that are preserved under continuous deformations, which he called "geometry of position" (Geometria Situs) and "position analysis" (Analysis Situus). Leibniz was the first to use the term analysis situs, which would later be used in the 19th century to refer to what is known as topology.

Eponymy

In addition to the different mathematical concepts that bear his name, one must:

• Lunar crater Leibnitz bears this name in his memory.
• The asteroid (5149) Leibniz also commemorates its name.