Gottfried Wilhelm Leibniz

Gottfried Leibniz (1646-1716) was a German mathematician and philosopher. As a mathematician, his most famous contributions were the creation of the modern binary system and differential and integral calculus.

As a philosopher, he was one of the great rationalists of the 17th century, along with Descartes and Spinoza, and is renowned for his metaphysical optimism.Leibniz

Denis Diderot, who disagreed with Leibniz on several points, commented:”Perhaps there has not been a man who has read, studied, meditated and written as much as Leibniz…. What he has composed about the world, God, nature and soul is of the most sublime eloquence.

More than a century later, Gottlob Frege expressed a similar admiration, declaring that”in his writings Leibniz showed such a profusion of ideas that in this respect, he is virtually a class of his own.

Unlike many of his contemporaries, Leibniz does not have a single job that allows him to understand his philosophy. Instead, to understand his philosophy, it is necessary to take into account several of his books, correspondence and essays.


Gottfried Wilhelm Leibniz was born on July 1, 1646 in Leipzig. It was born in the Thirty Years’ War, just two years before this conflict ended.

Gottfried’s father was Federico Leibniz, who was a professor of moral philosophy at the University of Leipzig, as well as a lawyer. The mother was the daughter of a law professor and was named Catherina Schmuck.

A movie, an interview with Maria Rosa Antognazza, author of Leibniz: An Intellectual Biography

Of all the thinkers of the century of genius that inaugurated modern philosophy, none lived an intellectual life more rich and varied than Gottfried Wilhelm Leibniz (1646–1716).


Gottfried’s father died when he was still a child; he was only six years old. From that moment on, both his mother and uncle took care of his education.

His father had a large personal library, so Gottfried was able to access it from the early age of seven and devote himself to his own education.

The texts that interested him most in the beginning were those related to the so-called Fathers of the Church, as well as those related to ancient history.

It is said that he had a great intellectual capacity, since at the young age of 12 he was already fluent in Latin and was in the process of learning Greek. When he was only 14 years old, in 1661, he enrolled at the University of Leipzig with a law degree.

At the age of 20 Gottfried completed his studies and was already a professional specialized in philosophy and scholastic logic, as well as in the classical field of law.

Teaching motivation

In 1666 Leibniz prepared and presented his habilitation thesis at the same time as his first publication. In this context, the University of Leipzig denied him the possibility of teaching at the university.

Leibniz then handed this thesis over to another university, Altdorf University, from which he earned a doctorate in just 5 months.

Later, this university offered him the possibility of teaching, but Leibniz rejected this proposal and instead dedicated his working life to serving two German families that were very important to the society of the time.

These families were the Schönborn, between 1666 and 1674, and the Hannover, between 1676 and 1716.

First jobs

Leibniz got his first work experience from an alchemist’s job in Nuremberg.

At that time he contacted Johann Christian von Boineburg, who had worked with Juan Felipe von Schönborn, who was the electoral archbishop of Mainz, Germany.

Boineburg initially hired Leibniz as his assistant. Later he introduced him to Schönborn, with whom Leibniz wanted to work.

In order to get Schönborn’s approval and for him to offer him a job, Leibniz prepared a writing dedicated to this character.

Eventually this action brought good results, as Schönborn contacted Leibniz with the intention of hiring him to rewrite the legal code for his electorate. In 1669 Leibniz was appointed an adviser to the Court of Appeal.

The importance that Schönborn had in Leibniz’s life was that it was thanks to him that he became known in the social sphere in which he developed.

Diplomatic Actions

One of the actions Leibniz carried out in the service of Schönborn was to write an essay in which he presented a series of arguments in favour of the German candidate for the Polish Crown.

Leibniz had proposed to Schönborn a plan to revitalize and protect German-speaking countries after the devastating and opportunistic situation left by the Thirty Years’ War. Although the elector listened to this plan with reservations, Leibniz was later summoned to Paris to explain the details of the plan.

In the end, this plan was not carried out, but it was the beginning of a Parisian stay in Leibniz that lasted for years.


This stay in Paris allowed Leibniz to be in contact with several well-known personalities in the field of science and philosophy. For example, he had several conversations with the philosopher Antoine Arnauld, who was considered the most relevant of the moment.

He also had several meetings with the mathematician Ehrenfried Walther von Tschirnhaus, with whom he even developed a friendship. In addition, he met the mathematician and physicist Christiaan Huygens, and had access to the publications of Blaise Pascal and René Descartes.

It was Huygens who acted as a mentor in the next path Leibniz took, that of strengthening his knowledge. Having been in contact with all these specialists, he realized that he needed to expand his areas of knowledge.

Huygens’ help was partial, as the idea was for Leibniz to follow a self-education program. This program had excellent results, even discovering elements of great importance and transcendence, such as his investigations linked to the infinite series and his own version of differential calculus.


