It was the Greeks who first grappled with the idea of infinity, such as described in the well-known paradoxes attributed to the philosopher Zeno of Elea in the 5th Century BCE. The most famous of his paradoxes is that of Achilles and the Tortoise, which describes a theoretical race between Achilles and a tortoise. Achilles gives the much slower tortoise a head start, but by the time Achilles reaches the tortoise’s starting point, the tortoise has already moved ahead. By the time Achilles reaches that point, the tortoise has moved on again, etc, etc, so that in principle the swift Achilles can never catch up with the slow tortoise.
Paradoxes such as this one and Zeno’s so-called Dichotomy Paradox are based on the infinite divisibility of space and time, and rest on the idea that a half plus a quarter plus an eighth plus a sixteenth, etc, etc, to infinity will never quite equal a whole. The paradox stems, however, from the false assumption that it is impossible to complete an infinite number of discrete dashes in a finite time, although it is extremely difficult to definitively prove the fallacy. The ancient Greek Aristotle was the first of many to try to disprove the paradoxes, particularly as he was a firm believer that infinity could only ever be potential and not real.
Democritus, most famous for his prescient ideas about all matter being composed of tiny atoms, was also a pioneer of mathematics and geometry in the 5th – 4th Century BCE, and he produced works with titles like “On Numbers“, “On Geometrics“, “On Tangencies“, “On Mapping” and “On Irrationals“, although these works have not survived. We do know that he was among the first to observe that a cone (or pyramid) has one-third the volume of a cylinder (or prism) with the same base and height, and he is perhaps the first to have seriously considered the division of objects into an infinite number of cross-sections.
However, it is certainly true that Pythagoras in particular greatly influenced those who came after him, including Plato, who established his famous Academy in Athens in 387 BCE, and his protégé Aristotle, whose work on logic was regarded as definitive for over two thousand years. Plato the mathematician is best known for his description of the five Platonic solids, but the value of his work as a teacher and popularizer of mathematics can not be overstated.
Plato’s student Eudoxus of Cnidus is usually credited with the first implementation of the “method of exhaustion” (later developed by Archimedes), an early method of integration by successive approximations which he used for the calculation of the volume of the pyramid and cone. He also developed a general theory of proportion, which was applicable to incommensurable (irrational) magnitudes that cannot be expressed as a ratio of two whole numbers, as well as to commensurable (rational) magnitudes, thus extending Pythagoras’ incomplete ideas.
Perhaps the most important single contribution of the Greeks, though – and Pythagoras, Plato and Aristotle were all influential in this respect – was the idea of proof, and the deductive method of using logical steps to prove or disprove theorems from initial assumed axioms. Older cultures, like the Egyptians and the Babylonians, had relied on inductive reasoning, that is using repeated observations to establish rules of thumb. It is this concept of proof that give mathematics its power and ensures that proven theories are as true today as they were two thousand years ago, and which laid the foundations for the systematic approach to mathematics of Euclid and those who came after him.