He studied geometry under the guidance of Christiaan Huygens and in 1676 Leibniz completed his discovery of Differential Calculus (independently of Sir Isaac Newton). Prior to this discovery of Calculus, Gottfried Leibniz wrote extensively on a number of subjects such as logic, truth, reason, philosophy and metaphysics which culminated in his Monadology (Philosophical Investigations, 1670).
Gottfried Leibniz's Monadology is largely correct, matter and universe are One. But we can now better understand his Monad as a Spherical Wave Motion of Space that determines the size of our finite spherical universe within an infinite Space, and thus interacts with ALL other matter within our universe.
After completing his philosophical and legal education at Leipzig and Altdorf, Gottfried Leibniz spent several years as a diplomat in France, England, and Holland, where he became acquainted with the leading intellectuals of the age. He then settled in Hanover, where he devoted most of his adult life to the development of a comprehensive scheme for human knowledge, comprising logic, mathematics, philosophy, theology, history, and jurisprudence. Although his own rationalism was founded upon an advanced understanding of logic, which Leibniz largely kept to himself, he did publish many less technical expositions of his results for the general public.
La Monadologie (Monadology) (1714) is a highly condensed outline of Leibniz's metaphsics. Complete individual substances, or monads, are dimensionless points which contain all of their properties—past, present, and future—and, indeed, the entire world. The true propositions that express their natures follow inexorably from the principles of contradiction and sufficient reason.
Along with Newton, Leibniz is also given credit to the presentation of “infinitesimal calculus”, there was friction between Newton and Gottfried Leibinz on the invention of calculus. Till today the product rule of differential calculus is still called “Leibniz’s law”.
He never married and is known to complain about money at times, his sister’s stepson inherited what he left behind after his death. Leibniz was fond of documenting all this mathematical pursuits and is known to have backdating and tweaking his journal too which fuels the debate centered on ‘Calculus and Newton’. He died in Hanover.
In philosophy, Leibniz is mostly noted for his optimism, e.g. his conclusion that our Universe is, in a restricted sense, the best possible one that God could have created. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th Century advocates of rationalism. The work of Leibniz also anticipated modern logic and analytic philosophy, but his philosophy also looks back to the scholastic tradition, in which conclusions are produced by applying reason to first principles or a priori definitions rather than to empirical evidence.
Leibniz occupies a prominent place in the history of mathematics and the history of philosophy. Leibniz developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. Leibniz also developed the binary number system, which is at the foundation of virtually all digital computers.
Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Ten who was never the greatest living algorist or theorem prover.
Leibniz’s contribution to metaphysics is his theory of monads, such as stated in the Monadologie. Monads (Greek unit) are the metaphysical field as the atoms to the physical / phenomenal monads are the ultimate elements of the universe. Monads are centers of force, the substance is force, while space, matter and motion are merely phenomenal.