The reason why Leibniz was summoned to Paris did not take place (the implementation of the plan mentioned above), and Schönborn sent him and his nephew to London; the reason was a diplomatic action before the government of England.

In this context, Leibniz took the opportunity to interact with such illustrious figures as the English mathematician John Collins and the German-born philosopher and theologian Henry Oldenburg.

During these years he took the opportunity to present to the Royal Society an invention he had been developing since 1670. It was a tool through which it was possible to make calculations in the field of arithmetic.

This tool was called stepped reckoner and differed from other similar initiatives in that it could perform all four basic mathematical operations.

After witnessing the operation of this machine, the members of the Royal Society appointed him an external member.

After this achievement, Leibniz was preparing to carry out the mission for which he had been sent to London, when he learned that the elector Juan Felipe von Schönborn had died. This made him go directly to Paris.

Family Hannover

The death of John Philip von Schönborn meant that Leibniz had to find another occupation and, fortunately, in 1669, the Duke of Brunswick invited him to visit the House of Hanover.

At that time Leibniz refused this invitation, but his relationship with Brunkwick continued for several more years through an exchange of letters since 1671. Two years later, in 1673, the Duke offered Leibniz a position as secretary.

Leibniz arrived in Hanover at the end of 1676. Previously it was to London again, where he received new knowledge, and there is even information that states that at that time he saw some documents of Isaac Newton.

However, most historians state that this is not true and that Leibniz came to his conclusions independently of Newton.

Long-term service

Once at the House of Brunswick, Leibniz began working as a private justice counselor and served three of the house’s rulers. His work revolved around political advice, history and also as a librarian.

I also had the opportunity to write about the theological, historical and political issues related to this family.

While in service to the House of Brunswick, this family grew in popularity, respect, and influence. Although Leibniz was not very comfortable with the city as such, he did recognize that it was a great honor to be part of this dukedom.

For example, in 1692, the Duke of Brunswick was named hereditary elector of the Roman Germanic Empire, which was a great opportunity for promotion.


While Leibniz dedicated himself to providing his services to the House of Brunswick, they allowed him to develop his studies and inventions, which were in no way linked to obligations directly related to the family.

Then, in 1674 Leibniz began to develop the conception of calculus. Two years later, in 1676, he had already developed a system that was coherent and that saw the light of day in 1684.

1682 and 1692 were very important years for Leibniz, as his documents were published in the field of mathematics.

Family History

The Duke of Brunswick of that time, Ernest Augustus, proposed to Leibniz one of the most important and challenging tasks he had; to write the history of the House of Brunswick, beginning in the times linked to Charlemagne, and even before this time.

The duke’s intention was to make this publication favourable to him in the context of the dynastic motivations he possessed. As a result of this task, Leibniz traveled throughout Germany, Italy, and Austria between 1687 and 1690.

The writing of this book took several decades, which caused the discomfort of the members of the House of Brunswick. In fact, this work was never finished and two reasons for it are attributed:

First of all, Leibniz was characterized as a meticulous man and very dedicated to detailed research. Apparently, there were no really relevant and truthful data on the family, so it is estimated that the result would not have been to their liking.

Second, Leibniz at that time was dedicated to producing a lot of personal material, which prevented him from dedicating all the time he had at his disposal to the history of the House of Brunswick.

Many years later it became clear that Leibniz had indeed managed to compile and develop a good part of the task assigned to him.

In the 19th century, Leibniz’s writings were published in three volumes, even though the heads of the Brunswick House would have been comfortable with a much shorter and less rigorous book.

Dispute with Newton

During the first decade of the 1700s, the Scottish mathematician John Keill indicated that Leibniz had plagiarized Isaac Newton in relation to the conception of calculus. This accusation took place in an article written by Keill for the Royal Society.

This institution then carried out extremely detailed research on both scientists to determine who had been the author of this discovery. It was eventually determined that Newton was the first to discover calculus, but Leibniz was the first to publish his dissertations.

Final years

In 1714, George Louis of Hanover became King George I of Great Britain. Leibniz had a lot to do with this appointment, but George I was adverse and demanded that the show at least one volume of his family’s history, otherwise he would not be reunited with him.

In 1716 Gottfried Leibniz died in the city of Hanover. An important fact is that George I did not attend his funeral, which shows the separation between the two.

Main contributions In mathematics


Leibniz’s contributions in mathematics were several; the most known and controversial is infinitesimal calculus. Infinitesimal calculus, or simply calculus, is a part of modern mathematics that studies limits, derivatives, integrals and infinite series.

Both Newton and Leibniz presented their respective theories of calculus in such a short period of time that plagiarism was even mentioned.

Today both are considered co-authors of the calculation, however, Leibniz’s notation was eventually used for its versatility.

