Leibniz was a mathematician, scientist, lawyer, diplomat engineer, inventor, historian and philosopher. Like Pascal he invented a calculating machine and independently of Isaac Newton, he invented differential calculus. Much of the notation and vocabulary used today in calculus comes from his work and is considered to be more satisfactory than Newton’s. Leibniz is the third of the great rationalist philosophers after Descartes and Spinoza.
As with Descartes and Spinoza, Leibniz's philosophy proceeds from the Aristotelian notion of substance, however, where Descartes maintained that reality consists fundamentally of two substances and Spinoza held that there was only one, Leibniz argued a case for infinitely many substances.
He called these substances 'monads'. A monad is somewhat similar to Democritus' atoms, yet more like the geometrical points of Pythagoras.
Like atoms, monads are the indivisible elements of reality of which all material things are constituted, however, they themselves are not extended or composed of matter.
Leibniz maintained that a monad is a psychological entity, which when embodied in human beings, is a 'soul.' Monads are unified, independent substances consisting of energy, rather than matter, and different from each other in that they possess varying degrees of consciousness. The individual monad can only be created or annihilated by God and each monad carries within itself the potentiality of all it would ever be.
For Leibniz each monad is like a living mirror of the universe. It unfolds its being in a way which is harmonious with the unfolding of every other monad, but without affecting or being affected by any other. Likewise, as a result of his discovery of calculus, in which the calculations involve diminishing values right down to the infinitesimal, Leibniz came to see that all things consisted of infinitesimally small points which had neither space nor time attributes (monads).
One of Leibniz's other central ideas was his scientia generalis, his own version of the scientific method. It emphasized rational analysis and the reduction of concepts to their simplest element. These simplest elements could be expressed as definitions, which in turn could be expressed as truths that do not follow of logical necessity.
For Leibniz there were three types of truth:
It was pointed out to Leibniz, however, that not all truths fall into these three categories. For instance Euclid's axioms "The whole is greater than the part" and "Things which are equal to the same thing are also equal to each other" (i.e. if A=B and B=C, then A=C). While Leibniz was willing to concede this point, he noted that neither of these axioms is a definition or an identical proposition. They somehow manage to fall between the two and he maintained that such axioms had to be accepted if science was to move forward.
He also suggested a method of validating such axioms by means of the principle of contradiction, which stated that such truths were in fact logically necessary, because to maintain their opposite would lead to a contradiction.
Of course, just because something is logically possible doesn't mean that it happens. Therefore, in order to account rationally for what actually exists, a second principle was required. Science required a sufficient reason for something to take place, hence Leibniz's Principle of Sufficient Reason, which states that "nothing happens in the world without there being a sufficient reason why it should happen in this way and no other."
Leibniz would go on to use this principle of sufficient reason to demonstrate the existence of God along with other metaphysical and theological positions. In his ontological argument he shows that God, the most perfect being, exists; since he is perfect, it is inconceivable that he might have made things better than he has; therefore, this world is the best of all possible worlds.
Unlike most of the great philosophers of his time Leibniz did not write a magnum opus. Although he wrote trunk loads of essays and papers, he did not publish much during his lifetime. Many of his writings have not yet been published.