XVII. Иррациональности в математике 839 и догмат (к стр. 59).

It can be explained with one example that super-rational synthesis, which is what dogma is, is not something completely unprecedented and unexpected in science. It is carried out very often, for example in the construction of so-called irrational numbers. According to the original definition, “number” is an integer, and then a rational number, i.e. a fraction. However, the solution of some geometric problems often leads to a ratio of quantities - for example, segments - that is not expressed in numbers. Arithmetic operations corresponding to these actions are impossible, because this combination of arithmetic symbols is devoid of any meaning. What does √2 mean, for example? – This means that and only that in the course of the decision we stumbled upon a wall. We were looking for a certain number, but it turned out that there was no number that would satisfy the conditions of the problem: √2 is a symbol of arithmetic impossibility. Why? – But because in general √a means a number x that, when squared, gives a, that is, a number that satisfies the equation: x2 = a. But it is easy to prove that there is not and cannot be a number x whose square would be equal to 2. The same is true in geometry. We can, for example, try to find out how many times one segment is longer than another. In some cases we will define this “what time?” , i.e. let's find the number that characterizes it; and in others there will not be such a number at all, and then the very question “how much” does not make sense. Thus, the diagonal of a square is neither longer nor shorter than a side of the same square. The diagonal and the side are incomparable with each other in the direction “at what time”, as they say, “incommensurable”. Whatever number we take to characterize this ratio will turn out to be unsuitable. If a side has its own number, then the diagonal does not, and vice versa. Between one and the other there is some kind of abyss, impassable for numbers. The length of the diagonal is transcendental ( - I use this word in a general sense, and not as a mathematical terminus technicus -) in relation to the length of the side. This fact was first discovered by Pythagoras; As you know, the geometer himself was horrified by the depth of the fact he discovered and the consequences that flow from it. After all, this one fact has once and for all caused irreparable breaches to all rationalism. Combinations of symbols such as √2 in the Middle Ages were called “numeri ficti, fictitious numbers” or, in the Liber abaci of Leonard of Pisa, dating back to 1202, “numeri surdi, blind numbers” and were not considered numbers at all. For the first time, only in the Arithmetica integra of Michael Stiefel, published in Nuremberg in 1544, they were given the conventional meaning of numbers and the corresponding name “numeri irrationales”, and Stiefel declares that “irrationalis numerus non est verus numerus”, i.e. that “an irrational number is not a true number.” And even now in many algebra textbooks it is important to state that although it is impossible to extract the square root of 2, but still, etc. In the range of operations that arithmetic knows, there is no way out of this difficulty. These operations lead to a result that no longer makes sense unless their circle is broken; and if it is not broken, then this combination violates the integrity of the circle itself, producing internal destruction and devastation. So - and in general: rational operations lead to such combinations that no longer have a place among their producers and which require a rupture in the rational realm in order to be born into a new, hitherto unprecedented and unimaginable world. The solution in algebra is achieved only by the creation of other-worldly, transcendental arithmetic entities for the range of given operations, which are already inexpressible in finite symbols, but are postulated by them, they are justified and they are given a new, higher meaning. However, as soon as we want to think of these new arithmetic entities in terms of the old ones, as soon as we want to pour new wine into old wineskins, this results in the decomposition of the symbol of the new entity into component elements that are incompatible with each other in the field of old concepts, and the essence itself evaporates. I'll tell you more clearly what's going on here. – Irrational “numbers” for a long time were a vague absurdity, which everyone unconsciously used for the sake of practical necessity, and of which, however, no one was aware. But by the 70s of the 19th century, getting out of this situation became an urgent necessity. The question was ripe, and the answer to it was given almost simultaneously by several researchers, among whom, as the largest, the names of G. Cantor, K. Weierstrass, C. du