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There is a unique civilization in the world (Chinese), which, developing a systematic reflection on the ways of forming concepts characteristic of it, as well as on the deductive reasoning techniques that correspond to these methods, has formed a system of corresponding logical and methodological views on a fundamentally different linguistic basis than in other civilizations that created logic. While ancient Greece and India, the founders of the original logical traditions, share the same Indo-European linguistic foundation, the edifice of Chinese logical and methodological thought was built on a completely different linguistic foundation. The most significant feature of this otherness is the primordial difference between phonetic and ideographic writing that separates Indo-European and Chinese civilizations. This linguistic fact only most clearly reflected that radically different from the Western (in particular, ancient) attitude of consciousness, which contributed to the formation in China of a style of thinking radically different from the Western one. Later, at the level of theoretical reflection, this attitude was materialized in the exceptional originality of the very type of theorizing characteristic of Chinese logical and methodological thought. First of all, this is reflected in the striking dissimilarity of theoretically conscious ways of forming concepts with the standard European interpretation of concepts.

The peculiarity of Chinese logical and methodological techniques begins with a very special interpretation of concepts. Since the meaning of concepts is in the generalization of the individual, the ways in which concepts are formed are determined by the nature of the generalization used. The specificity of the Chinese generalization lies in the fact that it is based not on the intuition of a class (sets, aggregates, etc.), but on the idea of construction1.

The obvious (but far from the only) basis of this Chinese attitude to the constructibility of objects of discourse is such an important constructive feature of Chinese hieroglyphic writing as its characteristic arrangement of the whole (composite character) from primary evidence (pictograms) through intuitively transparent operations of their spatial juxtaposition. Just in case, let me remind you what this is about.

The pictographic origins of Chinese hieroglyphics are indisputable, and the Chinese cultural bearers themselves have always been aware of hieroglyphs as images.-

1 For the fundamental opposition between class and construction in logic and mathematics, see, for example: [Smirnov, 2001, pp. 402-437; Weil, 2005; Geiting, 1965].

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specific items 2. Pictograms, i.e. simple pictorial signs, are the oldest representations of objects schematized according to the requirements of calligraphy and the general system of written signs, such as the stylized image of a hunting dog mentioned above, drawings of a tree (mu ), a mountain (shan ), a water stream (chuan ), the sun (zhi ), the moon (yue ) etc.

Ideograms (the traditional Chinese term for this category of signs refers here to "combining meanings") they belong to the class of two-part signs, in which a complex sign consists of two simple ones, and each of the "summands" retains its value as part of the "sum". For example, the character Xiu "rest" is a combination of the characters Ren" man "and mu" tree", thus depicting"a man under a tree". In other words,"person" +"tree" ="rest". Similarly, combining the pictograms"sun" and" moon "in one sign creates an ideogram light", which is a juxtaposition of stylized images of both luminaries ("sun" +"moon" ="light"). Another expressive example: the characterdun"east" is obtained (according to traditional etymology) by combining images of the sun and a tree, representing the image of the sun shining through a tree ("sun" +"tree" ="east"). The number of such examples could be significantly multiplied.

I emphasize the theoretical awareness on the part of Chinese traditional philological science of the constructive and mathematical nature of such a combination of meanings, when the concatenation of two simple signs into one complex sign leads to a synthesis - a kind of "addition" - of the meanings represented by these simple signs into one composite (so to speak, "total") meaning denoted by the resulting complex sign. This awareness is reflected, in particular, in the Chinese term itself, traditionally assigned to the ideographic category of signs - "[characters], combined meanings" (hui yi ) [see: Xu Shen, 1994, p.1332].

The deep form of constructiveness inherent in Chinese writing is already found at the basic level of the semantics of a single pictogram. From the point of view of semantics, there is a real gap between phonetic and ideographic writing. In the terminology of the American logician and philosopher C. S. Peirce (1839-1914), who subdivided signs into iconic (formed on the basis of the similarity of the signifier and the signified), index (created by the contiguity of the signified and the signifier) and symbolic (generated by establishing a connection between the signified and the signifier by conditional agreement), pictograms of popad-

2 In the words of Confucius: "When we look at the character quan (dog), it is as if [we see that someone is] I drew a dog" [Xu Shen, 1994, p. 844]. The following early forms of the hieroglyph confirm the validity of this judgment::

These are the oldest examples of Chinese writing known to date, dating back to the Shang-Yin Dynasty (mid-2nd millennium BC) and representing inscriptions on fortune-telling bones. Since that time, the ways of writing and the appearance of hieroglyphs have repeatedly changed, until the modern style of writing them developed in the first century. The composition of Chinese characters by this time also lost its original uniformity and (according to modern linguistic concepts) is reduced to the following three categories of characters:: 1) pictograms and ideograms; 2) phonoideograms (two-part signs in which one part indicates the meaning, and the other-the sound of the whole formed by them); 3) transcriptions (hieroglyphs that act as a phonetic record of homonymous words - the commonality of sound determines the use of the hieroglyph "not in its meaning").

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go to the category of iconic signs. At the same time, the basis of the usual sound-letter writing is formed by signs-symbols. In contrast to symbolism-the conventionality of phonetic writing, due to the factor of conventionality of sound mediation, when the sound, so to speak, mediates between the sign and its meaning, the pictogram directly - by its graphic structure - depicts the represented object, or rather its vision and understanding by the creators of iconic writing. The fundamental difference between phonetic and ideographic writing is indicated by the fact that in a certain sense we are talking about the difference between a language and a non-language, respectively. After all, the conventional sign (i.e. symbolic) function characteristic of phonetic writing in the case of ideography is opposed by the iconic, i.e. pre-linguistic pictorial function of writing integrated into the language. Therefore, the semantic distinction between a symbol and an icon (conventionality of a sign vs its "naturalness") is fundamental for any serious analysis of the logical and methodological aspect of Chinese culture.3

Since in many cases the pictography is not just a naturalistic reproduction of pre-set objects, but rather a projection of a reality that has yet to be created, very often the pictogram is not just a picture, but first of all a kind of "drawing" of the object depicted on it. That is, we are again talking about construction.

