“Some Americans could not by any means count to 1000”: the cognitive effects of the lack of names for numbers in exotic languages from the perspective of linguistic theorists before Humboldt

Gerda Haßler
Universität Potsdam

The limited number word vocabulary in some languages for quantities above a specific amount has for some time been a much-debated topic. A study published in 2008 (Butterworth, Reeve, Reynolds, Lloyd 2008), which attracted much attention, found the development of numerical cognition to be independent from the presence of number words. Test participants from two Australian Aboriginal communities, both of whose languages only have words for ‘one’, ‘two’, ‘few’ and ‘many’, performed just as well in a counting test as a comparable group of Aboriginal people who spoke only English. From these results it was concluded that abstract concepts for numbers are based on innate mechanisms and not on socially learned words.

These findings appear contrary to the position propagated since the 1990s in which the discussion about the linguistic relativity of thought[1] was revived, based on the specific example of number words. In the history of the theory of the linguistic worldview, usually associated with Wilhelm von Humboldt, there are several conceptual fields that repeatedly attract attention as potential targets of linguistic influence, such as space-time relationships, kinship or religious terms. Perhaps numbers did not belong to these categories due to the non-linguistically motivated process of counting. Nevertheless, the lack of number words was noticed particularly in exotic languages, causing communicative difficulties and encouraging speculation about cognitive effects. In the following text, we want to explore the assumption of cognitive effects of the lack of number words prior to Wilhelm von Humboldt.

1. References to the lack of number words in indigenous languages as a deficit

References to the lack of number words in indigenous languages can already be found in the writings of travelers and missionaries in the 16th century. In 1578, after Jean de Léry (1534-1611) pointed out the lack of number words in indigenous languages in Brazil and the resulting communication problems, observation of this phenomenon became a common theme, which was confirmed repeatedly in the writings of missionaries and travelers.

Charles-Marie de La Condamine (1701-1774) also took up Léry’s idea in his Relation abrégée d’un Voyage fait dans l’intérieur de l’Amérique Méridionale (1745), but he developed these linguistic considerations further. He recorded the most frequent words in the different Indian languages, because he thought their comparison would provide information about the migration of the tribes over the continent. Moreover, the comparison with African, European and Asian languages should help to reveal the origin of the American peoples (La Condamine 1745: 55). Near a recently established mission he encountered a native tribe called the Yameos, whose language was extremely difficult to learn and hard to pronounce. They held their breath when speaking and uttered almost no vowel sounds. If written, their words would require nine to ten syllables. However, when speaking the Yameos only appeared to pronounce three or four:

Leur langue est d’une difficulté inexprimable, & leur maniere de prononcer est encore plus extraordinaire que leur langue. Ils parlent en retirant leur respiration, & ne font sonner presque aucune voyelle. Ils ont des mots que nous ne pourrions écrire, même imparfaitement, sans employer moins de 9 ou 10 syllabes ; & ces mots prononcés par eux semblent n’en avoir que trois ou quatre.
(La Condamine 1745: 66)

La Condamine illustrated his intuitive feeling for the degree of difficulty of this obviously highly agglutinative language, which was also difficult to linearize graphically, using number words. In this language, poettarrarorincouroac means ‘three’. Fortunately for all those dealing with this people and having to learn their language, La Condamine remarked, they could only count to three:

Poettarrarorincouroac signifie en leur langue le nombre Trois : heureusement pour ceux qui ont affaire à eux, leur arithmétique ne va pas plus loin.
(La Condamine 1745: 67)

It seems reasonable to view the awkwardness of the expression of small numbers and the lack of words for larger numbers as a cognitive constraint: the arithmetic of these people stops at three. Based on his experience and on reading travelogues, La Condamine noticed that this language was not an isolated case. The language spoken in Brazil – he was obviously referring to Tupí – was spoken by less primitive peoples, but it still had the same deficit: as soon as they wanted to count beyond the number three, they had to fall back on Portuguese:

La langue Brasilienne parlé par des peuples moins grossiers, est dans la même disette, & passé le nombre Trois, ils sont obligé, pour compter, d’emprunter le secours de la langue Portugaise.
(La Condamine 1745: 67)

The lack of number words above three is apparently viewed here not only as a communicative obstacle, but at the same time as a cognitive deficit. Having to express such a simple number as three with a decasyllabic word was interpreted as a circumstance that made it harder to think in numbers and that made communication more difficult. Apparently, the tribes with few number words were considered capable of operating with larger numbers because they could always fall back on Portuguese to compensate for the deficit.