Meret, E. Heine, R. Dedekind, G. Kossak, S. Pincherle, O. Biermann, J. Tannery, M. Pasha, V. Rössel and others can be indicated. independent of each other and because - which is very understandable - they differ significantly in their appearance. But, in essence, they all say the same thing. Therefore, I dwell a little on one of them - on the method of G. Cantor - the same Cantor about whom you and I have had the opportunity to talk so many times in the expanse of slowly stirring bread, near the edge of a birch grove and at home, in front of a blazing oven. Do you remember? sometimes, waking up at night, we got involved in a quiet conversation, and time slipped unnoticed until the clock on the bell tower reminded us of the approaching morning... In order to understand Cantor’s construction, I will ask you one thing: forget everything you have heard so far about irrational numbers, and keep in mind that you need to create a completely new object of thought. G. Cantor takes an infinite set of numbers a 1, a 2, a 3, a 4,... a n, ... and n+m,..., arranged in the order of writing, so that after each number comes the closest to it and before each, except the first, there is the closest one that precedes it. This series of numbers is considered by Cantor as a single object α. Let us symbolically denote this by enclosing the entire group in brackets, so that it becomes possible to write the equality that serves as the definition of α, namely: α=(a 1, a 2, a 3, … a n,… a n+m, …) It means that α is nothing more than an infinite group, conceivable in unity. In certain cases, the numbers a 1, a 2,.. a n,... may turn out to be such that the series α, as they say, “will converge”, i.e., will have the following property: no matter how small the number σ we take, there will always be an n so large, i.e., a term or element, and n so distant that the difference between it and any subsequent term an+m, no matter how large this m, is in absolute value (i.e., without taking into account the sign of the difference) will be less than σ; symbolically: an+m – and n < σ, where σ is arbitrarily small. In other words, the further we take any term a n, the less the difference becomes between it and all those following it, and, moreover, it can be brought as close to zero as desired, although, generally speaking, it will never become zero. This, for example, would be a group of numbers. In fact, for it the absolute value of the difference between the (n+m)th and nth terms will be: The more m, the smaller the value of 1/2 m, and therefore the larger the numerator. But he is always n. It is clear that no matter how small some number σ given to us may be, it is always possible to choose n so large that there is 1/2n < σ and therefore, even more so, so that the difference between the (n+m)th and nth terms, i.e. there was convergence, “converges”, then the series a 1, a 2, a 3,... receives the name of the main series, Fundamentalreihe, and the entire group, as a single object α, receives the name of an irrational number. Thus, irrationality, in the field of finite arithmetic based on reason, is nonsense from the point of view of “numbers,” that is, numbers in their own, final sense, obtained as a combination of a finite number of basic symbols (1, 2, 3, 4, 5, ... n, ...). No combination of these symbols, finite, immanent in reason, can provide an image for irrationality or even anything “similar” to it. Irrationality is certainly transcendental, absolutely incomprehensible for the rational realm. And once and for all, finally and irrevocably, we must abandon the intention to present irrationality in the form of a finite combination of rationalities. But, using rational symbols as a formless substance, we can, with the help of completely new constructive definitions, introduce into the aggregate of rational numbers, which is qualityless as a whole, a new essence that organizes it. Then the irrational number will be imprinted and embodied in this “substance”. Each rational number separately, each element, each atom of this aggregate in itself, in its original meaning, has nothing in common with the whole embodied in it, just as the aesthetic idea of ​​a statue has nothing in common with the marble crystals that make up the statue, or the meaning of a poem with the sounds of individual words. But the infinite, - more precisely: super-finite, - the totality of them completely reflects this whole, this idea. In the Cantor basic series, which depicts, embodies, represents and is, according to the definition, an irrational number α, each of the elements a1, a2,… аn, … as long as we are just entering the realm of irrationality, it has nothing to do with α and it is even absurd to ask in what relationship these essentially incomparable symbols are located, of which α is transcendental for every ai (where i = 1, 2, 3,... n,...). But the set of