Chrysippus - Mathematics

Mathematics

Chrysippus regarded bodies, surfaces, lines, places, the void and time as all being infinitely divisible. He determined one of the principal features of the infinite set: since a man and a finger have an infinite number of parts as do the universe and a man, it cannot be said that a man has more parts than his finger, nor that the universe has more parts than a man.

Chrysippus also responded to a problem first posed by Democritus. If a cone is divided by a plane parallel to its base, are the surfaces of the segments equal or unequal? If they are equal, then the cone becomes a cylinder; if they are unequal, then the surface of the cone must be stepped. The reply of Chrysippus was that the surfaces are both equal and unequal. Chrysippus was, in effect, negating the law of excluded middle with respect to the equal and unequal, and thus he may have anticipated an important principle of modern infinitesimal calculus, namely, the limit and the process of convergence towards a limit.

Chrysippus was notable for claiming that "one" is a number. One was not always considered a number by the ancient Greeks since they viewed one as that by which things are measured. Aristotle in his Metaphysics wrote, "... a measure is not the things measured, but the measure or the One is the beginning of number." Chrysippus asserted that one had "magnitude one" (Greek: πλῆθος ἕν), although this was not generally accepted by the Greeks, and Iamblichus wrote that "magnitude one" was a contradiction in terms.