It was also Leibniz who gave the name to this study and who gave it the symbolism used today: ∫ and dy = y²/2.

Binary system

In 1679, Leibniz devised the modern binary system and presented it in his work Explication de l’Arithmétique Binaire in 1703. The Leibniz system uses the numbers 1 and 0 to represent all numerical combinations, unlike the decimal system.

Although his creation is often attributed to him, Leibniz himself admits that this discovery is due to an in-depth study and reinterpretation of an idea already known in other cultures, especially China.

Leibniz’s binary system would later become the basis of computing since it is the one that governs almost all modern computers.

Calculating machine

Leibniz was also an enthusiast in the creation of mechanical calculator machines, a project that was inspired by Pascal’s calculator.

The Stepped Reckoner, as he called it, was ready in 1672 and was the first to allow for addition, subtraction, multiplication, and division. In 1673 he introduced her to some of his colleagues at the French Academy of Sciences.

The Stepped Reckoner incorporated a stepped drum gear device, or”Leibniz wheel”. Although Leibniz’s machine was not practical due to technical failures, it laid the foundation for the first mechanical calculator to be marketed 150 years later.

Additional information about the Leibniz calculator machine is available from the Computer History Museum and the Encyclopædia Britannica.

In philosophy

It is difficult to encompass Leibniz’s philosophical work, since, although abundant, it is based mainly on diaries, letters, and manuscripts.

Continuity and sufficient reason

Two of the most important philosophical principles proposed by Leibniz are the continuity of nature and sufficient reason.

On the one hand, the continuity of nature is related to infinitesimal calculus: an infinite numeric, with infinitely large and infinitely small series, which follow a continuity and can be read from front to back and vice versa.

This reinforced in Leibniz the idea that nature follows the same principle and therefore”there are no jumps in nature”.

On the other hand, sufficient reason refers to the fact that’nothing happens without a reason’. In this principle it is necessary to take into account the subject/predicate relationship, i.e. A is A.


This concept is closely related to that of fullness or monads. In other words,’monad’ means that which is one, has no parts and is therefore indivisible.

They are about the existing fundamental things (Douglas Burnham, 2017). Monads are related to the idea of wholeness, because a full subject is the necessary explanation of everything it contains.

Leibniz explains God’s extraordinary actions by establishing Him as the whole concept, that is, as the original and infinite monad.

Metaphysical optimism

On the other hand, Leibniz is well known for his metaphysical optimism. “The best of all possible worlds” is the phrase that best captures his task of responding to the existence of evil.

According to Leibniz, of all the complex possibilities within God’s mind, it is our world that reflects the best possible combinations and to achieve this, there is a harmonious relationship between God, the soul, and the body.

In Topology

Leibniz was the first to use the term analysis situs, which was later used in the 19th century to refer to what is now known as topology.

In an informal way, it can be said that the topology is responsible for the properties of the figures that remain unchanged.

In Medicine

For Leibniz, medicine and morality were closely related. He considered medicine and the development of medical thought as the most important human art, after philosophical theology.

He was part of scientific geniuses who, like Pascal and Newton, used the experimental method and reasoning as the basis of modern science, which was also reinforced by the invention of instruments such as the microscope.

Leibniz supported medical empiricism; he thought of medicine as an important basis for his theory of knowledge and the philosophy of science.

He believed in the use of body secretions to diagnose a patient’s medical condition. His thoughts on animal experimentation and dissection for the study of medicine were clear.

He also made proposals for the organization of medical institutions, including ideas on public health.

In religion

His reference to God becomes clear and habitual in his writings. He conceived of God as an idea and as a real being, as the only necessary being, who creates the best of all worlds.

For Leibniz, since everything has a cause or reason, at the end of the investigation there is only one cause from which everything is derived. The origin, the point where everything begins, that”seized cause”, is for Leibniz the same God.

Leibniz was very critical of Luther and accused him of rejecting philosophy as the enemy of faith. In addition, he analyzed the role and importance of religion in society and its distortion by becoming only rites and formulas, which lead to a false conception of God as being unjust.


Leibniz wrote mainly in three languages: scholastic Latin (ca. 40 %), French (ca. 35 %) and German (less than 25 %).

The Theodicy was the only book he ever published. It was published in 1710 and its full name is Theodicy’s Essay on the Goodness of God, the Freedom of Man and the Origin of Evil.

Another of his works was published, although posthumously: New Essays on Human Understanding.

Apart from these two works, Lebniz wrote especially academic articles and pamphlets.


The Theodicy contains the main theses and arguments of what began to be known as”optimism” as early as the eighteenth century (…): a rationalist theory on the goodness of God and his wisdom, on divine and human freedom, on the nature of the created world and the origin and meaning of evil.

