Infinite

INFINITE. — That which has no end or limit. If that whose limit is denied is something negative and imperfect, there is a negative i., which coincides with the indeterminate or the undefined (potential i.); if instead the limit of something positive and perfect is denied, there is positive i. (actual i.). Potential i. is that which can never be exhausted either in increase or in division; actual i. is that which contains in itself, at least in a certain order, the fullness of being and of determination. I., whether actual or potential, 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 restrictions: prime matter, pure potency and indetermination, as potential i., and God, pure act, absolute perfection and fullness, as actual i.

The notion of i. is fundamental in philosophy and in science, ever since Anaximander made i. (ἀπειρον) the principle from which everything derives; since, as Aristotle observes (Phys., III, 4, 205 a 1-b 15), all the best philosophers of antiquity spoke of i. as the principle of beings: from the Ionian physicists, with the infinity of the single original matter, to Melissus, who extends to i. 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 the obscure doctrine of the even number as the root of i., to Plato, who transforms the Pythagorean i. into the indeterminate dyad of the great and the small. But the antinomies of the finite and of i. did not take long to present themselves to the Greek mind, as appears from the famous arguments of Zeno against motion and divisibility (Aristotle, Phys., VI, 9, 239 b 5-40 a 18). The solution of this antinomy drove Aristotle to the clear distinction between actual i. and potential i. and to the denial of any actual i. in matter (ibid., III, 5-7, especially 206 a 9-25); nevertheless he recognized, at least incidentally, also the positive actual infinity in the Prime Mover, i. in time and in power (ibid., VIII, 10, 267 b 17-26), the best among all beings (Met., XII, 7, 1072 b 24-30). The deepening of the notion of actual i. in all its positivity was, however, the work of Christian philosophy, which recognizes infinity as one of the divine attributes, according to the dogmatic constitution of the Council of the Vatican (Denz-U, 1782).

In modern philosophy the problem of the genesis and the real value of the concept of i. was explicitly posed. Indeed, while sensualism 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 the intuition of i. precedes it. According to traditional philosophy, on the other hand, 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 fixes not only the positive concept of entity and perfection, but also the negative one of distinction and consequently of limit. With the negation of this limit the concept of i. is born. Nevertheless, in the absence of direct intuition, it is not possible to pass from the idea of i. to the affirmation of its reality, nor even of its possibility; in this lies the internal weakness of the Anselmian argument, which from the concept of divine infinity argues to its real existence.

In recent times the development of mathematics has brought back into prominence the antinomies of the infinitely great and the infinitely small, first with the infinitesimal calculus and then with Cantor's transfinite numbers. Nor has the antinomy between the finite and i. in physical space been any less alive in every age, having returned to discussion on the foundations of the Copernican system, of Newtonian mechanics, of the most modern results of astronomy and of recent physical theories, in particular of the theory of relativity, which posits space as finite but not limited.

These are the principal historical developments and the problems of the notion of i., for the treatment of which v. ATTRIBUTI DIVINI; CONTINUO; LIMITE; MOTO; RELATIVITÀ; UNIVERSO.

Filippo Selvaggi