Introduction
The numeral systems in natural languages are special subsystems that are vulnerable to outside impacts. For example, language contact has fundamentally changed the numeral systems of many languages, demonstrated in the decimal undercounting system that is prevalent in most of the civilised world. In a decimal undercounting system, numbers are perceived as points, from smaller ones to larger ones, based on ten-interval in the number line. For example, the number 17 is perceived as the seventh point after ten in the number line. Linguistically the number 17 is expressed as ten seven in Chinese or seven-teen in English or other forms in other languages.Footnote 1 However, cross-linguistically, the numeral systems of natural languages are not so uniform and regular, featuring many peculiarities and oddities in their numeral formation, especially in the earlier development of numeral systems. As early as the nineteenth century, German historical linguists discovered a special form of counting in Old Turkic texts. It is a different counting system, in which, say, yeti yeɡirmi— ‘seven twenty’—refers to, intuitively, the seventh point within the interval from ten to 20 or seven on the way to 20, that is 17, but not to 27 by addition or 13 by subtraction. It is to count by looking forwards (anticipating) in the number line, compared to undercounting which counts by looking backwards in the number line. This form of counting is named as Oberstufenzählung (over-step-counting) in historical linguistics.Footnote 2 In general linguistics, it is more commonly referred to as overcounting, translated from Oberstufenzählung, while undercounting is translated from Unterstufenzählung (under-step-counting),Footnote 3 although it has also been given other different terms, such as grading method of counting,Footnote 4 anticipatory counting,Footnote 5 anticipative counting,Footnote 6 going-on counting,Footnote 7 and 響數法(xiǎng shù fǎ).Footnote 8
The discovery of overcounting has been an important contribution of historical comparative linguistics during the nineteenth century and has been widely known and studied in Altaic linguistics and philology.Footnote 9 It has been observed that in Old Turkic, overcounting applied globally to 11–19…21–29…91–99 and larger numbers ending in them such as 111–119, etc. The Orkhon Inscriptions featured overcounting only up to 40, for example, bir qïrq ‘one forty = 31’ in Kül Tegin Inscription, while overcounting for larger numbers can be found in later Uighur Buddhist texts, for example, iki yüz bir toquz on ‘two hundred one nine ten = 281’, iki yüz iki toquz on ‘two hundred two nine ten = 282’, and iki yüz üč toquz on ‘two hundred three nine ten = 283’ in Hasen—Jātaka (兔王本生). Overcounting for numbers 91–99 is not formed on the basis of yüz ‘hundred’ because iki yüz means 200 not 92. Old Turkic used a special formation by adding örki ‘upward’ to the digits, for example, tokuz örki ‘nine upward = 99’, which can be found in Altun Yaruq Sudur (金光明经). Although overcounting has been shown as the prevalent way of counting in Old Turkic, it has been basically lost in modern Turkic languages, having given way to the decimal undercounting system and only remaining in the Siberian Yakut and Modern West Yugur in China. In Yakut, overcounting can count up to 100, while in Modern West Yugur overcounting only applies to 11–19 (and 21–29 for older people), as shown below in the data (1) for YakutFootnote 10 and (2) for Modern West Yugur.Footnote 11 In addition, overcounting may also exist in the Tungusic languages of the Altaic family, as shown in (3) for some dialects of Evenki, whose structure is the same as that of Yakut.Footnote 12
Overcounting seems to be a rare and unfamiliar manner of counting: Menninger comments that it is ‘a remarkable manner of counting, which once prevailed in two areas of the world, the Germanic north of Europe [in Old Norse] and ancient Mexico [in Mayan languages]’.Footnote 14 However, linguistic fieldwork and documentation during the twentieth century have shown that overcounting is more widespread than earlier believed. In addition to Altaic, Indo-European, and Mayan, it existed, or exists, in Sino-Tibetan, Ural, Austronesian, Niger-Congo, and Dravidian language families. Considering the geographical distance between these language families, the widespread existence of overcounting may not be due to language contact, but instead be the earliest common form of human counting, which has been altered by the undercounting system.
