Gottfried Leibniz (1646-1716)

Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals."

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. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. (And his political influence may have been huge: he was a special consultant to both the Holy Roman and Russian Emperors, and was helped arrange for the son of his patron Sophia Wittelsbach, only distantly in line for the British throne, to be crowned King George I of England.)

Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the symbols ∫, df(x)/dx; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He invented more mathematical terms than anyone, including "function," "analysis situ," "variable," "abscissa," "parameter," and "coordinate." His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation ("calculus"), because with "symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished."

Leibniz' thoughts on mathematical physics had some influence. He developed laws of motion that gave different insights from those of Newton. His cosmology was opposed to that of Newton but, anticipating theories of Mach and Einstein, is more in accord with modern physics. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves.

Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π: 

        π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...