Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz a German philosopher, mathematician, diplomat and universal scholar (1646-1716), who invented the calculus independently of Newton, devised a system of binary arithmetic, and constructed a comprehensive metaphysics of monads in which the actual world is the best of all possible worlds. He was one of the last thinkers to command the whole range of contemporary learning.

Life and Public Career

Gottfried Wilhelm Leibniz was born in Leipzig in 1646, the son of a professor of moral philosophy. He was a precocious student, reading widely in his father's library, and he took degrees in philosophy and in law. He entered the service of the elector of Mainz and travelled on diplomatic missions, including one to Paris, where he met the leading mathematicians and philosophers of the day.

In 1676 he became librarian and court counsellor to the duke of Hanover, a position he held for the rest of his life. He served the Hanoverian house as historian and diplomat, and his researches into its genealogy helped to support its eventual claim to the British throne. He also founded and presided over the Berlin Academy of Sciences.

His career combined scholarship with diplomacy and with practical invention. He designed calculating machines, proposed reforms of law and of language, and corresponded with hundreds of scholars across Europe. He died in Hanover in 1716, largely forgotten by the court he had served, and his funeral was attended by few.

The Calculus and Mathematics

Leibniz's greatest mathematical achievement was the invention of the differential and integral calculus, developed independently of Newton and published in 1684 and 1686. His notation, with the symbols dy and dx and the long s of the integral, proved more convenient than Newton's and is the one still used. A bitter priority dispute with the followers of Newton clouded his last years.

He also devised the binary system of numeration, using only the digits zero and one, which he believed to be of great philosophical significance and which is today fundamental to computing. He worked on determinants, on the geometry of curves, and on the foundations of logic, seeking a universal characteristic and a calculus of reasoning.

His dream of a language in which all concepts could be analysed into simple elements, and all reasoning reduced to calculation, anticipated developments in symbolic logic by more than a century. Although he did not complete this programme, his logical writings, largely unpublished in his lifetime, show a depth that later scholars have only gradually recovered.

Monads, Harmony and the Best World

The central concept of Leibniz's mature metaphysics is the monad, a simple, indivisible and immaterial substance that has no parts and cannot be acted on from outside. Monads are perceiving and appetitive beings, each mirroring the whole universe from its own point of view, and they are the ultimate constituents of reality. There are no material atoms, and bodies are appearances grounded in collections of monads.

Because monads have no windows through which anything could enter, they do not interact causally. Their apparent agreement is due to a pre-established harmony, arranged by God at creation, so that the states of one correspond to the states of others like two clocks running in perfect synchrony. This doctrine replaced the occasionalism of Malebranche and the interactionism of Descartes with an elegant alternative.

From the goodness of God, Leibniz argued, it follows that the world he chose to create is the best of all possible worlds, since a perfectly good and wise being would select the optimum among the infinite possibilities. The presence of evil is compatible with this, he held, because the whole is more perfect than any alternative whole, though the argument was mocked by Voltaire and gave optimism a bad name.

Logic, Knowledge and Contingency

Leibniz distinguished truths of reasoning, which are necessary and whose contrary implies a contradiction, from truths of fact, which are contingent and grounded in the principle of sufficient reason. Necessary truths rest on the law of identity and are knowable by analysis; contingent truths are known by experience and, in the case of future events, only by God, who sees them in the complete concept of each individual.

He held that every individual substance has a complete concept that includes all its properties and all its history, so that the past, present and future of an individual are contained in its notion. This doctrine of complete concepts grounded his account of free will and of the relation between God's foreknowledge and human freedom, which he tried to reconcile more fully than had his predecessors.

His epistemology allowed for innate ideas and dispositions, against the empiricism of Locke, though he argued that the mind is like a block of marble whose veins determine what can be carved from it. His dialogue New Essays on Human Understanding was a detailed response to Locke, published only after his death.

Influence and Legacy

Leibniz's philosophical influence was transmitted chiefly through his published works and through the wide correspondence that carried his ideas across Europe. The Theodicy gave the problem of evil a name and set the terms of debate for the century that followed, and his monadology became a central point of reference for German idealism and for later metaphysics.

His mathematical legacy is fundamental. The calculus he helped to create became the instrument of modern physics and engineering, and his notation remains in universal use. His binary arithmetic and his work on logic pointed toward the formal systems of the nineteenth and twentieth centuries, and his calculating machines were among the first steps toward automated computation.

He is remembered as a thinker of encyclopedic range who tried to reconcile the new science with reason and with faith, and who left his mark on mathematics, logic, law, history and diplomacy. The vision of a universal science and a rational language, together with the picture of a harmonious universe of perceiving monads, gives his thought a grandeur that still attracts and troubles its readers.

See also

• Monadology
• Calculus
• Pre-established harmony
• Rationalism