Considering the evolution of the oldest counting systems, the author finds its similarity with the evolution of counting in children. On the basis of this typological similarity, relying on the ideas of L. S. Vygotsky, as well as on the work of the American researcher D. Schmandt-Besser on counting systems in the ancient Near East, the author reconstructs the processes in the evolution of consciousness that led to the emergence of abstract counting, writing and, as a result, conceptual thinking.
Keywords: ancient counting technologies, abstract counting, concrete counting, evolution of mental activity.
The rudimentary state in which the problem of concrete counting still exists is mainly due to the fact that until recently it was of interest only to historians of mathematics. However, the one-line evolutionist approach, which is determined to see in each phenomenon only another step in progressive development, is counterproductive when studying archaic phenomena of culture and science. Using it, the researcher can see in prehistoric and even early historical forms of scientific knowledge only the same science, only in a more primitive, rudimentary form. The idea that he is dealing with something qualitatively different from science will not even occur to him, and if he does, he will not know what to do with this discovery - his approach does not accept otherwise.
B. A. Frolov's book "Numbers in Paleolithic Graphics" (Frolov, 1974) is a work that translates this problem into a historical and cultural plane and finally provides a definite breakthrough in its development. The problem posed in it required extensive interdisciplinary research in the future with the participation of specialists in non-archaeology, but this did not happen. Meanwhile, the results of B. A. Frolov's research not only had numerous ethnographic parallels (as later pointed out by V. V. Ivanov [Ivanov, 1999]), but also found support in some ideas of L. S. Vygotsky [Vygotsky, 1996].
The fact that the Upper Paleolithic was too far removed from our epoch and from historical time in general, both in terms of calculation and technology, and in terms of cultural typology, was probably a factor that did not favor such studies, which created a certain cognitive gap. The situation changed only with the appearance of D. Schmandt-Besser's works concerning the evolution of counting systems in the ancient Near East and raising a number of topical issues of the genesis of writing, abstract counting, etc. [Schmandt-Besserat, 1978; Schmandt-Besserat, 1980; Schmandt-Besserat, 1981; Schmandt-Besserat, 1982(1); Schmandt-Besserat, 1982 (2); Schmandt-Besserat, 1983; Schmandt-Besserat, 1984]. The works of D. Schmandt-Besser directly addressed the problems of the Neolithic revolution and the subsequent evolution of the economy and society up to the appearance of the first states in the Two Rivers.
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THE OLDEST COUNTING DEVICES AND THE PROBLEM OF THE ORIGIN OF ABSTRACT COUNTING IN THE WORKS OF D. SCHMANDT-BESSER
The hypothesis of D. Schmandt-Besser about the origin of abstract counting was the result of studying the development of counting devices in the ancient Middle East between 10,000-3,000 BC, in particular bone accounts, clay three-dimensional signs and digital notations on clay tablets. The study of the same counters formed the basis of her hypothesis about the origin of Middle Eastern writing, i.e. both hypotheses are closely related. It should be noted at once that these hypotheses have split the scientific community: for example, S. Lieberman, P. Michalovsky and J. P. Blavatsky. Alden was strongly opposed, while M. Powell, on the contrary, was in favor (see [Powell, 1981]). These hypotheses were also supported by I. S. Klochkov ([Klochkov, 1984]). Objective evidence against the origin of Proto-Sumerian pictography from three-dimensional signs was put forward by A. A. Wyman ([Wyman, 1976]) and M. Green ([Green, 1981]). However, in this article we are interested not in the genesis of pictography, but in three-dimensional signs as a counting system. Here D. Schmandt-Besser has a much stronger position.
Putting forward her hypotheses, the American researcher assumes, as is customary in the history of mathematics (see: [Danzig, 1959; Smith, 1951; Kramer, 1970; Flegg, 1983]), the existence of three main stages in the evolution of counting: 1) one-to-one correspondences; 2) concrete invoice and 3) abstract invoice.
