The Greek term Apeiron, meaning originally "boundless" rather than "infinite," was used by Anaximander for the ultimate source of his universe. He probably meant by it something spatially unbounded, but since out of it arose the primary opposite substances (such as the hot and the cold, the dry and the wet) it may have been regarded also as qualitatively indeterminate. Aristotle, summarizing the views of certain early Pythagoreans (Metaphysics A, 5), puts the pair Peras ("Limit") and Apeiron ("Unlimited") at the head of a list of ten opposites. Peras is equated with (numerical) oddness, unity, rest, goodness, and so on; Apeiron is equated with evenness, plurality, motion, badness. The two principles Peras and Apeiron constituted an ultimate dualism, being not merely attributes but also themselves the substance of the things of which they are predicated. From the Pythagoreans on, the opposition of Peras and Apeiron was a standard theme in Greek philosophy.
Parmenides (fr. 8, 42ff.) seems to have accepted Limit and rejected the Unlimited for his One Being. The later Pythagoreans removed unity from the list of identities with Peras and argued that unity was the product of the imposition of the Peras upon the Apeiron, or else it was the source of both of them. Plato in the Philebus regards Peras and Apeiron as contained in all things, and supposes that it is through limit that intelligibility and beauty are manifested in the realm of Becoming. Exactly how the Ideas fit into this scheme is controversial, but in the doctrine of ideal numbers which Aristotle attributes to him Plato seems finally to have identified a material principle with the Apeiron and a formal principle with the Peras. Both principles apply to the ideal as well as to the sensible world. This leads in due course to the doctrine in Proclus (Elementa 89–90) that true being is composed of Peras and Apeiron, and beyond being there is a first Peras and a first Apeiron. The Christian writer known as Dionysius the Areopagite identified this doubled First Principle with God.
The concept of infinity, for long wrongly regarded as contrary to the whole tenor of Greek classicism, was in fact a Greek discovery, and by the fifth century BCE the normal meaning of Apeiron was "infinite." Infinite spatial extension was implied in the doctrines of Anaximander, Anaximenes, and Xenophanes and was made explicit by the Pythagoreans (see Aristotle, Physics IV, 6). Denied by Parmenides, it was reasserted for the Eleatics by Melissus (frs. 3–4) and adopted by the Atomists. Plato, however (in the Timaeus ), and Aristotle (Physics III) insisted upon a finite universe, and in this they were followed by the Stoics and most subsequent thinkers until the Renaissance. Aristotle had, however, admitted that infinity could occur in counting and he stated the concept clearly for the first time. He also accepted infinite divisibility (Physics VI), which had been "discovered" by Zeno and adopted wholeheartedly by Anaxagoras. It was rejected by the Atomists. Plato rejected it in the Timaeus, although he seems to have admitted it at the precosmic stage in Parmenides 158b–d, 164c–165c. Aristotle accepted infinite divisibility for movements, for magnitudes in space, and for time. The concept of a continuum so reached has been a basic concept in physical theory ever since. The mathematical concept of infinitesimal numbers associated with infinite divisibility and also with the doctrine of incommensurables remained important until the development of calculus in modern times.
Mondolfo, R. L'infinito nel pensiero dell'antichita, 2nd ed. Florence, 1956.
Solmsen, Friedrich. Aristotle's System of the Physical World. Ithaca, NY: Cornell University Press, 1960. Ch. 8.
G. B. Kerferd (1967)