Разъяснение и доказательство некоторых частностей, в тексте предполагавшихся уже доказанными. XV. Некоторые понятия из учения о бесконечности 829 .

In motu quiesco. Quiet in bad weather. Infinite, infinity is heard very often in ordinary conversation; but one has only to ask for an explanation of these words to be met with a perplexed look. However, if in the general public there is only misunderstanding, then among people engaged in mental work, there is often a perversion in understanding and even complete confusion on this score. Very strong and subtle minds were often not free from ambiguities and omissions on the issue of the concept of infinity. Lack of space, unfortunately, does not allow us to give a number of instructive examples, but the reader will figure out for himself from the further presentation who could be referred to here. However, the difficulties depend only partly and even in a very insignificant part on the abstractness of the question; the main reason here is biased thinking, the reluctance or inability to look at the object of research directly. Starts studying with confidence that it is already known; and imaginary knowledge, in Cantor’s words, the “horror infiniti” that reigns in society, makes itself felt. The main mistakes that are made quite often in reasoning about the infinite appear as a result of neglecting the basic and completely elementary distinction between actual and potential infinity. Therefore, I will have to dwell on this division in more detail than I would like. For now, however, a preliminary definition of infinity will be given; We will not build on it, since it does not rely on fairly simple concepts, although in itself it is true. Every quantum, or, as N.I. Lobachevsky suggests saying, every “colic,” by its very definition, can be twofold. It can be given and unchangeably and firmly established, completely definite, and then it will represent what is called permanent or constant. It may also not be specific, it may change, becoming larger or smaller. In this last case, quantum is called a variable. So, actual infinity is a special case of a constant, and potential - a variable “colic”, and this is their deepest fundamental difference - if you like, their essential opposite. Let's explain this in more detail. Let us have a variable, and let it change not in some way, but in a certain way, so that it becomes more than any constant “colic” of the same kind, or less. In each state this variable is finite; but in our understanding, the totality of these states differs from the totality of any arbitrarily selected states. In this sense, we say that our quantum is potential infinity, potential - due to the fact that it can become more than any other quantum. Thus, potential infinity does not denote any quantum taken within itself, but only a special way of considering quantum, namely, in connection with the nature of its special change. Potential infinity, according to Cantor, is not an idea, but only an auxiliary concept; it is ens rationis, according to Stöckl’s happy expression. In a word, potential infinity is what the ancients called ἄπειρον, the scholastics - syncategorematice infinitum or indefinitum, the new philosophers - bad or, more precisely, simple infinity, schlechte Unendlichkeit. So, this never-ending potential infinite is a variable finite quantity, quantum, increasing over all boundaries, or, conversely, falling below any finite limit. Such, for example, are differentials, already characterized by Leibniz, precisely for this property, as pure fictions. In view of this, it is clear that talking about a complete potential infinity, which, according to Cantor, Fontenelle did, is a contradictio in terminis. Unfortunately, countless, legion, authorities of all specialties have grasped this simple truth too firmly and, forgetting about the word potential, began to declare in different voices that “complete infinity” is something absurd. From here comes the old aphorism: “Numerus infinitus repugnat”, hence Tongiorgi’s statement: “Multitudo actu infinita repugnat” and other similar ones. This apparently quite innocent omission gave rise to more than one gross error, and, by the way, Kant’s first “antinomies of pure reason” are based on it. The so-called arguments against complete infinity and many considerations of positivism are based on this same omission, as we will see. Let us now consider another kind of infinity - actual infinity. For this purpose, we return to our starting point, to the concept of quantum, namely, quantum of constant, and we will enrich the content of this concept of constant with a new feature. Some constant may be such that it stands in a series of other constants of the same kind, that is, more than some finite constants and less than others. Then she herself will be finite. But it may happen that it does not stand among other constants, because it is greater than any finite constant, no matter how great we take it. Then we will say that our quantum is actual infinity, infinity in actu, actualiter, and not only in potentia. So, for example, in the dialogue “Bruno” Schelling 830 brilliantly reveals that every concept is infinity, because it unites a set of representations that is not finite; but since the scope of the concept, in essence, is completely defined and given, then this infinity cannot be anything other than actual infinity. Every judgment, every theorem carries within itself actual infinity, and this is the whole power of logical thinking, as Socrates pointed out. Let's take more specific examples. For example, turning to space, we can assert that all points inside some closed surface form an actual-infinite set. In fact, each of them is completely defined, which means that all of them are also completely defined; but, however, their number exceeds any of the numbers in the series: 1, 2, 3... n,... and more than each of these numbers. In the same sense, we can say that the power of God is actual and infinite, because it, being determined (there is no change in God), is at the same time greater than any finite power. 