Gottfried Wilhelm Leibniz was a prominent German philosopher and mathematician. Though Leibniz was a polymath who contributed many works to many different fields, he is best known for his contributions to math, in which he invented differential and integral calculus independently of Sir Isaac Newton. In philosophy, Leibniz is known for his contributions on a wide range of subjects, including “optimism”—the idea that the current world is the best of all possible worlds, and was created by a freely thinking God who chose this for a good reason.

### Fast Facts: Gottfried Wilhelm Leibniz

**Known For:**Philosopher and mathematician known for a number of important contributions to mathematics and philosophy, such as the modern binary system, a widely used calculus notation, and the idea that everything exists for a reason.**Born:**July 1, 1646 in Leipzig, Germany**Died:**November 14, 1716 in Hanover, Germany**Parents:**Friedrich Leibniz and Catharina Schmuck**Education:**Leipzig University, University of Altdorf, University of Jena

### Early Life and Career

Gottfried Wilhelm Leibniz was born in Leipzig, Germany on July 1, 1646 to Friedrich Leibniz, a professor of moral philosophy, and Catharina Schmuck, whose father was a law professor. Though Leibniz attended elementary school, he was mostly self-taught from the books in his father’s library (who had died in 1652 when Leibniz was six). While young, Leibniz immersed himself in history, poetry, math, and other subjects, gaining knowledge in many different fields.

In 1661, Leibniz, who was 14, began studying law at the University of Leipzig and was exposed to the works of thinkers such as René Descartes, Galileo, and Francis Bacon. While there, Leibniz also attended summer school at the University of Jena, where he studied mathematics.

In 1666, he finished his law studies and applied to become a doctorate student in law at Leipzig. Because of his young age, however, he was refused the degree. This caused Leibniz to leave the University of Leipzig and earn the degree the following year at the University of Altdorf, whose faculty were so impressed with Leibniz that they invited him to become a professor despite his youth. Leibniz, however, declined and opted instead to pursue a career in public service.

#### Leibniz’s Tenure in Frankfurt and Mainz, 1667-1672

In 1667, Leibniz entered the service of the Elector of Mainz, who tasked him to help revise the *Corpus Juris*—or body of laws—of the electorate.

During this time, Leibniz also worked to reconcile Catholic and Protestant parties and encouraged Christian European countries to work together to conquer non-Christian lands, instead of waging war on each other. For example, if France left Germany alone, then Germany could help France in conquering Egypt. Leibniz’s action was inspired by France’s king Louis XIV, who seized some German towns in Alsace-Lorraine in 1670. (This “Egyptian Plan” would be ultimately passed on, although Napoleon unwittingly used a similar plan over a century later.)

#### Paris, 1672-1676

In 1672, Leibniz went to Paris to discuss these ideas more, staying there until 1676. While at Paris, he met a number of mathematicians like Christiaan Huygens, who made many discoveries in physics, mathematics, astronomy, and horology. Leibniz’s interest in mathematics has been credited to this period of travel. He quickly advanced in the subject, figuring out the core of some of his ideas on calculus, physics, and philosophy. Indeed, in 1675 Leibniz figured out the foundations of integral and differential calculus independently from Sir Isaac Newton.

In 1673, Leibniz also made a diplomatic trip to London, where he showed a calculating machine that he had developed called the Stepped Reckoner, which could add, subtract, multiply, and divide. In London, he also became a fellow of the Royal Society, an honor awarded to individuals who have made substantial contributions to science or math.

#### Hanover, 1676-1716

In 1676, upon the death of the Elector of Mainz, Leibniz moved to Hanover, Germany, and was placed in charge of the library of the Elector of Hanover. It Hanover—the place that would serve as his residence for the rest of his life—Leibniz wore many hats. For instance, he served as a mining engineer, an advisor, and a diplomat. As a diplomat, he continued to push for the reconciliation of the Catholic and Lutheran churches in Germany by writing papers that would resolve the views of both Protestants and Catholics.

The last part of Leibniz’s life was plagued by controversy—with the most notable being in 1708, when Leibniz was accused of plagiarizing Newton’s calculus despite having developed the math independently.

Leibniz died in Hanover on November 14, 1716. He was 70 years old. Leibniz never married, and his funeral was only attended by his personal secretary.

