Infinite Division

This theory is exposed in Plato's dialogue Timaeus and was also supported by Aristotle. Andrew Pyle gives a lucid account of infinite divisibility in the first few pages of his Atomism and its Critics. There he shows how infinite divisibility involves the idea that there is some extended item, such as an apple, which can be divided infinitely many times, where one never divides down to point, or to atoms of any sort.
Until the discovery of quantum mechanics, no distinction was made between the question of whether matter is infinitely divisible and the question of whether matter can be cut into smaller parts ad infinitum.
As a result, the Greek word átomos, which literally means "uncuttable", is usually translated as "indivisible". Whereas the modern atom is indeed divisible, it actually is uncuttable: there is no partition of space such that its parts correspond to material parts of the atom. In other words, the quantum-mechanical description of matter no longer conforms to the cookie cutter paradigm. This casts fresh light on the ancient conundrum of the divisibility of matter. The multiplicity of a material object — the number of its parts — depends on the existence, not of delimiting surfaces, but of internal spatial relations (relative positions between parts), and these lack determinate values.
Physical space is often regarded as infinitely divisible: it is thought that any region in space, no matter how small, could be further split. Similarly, time is infinitely divisible.