831 The author of the book “On the Heavenly Hierarchy,” a book attributed to St. Dionysius the Areopagite. “And that, in my opinion,” he says, “is worthy of careful reflection, what Scripture says about the Angels, that is, that there are thousands of them and tens of them, multiplying the numbers by themselves, we have the highest. Through this it clearly shows that the types of celestial beings are innumerable for us; because the blessed army of premium minds is countless. It surpasses the small and insufficient counting of the numbers we use, and is precisely determined by their superior understanding alone” 832. In the concept of actual infinity considered here, it is not difficult to recognize what the ancients knew under the name ἀφωρισμένον, among the scholastics - under the name kategorematice infinitum, among the new philosophers - positive, proper infinity. As Goethe puts it, 833 “this is a closed infinity, more appropriate to man than the starry sky,” and the latter, of course, is understood precisely as a certain possibility of rushing further and further, never being able to produce a synthesis and settle down on the whole. Here we are faced with a new consideration. For potential infinity to be possible, infinite change must be possible. But, after all, for the latter, a “region” of change is necessary, which itself cannot change, since otherwise it would be necessary to require a region of change for the region, etc. It, however, is not finite and, therefore, must be recognized as actually infinite. Consequently, any potential infinity already presupposes the existence of actual infinity as its super-finite limit 834; every infinite progress already presupposes the existence of an infinite goal of progress; every infinite improvement requires recognition of infinite perfection. He who denies the actual-infinite in any respect thereby denies potential infinity in the same respect, and positivism carries within itself elements of its own decomposition - so to speak, with positivism self-poisoning occurs with the products of its own activity. Based on the definition of actual infinity revealed above, we can conclude that such infinity can be conceivable in two modifications. Firstly, being greater than any finite quantum, it itself may turn out to not have another quantum, also infinite, which would be greater than it; in other words, here it turns out to be incapable of being less than anything else. This is actual infinity, incapable of increase, the absolute maximum; both in general and in Cantor, it is called Absolutum. Secondly, and those who spoke about infinity did not notice this, from the definition of actual infinity the possibility of its second modification follows. Actual infinity, precisely, can have above it other quanta, larger than itself; then it will be capable of increasing, will be increasing by actual infinity. In order to avoid once and for all confusion of words and lengths, Kantor gives it the name of super-limb, Ueberendlichkeit. From these formal considerations let us move on to real ones. We encounter actual infinity, or at least we can hope to encounter it, in three different areas. Firstly, since this actual-infinite is realized in the highest perfection, in a completely independent, extra-worldly being, in a word - in Deo sive natura naturante, and Cantor understands the last expression not in the sense of pantheism, but in the original sense that Thomas Aquinas and other theologians gave it. Here the infinite is the absolute maximum and is what was previously called Absolutum or absolute infinity. Secondly, the actual-infinite can be presupposed in concreto, in the dependent world, in the creature, in natura naturata. Here Cantor calls it Transfinitum. Finally, thirdly, the actual-infinite can be in abstracto in the spirit, since it has the ability to cognize the Transfinitum in nature and, to a certain extent, the Absolutum in God. In this latter case, infinity is called infinity symbols. In particular, if we are talking specifically about the knowledge of Transfinitum, these symbols are called transfinite numbers and transfinite types. The last two types of infinity are augmentable infinities. 835 Complete bibliography on scientists. oh infinite not available, some are indicated in: Vivanti, - Lista bibliografia della teoria degli aggregati 1893–1899 (“Bibliotheca mathematica”, 3 1, 1900, p. 160), where 70 articles and books are named; – L. Couturat, – De l’Infini Mathématique, Paris, 1896, pp. 657–660; – “Encyclopädie d. Math. Wissensch.”, Lpz., 1898–1904, Bd. 1 1 , SS. 184 ff. – In addition to the numerous works of the founder of modern times. uch. about the act. endless G. Cantor (Grundl. einer allgemeinen Mannigfaltigkeitslehre, Lpz. 1883; - articles in “Math. An.” Bdd. 46, 49; the most important articles from other journals collected in “Acta Math.”, 1883, 24), next. name in particular: Killing, – Ueber transfiniten Zahlen (“Math. An.”, Bd. 48); – Veronese, – Intorno ad alcune osservazioni sui segmenti infiniti e infenitesimali attuali (id., Bd. 47); – Lindelöf (“Comptes Rendus”, 1903(2), 37); – Evellin, – Infini et Qpantité. Etude sur le concept de l'infini en philos. et dans les sciences, Paris. 1880; – Ger. Hessenberg, – Grundbegriffe der Mengenlehre, Göttingen, 1906 (Sonderabdr. aus den “Abhandl. der Fries’schen Schule”, Bd. 1, Hft. 4). – His, – Das Unendliche in die Mathematik (id. Bd. 1, Hft. 3). – E. Zermelo, – Beweis, dass jede Menge wohlgeordnet werden kann (“Math. An.”, Bd. 59, 1904, SS. 514–516). – Arth. Schoenflies , – Die Entwickelung d. Lehre von den Punktmannigfaltigkeiten (“Jahresbericht d. Deutschen Mathematiker-Vereinigung”, Lpz., 1900, Bd. 8, Hft. 2). – Couturat (see above). – Borel,. – Ha pyc. language: I. [I.] Zhegalkin, - Transfinite numbers, M., 1907. - H. Weber and S. Wellstein, - Encyclopedia of elementary. mathematics, trans. with him. edited by V. Kogan, ed. “Mathesis”, T. 1, Odessa, 1907. – A. Puankare, – Science and method. – A. B. Vasiliev, – Intro. in analysis, vol. II., ed. N. N. Iovleva, Kazan, 1908. – Flor.. – “New. ideas in mathematics,” Sat. No. 1, ed. “Image.”, St. Petersburg, 1913. – Bolzano. Schelling, - Bruno (Russian translation, ed. together with “Phil. study on the essence of the people of St.”). H. Poincaré, – La Science et l’Hypothèse, pp. 20 s. there is p.p. – Similarly, J. Tannry asserts that “the concept of a whole number already contains the concept of infinity” (J. Tannry, - Pure Mathematics. “Method in the Sciences”, translated from the 2nd French edition by I. S. Yushkevich and I. K. Brusilovsky, “Education” publisher, St. Petersburg, 1911, p. 35). Dion. Areop. , – Oh heavenly. Hier. 14 (Rus. p. 1898, p. 52). Con. Gutberlet, – Das Problem d. Unendlichen (“Zeitschr. f. Philos. u. philos. Kr.”, Bd. 88, 1886. S. 215). This excursion – abbr. separation from “About symbol. endless." . - In addition decree. works on history Mon. oh endless Indication: Tannery, – Hist. du concept de l'infini au IV siècle (“Rev. philos.”, XIV, 618).