1. One-to-one correspondence. Counting in one-to-one matches consists of matching the counted items with an identical number of counting chips. The historians of mathematics mentioned above suggest that counting originally consisted only in the repeated addition of a single unit, without the idea of an aggregate quantity. Even in historical times, tribes such as the Ceylon Veddhas were at a stage not much higher than this. For example, they considered coconuts to match each coconut to a stick. Each stick added corresponded to a score of "one more" until the collection of coconuts was exhausted. Then they would just point at the resulting pile of sticks, saying, " That's a lot." In other words, there are no numerical concepts at this stage. Collections of objects are understood as a series of individual, unrelated things rather than as a connected whole.
2. A specific invoice. It is considered that at this second stage the concept of a set, or set, has already been achieved, but numerical concepts and counted objects (denotates) have merged into a single semantic block. As a result, many different things with the same numerical concept have different numerical expressions, which are called specific numbers. As a result, in a particular invoice, the numeric words that express the concepts "one", "two", "three", etc. differ depending on the invoice object: whether they count people, canoes, or trees. Relicts of this classification scheme are more or less fully preserved in many languages of the world. In particular, the author refers to the work of I. M. Diakonoff (1983), who studied the language of one of the ethnic groups of the Amur Nivkhs (Gilyaks) from this point of view. This language has 24 counting classes (an order of magnitude more than, for example, in Chinese). I. M. Diakonov finds numerous typological parallels to this in the ancient history of Mesopotamia and, according to D. Schmandt-Besser, the proposed archaeological material supports the hypothesis of the Soviet orientalist.
3. Abstract invoice. In this third and final stage, the concepts of numbers can be abstracted from the items considered. Thus, abstract numbers appear that can be applied universally, like our "one", "two", "three", etc. D. Schmandt-Besserat notes [Schmandt-Besserat, 1984, p. 57-58] that in a number of societies the words for expressing abstract numbers come from a specific numbering, it has the most frequent use.
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This particular remark is important for her hypothesis, which is as follows. All these successive stages can be clearly traced in artifacts found during excavations in the Middle East and identified as counting devices.
1. Counting dice used for counting in one-to-one matches. These are animal bones and horns bearing a series of notches found in Mesolithic sites of the Middle East around 10,000 BC. According to D. Schmandt-Besserat, they are the oldest artifacts interpreted by scientists as counting devices (Schmandt-Besserat, 1984). She believes that here we are dealing with the purest case of counting in one-to-one correspondences, since, in her opinion, there is nothing in the counting tags that would indicate any concept of a set.
2. Signs used for a specific invoice. These are three-dimensional signs that were modeled in clay in specific, systematically repeated forms. While a series of notches could only be understood by those who made them, groups of three-dimensional signs could be used for much broader communication. In particular, by comparing the imprints of three-dimensional signs with the signs of proto-Sumerian writing, the American researcher was able to show that each such sign form could symbolize a certain unit of quantity, weight or volume of a particular product [Schmandt-Besserat, 1978; p.44-45; Schmandt - Besserat, 1984, p. 54]. Thus, a sphere could mean a large measure of grain, a cone-a small measure of grain, while an oval probably represented a vessel of oil. At the same time, on the one hand, D. Schmandt-Besser managed to show that these signs were used for counting in one-to-one correspondences, i.e. one oil vessel was represented by one oval, two-by two ovals, etc. On the other hand, this sign system clearly included certain elements of abstraction. First, units of measures of real property were replaced by clay symbols. Secondly, signs abstracted data from its context, creating prerequisites for arbitrary manipulation of property. However the signs remained specific in several ways: a) the counting chips were voluminous, tangible, and manipulable; b) the signs represented the concept of quality and two concepts of quantity fused together; c) the signs represented the set as it is in nature: in one-to-one correspondences.
3. Switch to an abstract account. Then there is a change in several more stages of development of counting technologies. First, the three-dimensional signs are replaced with their images printed on an envelope made of clay tablets. At this stage, there is still no change in the practice of counting. It occurs at the next stage, characterized by the technique of drawing ideograms with the sharp end of the stylus (simply put, writing). At the same time, to indicate the number of measures of property (indicated by pictograms), numbers are used - symbols that, according to D. Schmandt-Besserat (1984, p.57), express abstract numbers. For the first time such notations appear in the pictographic tablets of Uruk IVa, around 3100 BC. e. At the same time, units of grain measures were taken as units of digital notation as, apparently, the most commonly used.