This theory is often summed up by the famous and often misunderstood Leibnizian thesis that this world, despite the evil and suffering it contains, is”the best of all possible worlds”. (Caro, 2012).

Theodicy is the Leibzinian rational study of God, with which it seeks to justify divine goodness by applying mathematical principles to Creation.


Leibniz acquired a great culture after reading the books in his father’s library. He had a great interest in the word, he was aware of the importance of language in the advances of knowledge and the intellectual development of man.

He was a prolific writer and published numerous pamphlets, including De jure suprematum, an important reflection on the nature of sovereignty.

On many occasions, he signed with pseudonyms and wrote about 15,000 letters sent to more than a thousand recipients. Many of them have the extension of an essay, more than letters were dealt with on different subjects of their interest.

He wrote a lot during his life but left many unpublished writings, so much so that his legacy is still being edited today. Leibniz’s complete work already exceeds 25 volumes, with an average of 870 pages per volume.

In addition to all his writings on philosophy and mathematics, he has medical, political, historical and linguistic writings.

Other contributions

He discovered infinitesimal calculus, independently of Newton, and its notation has been used ever since. He also discovered the binary system, the foundation of virtually all current computer architectures.

Together with René Descartes and Baruch Spinoza, he is one 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 information science.


Leibniz became well known in the learned world of England in the late seventeenth and early eighteenth centuries.

His residence in Paris brought him into contact with the great men of the court of Louis XIV, as well as with all the writers who distinguished themselves in the world of science or theology at that time.

It was, however, in his own country that he came to be recognized as a philosopher. The multiplicity of his interests and the variety of tasks he set himself were not conducive to the development of his philosophical doctrines.

It was thanks to the efforts of his follower Christian Wolff, who reduced his teachings to a more compact form, that he was able to exert the influence he achieved on the movement known as the German Enlightenment.

In fact, until Kant began to publicly expose his critical philosophy, Leibniz was the dominant mind in German philosophy.

His influence was, globally speaking, healthy. He did his part to stop the wave of materialism and helped preserve spiritual and aesthetic ideals until such time as they could be treated constructively, as they were by the greatest thinkers of the nineteenth century.

Movie Gottfried Wilhem Leibniz


This work allows us to know the life of this famous thinker of those times, where he had participation and dispute with other philosophers for his thoughts and ideas, but this did not discourage his principles and thoughts.

Thanks to his way of thinking we used much of his knowledge in various disciplines. We see that Liebniz’s contributions are many and that he has contributed in almost all disciplines but with greater strength in philosophy. This work is a compilation of Leibniz’s work and biography.

He invented infinitesimal calculus

The invention of infinitesimal calculus is attributed to both Leibniz and Isaac Newton.

According to Leibniz’s notebooks, a fundamental 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 Latin summa, and the letter d to refer to differentials, from Latin differentia.

This ingenious and suggestive notation for calculus is probably his most enduring mathematical legacy. Leibniz didn’t publish anything about his Calculus until 1684.

The differential calculus product rule is still called the Leibniz rule for the derivation of a product.

In addition, the theorem that tells when and how to differentiate under the integral symbol is called the Leibniz rule for the derivation of an integral.

Main contributions Gottfried Wilhelm Leibniz

  1. He discovered that every number can be expressed by a series of zeros and a few.
  2. It is due to him the diffusion of the point in the multiplication.
  3. Obtained series of the circular and hyperbolic tangent arc by calculating the elliptical and hyperbolic sectors developed in series.
  4. Worked with complex numbers, but never understood their nature.
  5. He offered several arguments to show that the logarithms of negative numbers do not exist.
  6. Discovered the inverse relationship between tangent drawing methods (differentiation) and quadratures (integration).
  7. He generalized the concept of a differential to the case of a negative and fractional exponent.
  8. Introduced the catenary equation.
  9. He solved first-order equations.
  10. Refined combinatorial symbolism with the help of the index system.
  11. He found a serial expression for.
  12. The first criterion for establishing the convergence of a series is owed to it.
  13. Obtained the formula of the multinomial coefficients but did not publish it.
  14. He is owed the expression”transcendent quantities”.
  15. Introduced the notation currently used in differential and integral calculus
  16. He used infinitely large numbers as if they were ordinary numbers.
  17. He used the term”imaginary” for complex numbers.
  18. He laid the first foundations of symbolic logic.
  19. He introduced combinatorics as a mathematical discipline.
  20. Generalized the binomial and multinomial theorem.
  21. First reference in the West to the determinants.
  22. He demonstrated Fermat’s”little theorem”.
  23. It is considered the initiator of geometric calculation and topology.

A great book I recommend is this.

Discourse on Metaphysics
  • Gottfried Wilhelm Leibniz Fre
  • Publisher: Createspace Independent Publishing Platform
  • Pasta blanda: 30 pages

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