As an example, consider the structure of the grapheme "Tsar" (Van ):

Figure 1

According to the canonical interpretation of this pictogram [Dong Zhongshu, ts. 11, p. 6], the "King" is the only person who connects Heaven, Earth and Man (represented in Figure 1 by three horizontal lines). Thus, there is a generalization of the three levels of being (upper-Heaven, lower-Earth, and the center of the upper and lower-Man) by reducing them to a highlighted-again central-part of one of the generalized levels (human), visualized by a vertical line [. As you can see, the very graphic device of the "Tsar" icon provides instructions for creating the corresponding reality.

As applied to the problem of generalization, the consequence of the substitution of the idea of a set by the idea of a construction, which I mentioned above, is a fundamentally different understanding of the generalization procedure itself than in traditional logic (in the sense of sub-generalization).-

3 The confusion of the concepts of iconicity (imagery) and symbolism (symbol), unfortunately common in the Sinological literature, is a serious obstacle both to understanding the specifics of the Chinese cognitive-theoretical attitude in general, and to identifying, describing and analyzing the originality that distinguishes the logical and methodological thought of ancient China in particular. Moreover, ignoring the semantic features of the iconic reference, which contrast with the conventional way of designating that is characteristic of symbolism ("blind implication", not supported by imagery), usually leads to an untenable conclusion about the allegedly negative impact of the clarity inherent in Chinese writing on the level of abstractness of concepts expressed using this type of writing. The confusion I have just noted largely blocks the perception of the very problem of the existence of logical and methodological theories in Chinese antiquity, forcing us to perceive Chinese theorism as a manifestation of primitive magical and mythological thinking, or at best as a kind of protoscience.

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the formal scheme according to which individual entities are subordinated to the concept in the process of forming the latter. From the classical point of view, the distinctive feature of a concept is its "generality" - in the sense of the generality of a feature, equally inherent in all possible "carriers" of this feature, which form the scope of this concept. On the contrary, in our case, generalization - the transition from the individual to the universal-is understood as the result of replacing many singularities with one of the same singularities. Here generalization consists in choosing - for some reason - one of the generalizable individual objects as a representative (representative) of all generalizable objects. This type of generalization is commonly referred to as representative abstraction. As a result, a concrete representation begins to play the role of a concept. Thus, the traditional concept of generalization through the abstraction of features (hypostatized later in the form of some independent ideal entities) is replaced by the understanding of generalization as a representative abstraction.

It cannot be said that the specificity of the Chinese understanding of generality and the unusual Chinese interpretation of the central epistemological procedure (generalization) did not attract the attention of scientists. On the contrary, it was in her that the leading Western sinologists saw the quintessence of Chinese mental originality. The abundance of M. Granet's insightful conjectures on this subject crystallized into a clear contrast formulated by G. Wilhelm, on the one hand, to the" vertical "gens-specific ordering characteristic of European science, and on the other-to the one-level "correlative" style of thinking, in which there is no subsuming of less general concepts under more general concepts, but rather their alignment within the same framework. or other classification scheme 6.

The above works contain a wealth of factual material and many valuable observations. Thus, A. I. Kobzev quite correctly qualified the Chinese version of generalization as a representative abstraction. However, its main pathos was to emphasize the arbitrarily conventional, value-normative, axiological, etc. 7, i.e., the character of the Chinese version of the generalization devoid of general significance, which indicated insufficient consideration of the specifics of the Chinese generalization, as a result of which it did not go beyond the correct qualification.

Depending on the way the volumes of concepts are constructed, I distinguish two types of generalization: generalization-reduction (yue ) and generalization-reduction (tun ) [see: Krushinsky, 2002; Krushinsky, 2003].

4 That is, hierarchizing concepts - establishing a kind of "subordination" for them - according to the degree of their generality using the class inclusion relation.

5 Also referred to as" coordinating "(in contrast to" subordinating") and" associative " thinking [see: Needham, 1956, p.280-281].

6 In Russian Sinology, V. S. Spirin [Spirin, 1961] raised the problem of the peculiarities of the Chinese version of generalization. Following this, A. I. Kobzev thoroughly worked on it [Kobzev, 1986; Kobzev, 1994].M. V. Isaeva also addresses this issue in her relatively recently published monograph [Isaeva, 2000, pp. 136-139].

7 In his opinion, generalization takes place "not by abstracting the ideal properties of different objects of a certain set, but by highlighting one object, or the name of this object ... as a representative of the entire set of homogeneous (tong lei) objects: this is how the North Star unites... the remaining stars ("Lun Yu"), and the hub - spokes in the wheel ("Tao te ching")". [Chinese Philosophy..., 1994, p. 79]. That is, the representation of "a class of objects is carried out by isolating the most characteristic (valuable, main, etc.) object from it as a representative without abstracting the properties of the class" [Chinese Philosophy..., 1994, p. 308].

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I have already written about the concept of generalization-reduction [Krushinsky, 2002, pp. 174-179], so I will not dwell on it now and will focus on generalization-reduction. I will start with the fundamental difference between this type of generalization and the already considered generalization-reduction. Despite the common goal (the formation of concepts) and their equally inherent constructiveness (defining the scope of a concept by means of a certain construction), the types of construction in these two types of generalization differ significantly from each other. If the reduction generalization is based on the principle of mathematical induction (transition from n to n + 1), then the construction inherent in the reduction generalization is of an algebraic nature. In the case of generalization-reduction, we are talking about exploiting some characteristic properties of the simplest algebraic structures 8 (for example, the properties of opposite elements of a finite additive group gn of residues modulo n and the exact upper face of the lattice L). At the same time, taking into account the constructivist, algorithmic style of Chinese thinking in the case of generalization-reduction under consideration requires a preliminary analysis of such a necessary condition for this logical operation as numerical encoding. The problem of numerical coding usually appears in the synological literature under the name of the very "correlative thinking" that was mentioned on the page above. The fact is that outwardly correlative thinking looks like such a method of organizing knowledge, the brightest distinguishing feature of which is operating with numbers and appealing to arithmetic laws in contexts that are apparently far from mathematics and natural science. After all, as already mentioned, the peculiarity of the Chinese logical and methodological approach is that the main sense-organizing load here is not the hierarchy of concepts according to the degree of their generality using the class inclusion relation, but the introduction and mutual coordination of concepts through the so-called "classification schemes"9. But since classification schemes are simply different ways of encoding generalizable objects and mutually related concepts, using a number of selected natural numbers (usually not exceeding the first ten), the logical and methodological primacy of classification schemes reveals itself in an effort to digitize everything and everything.10

8 Of course, there is no need to talk about any awareness - especially an abstract representation - of these structures in a general form (at least in the usual sense), which did not in the least prevent the constant and widespread use of these structures.