A few comments should be made here. First, it is unclear both the grounds for a particular invoice and the criteria by which the invoice was assigned to a specific invoice using three-dimensional characters, and not to one-to-one correspondences. In fact, in the real technique of counting with the help of three-dimensional signs, as D. Schmandt-Bessera reconstructs it, one-to-one correspondences are present. The argument that there is no concept of a set or set (as opposed to three-dimensional signs) in the counting bones assigned to one-to-one correspondences does not work for the following reasons. B. A. Frolov [Frolov, 1974] convincingly showed that already in the Upper Paleolithic numerical notations incised on counting bones, there is an idea of a standard set based on a common numeric
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the concept. At the same time, there is no reason to refer these numerical notations to anything other than one-to-one correspondences: the very process of notching involves just "adding one" and nothing more. In addition, the process of transition from one-to-one correspondences to abstract counting is not as transparent as it is portrayed by D. Schmandt-Besser and the historians of mathematics cited by her. It seems that there is a mixture of several levels of signification. A satisfactory solution to this problem requires a change in the research paradigm, which allows us to look at the problem in a completely different aspect.
PALEOPSYCHOLOGICAL ASPECTS OF STUDYING THE OLDEST COUNTING SYSTEMS
Studying the most ancient systems of counting, one cannot ignore the specifics of the consciousness that gave rise to them, and not pay attention to its striking difference from the consciousness of a modern European person. The use of ethno-cultural parallels makes us turn to the phenomenon of children's counting and, more broadly, children's consciousness (the possibility of such an approach was pointed out by L. S. Vygotsky and A. R. Luria (see: [Luria, 1974]).
Mastering a digital account for children in general terms is as follows. Children start counting with the help of counting sticks, fingers, etc., i.e. in one-to-one correspondences in pure form, without the idea of any sets. Then, in the practice of forming groupings, "piles", the child begins to form numerical concepts (of course, not without the help of adults). Then the fun begins. When the numerical concepts are fully formed: the child consistently and accurately identifies certain sets of counting chips with certain numbers, it is enough to show him the numbers denoting the numbers of the natural series up to ten and let them memorize (as one learns the alphabet), as the child begins to use the digital counting system within the first ten, and does it simply and quite naturally. At first, the child uses both counting technologies simultaneously, but if you immediately block the child's channel for building one-to-one correspondences (simply put, prohibit the use of fingers), he / she easily switches to the digital counting system and never returns to one-to-one correspondences.
Thus, first of all, we can say that stable numerical concepts are formed directly in the practice of counting in one-to-one correspondences. Secondly, as a result of this, in children, the transition from one-to-one correspondences to digital notation and, consequently, to arithmetic counting is carried out in leaps and bounds, without any preliminary stages. In this sense, the stage of a specific invoice looks like an extra link. As B. A. Frolov [Frolov, 1974] has already shown, stable numerical concepts began to form, and probably were formed (at least partially) already in the Upper Paleolithic; i.e., the Schmandt-Besser assumption that carriers of one-to-one correspondences are completely incapable of any abstraction is groundless. Moreover, according to V. V. Ivanov, " the earliest stages of the attitude to number in primitive tribes and in children in modern society are characterized by ability... to determine the number of large groups of objects by eye with amazing accuracy and speed" [Ivanov, 1999, p. 466], which is consistent with the research of Zh. Piaget [Piaget, 1969, p. 325]. That is, a Paleolithic person only needed to look at the sequence of notches to know their exact number. What prevented us from immediately switching to abstract counting (using the terminology of D. Schmandt-Besser)? This question can only be answered by understanding the psychosemantic foundations of a particular account. This raises the following question: can a digital account used by children aged 6 to 9 be considered abstract?
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As it has already been established, children who have mastered counting in one-to-one correspondences perfectly do not have any problems when switching to digital notation, since there are no problems with abstracting numbers from the subject. In this case, what causes these problems? Observations show that the greatest difficulties are with establishing coordination between the concepts of numbers (quantitative properties of objects) and quality concepts (qualitative properties of objects): a child easily puts shovels with diggers. Interestingly, this is in good agreement with the following fact experimentally established by L. S. Vygotsky [Vygotsky, 1996, pp. 260-262]. In the process of learning spontaneous concepts (for example, "brother"), the child understands the subject better than the concept itself. In the process of mastering scientific concepts (for example, "exploitation"), the child is much more aware of the concept than the object or process presented in it. The child easily performs logical operations with them, but hardly correlates them with the subject area. The fact is that the child's thinking is not conceptual. The child only imitates the conceptual grid communicated by adults, but imitates by means that are fundamentally different from adult thinking: these are two processes that coincide only at the end point. So abstract counting and using numeric notation are two different things.