In any sense, the discovery of overcounting in Old Turkic should be a proud and important contribution made by historical comparative linguistics during the nineteenth century, which may be able to reveal the true situation of how our ancestors counted. However, the initial interpretation of overcounting in Old Turkic by Thomsen and Radloff was mistaken, giving rise to serious difficulties in reconstructing the old Turkic history. It was not until 1898 that Bang and Marquart finally revealed the true semantics of Old Turkic numerals. This article provides an overview of the history of the discovery and interpretation of overcounting in Orkhon Inscriptions by historical linguists during the nineteenth century. Since Bang and Marquart did not explicitly spell out what kind of evidence they had for reaching the correct values of overcounted numerals, this article will present a series of arguments to prove the existence of overcounting in Old Turkic from the perspectives of language per se, historical facts, logical reasoning, and bilingual translation.
Interpreting the overcounted numerals in the Orkhon Inscriptions
The numerals in the Orkhon Inscriptions include those for digits and round integer numbers (tens, hundreds, and thousands) such as yeti yaš-da ‘7 years old’, yeɡirmi kün ‘20 days’, äliɡ yïl ‘50 years’, yeti yüz är ‘700 people’, äki biŋ ‘2,000’, beš tümän sü ‘50,000 army’, and so on. These numerals are monomorphemic or multiplicative, featuring a simple structure and transparent semantics. In addition, there are two forms of juxtaposed compound numerals in the Orkhon Inscriptions. One of them takes on the form of decade + digit connected with artuqï ‘more, in addition to’, in-between, such as qïrq artuqï yeti ‘forty more seven = 47’.Footnote 15 This type of additive numeral is common, featuring a simple structure and transparent semantics too. The other type takes on the form of digit + decade without any linking morpheme in-between, for example, bir yeɡirmi ‘one twenty’. The latter form of juxtaposed numerals is also widely attested in European languages, for example, the German acht-zehn ‘eight-ten = 18’ and classical Greek oktō-kai-deka ‘eight-and-ten = 18’. Both Thomsen and Radloff, the two pioneers in deciphering the Orkhon Inscriptions, analysed them as ordinary additive numerals, so the value expressed by bir yeɡirmi is 21, and so on.
When translated as additive numerals, the values of some of these juxtaposed numerals cannot be verified by history, and they do not provide a key figure in reconstructing the history of Turkic people. Consider the following sentence (4) from the texts of the Kül Tegin and the Bilgä Qaγan Inscriptions;Footnote 16 (5) is the French translation by Thomsen, and (6) and (7) are the German translation by Radloff.
This does not seem to leave any historical evidence, and whether ‘my father Qaγan’ left with 17 or 27 people (after being defeated by Tang's army) seems a rather trivial fact and does not seem particularly important to the Turkic history. Today we know that Qaγan left with 17 people, which is known to us from the true meaning of the numeral yeti yeɡirmi ‘seven twenty’.
However, the interpretation of some numerals is especially important to Turkic history, particularly those stating the ages of Turkic leaders and the dates of important events. If misinterpreted, they might bring considerable inconsistency, or even contradiction, in reconstructing Turkic history. And there are even some numerals, if interpreted as additive, that would be inconsistent with common sense. For example, both the Kül Tegin and Bilgä Qaγan Inscriptions contain the dates of the deaths and funerals of the protagonists, as shown below.
In both sentences, Thomsen translated all the numerals as additive, that is, yeti yeɡirmi ‘seven twenty’ into vingt-septième ‘27th’, altï otuz ‘six thirty’ into trente-sixième ‘36th’, and yeti otuz ‘seven thirty’ into trente-septième ‘37th’, as shown below.