9 The concept of classification scheme, which is fundamental to Chinese traditional thought, was introduced by Granet [Granet, 2004]. For contemporary Russian studies that discuss this concept or related issues, see [Karapetyants, 1989] and, especially, [Ageev, 2005].

10 A brief introduction to the history of the issue. The famous French sinologist M. Granet was the first Western scientist to take seriously the peculiarity of the Chinese position on numbers. In "Chinese Thinking" (1934), he consistently and convincingly demonstrates the leading role of numbers in Chinese culture, illustrating this demonstration with a number of vivid examples. However, M. Granet's remarkable achievements in understanding the importance and revealing the peculiarities of Chinese numerical mechanics are combined with an extremely unsuccessful interpretation of the peculiarity that reveals the use of numbers by Chinese scientists of the ancients: I mean the disorienting emphasis on their allegedly insufficient quantiativity. It is clear that the functioning of numbers within the framework of Chinese classificationism is very far from the usual role of numbers in Modern European quantitative natural science. However, this indisputable fact still does not imply that they are" extra-mathematical", as M believed. Most of the researchers who have addressed this issue to one degree or another have followed him. For example, J. Needham, with his pathetic (but rather superficial) denunciation of Chinese " empty symbolism "(not to be confused with the "mathematized hypotheses" of modern natural science!), as well as the "illusory realm of numerology", where numbers do not serve as an empirically based and rationally quantitative aid in the study of natural science phenomena, but, on the contrary They claim unnatural dominance over them (see Needham, 1956, pp. 325-326).

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The most well-known manifestations of this tendency are the ubiquitous and constant references to various classifications that are characteristic of Chinese traditional discourse. The basic classifications are the binary classification, conceptualized using the concepts of Yin and Yang (), the triadic classification of Sancai (, literally - "Three Materials"), understood as the trinity of Heaven, Earth and Man (Top, Bottom and Middle, respectively) and, finally, the fivefold classification, denoted by the term Five Elements (Wu xing )11.

In essence, we are talking about dividing the whole of both real and only conceivable reality into two, three or five classes of things. It is important to understand that this practice does not involve the use of numbers for the purposes of banal counting of objects or utilitarian measurement of quantities, which seems to be so certain of those researchers of Chinese traditional thought who from the very beginning choose a methodologically inadequate approach to Chinese classificationism-otherwise they would not consider it just as a Chinese version for various reasons failed science (and, of course, experimental mathematized natural science acts as the standard of science itself)12. But it should be noted that their chosen perspective of consideration excludes the possibility of an adequate approach to the phenomenon under study: after all, in this case we are not talking about natural science at all (in the rudimentary form of "cosmological speculations" or some other). Moreover, it is not at all about empirically established facts or verifiable hypotheses. Classificationism is an essential part of the Chinese conceptual framework. The latter is not extracted from experience, but on the contrary, is a necessary condition for the conceptual development of any experience. In other words, the tasks of Chinese classification - in accordance with its epistemological status - are by no means natural-scientific, but logical and methodological. They are reduced, firstly, to a kind of "finalizing" potentially infinite subject areas by replacing arguments about an infinite number of objects with arguments in a suitable module (for example, modulo 2 for Yin-Yang or modulus 5 for the Five Elements) and, secondly, to structuring the finite set of objects obtained in this way or concepts in accordance with the algebraic features of a given number.

I'll start in order - with the first of the two points listed above. Take, for example, the binary classification and its corresponding binary oppositions. The concept of Yin-Yang sets the minimum and therefore the most important classification scheme, denoted and simultaneously visualized by the number two () 13. This universal binary classification reduces the entire infinite variety of things to exactly two things-Yin and Yang 14.

The situation is similar with the threefold and fivefold classifications - through these classification schemes, the discourse is radically simplified, since the entire universe of consideration is reduced to three, respectively five, objects.

Thus, the classificationism described above, on the one hand, is the first step in the direction of generalization-reduction, since it reduces the entire uni-

11 The five elements are the element of Water, the element of Fire, the element of Wood, the element of Metal, and the element of Soil.

12 See, for example, [Graham, 1989, p. 315-382], where the problems of classificationism and "correlative thinking" are categorized as "cosmology".

13 The opposition of Heaven and Earth is the most important "cosmological" exemplification of the Yin-Yang concept pair: "Split in two to represent the Duality [of Heaven and Earth] "(Feng er, wei er, and xiang liang ). [Zhou And zhengyi, 1935, p. 68].

14 The oppositions associated with this dichotomy are not only a Chinese phenomenon. Systems of binary oppositions (not necessarily coinciding with the Chinese one) are found both in primitive peoples and in ancient natural philosophy. Of course, in the latter case, not as a relic of mythological thinking, but as its conscious processing.

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version of reasoning to a finite and additionally digitized space (consisting of two, three or five positions). On the other hand, it obviously represents the beginning of numerical coding of discourse objects, since each of the classification categories - respectively, each thing included in this category - is assigned a certain number.