### Legacy

Leibniz was considered a great polymath and he made many important contributions to philosophy, physics, law, politics, theology, math, psychology, and other fields. He may be most well known, however, for some of his contributions to math and philosophy.

When Leibniz died, he had written between 200,000 to 300,000 pages and more than 15,000 letters of correspondence to other intellectuals and important politicians—including many notable scientists and philosophers, two German emperors, and Tsar Peter the Great.

### Contributions to Math

#### Modern Binary System

Leibniz invented the modern binary system, which uses the symbols 0 and 1 to represent numbers and logical statements. The modern binary system is integral to the functioning and operation of computers, even though Leibniz discovered this system a few centuries prior to the invention of the first modern computer.

It should be noted, however, that Leibniz did not discover binary numbers themselves. Binary numbers were already used, for example, by the ancient Chinese, whose use of binary numbers was acknowledged in Leibniz’s paper that introduced his binary system (“Explanation of Binary Arithmetic,” which was published in 1703).

#### Calculus

Leibniz developed a complete theory of integral and differential calculus independently of Newton, and was the first one to publish on the subject (1684 as opposed to Newton’s 1693), though both thinkers seem to have developed their ideas at the same time. When the Royal Society of London, whose president at the time was Newton, decided who developed calculus first, they gave credit for the *discovery* of calculus to Newton, while credit for the publication on calculus went to Leibniz. Leibniz was also accused of plagiarizing Newton’s calculus, which left a permanent negative mark on his career.

Leibniz’s calculus differed from Newton’s mainly in notation. Interestingly, many students of calculus today have come to prefer Leibniz’s notation. For example, many students today use “dy/dx” to indicate a derivative of y with respect to x, and an “S”-like symbol to indicate an integral. Newton, on the other hand, placed a dot over a variable, like ẏ, to indicate a derivative of y with respect to s, and did not have a consistent notation for integration.

#### Matrices

Leibniz also rediscovered a method of arranging linear equations into arrays or matrices, which makes manipulating those equations much easier. A similar method had first been discovered by Chinese mathematicians years earlier, but had fallen into abandonment.

### Contributions to Philosophy

#### Monads and Philosophy of Mind

In the 17^{th }century, René Descartes put forward the notion of dualism, in which the non-physical mind was separate from the physical body. This sparked the question of how exactly the mind and body are related to one another. In response, some philosophers said that the mind could only be explained in terms of physical matter. Leibniz, on the other hand, believed that the world is made of “monads,” which are not made of matter. Each monad, in turn, has its own individual identity, as well as its own properties that determine how they are perceived.

The monads, furthermore, are arranged by God—who is also a monad—to be together in perfect harmony. This laid down Leibniz’s views on optimism.

#### Optimism

Leibniz’s most famous contribution to philosophy may be “optimism,” the idea that the world we live in—which encompasses everything that exists and has existed—is the “best of all possible worlds.” The idea is based on the assumption that God is a good and rational being, and has considered many other worlds in addition to this one before choosing this one to come into existence. Leibniz explained evil by stating that it may result in a greater good, even if an individual experiences negative consequences. He further believed that everything existed for a reason. And humans, with their limited viewpoint, cannot see the greater good from their restricted vantage point.

Leibniz’s ideas were popularized by the French writer Voltaire, who did not agree with Leibniz that humans are living in the “best of all possible worlds.” Voltaire’s satirical book *Candide* ridicules this notion by introducing the character Pangloss, who believes that everything is for the best despite all of the negative things going on in the world.

### Sources

- Garber, Daniel. “Leibniz, Gottfried Wilhelm (1646–1716).”
*Routledge Encyclopedia of Philosophy*, Routledge, www.rep.routledge.com/articles/biographical/leibniz-gottfried-wilhelm-1646-1716/v-1. - Jolley, Nicholas, editor.
*The Cambridge Companion to Leibniz*. Cambridge University Press, 1995. - Mastin, Luke. “17th Century Mathematics - Leibniz.”
*The Story of Mathematics*, Storyofmathematics.com, 2010, www.storyofmathematics.com/17th_leibniz.html. - Tietz, Sarah. “Leibniz, Gottfried Wilhelm.”
*ELS*, Oct. 2013.