There are two more factual objections to the interpretation of a particular account proposed by D. Schmandt-Bessera and the historians of mathematics cited by her (Danzig, 1959; Smith, 1951; Kramer, 1970; Flegg, 1983):
A) The existence of countable classes does not negate the existence of universal numerical concepts; in turn, universal numerical concepts do not themselves make numbers abstract. This is clearly seen in the material of the Chinese language, where the word denoting a number is always accompanied by a special counting word denoting the corresponding counting class. Thus, at the level of language meanings, the concept of number and the concept of quality exist separately, but at the level of deep semantics, they exist only together, within an undifferentiated semantic block, which follows from the fact that they do not meet separately in speech;
B) The gradual abstraction of counting from the concrete to the abstract, which is assumed in the Schmandt-Besser hypothesis, means the following. In the beginning, we must have the maximum number of counting classes, which will gradually decrease in the process of abstracting, until, finally, they do not disappear altogether. In reality, the situation is just the opposite: judging by the description of D. Schmandt-Besser, the number of counting classes embodied in three-dimensional signs increased very quickly over time, until suddenly it was replaced by a digital system (created on the basis of signs denoting grain measures).
What's the big deal here? We will not understand the essence of a particular invoice until we treat it as a stage in the development of accounting technologies. As will be shown below, the semantic core of a particular account as a phenomenon is located on a completely different plane.
Let us turn to the reconstruction of the semantics of numbers in the Anau culture (Southwestern Turkmenistan, 5th-3rd millennium BC) (see [Vyrshchikov, 2001]). Here, each concrete number is an element of a large, undifferentiated, or weakly differentiated semantic block (as opposed to an abstract number that is completely abstracted from any concrete relationships). This is reflected in one of the main features of pre-conceptual thinking - the lack of separation of the abstract and concrete. In this regard, it is useful to recall once again the conclusions of L. S. Vygotsky:
"The child, apparently, does not enter the realm of abstract concepts, starting from special types and rising higher and higher. On the contrary, he first uses the most general concepts. He arrives at the middle rows not by abstracting them, but by defining them. The development of ideas in a child goes from undifferentiated to differentiated, and not back " [Vygotsky, 1996, p. 178].
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Thus, the process of increasing the number of counting classes (types of three-dimensional signs), recorded by D. Schmandt-Besser, is a differentiation of large semantic blocks. Here we seem to see in action the very differentiation of consciousness without which the transition to abstract counting would be impossible.
Here it is interesting to recall the connection between the system of three-dimensional signs and sign systems associated with ceramic ornaments, signs on figurines, seals [Vyrshchikov, 2001, pp. 57-60]. Obviously, sign systems of this type were not created for counting purposes. There is reason to believe that three-dimensional signs were also originally created for other purposes, and were adapted for counting only later, in the process of conducting a productive economy.
It is hardly legitimate to identify a digital account with an abstract one. The transition to a digital system of registration of numbers is a process that leads to generalization of numerical properties of objects, i.e. to abstraction. However, it is only a matter of possibility: the abstraction of a number from an object is not in itself a transition to abstract counting; the latter presupposes a qualitatively different way of thinking - conceptual thinking. L. S. Vygotsky remarked on this point:
"A pre-concept is an abstraction of a number from an object and a generalization of the numerical properties of the object based on it. A concept is an abstraction from a number and a generalization based on it of any relations between numbers. Abstraction and generalization of thought are fundamentally different from abstraction and generalization of things "[Vygotsky, 1996, p. 279].
The process of differentiation of large semantic blocks, which was mentioned above, creates conditions for the transition from concrete to abstract counting - in the field of counting, from pre-concepts to concepts-in the field of thinking.
