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a term used by ancient Greek philosophers to designate problems that were difficult or impossible to solve. These problems are most frequently connected with contradictions that exist between the facts of observation and experience, on the one hand, and attempts to analyze them intellectually, on the other hand.
The most well-known aporias originated with Zeno of Elea, who lived in the fifth century B.C. They are expounded in various later redactions that are often mutually contradictory, since the authentic arguments of Zeno himself have not been preserved. The aporia “against the plurality of objects” poses the question of the possibility of intellectually conceiving things in a plural form. The opinion that such a conception is impossible because of its contradictory nature is ascribed to Zeno. If a thing is a plurality, it is an infinite plurality, since a third thing is needed in order to divide two things, and so forth; but then, a thing of finite dimensions must either have infinite dimensions (if its components have dimensions) or not have dimensions (if its components do not have dimensions). In this aporia the “paradox of measure” appears, which demonstrates the difficulty of a logically noncontradictory conception of prolonged dimensions in the form of totality of zero-dimensional points. Another version of this aporia demonstrates a contradiction between assertions about the finite and infinite quality of the plurality of things that really exist. Both assertions, moreover, may be considered justifiable.
“The Dichotomy,” “Achilles,” “The Arrow,” and “The Stadium” are aporias devoted to the difficulties connected with the concept of motion. “The Dichotomy” states that before a moving body can traverse an entire course, it must traverse half the course; before that, a fourth, and so on. But inasmuch as this process of intellectual division is infinite, the motion can never begin. Another variant of this same aporia concludes that the motion can never come to an end. This contradiction poses the question of the correctness of the representation of the concepts of space, time, and motion by means of mathematical abstractions such as point and segment as well as the disputable nature of various abstractions about infinity. In one of the most popular aporias, “Achilles,” there is an analysis of the contradiction between the obvious facts of sensory experience and the reasoning according to which the swift Achilles cannot overtake the tortoise. The reason for this is that while Achilles is trying to catch up to the tortoise, the tortoise also manages to advance a certain amount, and while Achilles is covering this amount, the tortoise is crawling just a little bit farther, and so on. The aporia entitled “The Arrow” demonstrates the difficulties in representing motion that have arisen from the adoption of “atomistic” conceptions. If we consider that space, time, and the very process of motion itself are composed of certain “indivisible” elements, then within one such “indivisible” element a body cannot move (otherwise an “indivisible” element would be “divided”), and hence it cannot even move in general (a sum of “rests” cannot add up to motion). This means that the flying arrow is “in fact” at rest.
Zeno’s aporias underscore the relative and contradictory character of mathematical descriptions of the real processes of motion, the invalidity of the pretensions to “adequacy” (“isomorphism”) of any and all mathematical representations of physical processes, and finally the disputable nature of generally held opinions concerning the one-to-one definiteness of the concepts that figure in these representations—such as, for example, the natural series of numbers. In particular, the logical conflicts that are established in “The Dichotomy” and “Achilles” may be explained by the invalidity of the “obvious” assumption that the successions of points figuring in these aporias will give one and the same natural series as their mental images—that is, the numbers designating these points. Confidence in the indisputability of this supposition was undermined by the development of the so-called nonstandard, or mutually nonisomorphic models of number theory.
Not a single one of the solutions proposed to resolve the contradictions of aporias can, at the present time, be considered universally accepted. The problems connected with aporias continue to be discussed intensively; included in this are the works of Soviet scholars. The influence of Zeno’s aporias can be clearly traced, for example, in the theses of ancient skepticism and in I. Kant’s so-called antinomies of pure reason. In general, the analysis of aporias, which are a kind of negative expression of the dialectics of the interrelationship between the real world and its reflection in thought, has had an important effect on the subsequent development of logic and the theory of knowledge.
REFERENCESIanovskaia, S. A. “Aporii Zenona Eleiskogoisovremennaianauka.” In the book Filosofskaia entsiklopediia, vol. 2. Moscow, 1962. Pages 170–174.
Ianovskaia, S. A. “Preodoleny li v sovremennoi nauke trudnosti, izvestnye pod nazvaniem ’Aporii Zenona’?” In the collection Problemy logiki. Moscow, 1963. Pages 116–136.
Petrov, Iu. A. Logicheskie problemy abstraktsii beskonechnosti i osushchestvimosti. Moscow, 1967.
Fränkel, A., and Y. Bar-Hillel. Osnovnaia teoria mnozhestv. Moscow, 1966. Pages 23, 26–27. (Translated from English; contains a bibliography and notes by the editor.)
IU A. GASTEV, V. A. KOSTELOVSKH, and IU. A. PETROV