INFINITE. — That which has no end nor limit. If what is denied a limit is something negative and imperfect, one has a negative i., which coincides with the indeterminate or indefinite (potential i.); if instead the limit of something positive and perfect is denied, one has the positive i. (actual i.). Potential i. is that which can never be exhausted either in growth or in division; actual i. is that which contains within itself, at least in a certain order, the fullness of being and determination. Both actual and potential i. can be subdivided into relative and absolute. Relative i. concerns a particular genus of being, such as mathematical i. and, in Thomist doctrine, the angelic essence (Sum. Theol., q. 50, a. 2, ad 4). Absolute i., on the other hand, refers to all being without restriction: prime matter, pure potency and indetermination, as potential i., and God, pure act, perfection and absolute fullness, as actual i.
The notion of i. is fundamental in philosophy and science, from the time when Anaximander made the i. (ἄπειρον) the principle from which all things derive; since, as Aristotle observes (Phys., III, 4, 203 a 1-b 15), all the best philosophers of antiquity spoke of the i. as the principle of beings: from the Ionian physicists, with the infinity of the single original matter, to Melissus, who extends the i. to the immobile sphere of Parmenides, to Anaxagoras and the atomists, who posit an infinite number of elements in an infinite space; from the Pythagoreans, with their obscure doctrine of the even number as the principle of the i., to Plato, who transforms the Pythagorean i. into the indeterminate dyad of the great and small. But the antinomies of the finite and the i. soon presented themselves to the Greek mind, as appears from Zeno’s famous arguments against motion and divisibility (Aristotle, Phys., VI, 9, 239 b 5-40 a 18). The solution of this antinomy led Aristotle to a clear distinction between actual and potential i. and to the denial of any actual i. in matter (ibid., III, 5-7, especially 206 a 9-25); yet he acknowledged, at least incidentally, actual positive infinity in the Unmoved Mover, i. in time and in power (ibid., VIII, 10, 267 b 17-26), the best of all beings (Met., XII, 7, 1072 b 24-30). The deepening of the notion of actual
i. in all its positivity, however, was the work of Christian philosophy, which recognizes infinity as one of the divine attributes, according to the dogmatic constitution of the Vatican Council (Denz
U., 1782).In modern philosophy the problem of the genesis and real value of the concept of i. was explicitly posed. For while sensism and empiricism (Locke, Hobbes) simply denied the notion of actual and positive i., ontologism on the contrary demanded for the human intellect the immediate intuition of the Infinite Being, since the finite cannot represent i., nor even be conceived as such, unless preceded by the intuition of i. According to traditional philosophy, however, the human intellect has the concept of i., but only indirectly, analogously, and through a process of double negation, inasmuch as, from the consideration of the various degrees of reality, it forms not only the positive concept of entity and perfection, but also the negative concept of distinction and consequently of limit. With the negation of this limit arises the concept of i. Yet, lacking direct intuition, it is not possible to pass from the idea of i. to the affirmation of its reality, nor even of its possibility; herein lies the internal weakness of the Anselmian argument, which from the concept of divine infinity infers its real existence.
In recent times the development of mathematics has brought to the fore once more the antinomies of the infinitely great and the infinitely small, first with the infinitesimal calculus and then with Cantor’s transfinites. Nor has the antinomy between finite and i. in physical space been less alive at all times, resurfacing in discussion on the basis of the Copernican system, Newtonian mechanics, the most modern results of astronomy, and the recent physical theories, in particular the theory of relativity, which posits space as finite but unbounded.
These are the principal historical developments and problems of the notion of i., for the treatment of which see DIVINE ATTRIBUTES; CONTINUUM; LIMIT; MOTION; RELATIVITY; UNIVERSE.