Gottfried Wilhelm Leibniz (1646 - 1716), the eminent philosopher, physicist, and mathematician, played a crucial role in advancing modern science (Guzmán Martínez, 2018). His legacy endures as one of the pillars of the rationalist tradition of modernity. According to Guzmán Martínez (2018), Leibniz, with his profound knowledge in mathematics and physics, not only explored the mysteries of nature but also the intricate phenomena affecting humanity.
Biography
Gottfried Wilhelm Leibniz, born on July 1, 1646, in Leipzig, Germany, grew up in a devout Lutheran family during the final years of the Thirty Years' War, which had left the country in ruins (Guzmán Martínez, 2018). His early education at the Nicolai School was complemented by self-directed learning in his father's personal library, inherited from a professor of moral philosophy at the University of Leipzig. According to Guzmán Martínez (2018), by the age of 12, he was already proficient in Latin and studying Greek.
In Easter 1661, he enrolled at the University of Leipzig as a law student, where he immersed himself in the thought of revolutionary scientists and philosophers like Galileo, Francis Bacon, Thomas Hobbes, and René Descartes (Belaval & Look, 2024). He aspired to reconcile these modern thinkers with the Aristotelian scholastics. His bachelor's thesis, "De Principio Individui," published in May 1663, was partly inspired by Lutheran nominalism (the theory that universals have no reality but are mere names) and emphasized the existential value of the individual, which should not be explained solely by matter or form but by its entire being. According to Belaval & Look (2024), this concept laid the foundation for his future theory of the "monad."
In 1666, he wrote "De Arte Combinatoria," in which he formulated a model that is the theoretical ancestor of some modern computers: all reasoning, any discovery, whether verbal or not, is reducible to an ordered combination of elements, such as numbers, words, sounds, or colors (Belaval & Look, 2024). After completing his legal studies in 1666, he applied for a doctorate in law but was rejected due to his age and subsequently left his hometown forever. According to Belaval & Look (2024), in Altdorf, the university town of the free city of Nuremberg, his dissertation "De Casibus Perplexis" immediately earned him the title of doctor as well as an immediate offer of a professorship, which he declined.
During his stay in Nuremberg, he met Johann Christian Freiherr von Boyneburg, one of the most distinguished German statesmen of the time (Belaval & Look, 2024). Boyneburg took him into his service and introduced him to the court of the Elector Prince, Archbishop of Mainz, Johann Philipp von Schönborn, where he dealt with legal and political matters. King Louis XIV of France posed an increasing threat to the Holy Roman Empire. To divert the king's interests elsewhere, the archbishop hoped to propose an expedition to Egypt using religion as a pretext and expressing the hope that the project would promote church reunion. According to Belaval & Look (2024), Leibniz, in view of this reunion, worked on the Demonstrationes Catholicae.
His research led him to locate the soul in a point, a new step towards the monad, and to develop the principle of sufficient reason (nothing exists or occurs without a reason) (Belaval & Look, 2024). His meditations on the difficult theory of the point were related to problems encountered in optics, space, and motion; they were published in 1671 under the general title "Hypothesis Physica Nova." According to Belaval & Look (2024), he asserted that motion depends, as in Johannes Kepler's theory, on the action of a spirit (God).
In 1672, the Elector Prince sent the young jurist on a mission to Paris, where he arrived in late March (Belaval & Look, 2024). In September, he met with Antoine Arnauld, a Jansenist theologian known for his writings against the Jesuits (Jansenism was an unorthodox Roman Catholic movement that engendered a rigorous form of morality). He sought Arnauld's help for the church reunion. Soon he was left without protectors due to the deaths of Freiherr von Boyneburg in December 1672 and the Elector Prince in February 1673; however, he was now free to continue his scientific studies. In search of financial support, he built a calculating machine and presented it to the Royal Society during his first trip to London in 1673. By the end of 1675, he had laid the foundations of integral and differential calculus. Belaval & Look (2024) mention that with this discovery, he ceased to consider time and space as substances, another step closer to monadology.
In his exploration of the concepts of extension and motion, he concluded that they contained an imaginary element (Belaval & Look, 2024). Although the laws of motion could not be discovered by studying their nature, he argued that extension and motion could provide a means to explain and predict phenomena. Contrary to Descartes, he posited that this world could be a well-ordered dream. If visible motion depended on the imaginary element present in the concept of extension, it could no longer be defined solely by local motion; it had to be the result of a force. By criticizing the Cartesian formulation of the laws of motion, he became the founder of a new formulation: dynamics, which replaced kinetic energy with the conservation of motion. Additionally, according to Belaval & Look (2024), he believed that light followed the path of least resistance and could demonstrate the order of nature towards an objective or final cause.
