Calculus
Leibniz developed calculus between 1675 and 1676 — independently of Newton, who had developed it a decade earlier but published nothing. Leibniz published first (1684) and his notation — dy/dx for derivatives, ∫ for integrals — is what every student learns today. The Newton-Leibniz priority dispute, which consumed European mathematics for decades, is now regarded as a case of genuine independent invention.
Binary Arithmetic
Leibniz developed the binary number system in 1679, inspired by the I Ching's hexagrams. He saw binary as philosophically significant — creation ex nihilo, 1 and 0, something and nothing. The practical implications (digital computing) would not be realized for 250 years, but the mathematics was complete and correct.
The Monadology
Leibniz's metaphysics: the universe consists of simple, indivisible substances called monads — spiritual units with no windows to the outside world. Their apparent interaction is pre-established harmony, arranged by God. Every monad contains a complete description of its entire history and future. It is a peculiar, self-consistent system that influenced Kant, Husserl, and Whitehead.
Best of All Possible Worlds
Leibniz argued that God, being perfectly rational and good, must have created the best of all possible worlds — the world where the greatest good is realized given all constraints. Voltaire savaged this argument in Candide (1759): if this is the best possible world, what would the worst look like? The argument remains philosophically contentious but historically important for theodicy.
Frequently Asked Questions
Did Leibniz invent calculus?
Leibniz independently invented calculus between 1675 and 1676, publishing in 1684 — the first publication of calculus, though Newton had developed it earlier without publishing. Leibniz's notation (dy/dx, ∫) became the universal standard. Modern consensus is that both invented calculus independently.
What is Leibniz's binary system?
Leibniz developed the binary number system (base 2, using only 0 and 1) in 1679. He saw philosophical significance in it. Modern digital computers operate entirely in binary — every calculation, every piece of stored data uses the number system Leibniz invented 250 years before the first computer.
What is the best of all possible worlds argument?
Leibniz argued that God, being omniscient and perfectly good, chose to create the world that maximizes total goodness — the best of all possible worlds. This doesn't mean no suffering exists, but that any alternative world would have more. Voltaire mocked this in Candide; Leibniz's argument is an answer to the problem of evil.
