Antecedents of Mathematical Logic

A striking aspect of Leibniz's thought was the recurring notion of a universal symbolic language. In 1666 he published an article entitled Dissertatio de arte combinatoria, with subtitle "General Method in Which All Truths of the Reason Are Reduced to a Kind of Calculation." This early work establishes the theme of the gigantic project which was Leibniz's lifelong goal. The project involved bringing together all knowledge in a single compendium, with each division of the arts and sciences reduced to its primary propositions and related to other subjects in such a way that any portion or desired fact could be extracted at will, and from which the whole body of human knowledge could be reconstructed. It would provide a tool for learning without a teacher and would point up areas in which further investigation was needed.

The most remarkable feature of the plan was the lingua characteristica, a system of symbols representing logical ideas which would constitute a universal language of reasoning and would facilitate thought in the same way that mathematical symbols facilitate calculation. In the Chinese ideogram, which represents a concept rather than a sound, Leibniz saw a possible model for his "alphabet of thoughts."

Although he was unable to bring to fruition either his grand design for an encyclopedia of knowledge or the symbolic language into which it was to be translated, Leibniz's ideas were embodied in the mathematical logic developed by George Boole and Giuseppe Peano in the 19th century and by Alfred North Whitehead and Bertrand Russell in the 20th, and these ideas foreshadowed modern cybernetics and computer theory.

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Contribution to Philosophy

Leibniz voluminous notebooks indicate that during the years at Hanover Leibniz's thought was increasingly dominated by the development of a comprehensive cosmic philosophy. He composed no complete exposition of his philosophical theories, but to any of his correspondents who inquired about them he freely expounded phases of his "new system," and on three important occasions he took issue with exponents of differing views in extended polemical essays which brought out the essentials of his own philosophy.

In his Théodicé, written in reply to an attack upon his views in Pierre Bayle's Dictionnaire historique et critique (1699), Leibniz defines God as "infinite possibility" and the world (actuality) as "compossibility" in that it contains the greatest number of stimultaneous possibilities; it is therefore the best of all possible worlds. In defining "substance," he proceeds from the traditional postulate that all predicates are contained in their subjects, to the designation as substances of all words which can be used only as subjects.

In a criticism of John Locke's Essay on Human Understanding (1690) Leibniz refuted Locke's major premise that the senses are the source of all understanding by adding "except the understanding itself," distinguishing three levels of understanding: the self-conscious, the conscious, and the unconscious or subconscious. And in an essay known as the "Monadology," he more specifically defines the ultimate elements of the universe as individual precipient centers of possibility or force, which he calls "monads." Each unit perceives the universe from its own point of view and interprets what it perceives according to its own level of understanding, but there is no interaction or intercommunication among the units and therefore no operation of cause and effect.

In the famous exchange of letters (1715-1716) with Samuel Clarke, Leibniz describes space and time as merely systems of relationship or order, calling Newton's treatment of them as absolute entities a reversion to medieval notions.

Such ideas as these, characteristic of Leibniz's application of logic to the problems of metaphysics, found little response among the philosophers of his time, who were more receptive to the patterns of Locke's empiricism. But when Leibniz's Nouveaux essais sur l'entendement humain was finally published in 1765, Locke's influence was receding, and Leibniz's work became a major factor in the formation of the transcendental philosophy of Immanuel Kant.

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Biography

Gottfried Wilhelm von Leibniz (1646-1716) was a German mathematician and philosopher. Known as a statesman to the general public of his own times and as a mathematician to his scholarly contemporaries, he was subsequently thought of primarily as a philosopher.
Gottfried Wilhelm von Leibniz was born in Leipzig on June 23, 1646. His father, who was professor of moral philosophy at the University of Leipzig, died in the boy's sixth year. As a result, his early education was somewhat haphazard, but through his own industry he was ready for the university at the age of 15. He pursued the course in law in preparation for a political career and also studied theology, mathematics, and the new natural philosophy of the Enlightenment, receiving his bachelor's degree in 1663.
After 3 years of further study at Leipzig, Leibniz transferred to the University of Altdorf, where he received his doctorate in law in 1667. He declined the offer of a professorship there and accepted instead a position in the service of the elector of Mainz.

Early Travels
At this time Louis XIV's aggressive activities were a serious threat to the German states, and in a pamphlet published in 1670 Leibniz proposed a defensive coalition of the northern European Protestant countries. At the same time, to give the German principalities, recently weakened by the Thirty Years War, a respite for economic recovery, he conceived a plan whereby Louis might gain Holland's valuable possessions in Asia by way of a "holy war" against non-Christian Egypt. Leibniz was invited to Paris to present his plan; although it was not adopted, his 4-year stay in the French capital, with visits to London in 1673 and 1676, was crucial for his intellectual development.
Before coming to Paris, Leibniz had devised a calculating machine based on the principles of an earlier one invented by Blaise Pascal but capable of performing much more complicated mathematical operations. His demonstrations of this machine before the Académie Royale des Sciences and the Royal Society of London aroused much interest and led to fruitful relations with members of these groups and to his election to membership in the Royal Society shortly after his first London visit.
Especially important as a stimulus to Leibniz's interest in mathematics was his contact in Paris with the Dutch mathematician Christiaan Huygens, which resulted in Leibniz's developing both the integral and the differential calculus during the years of his residence there.

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