Now let's talk about the spatial structure defined by the classification schemes under consideration. In the case of China, the key role of individual numbers (in contrast to natural series that are almost impersonal, like points on a straight line) as a tool for two-dimensional (rather than exclusively linear) ordering is well known.15 Indeed, in the case of classification schemes, numbers act as real individuals (a kind of eidos), differing from each other by unique sets of features, in particular, by their figured properties (i.e., their organization on the plane). For example, the number " two " (), exemplified by Heaven and Earth, as well as the number "three" (), spatially interpreted as an image of the Top, Middle and Bottom (where the Top is the Sky, the Middle is a Person, and the Bottom is the Earth [see: Xu Shen, 1994, p. 23]), were realized Chinese thought precisely as a two-and three-tiered vertical ordering.

The number "five "was represented by a figure of the following type: and was also interpreted in" cosmological " terms 16.

The main advantage of such a spatial representation of numbers (i.e., it is not limited by linearity) is the visualization of algebraic structures (group and lattice) associated with the classification schemes represented by these numbers. For example, looking at a diagram formed by two opposing features gives the impression of minimal symmetrical placement. Since this placement is the smallest possible, the difference between the symmetrical objects should be maximum 17. Such a maximum difference can only be the relation of mutual opposition. The impression of a graphical representation of a binary scheme (which is an image of symmetry) develops into confidence when we turn to the numerical representation of the opposition of Yin and Yang, represented by the named diagram. As we know, the graphic opposition of the upper (Yang) and lower (Yin) lines of the diagram in the area of numbers corresponds to the Odd-Even opposition. But the last opposition is an example of one of the most important algebraic structures - the additive group, i.e., the residue group modulo " twoG218. The fact is that the mathematical concept of a group is essentially an explication of the general principle that lying on a group is a group.-

15 Here it is enough to refer to Granet's fundamental "Chinese Thinking" already mentioned above [see: Granet, 2004, pp. 103-234].

16 The author of the most authoritative explanatory dictionary of antiquity, Xu Shen (c. 58-147), explains the ancient inscription of the five elements as follows: "Five elements. [Formed] from a binary. Yin and Yang intersect [in space] between Heaven and Earth (Wuxing ye, cong er, yin yang zai tiandi jian jiaou ye )" (Xu Shen, 1994, p. 1282).

17 After all, their opposition is not mediated by anything - accordingly, there can be no gradualness and gradation in the transition from one of them to the other.

18 G2 = (H, N), where H represents even (Yin) and H odd (Yang) numbers, the group operation f is an addition operation defined as follows: H + H = H = H + H, H + H = H = H + H; the neutral element (that is, the zero of the additive group) is H, and the opposite (opposite in additive notation) element for H is H itself and similarly for N. In other words, G2 is the group of residues (0, 1) from dividing integers by two. The group-theoretic nature of the Yin - Yang pair was pointed out by the modern Chinese researcher Yi Jing in the late 80's of the last century (Dong Guangbi, 1987, pp. 90-91).

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based on the simplest case of symmetry - mirror symmetry 19. The principle is that each of the two objects connected by a symmetry relation (for example, point A and point A*) is a mirror double of the other (i.e., point A will be a mirror double of point A* and vice versa). Moreover, this relation of mutual opposition between A and A* is specified by the following formal condition: for all objects connected to each other by the symmetry relation, the operation g is given in such a way that the result of this operation for mutually opposite objects A and A* is always zero: g (A, A*) = O 20.

As a result, the seemingly very primitive binary ordering of objects based on their mutual opposition (i.e., mirror symmetry), when examined more closely, in many cases reveals the features of a rather nontrivial algebraic structure that appears behind it, namely, the group structure. After all, it is quite obvious that the algebraic hypostasis of the binary Yin-Yang classification scheme (i.e., the Even-Odd pair) induces a group structure on binary oppositions that exemplify the binary classification scheme.

At first glance, the outline of the number" three", which represents the ternary classification, does not significantly differ from the image of the number" two " analyzed above: the difference between the diagrams looks purely quantitative. In fact, a seemingly insignificant difference of just one feature marks a difference between the two diagrams, no less than what separates the Even and Odd they represent. What is the graphical equivalent of the fundamental number-theoretic difference between Even and Odd? This is the presence or absence of a central figure in the shape layout. In a diagram made up of two figures (upper and lower lines), there is no such central figure.On the contrary, in a diagram, there is such a central figure (central line). According to Chinese general philosophical concepts, the relationship between the center and the periphery is always a relationship of domination and submission. Therefore, if the number two pattern visualizes the idea of symmetry, then the image of a triple is a graphical representation of the idea of hierarchy. A mathematical refinement of the latter is the concept of a lattice.

Reference to the numerical aspect of the diagram representing the ternary classification scheme confirms that it has a lattice structure of the residue ring modulo three Z 3 = (1, 3, 2), where the number 1 indicates the Sky, 2 indicates the Earth, and 3 indicates the Person 21. Note that the triple majorizes the other two elements a grid, being its largest upper face. The zero of the corresponding additive group G3 is the number 3 (Person). The numbers 1 (Sky) and 2 (Earth) are opposite to each other (because 1 + 2 = 3), and the number 3 is opposite to itself (3 + 3 = 3 modulo "three").

19 As is known, any two points A and A* are mirror symmetric with respect to the point O, if O is the midpoint of the segment AA*. According to the famous German mathematician G. Weyl, who gave a lot of thought to the problem of symmetry, "historically, the concept of a group, although in an implicit form, first appears in the phenomenon of symmetry; art, and above all the art of decorating surfaces with ornaments brought to high perfection by the ancient Egyptians, preceded science" (Weil, 1989, p.263).

20 If, for example, the point A is numbered with a positive number n, and the symmetric (i.e., opposite to it) point A* is numbered with the same number, but taken with the opposite sign, then it is clear that the sum of these numbers will necessarily be zero: n + (-n) = 0. Similarly, in the above-mentioned table In Even and Odd arithmetic, numbers of the same parity (i.e., either both are even or both are odd) are taken as the numerical codes of mutually opposite points A and A*, so their sum also always turns out to be zero: H + H = H = H + N.

