APEIRON (Greek ’απειρον [apeiron]—the limitless)—something infinite or indefinite.
According to Anaximander, the apeiron is a principle (’αρχη [arche]) and element of existing things (των ’οντων [ton onton]). Such a source cannot be something definite (περας [peras]) such as water or any of the other elements, because changes in the world would be impossible since everything would ultimately be something one. Anaximander thought that motion (κινησις [kinesis]) is eternal and said that only the limitless—the apeiron can be the infinite source from which all things arise by the separation of opposites, and to which they return in a process of destruction. Thus the world appears as a struggle of opposites that emerge from the limitless which, as Anaximander says, “inflict punishment on each other and penance for injustice, in accordance with the decision of time” (Simplikos, In Arist. Phys., 24, 13). Everything except the apeiron undergoes destruction. The apeiron encompasses (περιεχει [periechei]) everything (κυβερναι [kybernai]). Thus the limitless or infinity without beginning or end, immortal and indestructible, and thus divine, is the beginning and end of everything (Phys., 203 b) and is a material principle and element that penetrates and surrounds the existing beings that emerge from it.
For the Pythagoreans the highest principles of everything are the apeiron as the unlimited, and that which limits (Diels-Kranz, 44 R2). Number comes from the connection of these elements. A predominance of the unlimited element is found in the series of odd numbers, and a predominance of the limited element forms the series of even numbers (Met. 985 b–986 a 18). All things are knowable and arranged among themselves by number (Dielz-Kranz, 44 B 4). Therefore for the Pythagoreans the world is a harmony and is called a cosmos, that is, an order.
According to Plato, everything that exists is composed of unity and plurality, and posseses an element of limit (περας) and indefiniteness (apeiron) (Phil., 16 c). Indefiniteness is joined with the plurality that basically occurs in material reality. To describe this reality Plato presented two other factors besides apeiron and περας, namely unity which is the result of a mixture of the apeiron with the περας, and the cause (’αιτια [aitia]) of the mixture of these elements (Phil., 23). The apeiron is an indefiniteness of a plurality that is expressed in the contraries of what may be less or more (e.g., less or more hot). Περας is also a definiteness that is expressed in the quantitative determination of something. A mixture (μεικτον [meikton]) of these two factors cause a concrete being to come into existence. The unity of this being results from the introduction of a defined factor into an indefinite factor (ibid. 26). The reason (νους) is the cause of this process. The reason steers (κυβερναι) the world and brings order to it (διακοσμει [diakosmei]). At the same time it indicates the soul of the world (ibid. 30). This problematic is connected with Plato’s cosmology which is presented in the Timaeus. To provide an ultimate explanation for the problem of how the world arose, Plato creates the conception of the demiurge who with the help of mathematical beings models the world of ideas in indefinite matter (Tim., 29–33, 50–56).
According to Aristotle, indefiniteness (apeiron) is associated with material beings because they are characterized by divisibility. That which is divisible must be a magnitude or a plurality, and their attribute is infinity. That which is indivisible, e.g., substance, cannot be infinite. Thus infinity cannot exist as an actual being, whether as a substance, an attribute of substance, or a principle (Phys., 204 a). Therefore infinity exists only potentially and only for knowledge, but not in the sense of every being actualized (Met., 1048 b). Hence also infinity is matter which is potency capable of receiving different forms. Matter as infinite encompasses nothing but is encompassed. In the whole it is the potential factor. It is divisible both in the direction of subtraction (κατα διαιρεσιν [kata diairesin]) and addition (κατα προσθεσιν [kata prosthesin]) (Phys., 207 a).
In view of this divisibility Aristotle distinguishes various kinds of infinity—number, magnitude, motion, and time are potentially infinite: number by addition, in the sense that one can always conceive a greater number; magnitude by subtraction, since that which is continuous is infinitely divisible; motion in view of the fact that the magnitude encompassed by motion (change) is potentially infinite (Phys., 207 b); time by addition and subtraction in reference to motion, since time is the measure of motion with respect to “before” and “after” (ibid., 207 b, 219 b). Aristotle says in general terms, “the unlimited is the open possibility of taking more, however much you have already taken; that of which there is nothing more to take is not unlimited, but whole or completed” (ibid., 207 a).
R. Mondolfo, L’infinito nel pensiero dell’antichità classica, F. 1958; G. S. Kirk, J. E. Raven, M. Schofield, The Presocratic Philosophers. A Critical History with a Selection of Texts, C 1957, 1983² (Filozofia przedsokratejska. Studium krytyczne z wybranumi tekstami [Pre-Socratic philosophy. Critical Study with selected texts], Wwa-Pz 1999, 114–148; C. H. Kahn, Anaximander and the Arguments concerning the απειρον at Physics 203 b 4–15, in: Festschrift Enrst Kapp zum 70. Geburstag, H 1958, 19–29; B. Wisniewski, ’Απειρον d’Anaximandre et de Pythagore, SiFC 31 (1959), 175 –78; C. J. De Vogel, La théorie de l’απειρον chez Platon et dans la tradition platonicienne, RPFE 149 (1959), 21–39; F. Solmsen, Aristotle’s System of the Physical World, Ithaca (NY) 1960; L. Sweeney, L’infini quantitatif chez Aristotle RPL 58 (1960), 504–528; C. A. Carena, A proposito dell’Apeiron di Anassimandro, Rivista rosminiana di filozofia e di cultura 55 (1961), 39–40; F. Solmsen, Anaximander’s Infinite: Traces and Influences, AGPh 44 (1962), 109–131; W. Stróżewski, Wykłady o Platonie [Lectures on Plato], Kr 1992, 233–255; Reale I 81–88, 109–117; Reale II 159–188, 448–449.