Translating yeti yeɡirmi ‘seven twenty’ into vingt-septième ‘27th’ seems fine in terms of common sense because we do have the 27th day in a given month. However, it is obviously impossible to have trente-sixième ‘36th’ and trente-septième ‘37th’ as dates in a month. Thomsen surely noticed such striking values of these numerals for designating dates and attempted to give an explanation in two long footnotes. Thomsen explained that to comprehend such dates, it is necessary to compare the Chinese text in the same Inscription and to utilise the ancient Chinese sexagesimal calendar notation system. The Kül Tegin Inscription contains a passage written in Chinese stating when the monument was erected:
In the Chinese text (12), the numerals for the year, month, and date are followed by special terms used in the ancient Chinese sexagesimal calendar notation system, for example, ‘the seventh day’ is followed by Dīngwèi, which means that the seventh day of that month of that year is the Dīngwèi day, that is, the 44th day in the Chinese sexagesimal system. Thomsen thus speculated that the numbers 36 and 37 in fact refer to the 36th and 37th day in the sexagesimal cycle, only expressed in Turkic numerals altï otuz (supposedly 36) and yeti otuz (supposedly 37). This seems to be a reasonable explanation, but Thomsen encountered many difficulties. The Kül Tegin Inscription was erected in the twentieth year in the Kaiyuan Period of the Great Tang Dynasty (that is, ad732), and Thomsen found that the 37th day (corresponding to the Chinese Gēngzi day) does not occur in either July or September. Similarly, in the Bilgä Qaγan Inscription, the 36th day does not occur in October and the 37th day does not occur in May of the relevant year. In the end, Thomsen admitted that he did not understand exactly the meanings of the two numerals yeti otuz and altï otuz, which are presumably expressions for the numbers 36 and 37. He suspected that these were spelling mistakes or that the Turks were not good at counting by the Chinese sexagesimal cycle notation. He said:
Les chiffres forts qui se présentent ici (37 ici et dans II N 10; en ce dernier endroit, aussi 36) montrent qu'ils ne peuvent pas désigner le quantième de tel mois même, mais qu'ils indiquent le jour d'après sa place dans la semaine sexagésimale mentionnée plus haut. Il faut donc que, chez les Turcs, les singuliers caractères cycliques des chinois soient tout simplement remplacés par des nombres cardinaux. Cependant, l'identification exacte de cas dates avec le calendrier chinois, présente diverses difficultés qui ne s'expliquent que par la négligence des Turcs dans le maniement du calendrier.Footnote 22
‘The high figures presented here (37 here and in II N 10; in the last place, also 36) show that they cannot designate the date of a given month, but that they indicate the day according to its place in the sexagesimal system mentioned above. It is therefore necessary that, among the Turks, the singular cyclic characters of the Chinese are quite simply replaced by the cardinal numbers. However, the exact identification of dates with the Chinese calendar presents various difficulties which can only be explained by the negligence of the Turks in handling the calendar.’
In interpreting the Bilgä Qaγan Inscription, Thomsen encountered more severe doubts and difficulties. The Bilgä Qaγan Inscription contains the following two statements, describing the age and duration of Bilgä Qaγan's ruling and governance.
Thomsen translated them into (15) and (16).
According to his interpretation of the numerals, Bilgä became viceroy at the age of 24, after which he was viceroy for 29 years and Qaγan for another 29 years, so he was at least 82 years old. But Thomsen inferred from other historical sources that Bilgä Qaγan died in ad734 and lived for only 51 years, and so cannot have been the Qaγan for 29 years. That is why Thomsen put an exclamation mark behind vingt-neuf. Thomsen suggested that toquz yeɡirmi must be a spelling or arithmetic mistake, and commented that the meaning of this numeral should be 19. In fact, Thomsen had almost come to know the true semantic composition of Turkic numerals.Footnote 25
Another pioneer in interpreting the Orkhon Inscriptions, Radloff also translated the above juxtaposed numerals as additive, but he did not say anything about the striking values of 36 and 37 for dates in his monographs of 1895 and 1897.
Due to the authority of Thomsen and Radloff's interpretation of the Orkhon Inscriptions, later scholars, for example BangFootnote 28 and Barthold,Footnote 29 also translated these juxtaposed numerals as additives, failing to see their true semantic composition.
Soon the misinterpretation of the juxtaposed numerals in the Orkhon Inscriptions was corrected. Bang pointed out the systematic misinterpretation of these juxtaposed numerals in Orkhon Inscriptions, stating that:
ist yeti otuz (37) ohne jeglichen Zweifel ein Fehler für yeti yeɡirmi (27)…dass bir kïrk in I Ν 2 ein weiterer durch yeti otuz I Ν 1 veranlasster Fehler für bir otuz sei.Footnote 30
‘yeti otuz (37) should undoubtedly be an error of yeti yeɡirmi (27)… that bir kïrk in I Ν 2 is another error caused by yeti otuz I Ν 1 for bir otuz’.
It should be pointed out that Bang came to this conclusion through a comparative study of Turkic history, rather than basing it on the Turkic numerals themselves. It is certain that he did not realise, when writing that article, the semantic composition of the Turkic numerals, for he mistakenly thought that yeti yeɡirmi expresses 27 and failed to see that yeti otuz is not an error of yeti yeɡirmi; it is the correct numerical form expressing 27.