numbers a i, connected by the sign of convergence and the definition of “actions” on α as a single object, exactly depicts this transcendental essence of α. Subsequently, when α is completely examined, it turns out to be possible to transpose all ai in the form of α, although it is impossible, inversely, to transpose α in the form of ai; then the concept of “similarity” is established between a i and α, although this “similarity” is only a similarity of a hint, not a tautegory. This means that although α is transcendental for all ai, “incomprehensible” from the point of view of a i, but all ai are immanent for α, completely transparent to it. One can even say that from the point of view of ai it is impossible to see those transcendental roots of ai, that transcendental illumination of a i, which, however, is clear and obvious from the point of view of α. Immanence and transcendence in the realm of the essences of reason are similar to those in the realm of the essences of ontology: God is transcendental for the world, from the point of view of the world, but the world is not transcendental to God, but is entirely permeated by Divine energies - α and ai are different, but if α is considered in the series of all ai, then one can see that the difference or similarity of ai and α themselves change with a change in i. From the point of view of a formal legalist, rational, according to the law of identity, ai is not like α; but, for immediate consciousness, and i can hint at α, and, moreover, more transparent or cloudier, depending on the value of i. However, please note that here I am only presenting general results, but not the theory itself. From the concept of the equality of two irrationalities, α and another, analogous to it, β, obtained by different processes, it is established that the finite part of the symbols ai can be thrown out of α, that it is possible to select and remove an infinite group from the group (a 1, a 2,.. a n...), that it is possible, finally, to perform a pairwise permutation, a “transposition,” of an infinite set of elements a i, as long as the structure does not change series, as long as the elements do not move so as to be unable to return to the old arrangement by certain pairwise permutations - and yet α will not change. Moreover, even completely different sets (a1, a2, a3,.. an,..) and (b 1, b 2, b 3,.. b n,..) can express the same number α: completely different signs can express the same rational essence. So: having encountered an impossible combination of symbols, we were absolutely unable to solve the problem. We have stumbled upon a wall - the limitations of the most arithmetic entities embodied in these signs. There was only one thing left: either to abandon the task itself, or to rise above the plane of thinking that operates with “finite” symbols - to introduce a new idea, the idea of ​​actual - that is, synthesized - infinity and, with the help of it, to create, by a special creative act of the spirit, a completely new mental essence - irrationality. Was there a sequence of deductions here? Of course not ! We made a leap - a break in development; we have introduced something significantly new. We could not introduce it, limiting ourselves to those essences that are given, that is, “finite” essences, abandoning ourselves to the positivistic deprivation of the mind and resting on the impossibility of going beyond the boundaries of these symbols. We could also go higher; but this required an effort of will and a feat of reason - a very specific effort and humility before the object of study was required to create symbols of irrationality. Creating a new entity requires a free feat. Its freedom is expressed in the fact that we are given the opportunity to either remain with the “good” old, or rise to the “better” new. The feat lies in the fact that “natural forces” - the inertia and complacency inherent in the mind - push it to stagnation in the old, ultimately, in the “known”. It is necessary to overcome the complacency of reason, break the magic circle of its finite concepts and enter a new environment - the environment of the super-finite, inaccessible to reason and absurd for it. This is a reasonable feat in arithmetic. However, it would be a great mistake to see in this feat something exceptional and special. Modern mathematics, in its entirety, is built on the concept of a limit and a limiting process, which we have to deal with clearly every time the idea of infinity clearly appears, and without whose silent participation in the construction of science cannot be taken a single step. Irrationalities, some hints at the theory of which have been made here, are only the simplest and well-known case of a limiting process; but, in addition, there are many other similar applications of the basic concept of overcoming finitude. Thus, a transcendental analytic