Despite not having a revenue-generating position, in October 1676, he accepted employment with Duke John Frederick of Brunswick-Lüneburg, who had converted from Lutheranism to Catholicism (Belaval & Look, 2024). Appointed librarian from February 1677, he applied for the position of councilor, which was finally granted to him in 1678. In his quest for utility, he proposed that education become more practical and advocated for the creation of academies. Additionally, he worked on various mechanical devices, such as hydraulic presses, windmills, lamps, submarines, and clocks. According to Belaval & Look (2024), he devised a windmill-driven water pump, improving the exploitation of mines in the Harz Mountains, where he frequently worked as an engineer from 1680 to 1685.
He is considered one of the creators of geology based on his observations, including the hypothesis that the Earth initially melted (Belaval & Look, 2024). In March 1679, he perfected the binary numbering system and proposed the basis for analysis situs, now known as general topology. At the same time, he worked on his dynamics and philosophy, which became increasingly anti-Cartesian. Upon the death of John Frederick on January 7, 1680, his brother Ernest Augustus I succeeded him. At that time, France became increasingly intolerant, with harsh persecutions of Protestants between 1680 and 1682, paving the way for the revocation of the Edict of Nantes on October 18, 1685. Additionally, according to Belaval & Look (2024), threats on the borders increased, as in 1681, despite prevailing peace, Louis XIV took Strasbourg and claimed 10 cities in Alsace.
He played a crucial role as a patriot, serving both his prince and the empire (Belaval & Look, 2024). He suggested means to his prince to increase flax production and proposed a process for desalinating water. Additionally, he recommended the classification of archives and wrote a critical pamphlet against Louis XIV in both French and Latin. During this period, he continued to refine his metaphysical system, investigating the notion of a universal cause of all being. His goal was to reach a starting point that would reduce reasoning to an algebra of thought. In the mathematical field, he explored the ratio between a circle and a circumscribed square in 1681 and analyzed the resistance of solids in 1684. According to Belaval & Look (2024), in 1686, he published "Nova Methodus pro Maximis et Minimis," an exposition of his differential calculus.
His notable "Meditationes de Cognitione, Veritate et Ideis" defined his theory of knowledge (Belaval & Look, 2024). He argued that the ideas of God and humans are analogously related and that there is an identity between divine and human logic. Similarly, according to Belaval & Look (2024), he criticized the Cartesian version of the ontological argument for the existence of God and presented his own version.
In February 1686, he wrote the "Discours de Métaphysique," where he formulated his principle of the identity of indiscernibles (Belaval & Look, 2024). In the March publication of Acta, he revealed his dynamics in a piece titled "Brevis Demonstratio Erroris Memorabilis Cartesii et Aliorum Circa Legem Naturae." Additionally, in an unpublished text written in 1686, he generalized propositions stating that in every true proposition, whether necessary or contingent, the predicate is contained in the notion of the subject. This idea seemed to imply determinism that could undermine human freedom, just like his conception of monads, the individual soul-like substances that compose the universe, as they in some sense "contain" all their pasts and futures. According to Belaval & Look (2024), his proposed solution was to argue that although each monad already contains all its future actions, God can create those actions as "free."
In 1685, he was appointed historian of the House of Brunswick, and on this occasion, he was granted the title of Hofrat (Belaval & Look, 2024). His work was to demonstrate, through genealogy, that the princely house had its origins in the House of Este, an Italian princely family, which would allow Hanover to claim a ninth electorate. In search of these documents, he began to travel in November 1687, spending two years in Italy. His main stops were Rome, Florence, Modena, and Venice. On June 3, 1691, he arrived in Vienna, where he tried in vain to obtain a position from the emperor. Upon his return to Hanover, he was not well received. Meanwhile, Leibniz's calculus was attacked by Michel Rolle in 1691 and Jacob Bernoulli in 1692. According to Belaval & Look (2024), in his reply, he generalized the notion of the differential by introducing the difference as a constant ratio.
In 1694, in his "Specimen Dynamicum," he completed his system of dynamics, which he compared to the theory of the conservation of force and Newton's theory of gravitational attraction (Belaval & Look, 2024). Furthermore, in 1695, he published "Système Nouveau de la Nature et de la Communication des Substances." In his "Système Nouveau," he developed the concept of the monad, with a hierarchy of levels, from the basic level of perception (subconscious perception) to the higher levels of apperception (conscious perception) (Belaval & Look, 2024). According to Belaval & Look (2024), his universal harmony principle explained how every monad mirrors the entire universe from its unique perspective. Consequently, in 1698, he published his "Tentamen Anagogicum" in the Acta Eruditorum, which summarized his cosmological, physical, and metaphysical theories.