Here we should once again pay attention to the parallel processes of the emergence and development of digital numeracy and writing in the Middle East. As with the genesis of digital counting, the formation of writing does not yet mean the transition to conceptual thinking, but it creates an opportunity for this. As L. S. Vygotsky noted:
"...if all that we want to say consists in the formal meanings of the words we use, we would need to use many more words to express each individual thought than we actually do. But this particular case occurs in written speech... " [Vygotsky, 1996, p. 337].
The fact is that written speech means the absence of a real listener and, therefore, is characterized by monologue. At the same time, oral, dialogical speech occupies an intermediate position between written speech (absolute expansion) and internal speech (absolute predicativity). Consequently, the regular use of written, monologue-based speech (which is exclusively "speech for others") is a kind of exercise in separating "speech for others" from "speech for oneself"; and this differentiation is characterized by conceptual thinking. In this sense, "monologue is a higher, more complex form of speech, historically later developed than dialogue "(Vygotsky, 1996, p. 340).
Thus, the scribe who draws up a royal decree or registers the arrival/expenditure of property is forced to focus on the reader, who must understand what is written unambiguously, and, therefore, is forced to distract as much as possible from the situational context. Such a problem does not arise for the accountant: remember that to this day in Russia, he performs registration in the most simple and natural way for himself - by setting sticks or ticks (as in the Upper Paleolithic). Here, one-to-one correspondence is not hindered by either education or possession of an abstract account. The three-dimensional sign system as described by Schmandt-Besser,
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It is, of course, a more complex system, but computationally and technologically it is within the framework of one-to-one correspondences. For the transition to a digital account, even within the framework of special calculations, a qualitatively different incentive is required than accounting activities. In particular, as an option, it could be registration for the purpose of controlling accountants.
So, going back to a specific account, you need to say the following. A significant inaccuracy in the Schmandt-Besser hypothesis is the confusion of several levels of signedness. They are closely related and assume each other, but they have different psychosemantic nature and, as a result, are autonomous relative to each other. These are three levels: 1) the level of computational technologies; 2) the level of concepts and 3) the level of thinking.
1. It includes aspects directly related to the operation of signs. At this level, in the designated historical perspective, two stages are visible: one-to-one correspondences and digital counting. In natural language, this level roughly corresponds to the syntax of the language.
2. It consists of a system of concepts associated with the level of the counting system (1). In the historical perspective, this level contains two poles: pure one-to-one correspondences, where numerical concepts do not yet exist, and universal numbering, where the concept of number is completely separated from the concepts of quality. Between them are the special numbers described above, the spectrum of which can be very diverse. However, nowhere has it been proven that the stage of special numbering is inevitable under any historical circumstances. Let's recall that children master universal numbers, bypassing the stage of special numbering. It should be said, however, that children learn a ready-made numbering developed by adults, and do not invent their own; just as in the process of mastering a language (speech), they are guided by the language of adults. It is clear that the ancient people, being adults themselves, were deprived of such an opportunity. However, it is clear from the above example that the stage of special calculations is not necessary, and therefore unavoidable under any circumstances. In a language, this level corresponds to the level of language values. It is interesting that the proposed triad of generating counting "counting sign-numerical concept-thought" coincides with the triad of generating speech derived by L. S. Vygotsky: "word-meaning-thought".
3. This level in its historical development also shows similarities with the stages of development of the child's thinking identified by L. S. Vygotsky. Thus, pure counting in one-to-one correspondences, which is associated with the absence of any idea of a set in connection with a number, is in good agreement with the concept of syncretic thinking of a child, which was introduced by Klapared [Vygotsky, 1996, p.135].
A concrete account is an account in potential concepts and pseudo-concepts, while an abstract account is associated exclusively with the concept of a number. The unique poles of this multi-level process are, on the one hand, one - to-one correspondences, and, on the other, abstract counting, which is uniquely characterized by both the ultimate differentiation of the conceptual level (which is assumed by conceptual thinking) and digital counting technology. At the same time, a specific invoice fills the entire space between them, being characterized by exceptional diversity in both technological and conceptual aspects.
