S2EP6: Exploring the study methods of the (other) inventor of calculus

Hello and welcome to the Autodidactic podcast, season 2 episode 6. This week I’ll be talking about a German autodidactic who one of the great thinkers of the seventeenth and eighteenth centuries and is known as the last “universal genius”. He made deep and important contributions to the fields of metaphysics, epistemology, logic, philosophy of religion, as well as mathematics, physics, geology, jurisprudence, and history. Who? It is Gottfried Wilhelm Leibniz (1 July 1646–14 Nov 1716) a prominent German polymath and one of the most important logicians, mathematicians and natural philosophers of the Enlightenment. A contemporary Denis Diderot an eighteenth-century French atheist and materialist said: “When one compares the talents one has with those of a Leibniz, one is tempted to throw away one’s books and go die quietly in the dark of some forgotten corner” As a representative of the seventeenth-century tradition of rationalism, Leibniz developed, as his most prominent accomplishment, the ideas of differential and integral calculus, independently of Isaac Newton’s contemporaneous developments. Mathematical works have consistently favoured Leibniz’s notation as the conventional expression of calculus. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal’s calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in a device called the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of nearly all digital (electronic, solid-state, discrete logic) computers, including the Von Neumann machine, which is the standard design paradigm, or “computer architecture”, followed from the second half of the 20th century, and into the 21st. Leibniz made major contributions to physics and technology, and anticipated notions that surfaced much later in philosophy, probability theory, biology, medicine, geology, psychology, linguistics, and computer science. He wrote works on philosophy, politics, law, ethics, theology, and history. He wrote in several languages, primarily in Latin, French and German but also in English, Italian and Dutch. Leibniz was born in Leipzig on July 1, 1646, two years prior to the end of the Thirty Years War, which had ravaged central Europe. His father, Friedrich Leibniz, was a jurist and professor of Moral Philosophy at the University of Leipzig, and his mother, Catharina Schmuck, the daughter of a professor of Law. Leibniz’s father died in 1652, and his subsequent education was directed by his mother, uncle, and according to his own reports, himself. He was given access to his father’s extensive library at a young age and proceeded to pore over its contents, particularly the volumes of ancient history and the Church Fathers. His father’s library enabled him to study a wide variety of advanced philosophical and theological works—ones that he would not have otherwise been able to read until his college years. Access to his father’s library, largely written in Latin, also led to his proficiency in the Latin language, which he achieved by the age of 12. While young, Leibniz immersed himself in history, poetry, maths, and other subjects, gaining knowledge in many different fields. At the age of seven, Leibniz entered the Nicolai School in Leipzig. Although he was taught Latin at the elementary school, Leibniz had taught himself far more advanced Latin and some Greek. As he progressed through school he was taught Aristotle’s logic and theory of categorising knowledge. Leibniz was clearly not satisfied with Aristotle’s system and began to develop his own ideas on how to improve on it. In later life Leibniz recalled that at this time he was trying to find orderings on logical truths which, although he did not know it at the time, were the ideas behind rigorous mathematical proofs. In 1661, at the age of fourteen, Leibniz entered the University of Leipzig. It may sound today as if this were a truly exceptionally early age for anyone to enter university, but it is fair to say that by the standards of the time he was quite young but there would be others of a similar age. He studied philosophy, which was well taught at the University of Leipzig, and mathematics which was very poorly taught. Among the other topics which were included in this two year general degree course were rhetoric, Latin, Greek and Hebrew. He graduated with a bachelors degree in 1663 Leibniz then went to Jena to spend the summer term of 1663. At Jena the professor of mathematics was Erhard Weigel but Weigel was also a philosopher and through him Leibniz began to understand the importance of the method of mathematical proof for subjects such as logic and philosophy. Weigel believed that number was the fundamental concept of the universe and his ideas were to have considerable influence of Leibniz. By October 1663 Leibniz was back in Leipzig starting his studies towards a doctorate in law. He was awarded his Master’s Degree in philosophy for a dissertation which combined aspects of philosophy and law studying relations in these subjects with mathematical ideas that he had learnt from Weigel. A few days after Leibniz presented his dissertation, his mother died. Despite his growing reputation and acknowledged scholarship, Leibniz was refused the doctorate in law at Leipzig. It is a little unclear why this happened, but Leibniz was not prepared to accept any delay and he went immediately to the University of Altdorf where he received a doctorate in law in February 1667. Leibniz dedicated his working life to serving two German families that were very important to the society of the time. He had proposed to 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. Having been in contact with all these specialists, he realized that he needed to expand his areas of knowledge. Leibniz decided 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 and Leibniz was sent to London for a diplomatic mission to the government of England. 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. After witnessing the operation of this machine, the members of the Royal Society appointed him an external member. While in London, he learned that the elector Juan Felipe von Schönborn for whom he worked had died, and Leibniz had to find another occupation. Leibniz began working as a private justice counsellor for the House of Brunswick. 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. 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. Today historians believe both men made the discoveries completely independently and both are credited with the invention of calculus. The last part of Leibniz’s life was plagued by the controversy. 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 towards him. 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. George I did not attend his funeral, which shows the separation between the two. So what can we take away from Leibniz? Well he had two periods of intense self-learning. The first as a child in his fathers library, and the second from 1672 in Paris. Leibniz studied mathematics and physics under Christiaan Huygens a Dutch scientist in 1672 and it was Huygens who encouraged to read and gave him an extensive reading list. This list included, works of many famous mathematicians such as Pascal and Descartes. In both of these periods he read extensively. Like many of my previous podcasts about famous autodidactics reading was a keystone for Leibniz as well. Ray Bradbury a writer I featured recently called the library his college. For Leibniz reading was a critical method for extending his knowledge. Leibniz didn’t just read in a single area of knowledge, he read extensively in all areas he was interested in. In this podcast let’s focus on becoming more prolific readers. If you’re listening to this then you’re probably already doing a lot of study and like most people you’re short of time. Using what we know about Leibniz and his two periods of intensive study let me propose some things to help become voracious readers. First, make a list of books to read. Leibniz started both