21 I emphasize that this standard numbering of components of a ternary scheme is only one of such numbering.

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From the canonical interpretation I gave earlier of the initial outline of the number 5, it can be seen that it is understood as a representation of the fivefold scheme of the Five Elements 22. Five is an odd number, so the diagram that "portrays" it is centered (the figure has a center of symmetry). That is why the situation with the fivefold scheme is close to the situation described above with the ternary classification scheme. It (also known as a ternary scheme) combines a lattice structure with a group structure. Therefore, the fivefold scheme, on the one hand, is associated with the group of deductions modulo "five" G 5 = (1, 2, 3, 4, 5), where, according to standard numbering, the number 1 denotes Water, 2-Fire, 3-Wood, 4-Metal, 5-Soil 23. C On the other hand, the fivefold scheme is related to the natural linear ordering of these numerical codes, which is a lattice L 5 = (1, 2, 3, 4, 5) with the largest face equal to 5.

From the relations connecting the elements G5 with each other, it is clear that with this numbering, Water is opposite to Metal (since 1 + 4 = 5), Fire is opposite to Wood (2 + 3 = 5), and Soil, being the zero element of the group G5, is opposite to itself. I deliberately emphasize the obvious dependence of the antithesis relation between elements on the choice of element numbering. Although, as already mentioned, the numbering varies, the numbering of the central element (Soil) by the number "five "is invariant with respect to various variations - the element Soil is always encoded by the number 5 (or its analog - the number 10 - in the group of deductions modulo" ten " G10) and occupies the center with the spatial arrangement of the Five elements 24.

Further, I intend to show that the ultimate goal of structuring a binary, ternary, or fivefold set of objects or concepts in accordance with the algebraic features of the corresponding numbers (2, 3, 5)25 is to create the necessary prerequisites for the subsequent generalization of this structured set using one of the members of this set.

As for numerical coding, Leibniz had a somewhat similar idea with his famous project "universal characteristic". As you know, in it he suggests using prime numbers and decompositions of integers into prime factors to describe the world of statements. And in such a way that various questions about these statements can later be solved by simple arithmetic considerations (first of all, using the concept of divisibility) [see: Leibniz, vol.3, 1984, p. 533].

It is interesting that the divisibility relation, which Leibniz tried to rely on in the arithmetic of syllogistics, plays an important role in Chinese logic. First, the very name of the generalization-reduction logical operation discussed in this article is directly related to it. After all, the terminological meaning of the hieroglyph tun () is the designation of a mathematical procedure for finding a common multiple of two numbers, which is of key importance for manipulating fractions [see about this: Li Yan, vol. 1, 1954, pp. 28-32]. It is this special mathematical meaning of this hieroglyph (reduction of fractions to a common denominator!).-

22 The oblique cross is a representation of the four sides and the center (representing the fifth element).

23 " The first is Water, the second is Fire, the third is Wood, the fourth is Metal, and the fifth is Soil "[see Shangshu zhengyi, 1935, p. 188].

24 The element "Soil", which is numbered five or ten, is always the zero of the group of Five elements, which is confirmed by the following direct indication of the "emptiness" of both five and ten: "In [the system] The five elements five is empty..."(wu xing ye er wu Wei xu ). [cit. by: Hui Dong, 1989, pp. 337-338]. "Even though five and five [add up] give a ten, the ten is empty" (wu wu wei shi er shi wei xu ) [Hui Dong, 1989, p. 338].

That is, the ordering (for example, symmetrization or hierarchization) of objects according to one of a number of simple algebraic structures (such as a group, lattice, or ring).

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my translation of the tun hieroglyph by the word "ghost" has been corrected. Secondly, the algorithm for finding a common multiple (and in particular, the smallest common multiple, hereinafter referred to as the NOC) is directly used in some important cases to mathematically base the logical operation of generalization-reduction. So, in the case considered earlier, the traditional constructivist interpretation of the graphic structure of the pictogram " Tsar "(shown in Fig. 1), the graphical design is based on the corresponding numerical algorithm, which makes it possible to uniquely calculate the numerical code of the result of generalization-reduction from the numerical codes of generalized entities. However, the standard numbering 26 is not suitable for such a foundation, so in this case a completely different encoding is used: The sky is numbered by the number 9, the Earth by the number 6, and the Person by the number 8 [see: Krushinsky, 1999, pp. 149-150].

The algorithm that operates with the new numerical codes of the components of the ternary classification is the algorithm for finding the NOC numbers 9, 6, and 8. Therefore, the numerical code of the "King" (strictly speaking, the vertical component of the "King" icon) is the number 72 (NOC (9, 6, 8). Since the vertical line of the named pictogram is nothing more than a graphical representation of the generalization-the reduction of Heaven, Earth and Man, then its numerical code (number 72) it will just be the numeric code of the result of generalization-casting.

To better represent the algebraic structure of the numerical code of the "King" (which is a generalization-reduction of Heaven, Earth, and Man), which is given by the set of its divisors ordered with respect to divisibility, we represent this ordered set of divisors (the grid of divisors of the number 72) using the following diagram: 27:

2. L 72 - grid of divisors of the number 72.1

26 When the number 1 represents Heaven, the number 2 represents Earth, and the number 3 represents Man.

27 This scheme consists of two chains of multiples starting from one and converging to seventy-two. Any two numbers in the scheme have both the smallest common multiple and the largest common divisor (the smallest upper and largest lower faces of the lattice).

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From Fig. 2 it is clearly seen that the mathematical mechanism of "generalization-reduction" of Heaven, Earth, and Man (which ensures the logical universality of defining the scope of the concept of "King") is the reduction of the entire structured set of divisors of the number 72 to one (the largest!) of its components, 28.

However, when comparing Leibniz's project of "universal characteristic" with the ancient Chinese logical and methodological principles of numerical coding (found in the form of "correlative thinking") We must not lose sight of the fundamental difference between the two approaches. For Leibniz, the primary concepts are 29 and the generic relationships that connect them. The numbers and the divisibility relation that connects them, put in accordance with these generic relations, serve Leibniz only to recode the already existing conceptual relations in a more precise and convenient language of elementary number theory (so that, according to Leibniz, one can "not reason, but calculate").