Bang's comment was entertained by Marquart, who, in his Die Chronologie der alttürkischen Inschriften, subjected the entire chronology of the Inscriptions, which is based on addition, to a revision by reducing all the values of these juxtaposed numerals by 10. In this book, importantly, Marquart correctly pointed out that yeti otuz is not an error of yeti yeɡirmi, and that bir kïrk is not an error of bir otuz. They themselves just express the numerical values of 27 and 31, while yeti yeɡirmi and bir otuz express the numerical values of 17 and 21. Marquart said that the Turkic numerals altï otuz and yeti otuz for dates are not formed by the Chinese sexagésimale system, as suggested by Thomsen, but that they are incorrectly translated into 36 and 37.Footnote 31 This indicates that Marquart understood the semantic composition of the Turkic juxtaposed numerals like altï otuz and yeti otuz. After identifying the semantics of these numerals in Old Turkic, Marquart commented that many of the historical inconsistencies puzzling Thomsen were set right. In the preface to Marquart's book, Bang realised that the numeral such as tört yeɡirmi was not a spelling mistake, but the correct form, expressing vier auf zwanzig hin, der vor Zwanzig stehende Vierer etc. = 14 (four to twenty, the four in front of twenty etc. = 14). Bang also mentioned that Finnish features similar numerals, which led to cross-linguistic support of a new type of counting discovered in natural languages.Footnote 32, Footnote 33
Thanks to the works of Bang and Marquart, the mystery of Old Turkic juxtaposed numerals like altï otuz and yeti otuz in the Orkhon Inscriptions was completely solved and was established as another form of counting that is different from the decimal undercounting. In his 1899 edition of Die Altturkischen Inschriften der Mongolei, Radloff comprehensively revised the values of these Old Turkic juxtaposed numerals. Radloff acknowledged in the preface that Marquart's most important result is the correction of the Old Turk numerical expressions composed of digits and tens, which must be reduced by 10 in values. Radloff himself provided a proof for the overcounted interpretation of these numerals. He mentioned that his 1889 dictionary listed Turkic numeral names for months borrowed from a Chinese-Uighur dictionary,Footnote 34 in which the numeral for October onunč ay (the tenth month) is followed by bir yeɡirminč ay, which, of course, must necessarily mean the eleventh month. He admitted that:
Somit ist es nur meiner Unaufmerksamkeitund Vergesslichkeit zuzuschreiben, wenn ich nicht von Anfang an diesen Zahlausdrücken die richtige Deutung gegeben habe.Footnote 35
‘Thus, it is only due to my inattention and forgetfulness, if I have not given the correct interpretation of these numerical expressions from the beginning.’
In 1916 and 1924, Thomsen published a reinterpretation and German translation of the Orkhon Inscriptions, in which the values of overcounted numerals were all reduced by 10. But Thomsen did not explain the reason for such changes.Footnote 36, Footnote 37
Reconstructing the evidence for overcounting in Old Turkic
Since the publication of the works of Bang and Marquart, overcounting has been identified and established as an ancient form of counting. But the two authors did not say what kind of evidence they had to support deriving the correct values. This section will try to fill in this gap by presenting evidence to prove the existence of overcounting in Old Turkic from the perspectives of language per se, historical facts, logical reasoning, and bilingual translation. In order to make the arguments more comprehensive, this section uses not only the Orkhon Inscriptions but also the later Uighur texts.
As stated earlier, the Old Turkic numeral bir yeɡirminč ay was mentioned by Radloff as a key proof of overcounting.Footnote 38, Footnote 39 In fact, this striking numeral was first noticed by the German Orientalist Heinrich Julius Klaproth (1783–1835) who was, perhaps, the first Western scholar to study Gāo Chāng Guǎn Zá Zì.Footnote 40 In 1812, Klaproth published Abhandlung über die Sprache und Schrift der Uiguren in Berlin, in which he listed the names of months in Turkic and their German equivalents, including the two names for November. However, there was no comment on the interpretation of the old Turkic bir yeɡirminč ay ‘November’.Footnote 41
In the new edition of Abhandlung über die Sprache und Schrift der Uiguren published in Paris in 1820, Kraproth added a note under the entry bir yeɡirminč ay:
Bedeutet eigentlich den ein und zwantigsten Monat. Der eilfte Monat.-Bei Ulug-Beg, der die Uigurischen Monate giebt, steht wahrscheinlich durch einen Schreibfehler.Footnote 42.