During this period, he met Sophia Charlotte, Queen of Prussia, who had just married Frederick I, King of Prussia. From then on, he became one of her confidants (Belaval & Look, 2024). He helped her set up the Berlin Academy of Sciences and was its first president. However, the Hanoverian court was not happy with this situation and criticized his correspondence with the Princess Palatine, his investigation of German law, and his discussions on the reunification of the Church. Leibniz remained faithful to Sophia Charlotte, who was fascinated by philosophy and whom he made read the "Theodicy." She died suddenly in 1705, a blow from which he never fully recovered. Meanwhile, the controversy over the invention of calculus escalated into a dispute between Germany and England. In 1713, the Royal Society took a stand for Newton and declared Leibniz guilty of plagiarism. Leibniz vehemently denied this accusation (Belaval & Look, 2024).
His health began to fail in 1714, but he continued to work on his philosophical and scientific projects until his death on November 14, 1716 (Belaval & Look, 2024). Leibniz's passing marked the end of a prolific and varied career that spanned multiple disciplines, leaving a lasting legacy in mathematics, philosophy, and science.
Contributions to Philosophy and Science
Infinitesimal Calculus
Gottfried Wilhelm Leibniz, along with Isaac Newton, is recognized as one of the founders of calculus (Guzmán Martínez, 2018). It is reported that the first use of integral calculus appears in his notebooks from 1675. Leibniz used this calculus to find the area under the function y = x. Additionally, he introduced important notations in the field of calculus, such as the integral sign, an elongated "S" from the Latin "sum," and the "d" from the Latin "differentia," used for differential calculations. These contributions gave rise to Leibniz's Rule, specifically the product rule in differential calculus. He also contributed to the definition of mathematical entities known as "infinitesimals" and defined their algebraic properties, although these definitions presented many paradoxes at the time. However, according to Guzmán Martínez (2018), these definitions were reviewed and reformulated from the 19th century onwards, with the development of modern calculus.
Foundations for Epistemological and Modal Logic
Gottfried Wilhelm Leibniz, true to his mathematical training, defended the idea that the complexity of human reasoning could be translated into the language of calculations (Guzmán Martínez, 2018). Once understood, these calculations could be the solution to resolving differences of opinion and arguments. For this reason, he is recognized as the most significant logician of his time, at least since Aristotle. Among other contributions, he described the properties and method of linguistic resources such as conjunction, disjunction, negation, set, inclusion, identity, and the empty set, all of which are useful for understanding and making valid arguments and distinguishing them from invalid ones. According to Guzmán Martínez (2018), these contributions constitute one of the main foundations for the development of epistemic logic and also modal logic.
Philosophy of Mind and Body
In 1695, he published his perspective on the connection between mind and body (Cartwright, 2024). He called his theory the "new system of pre-established harmony." He argued that, although there is no physical connection between the mind and the body, the physical world was created by God in such a way that there is always a link between mental and physical events (experiences), as the mind can dictate an action of the body and vice versa. For Leibniz, "monads," a term derived from the Greek word for "unit," are the ultimate entities of reality. These simple substances combine to produce the world. Monads are the basic, individually isolated elements of reality that are unaffected by each other and are indivisible. According to Cartwright (2024), each monad is complete in itself and contains all the traces of the things that will happen to it.
This was a bold and novel theory, but one that proved difficult to fully explain (Cartwright, 2024). Like many other philosophers, Gottfried Wilhelm Leibniz tried to clarify his theory using analogies. He suggested that the body and mind are like two clocks that work independently but are also synchronized. There is a kind of pre-established harmony between these two "clocks." The cause that built the clocks/musicians/monads is God, and He knows everything that is planned for each monad. God is, in fact, the supreme monad and the only one that has a relationship with other monads; otherwise, monads never interconnect with each other. According to Cartwright (2024), this is a fundamental cause upon which all other thoughts of Leibniz's philosophy are built.
The Principle of Individuation
In his thesis "Disputatio Metaphysica de Principio Individui," which he wrote in the 1660s, he defends the existence of an individual value that constitutes a whole in itself but can be differentiated from the set (Guzmán Martínez, 2018). This thesis represents the first approach to the German theory of monads. By analogy with physics, he argued that monads in the mental realm are what atoms are in the physical realm. They are considered the ultimate elements of the universe and what gives substantial form to being. Among their properties are: they are eternal, do not decompose into simpler particles, are divisible, active, and subject to their own laws. Additionally, according to Guzmán Martínez (2018), they are independent of each other and function as an individual representation of the universe in itself.
Referencias
Belaval, Y., & Look, B. C. (2024). Gottfried Wilhelm Leibniz. En Encyclopedia Britannica. https://www.britannica.com/biography/Gottfried-Wilhelm-Leibniz
Cartwright, M. (2024). Gottfried Wilhelm Leibniz. World History Encyclopedia. https://www.worldhistory.org/Gottfried_Wilhelm_Leibniz/
Guzmán Martínez, G. (2018, octubre 22). Gottfried Leibniz: Biografía de Este Filósofo y Matemático. Psicología y Mente. https://psicologiaymente.com/biografias/gottfried-leibniz
Look, B.C. (2020). Gottfried Wilhelm Leibniz. En E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2020). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/spr2020/entries/leibniz/
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