In some important ways, the Chinese numeric encoding is characterized by the opposite situation. The goal of such coding is to standardize algorithmic operations on the encoded objects . Thanks to such standardization, all algorithms on objects of arbitrary (sic!) nature are reduced to elementary arithmetic operations and relations. Since - in contrast to Leibniz-Chinese scientists do not care about the objects themselves (as we know, they do not even have their own constant number) and not the relationships that connect them, but the operations that are performed on them30, depending on the application of an algorithm to any given object, the numerical code of this object may vary, to match the algorithm used. This eliminates a seemingly essential feature of the code - the ability to use any given object code to unambiguously restore the object hidden behind this code. But this weakening of the concept of "code "is fully repaid by success in, so to speak," globalizing " those error-free, thanks to their elementary nature, number - theoretic algorithms that - thanks to such a broad understanding of coding-can cover any form of discourse.

One passing remark. As we can see, in order to expand the scope of numerical algorithms, Chinese theorists were willing to sacrifice a lot: they did not particularly care about the substantive validity of their classifications - sometimes reducing quite heterogeneous things into one heading, 31 but they did not seem to be very concerned about the formal side of the matter, and did not bother to follow the sequence of numerical numbering of classification headings. What they were absolutely uncompromising about, however, was their unwavering adherence to the laws of arithmetic. Number-theoretic regularities are the real life nerve of Chinese "correlative thinking", the holy of holies of Chinese logical and methodological thought.

28 The situation with the construction of the scope of the concept of "King" is typical. In generalization-reduction, the representative is always either the largest element of the grid of divisors of a fixed number Ln (for example, the number 72 in the "King" case discussed above), or the zero of a finite additive residue group modulo n (examples will be given below).

29 The contents of which are quite traditionally conceived by Leibniz as sets of features, and volumes - as classes of objects that have these features.

30 After all, the numerical encoding of objects is focused on their standardization.

31 It is enough to recall the famous parody of H. Borges on Chinese classificationism - his description of the sections and subsections of a "certain Chinese encyclopedia", which is striking not only for its whimsical arbitrariness, but also for its outright paralogism [see: Foucault, 1977, pp. 31-32].

page 15

Both of these statements would have looked trivial if it were not for one curious misunderstanding - the continuing existence of a numerological myth dating back to M. Granet, which calls into question the functioning of numbers in the context of Chinese "correlative thinking" precisely as numbers, i.e., primary objects of mathematics.32 The methodological absurdity of statements about the possibility of the existence of "uncountable numbers" (what are they after this number?), "mathematical objects that are not related to each other according to the laws of mathematics" (then they should not be certified as "mathematical objects") and other annoying variations on the theme of the existence of numbers as certain numbers set almost a century ago, "symbols" or "emblems" that do not bind the user to anything are quite obvious. A weak, but at least some justification, here can serve only as a natural (but fatal in this case) for cultural historians, the limitations of their mathematical knowledge. But what is difficult to understand is the strange unwillingness to understand the significance of the fact that all so - called numerology is centered by two deliberately mathematical constructs-the "Drawing [from the yellow] River" (Hetu ) and "Writing [from the river]", which is well known to specialists in traditional Chinese culture Lo" (Loshu ).

It is puzzling how it is possible, having a magic square in front of your eyes (the mathematical principle of which is clear even to a schoolboy who already knows how to add the first ten numbers)33. continue to challenge the mathematical status of numbers in the context of Chinese " correlative thinking "and identify the phenomenon under discussion as a mathematically meaningless game with numbers - "numerology". After all, the key meaning of the space-number formations that are paradigmatic for all this "numerology "consists in the graphical representation not of some vaguely" symbolic "("emblematic"," aesthetic", etc.) meanings, but in the visualization of strictly mathematical, more specifically, number-theoretic, regularities. Namely, Het graphically represents the extension, so to speak, "completion" of the group of deductions modulo "five" G5 to the group of deductions modulo " ten " g1034. As far as Loshu is concerned, this graphically-numeric construction is concerned -

32 " Everyone (of the sages of ancient China. - A. K.) sought to manipulate numbers as symbols (des Emblemes. - A. K.) ... numbers in the eyes of the Chinese are remarkable for the fact that, like symbols (des Emblemes. - A. K.), are amenable due to their " capacity "(polyvalence. - A. K.) the most diverse effective manipulations" [Granet, 2004, p. 103]. "Numbers are just symbols (des Emblemes. - A. K.); the Chinese are wary of seeing them as abstract and shy signs of quantity " [Granet, 2004, p. 192]...Various phenomena of the natural and human world are described as if by an arbitrary and random set of numbers, so that the numbers themselves are taken in their qualitative, non-calculable aspect "[Malyavin, 2000, p. 299]. "The place of logic science ... In China, numerology (xiang shu zhi xue), i.e., a formalized theoretical system, the elements of which are mathematical or mathematical-like objects ..., connected, however, mainly not according to the laws of mathematics, but somehow differently-symbolically, associatively, factually, aesthetically, mnemonically, suggestively " [Kobzev, 2004, p. 131].

33 However, some Russian sinologists not only confuse themselves here, but also confuse others: on the one hand, they rightly qualify both diagrams as "graphical expressions of mathematical structures", on the other hand, they recklessly attribute the property of "magic" to both of them [see, for example, Chinese Philosophy..., 1994, p.399]. Or, even worse , both diagrams are immediately written down in "magic squares" [see, for example, Malyavin, 2000, pp. 302-303]. Meanwhile, the magic square is only Loshu. We must assume that our distinguished colleagues use in these cases the words "magic", "magic square" and the like in some meaning they know only by themselves, forgetting that they are talking about mathematical objects and their properties, and the words they use ("magic", "magic square") are also mathematical terms with a very strict meaning.

34 The mathematical structure and logical aspects of Het were analyzed by me in: [Krushinsky, 1999, p. 43-55].

page 16

the so-called magic (i.e. additive) square of order three 35-is specially designed to explicate the group structure of the fivefold classification in order to mathematically base the logical operation of generalization-reduction carried out within the framework of this classification 36.