‘… actually means the one and twentieth month. The eleventh month—by Ulug-Beg, who gives the Uyghur month name, is probably due to a spelling error.’
This means that Klaproth noticed the striking form of bir yeɡirminč ay in Turkic as expressing the eleventh month (November). Unfortunately, he did not identify that it is a special form of numeral formation, instead stating that bir yeɡirminč ay actually means one and twelfth month, and suggesting that the Old Turkic expression given by Ulug-Beg was probably a spelling mistake.
The Orkhon Inscriptions that Thomsen and Radloff initially interpreted include mainly the Kül Tegin and Bilgä Qaγan Inscriptions, both of which do not contain the numeral expression for November. If they did, the striking formation of Turkic numerals would have been worked out immediately by Thomsen and Radloff. However, the numeral expression for November appears in the Moyan Chor Inscription and many civil documents of the later Uighur period.
The ordinals in Old Turkic are aŋilki ‘first’, ikinti ‘second’, üč-unč ‘third’, tört-unč ‘fourth’, beš-unč ‘fifth’, altï-nč ‘sixth’, yeti-nč ‘seventh’, säkiz-inč ‘eighth’, toquz-unč ‘ninth’, on-unč ‘tenth’, bir yeɡirmi-nč ‘eleventh’, iki yeɡirmi-nč ‘twelfth, and so on. They frequently occur in later Uighur texts, used in pagination and to name the order of chapters, sheets, and some other items. For instance, the Uighur Buddhist drama Maitrisimit contains many instances of ordinals, used to name the order of chapters, sheets, and other items such as the order of man's virtues. For example, the tenth chapter is expressed in on-unč ülüš and the eleventh chapter in bir yeɡirmi-nč ülüš; the tenth sheet is expressed in on ptr and the eleventh sheet in bir yeɡirmi-nč ptr. Further see the following text from Maitrisimit:
The overcounted readings of some numerals can be verified by historical facts. For example, it is a well-known fact in Buddhism that Shakyamuni abandoned his wife and secretly fled from the city of Kapilwastu in order to pursue Buddhism at the age of 29, which is expressed in toquz otuz ‘nine thirty’ in Maitrisimit, as shown below. So toquz otuz expresses 29 by overcounting, not 39 by addition or 21 by subtraction.
The historical events experienced by the protagonists in the Orkhon Inscriptions are described in an order of increasing ages. In the Kül Tegin Inscription, the main events of Kül Tegin are arranged in the following order of increasing ages: 16 (altï yeɡirmi), 21 (bir otuz), 26 (altï otuz), 27 (yeti otuz), 30 (otuz), 31 (bir qïrq), and 47 (qïrq artuqï yeti). The actual contexts in which these numerals appear are listed below.
The overcounted readings of altï yeɡirmi, bir otuz, altï otuz, yeti otuz, and bir qïrq can give the above logically increasing order of ages, while an additive or subtractive interpretation of them will result in illogical chronological orders *26, 31, 36, 37, 30, 41, 47 and *14, 29, 24, 23, 30, 39, 47.
In the Bilgä Qaγan Inscription, the main events that happened to Bilgä Qaγan are arranged in the following chronological order of ages: 14 (tört yeɡirmi), 17 (yeti yeɡirmi), 18 (säkiz yeɡirmi), 20 (yeɡirmi), 22 (iki otuz), 26 (altï otuz), 27 (yeti otuz), 30 (otuz), 31 (otuz artuqï bir), 32 (otuz artuqï iki), 33 (otuz artuqï üč), 34 (otuz artuqï tört), 38 (otuz artuqï säkiz), and 39 (otuz artuqï toquz). The actual contexts in which these numerals appear are listed below. The overcounted readings of tört yeɡirmi, yeti yeɡirmi, säkiz yeɡirmi, iki otuz, altï otuz, and yeti otuz can give the above logically increasing order of ages, while additive or subtractive interpretation will result in an illogical age-increasing order, which is clear to see.