function cannot be expressed by any finite number of elements, whereas with respect to an algebraic function Weierstrass found that it can always be expressed as such. But then the beginning of overcoming finitude appears and it turns out, according to Poincaré’s theorem, that “every analytical function can be defined by means of a countable set of elements (x – a)" 840 . Thus, an analytic function stands in the same relation to an algebraic function as an irrational number stands to a rational number. The same area of ​​overcoming finiteness includes extremely interesting, from a theoretical-cognitive and ontological point of view, studies of signs of convergence and divergence of infinite series, in connection with the question of increasing and decreasing functions and the theory of definite integrals. Here are the “ideal functions” Π. du Bois-Reymond can again, in a certain sense, be equated to irrationality, but not among numbers and not among functions, but among integrals. Research by N. Abel, N. V. Bugaev, P. du Bois-Reymond, E. Borel, J. Hadamard, A. Poincaré and others 841, despite the specialty of the problems posed there and the methods used there, are of the greatest importance for philosophy, and one can only be surprised that so far almost no applications have been made of them 842. R. Dedekind, – Continuity and irrational numbers [1872]. Per. with him. with approx. S. O. Shatunovsky. Odessa, ed. "Mathesis". Ed. 2nd, 1909, - with accessory. articles “Doc. noun trans. numbers." – Weber. Wellstein. – Vasiliev, §§ 18–31. – J. Tannery, – Introduction to the theory of functions of one variable, 1913 (French 1st ed. became bibliographical edition). - His, - Theoretical course. and practical arithmetic. Per. A. A. Kotlyarevsky, ed. D. L. Volkovsky, M., 1913. – M. Volkov, – Evolution of the concept of number. St. Petersburg, 1899. - F. Klein, - Questions of elementary and higher mathematics, Part 1, Odessa, 1912, trans. edited by V. Kogan, ed. Mathesis, pp. 47–56. – F. Klein – Anwendung d. Differential-u. Integralrechnung auf die Geometrie. Eine Revision d. Principien, Lpz. 1901, 2nd Ausg. 1907. – A. Focc, – On the essence of mathematics. Per. I. V. Yashunsky. St. Petersburg, 1911. Publ. "Physics". – Ch. du Méray, – Nouv. Précis d'Analyse infini tésim. Paris, 1872 (he calls the main series “convergent variant”, and equal series - “equivalent”. - G. Cantor, - Ueb. die Ausdehnung eines Satzes aus der Theorie d. trigon. Reihen (“Mat. An.”, Bd. 5). – Pasch, – Einl. in d. Dif. u. Int.-rechn., Lpz. 1882. – B. Russel, – Principles of mathematics., 1902. – Heine, – Die Elemente d. Functionenlehre (“Crelle’s Journ.”, Bd. 74). – Weierstrass’s theory is not presented in his original works: Korsak, – Die Elemente d. – Grundl. für eine Theorie d. Fuktionen, Halle, 1880; 2-te Aufl., 1898 A. Pringsheim, – Irrationalzahlen u. Konvergenz unendlicher Processe (“Enc. d. Math. Wis.” Lpz. 1898–1904, Bd. 11, SS. 47 ff.). Natorp, – Die logischen Grundlagen d. exakten Wissenschaften, Lpz. 1910. – O. Stolz u. Borel, – Leq. s. la Th. d. fon.,, IV, pp. 54–55. N. H. Abel, – Note sur un mémorie de M. L. Olivier ayant pour titre “Remarque sur les séries infinies...” (“Oeuvres compl.” de N. H. Abel, nouv. éd. par L. Sylov et S. Lie, T. 1, p. 399–402). Abel proves that the criterion for convergence cannot be given as an equality, – [H. V.] Bugaev, – Infinite convergence. rows according to their external mind. M., 1863. – His own, – Introduction. in analysis and differential. isch., lit. Lek., ed. 2nd, [M.], 1898, pp. 143–144. – L. Euler, – De Infinities infinitis gradibus (“Acta Petrop.”, 1778). – P. du Bois-Reymond, – Die allgemeine Funktionentheorie, Tübingen, 1882. – E. Borel, – Leçons sur la théorie des Fonctions, Paris, 1898. – His, – Leç. s. les Fonctions entières, P., 1900. – His own, – L. s. l. séries divergantes, R., 1901. – His own, – L. s. l. séries à termes positifs, rec. et red. par R. d'Adhénar, P., 1902. His, L. s. l. Functions méromorphes. – H. Parfentiev, – Research on the theory of growth of functions, Kazan, 1910. Of the few attempts in this for example, and even then only in relation. irrationalities, - I know: the attempt of Solomon Maimon (about him, see: B. Yakovenko, - Philosophical concept of Sol. Maimon, “Questions of Phil. and Psych.”, 1912, books 114 (IV) and 115 (V); then: Benno Kerru, - System einer Theorie d. Grenzbegriffe. Eine Beitrag zur Erkenntnisstheorie. Her. von G. Kohn, Lpz. u. Wien, 1890. Posthumous work. - K. Zhakov, - Fundamentals of the evolutionary theory of knowledge (limitism). P. Florensky, - Limits of epistemology, Serg. Pos., 1913 (= “God. V.”, 1913, January). The newest transcendentalists also use the concept of limit, but, surprisingly, not to the extent that they could use it, not only without violating, but even strengthening the main lines of their constructions.