Let us recall the canonical numbering of the Five Elements presented in Hong Fan. For a number of reasons, it turns out to be unsuitable for the implementation of the main logical operation based on the algebraic structure of the fivefold classification scheme - generalization-reduction. Therefore, along with the standard numbering of the Five Elements, there is their alternative numbering, thanks to which the group-theoretic relations in this classification scheme are built into the structure required for the implementation of generalization-reduction and, in addition, acquire impressive visibility. I am referring to the "delineation" that the Five Elements receive in the context of the so-called Drawing of the Nine Palaces (Jiugong tu ), which is a variantsquare.:

Figure 3

Here, Fire (represented by the Li trigram) is assigned the number 9 (), Water (represented by the Kan ) is assigned the number 1 (), and Wood (represented by the Zhen and Xun ) is assigned the number no-

35 This magic square, which has the following form (in modern numerical notation)::

4

9

2

3

5

7

8

1

6

it is the most ancient of all the magic squares known to date. The earliest documented evidence of it is found in China [see, for example, Needham, 1959, pp. 55-61].

36 The purpose of this ancient magic square was still a mystery. The logical application of this spatial-numerical construction that I have identified (a generalization-reduction tool dedicated to the fivefold classification scheme) allows us to reveal the true foundations of the exceptional significance that Chinese traditional science has always recognized for it.

37 The image of the "Drawing of Nine Palaces" presented below is borrowed from [Yixue datsi-dian..., 1992, p. 508].

page 17

measure 3 () and 4 (), respectively, for Metal (Qian and Dui ) - numbers 6 () and 7 () respectively, the Soil (represented by the Kun trigrams )isnumbers 2 () and 8 () respectively, and the Soil located in the center (not marked with any trigram) is given the number 5 ()38, as we will see later, referring to the number 10.

It is clear that the relation of the mutual opposition of the elements of the group, which is basic for the group structure, looks completely different here than with the standard numerical encoding of the Five Elements. If there the mutual opposition (meaningfully interpreted as the" mutual generation " of elements) connected Metal and Water (4 and 1, respectively), Wood and Fire (3 and 2, respectively) in pairs, then here the group-theoretic mutual opposition (interpreted as the "mutual overcoming" of elements) takes place between Water and Fire, as well as Wood and Fire. Metal. The soil, being invariably the zero element of the group G5, is opposite to itself in any numerical encoding of the Five Elements and for this reason is considered as a term only for itself (and no more for any of the nine numbers contained in this magic square) - after all, the opposite element of the number 5 in the additive group G10 will be Again, the number 5. That is why in the" Drawing of Nine Palaces "there is a single five in the center - it is implied that it is taken here in two copies, so that the summation operation is determined on all the numbers of the"Drawing" without exception.39

As you know, a magic (additive) square of order n is a square table of size n x n, filled with various numbers or symbols in such a way that their sum in the columns, rows and main diagonals of the table is the same. The value of this sum is usually called the magic constant of the square and denoted by the letter S [see, for example: Chebrakov, 1995, pp. 57-58]. In our case, S = 15, however-minus the five, which is the zero of the group of Five Elements 40-the number 10 remains as a magic constant. But the number 10 is a neutral (i.e., zero) element of the residue group modulo ten G10, formed by the set of nine consecutive natural numbers that fills the cells of the magic square in question. Therefore, a constant property of centrally symmetric pairs of numbers, i.e. occupying the extreme middle (top and bottom of the second column and the beginning and end of the second row) positions of this magic square, as well as its corners 41, will be their property when pairwise summation invariably give ten, that is, zero of the group G10. I will explain this with the following diagram of a modified additive square, i.e.e. the original square, in which the domain of definition of the addition operation is bounded by mutually opposite elements of the group G10:

38 For the mutual correlation of elements and trigrams, see [Jing Fan, 1993, pp. 211-241].

39 "Five da five refers to the Soil" (wu wu wei tu ) and "five da five gives the ten" (Hui Dong, 1989, p. 338).

40 It is precisely the" emptiness "of the five (i.e., in modern terms, its neutrality with respect to addition in the additive group G5) that justifies the transition from 50 - the "number of the great expansion" (dayan zhi shu ) - to 55-the " number of Heaven and Earth (Tiandi zhi shu ) - in the context of reasoning about the Five Elements. "The number of Heaven and Earth is fifty-five. In [system] The five elements five is empty. Therefore, the number of large extensions is fifty" (Hui Dong, 1989, p. 337).

41 That is, such pairs of numbers that are opposite to each other: 9 and 1, 3 and 7, 4 and 6, 8 and 2.

page 18

4

9

2

3

5

7

8

1

6

Figure 4

Figure 4 (where an additive square is shown, with the restrictions on the addition operation fixed above) clearly shows the group structure G10, in which 10 is the zero of the group G10, and 3 and 7, 9 and 1, 4 and 6, 2 and 8, 5 and 5 are pairs of mutually opposite elements of G10. Now let the generalization-reduction-be based on the group structure G10. Then the generalization of objects encoded by elements of G10 (all or only part of them) proceeds as follows: the generalized objects are first divided into pairs with mutually opposite codes, and then these codes are summed in pairs, resulting in zero. We can say that the generalization here is zeroing out.

Why is that? The basis of this final line of thought is an appeal to considerations that already relate to the lattice properties of a naturally ordered set L 10 = (1, 2,..., 10), in which 10 majorizes (hence, in a sense, includes, contains) all other numbers belonging to this set. Thus, all elements of the residue group modulo ten G10 (associated with the fivefold classification scheme) are generalized by means of the largest element of the lattice L10 (also associated with the system of Five elements), that is, the number 10 encoding the element "Soil" and coinciding with the zero element of the group G10.

Now a number of concrete examples of generalization-reduction based on the fivefold classification scheme. The most transparent examples of generalization-reduction of objects using the fivefold scheme is, perhaps, setting the volume of the concept of " Thing "by reducing everything to one of the cardinal directions, as well as similar construction of the volume of the concept of" Year " by fixing one of its parts. I will start my analysis with the concept of "Thing".