The old Uighur Buddhist classics are mostly translated from Sanskrit, Tocharian, Tibetan, and Chinese. For example, the Uighur Xuan Zang Zhuan was translated from the Chinese Xuan Zang Zhuan ‘The life of Hiuen-Tsiang’ in the first half of the tenth century by the Uighur scholar Šingqu Säli. We compared both versions and found that about 200 decimal numerals (within the required intervals) in Chinese are translated into overcounted numerals in Uighur. For example, the following passage (47) is from the Chinese version which describes the quantity of sutras Xuan Zang (Hiuen-Tsiang) brought from India. In the Uighur version, it was translated as in (48).
Table 1 shows the correspondence between the Chinese undercounted numerals and Uighur overcounted numerals in (47) and (48), from which the difference between undercounting in Chinese and overcounting in Uighur can clearly be seen.
Table 1: Correspondence between some Chinese and Uighur numerals in The Life of Hiuen-Tsiang. The numerals in this table are from (47) and (48) cited in footnotes 45 and 46.
Conclusion
Overcounting, as a remarkable way of counting in a natural language numeral system, has potential value to the linguistic study of numeral systems, and some more general issues in anthropology, cognitive science, and the history of mathematics. In particular, overcounting in Turkic can provide us with ideal source material for a comprehensive and thorough investigation of human counting. Compared to other overcounting languages, the Turkic languages stand in a very unique and incomparable position thanks to the presence of overcounting in modern Turkic languages (West Yugur and Yakut) and the huge availability of continual historical written texts of Old Turkic stretching back more than 1,200 years, which can provide us with large-scale authentic data to investigate the linguistics of natural language numeral systems and some more general issues such as the origin and development of counting.
The purpose of this article is, of course, rather modest in that it reviews the history of the discovery and interpretation of overcounting in the Orkhon Inscriptions by historical linguists during the nineteenth century, and presents a series of arguments to systematically prove the existence of overcounting. Nevertheless, it is hoped that this article has shown that overcounting, discovered a long time ago, deserves due and sufficient attention from different disciplines with regard to its theoretical value and significance. For example, overcounting may reshape our standard compositional view towards form and meaning in linguistics, for the interpretation of, say, yeti yeɡirmi ‘seven twenty’ as 17 seems non-compositional. We know it expresses something like seven on the way to twenty, that is, 17. However, this is only our intuitive understanding of its meaning, and linguists owe us a formal account of how ‘seven twenty’ generates the value 17.Footnote 47
In a broader and more important sense, overcounting may tell us something about how counting originated in primitive minds and how human counting evolved throughout history. In fact, it has been hypothesised that numbers in primitive minds were not developed in a natural sequential order (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12…), but in a discontinuous order (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40…) with the interval numbers (11, 12, etc.) developed later.Footnote 48 This hypothesis may be confirmed in the counting system of Old Turkic. It can be speculated that the Old Turks initially developed the number concepts of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, and 50. Considering the fact that Old Turkic only featured overcounting in interval numbers in the Yenisei Inscriptions (the earliest written material available to us)Footnote 49 and the observed shift from overcounting to undercounting, but not in the opposite direction, it is plausible that primitive people overcounted, or at least the Old Turks did.Footnote 50 This issue and many others are very important and await further study. In any case, it should be kept in mind that when counting, ancient people may not have thought in the ways we do today, as Lévy-Bruhl commented:
On admet en général, sans examen, et comme une chose naturelle, que la numération part de l'unité, et que les différents nombres se forment par l'addition successive de l'unité à chaque nombre précédent. C'est là en effet le procédé le plus simple, celui qui s'impose à la pensée logique quand elle prend conscience de son opération. Omnibus ex nihilo ducendis sufficit unum. Mais la mentalité prélogique, qui ne dispose point de concepts abstraits, ne procède pas ainsi.Footnote 51
‘It is generally accepted, without examination, and as a natural thing, that counting starts from 1, and that different numbers are formed by successive addition of 1 to each preceding number. This is in fact the simplest process that requires logical thinking when one becomes aware of its operation. Omnibus ex nihilo ducendis sufficit unum. But the prelogic mentality, which does not have abstract concepts, does not do so.’
Acknowledgements
The author thanks the two reviewers of JRAS for very helpful comments. The work is supported by the China National Social Sciences Fund, project number 19FYYB024.
Conflicts of interest
None.