Probably, the forms of a particular account are determined by the specific cultural situation in which it originated and developed. Perhaps it was within the framework of the development of a particular account as a type of thinking that the high achievements of ancient Indian and ancient Chinese mathematics, ancient Egyptian engineering, etc. were realized. However, this issue is controversial and requires a separate study. It is interesting that each stage of counting matures inside the previous one, and there is an active inter-level interaction.
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In fact, numerical concepts are formed in the process of using one-to-one correspondences, which leads to the appearance of a specific account, during the development of which there is a differentiation of numerical concepts, which in turn leads to the appearance of universal numbers and the transition to a digital account. This creates the necessary conditions for abstracting already from the number and gradually moving to an abstract account.
Thus, concrete counting as a phenomenon does not relate to the history of mathematics, but rather to the history of consciousness.
THE PROBLEM OF THE GENESIS OF DIGITAL NOTATION IN THE CONTEXT OF SIGN ACTIVITY IN ARCHAIC CULTURES
Let us return, however, to the question of the origin of three-dimensional signs. From the above, it can be seen that the very parallelism in the formation of digital counting and writing systems in the Near and Middle East indicates that neither of them could have originated from the other (for this thesis, see also [Wyman, 1976; Green, 1981]). On the one hand, the number of Proto-Sumerian written signs that are clearly related to three-dimensional signs is relatively small. On the other hand, the counting system, as shown by D. Schmandt-Besserat (1984), also has an independent origin. At the same time, it should be remembered that the origin of volume signs is too early for them to have been originally created for accounting activities, and the very level of development of the economy in the 9th millennium BC made accounting and control operations meaningless. All this suggests that both systems of signs - counting and writing - are derived from some third, autonomous in relation to both. At the same time, there was a recoding - the former iconic repertoire was used to serve the iconic system (s) of a fundamentally different nature.
In this regard, interesting facts indicate the connection of three-dimensional signs with signs of ornaments, signs on figurines and other similar signs that do not have a direct object correlation. First of all, the similarity between the impressions of three-dimensional signs and the main forms of signs in the ornaments of Neolithic and Eneolithic cultures (as well as some Bronze cultures) is noteworthy. When projected onto a plane, a cone and tetrahedron give a triangle, a biconus - a rhombus, a sphere and disk - a circle, a half-disk-a semicircle, a cylinder - a rectangle - the main forms of ornamental signs. But, according to D. Schmandt-Besserat, it was the impressions of three-dimensional signs that served as prototypes for the set of symbols of Proto-Sumerian writing, which is associated not with pictorial, but with symbolic forms [Schmandt - Besserat, 1978, p.44-45]. However, such signs of Proto-Sumerian and Proto-Elam writing show great similarity with the corresponding signs of the Susianish ornamental tradition, and not only with them.
Moreover, the repertoire of signs-symbols, similar and even identical to Proto-Sumerian, is found throughout the entire space of ancient agricultural cultures from the Balkans to Harappa, without any connection with three-dimensional signs, but often revealing a connection with the symbols of the local pre-written tradition (for Asia Minor and the Balkans, see: [Ivanov, 1983, pp. 53-80; Marcotic, 1981]). As for the systems of pre-written symbols, the author's PhD thesis was devoted to the reconstruction and detailed characterization, including typological ones, of one of them (the Anau ornament, glyptics, and signs on figurines) (see [Vyrshchikov, 2001]). As the reconstruction shows, this sign system is designed for nonverbal communication of a very specific nature. One of the features of the described sign system is that signs encode not meanings in the usual sense of the word, but large semantic blocks, certain semantic dominants that can form semantic chains similar to those that L. S. Vygotsky called complexes. Exactly
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this feature made it possible to apply this sign system to various aspects of activity in a given culture - iconography, housing construction, rituals, mythopoetics, calendar, etc. - without structural adjustment.
The likely sacred function of three-dimensional signs is sometimes indicated by the places where they were found. For example, at the Togolok-21 complex (Margiana, 2nd millennium BC), egg-shaped clay balls were found piled up at the junction of two round altars - the big and small ones (Sarianidi, 1990, p.133).