The literal meaning of the Chinese word " thing "(passed from the classical literary language to modern Chinese) is" East and West " (dong xi ). According to this etymology, the concept of "Thing" was interpreted as everything that is located within the four cardinal directions.42 Let us recall the sequence of generalization-reduction steps that I indicated a few pages earlier.

The first one (you can say preliminary) the step of generalization-reduction consists here in, so to speak, "throwing" on the world a seine with an extremely rare mesh (only five cells!) - nevertheless, everything that is potentially infinite in number is captured by this network. The spatial aspect of the fivefold classification is taken: the four cardinal directions and the center. Projection of the element of Water on the area of the missile defense system-

42 "Things arise within the four cardinal directions, [therefore all of them] are abbreviated as 'East-West', just as the fixation of the four seasons is carried out by an abbreviated reference to 'Spring-Autumn' "[Qiyuan, vol.2, 1980, p. 1526].

page 19

spatial orientations will be North, elements of Fire-South, elements of Wood-East, elements of Metal-West, elements of Soil-Center.

In the second step (from which the actual generalization-reduction begins), five spatial positions are reduced to one - the central one. The reduction mechanism is standard-it uses the group structure of the Five Elements, visualized by an additive square (Fig. 4).

Comparing the spatial values of the elements assigned to the elements by the additive square, we get the following diagram:

Figure 5

We know that the number 10, which is the constant S of the magic square (shown in Fig. 4) and at the same time the zero element of the group G10, generalizes all the elements of this group, i.e. includes them as its components. Representing this group zero as a sum of mutually opposite elements, we get: 3 + 7 (East and West) = 9+1 (South and North) = 10 (Center). The choice between two formally equally possible representations (East and West or South and North) in favor of East and West, apparently, is no longer due to formal, but to substantive (i.e., not related to the sphere of logic) considerations 43, which I will not discuss in detail now.

To confirm that the case of the formation of the concept of "Thing" discussed above is not an isolated incident, but a regular logical technique, I will list some of the most famous similar cases. It is completely analogous (and this analogy is fully understood by Chinese scientists [see fig.: Qiyun, V. 2, 1980, p. 1526] the method of birth of one of the names of the year - "Spring and Autumn" (Chun qiu ).

Here the same pair of numbers (three and seven) is summed up as in the previous case, only now they no longer encode the cardinal directions, but the seasons of the year: Spring and Autumn, respectively. The mechanism that ensures the formal validity of reducing four seasons, apparently to two-but in fact, to one, artificially introduced central fifth season-is the same as in the previous case. I would like to emphasize that it is precisely such group-theoretic considerations that make it possible to explain in principle (and not by various ad hoc tricks) the usual way of designating the entire year in written monuments of ancient China by specifying only two of the four seasons-Spring and Autumn.

43 Obviously, this means the horizontal axis (in this coordinate system) on which the human world and everything that happens in it is located. In this case, turning to the vertical axis would already mean going beyond the ecumene into the cosmic sphere.

44 Like a very simplified explanation of the reasons for the appearance of this method of naming the year, appealing to the unlikely hypothesis of a longer (compared to the standard) duration of these seasons (when "spring" supposedly means the entire first half of the year, and "autumn" - the second).

page 20

Finally, an appeal to the previously introduced basic algebraic structures of the group G10 and the lattice R10 makes the mechanism of reducing the five senses to one of them completely transparent. Namely, to the body as an organ of touch, associated precisely with the central element, that is, the Soil 45. Accordingly-to vision and hearing, the sum of which in some sense gives the whole body. Here again, as in the previous cases illustrating the logical functioning of the fivefold classification, the whole is strictly regular (and not due to some value-normative, axiological, etc. arbitrariness) reduced to its largest part. In other words, there is an arithmetic algorithm that "glues" a set of objects into one of the components of this set. As usual, this arithmetic algorithm is based on the properties of the group and lattice structures of the classification scheme of the Five Elements.

The group structure of the numerical codes of the Five Elements is used here (as in all previous analogous cases) to represent the zero of the group G10 (coinciding with the exact upper face of the lattice L10). In the case of the five sense organs discussed now, the anatomical values of the five trigrams corresponding to the Five elements are taken, so that the projection of the element of Water on the area of human anatomy will be the ears, the elements of Fire-the eyes, the elements of Wood - the nose, the elements of Metal - the mouth, the elements of Soil - the body as

Comparing the anatomical values of the elements assigned to the elements by the additive square, we get the following diagram:

Figure 6

Representing the number 10-the zero of the group G10 - as the sum of mutually opposite elements encoding the corresponding sense organs, we get: 3 + 7 (Nose and Mouth) = 9+1 (Eyes and Ears) = 10 (The body as an organ of touch, which includes the other four receptors-Nose, Mouth, Eyes and Ears). When it is formally equally possible to represent the entire five senses in two ways through pairs of mutually opposite senses, only one such representation (Eyes and Ears)is actually used. 46 There are substantial reasons for this, which I will not discuss now, since they are outside the scope of issues discussed in this article.

45 The body, or rather the skin, is one of the elements of the fivefold set of human sense organs (body, mouth or tongue, nose, ears, eyes) that corresponds to touch, and at the same time - their entire totality. So to speak, "sum" in the sense of the whole human body. This remarkable observation belongs to A. I. Kobzev [Kobzev, 1994, p. 190].

46 Along with the direct meaning, dictionaries record a so-called "figurative" meaning of the binomial " Ears and eyes "(er mu ), which clearly implies the participation of all receptors in the perception process ("informant","spy").

page 21

The results of the analysis reveal the logical nature of the Chinese version of generalization and-more broadly-demonstrate and justify (contrary to widespread opinion) the mathematical meaningfulness and logical orientation of Chinese "correlative thinking" in its numerical representation. The last conclusion (the statement about the logical and methodological orientation of Chinese numerical constructions) is of particular importance - only the use of the concept of modern logic makes it possible for an adequate perception and fair assessment of traditional Chinese thought.

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