One gets the impression that both three-dimensional and planar signs were originally part of a sign system of the same type, the relics of which are preserved, for example, in some tribes of modern India (Mallebrein, 1993). To this day, the mountaineers of Central India use, in addition to planar signs, three-dimensional signs that resemble the egg-shaped ones that D. Schmandt-Bessera studied. However, unlike counting chips, they perform iconographic functions, or rather, syncretically combine iconographic, votive, protective, etc. functions. In general, referring to ethnographic materials (in particular, on tribal groups in India), it is easy to find that any socially significant type of activity in archaic societies is somehow connected with the ritual sphere (see, for example: [Naik, 1956; Elwin, 1969]).
Let us return, however, to the volume signs. When we talk about the genesis of the counting system restored by D. Schmandt-Besser, as we have already said, we do not mean an arbitrary act of "recoding", but a series of successive shifts that gradually led to the emergence of a qualitatively new one - this is how innovations usually mature in a traditional society.
Based on the above, we can outline several hypothetical stages of the formation of a digital account in the ancient Near East.
At the first stage, at the dawn of their existence, three-dimensional signs exist within the framework of universal sign systems and duplicate other types of sign repertoire. Indeed, ritual duplication is of great importance in archaic cultures (see, for example, Mallebrein 1993). Different types of images of the deity, different types of symbols, sacred objects, offerings to the altar, and the type of altar, temple, and often the surrounding area duplicate the same thing - the deity to whom they are dedicated. It is the deity who is the invisible core that ensures the syncretic unity of functions mentioned above.
The second stage is associated with the establishment of a stable connection (so far purely ritual, which occurred in the course of ritual duplication) between three-dimensional signs and offerings to the altar. It is difficult to say what motivated this, most likely the offering was duplicated by the production of the corresponding three-dimensional sign, i.e. both performed votive functions in this case.
At the third stage, the exact correspondence between the number and shape of three-dimensional signs and the quantity and qualitative composition of offerings according to the principle becomes significant.:
number of volume marks1= number of measures of offering1
number of volume marks2= number of measures of offering2
number of volume marksn= number of measures of offeringn.
Probably, this stage corresponds to the above-mentioned find from the Togolok-21 complex (egg-shaped clay balls stacked between two altars). In fact, from this moment on, a ready-made device appeared for recording the arrival/consumption of supplies for utilitarian and economic purposes. There remained only the appearance of an urgent need to bring this device into action.
The fourth stage is characterized by the transformation of temples into large, and then into giant storage facilities for supplies by the standards of that time (this state is attested in early written sources of the Middle East). Partially this pro-
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the process began at the previous stages due to the success of the productive economy: first, with the increase in the number of offerings to the temple, and secondly, with the fact that temples in early antiquity served as public repositories. One way or another, there was a need first for accounting, then for control over accountants, and eventually for control over supervisors. This need was the driving force behind the further evolution of the counting system based on operations with volume signs, which led to the emergence of digital notation.
In conclusion, we can say the following. In the process of studying the development of sign systems, as well as the development of culture in general, in ancient times, we often encounter a situation where evolutionary processes are not straightforward. Such a straightforward model of evolution assumes that an element of culture develops from the most primitive forms to fully developed ones in one capacity, and that is all. It is in this vein that models of cultural evolution are usually constructed. However, such a model of evolution is valid only for limited time periods. And most importantly, such a model does not allow us to study qualitative transitions, when a cultural form in the process of its development passes into a completely different qualitative state. Meanwhile, when studying ancient cultures, we are dealing with large time intervals, and it is impossible to avoid these problems. This work allows us to offer a different picture of the development of ancient cultures, where the evolution of cultural forms inevitably implies qualitative leaps with a change in the direction of development. That is, at a certain stage, due to changed historical conditions, certain cultural forms are attracted to perform functions that were not previously characteristic of them. At the same time, this form itself is qualitatively reconstructed (for the sign system, this is called recoding). In the long-term historical perspective, cultural evolution is a complex process that is accompanied by periodic changes in the dominant direction, while the most significant elements for subsequent stages of development are often those that were on the periphery of previous cultures. Such a view makes it possible to radically change the approach to reconstructing the development of cultural forms. Then, at the beginning of each evolutionary stage, there will often be not some primitive cultural form, but a form originally intended to perform completely different functions. In general, this will allow the researcher to see not only the historical perspective of the development of culture, but also its retrospective.
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