Similarly, in 1748 Pierre Louis Moreau de Maupertuis (1698-1759) formulated his general theory on linguistic relativity on the basis of a hypothetical reference to exotic languages. According to him, the signs by which men designated their first ideas had such great influence on their knowledge that these are worthy of the attention of philosophers. Here the issue is not the comparison of languages that classify the world in a similar way, i.e. what is called pain in France, is called bread in London. Rather, there are languages, especially those of far distant peoples, which are based on quite different ideas, so that translating from them is almost impossible:

On voit assez que je ne veux pas parler ici de cette étude des Langues dont tout l’objet est de savoir que ce qu’on appelle pain en France s’appelle bread à Londres : plusieurs Langues ne paroissent être que des traductions les unes des autres ; les expressions des idées y sont coupées de la même manière, & dès-lors la comparaison de ces Langues entre elles ne peut rien nous apprendre. Mais on trouve des Langues, sur-tout chez les peuples fort éloignés, qui semblent avoir été formées sur des plans d’idées si différents des nôtres, qu’on ne peut presque pas traduire dans nos Langues ce qui a été une fois exprimé dans celles-là. Ce seroit de la comparaison de ces Langues avec les autres qu’un esprit philosophique pourroit tirer beaucoup d’utilité.
(Maupertuis 1970 [1748] : 25-27)

In addition to the practical difficulties of communication and the development of people’s thinking, particularly given the diversity of completely differently structured exotic languages, an opportunity was seen to gain insight into the relationship of language and thought and the origin and development of man.

2. The recourse to exotic languages in the justification of the idea of number in the works of John Locke

In his Essay concerning Human Understanding (1690), John Locke (1632-1704) devoted an entire chapter to the idea of number. As the simplest and most universal idea (Locke 1894 [1690]: Book II, Chapter XVI, 1), numbers represent a kind of prototype for the abstracting and generalizing capacity of words. According to Locke, it is essential to have a designation for the concept of a number, which following its institutionalization enables operating with that number:

Names necessary to numbers. […] the idea of an unit, and joining it to another unit, we make thereof one collective idea, marked by the name two. And whosoever can do this, and proceed on, still adding one more to the last collective idea which he had of any number, and gave a name to it, may count, or have ideas, for several collections of units, distinguished one from another, as far as he hath a series of names for following numbers, and a memory to retain that series, with their several names: all numeration being but still the adding of one unit more, and giving to the whole together, as comprehended in one idea, a new or distinct name or sign, whereby to know it from those before and after, and distinguish it from every smaller or greater multitude of units.
(Locke 1894 [1690]: Book II, Chapter XVI, 5)

When we perform arithmetic operations we need number words to distinguish the numbers from one another. Man “is capable of all the ideas of numbers within the compass of his language, or for which he hath names, though not perhaps of more” (Locke 1894 [1690]: Book II, Chapter XVI, 5). Without number words “we can hardly well make use of numbers in reckoning, especially where the combination is made up of any great multitude of units; which put together, without a name or mark to distinguish that precise collection, will hardly be kept from being a heap in confusion” (Locke 1894 [1690]: Book II, Chapter XVI, 5).

Locke viewed the inability to count without corresponding number words as confirmed with regard to the indigenous Americans. He spoke with representatives of tribes that could not count to 1000. Otherwise, they seemed mentally alert and could reckon up to 20. Locke attributed the deficit of not being able to operate with large numbers or to conceptualize quantities of this size to the lack of corresponding number words, which had not developed in these languages because their speakers had no need for them. Several times, Locke gave as explanation for the absence of number words the lack of any need to designate them. Since the Indians were not familiar with trade and mathematics, they had not developed any further number words and would only refer to the hairs on their head in order to express large quantities.

Another reason for the necessity of names to numbers. This I think to be the reason why some Americans I have spoken with, (who were otherwise of quick and rational parts enough,) could not, as we do, by any means count to 1000; nor had any distinct idea of that number, though they could reckon very well to 20. Because their language being scanty, and accommodated only to the few necessaries of a needy, simple life, unacquainted either with trade or mathematics, had no words in it to stand for 1000; so that when they were discoursed with of those greater numbers, they would show the hairs of their head, to express a great multitude, which they could not number; which inability, I suppose, proceeded from their want of names.
(Locke 1894 [1690]: Book II, Chapter XVI, 6)

The Toupinambos had no names for numbers above 5; any number beyond that they made out by showing their fingers, and the fingers of others who were present.
(Locke 1894 [1690]: Book II, Chapter XVI, 6)

Ultimately, according to Locke, we Europeans would also be better at arithmetic if we had corresponding names for very large numbers. It is difficult to think beyond 18- or 24-digit numbers. If we had names for large numbers like those below — which are thought in a linear series, where we say first the units, then the millions, then billions, and so on — then our ability to conceptualize these numbers may be improved:

Nonillions	Octillions	Septillions	Sextillions	Quintillions
857324	162486	345896	437918	423147
Quatrillions	Trillions	Billions	Millions	Units
248106	235421	261734	368149	623137

(Locke 1894 [1690]: Book II, Chapter XVI, 6)

In English, however, it is common practice to repeat of millions and to say millions of millions of millions, which does not contribute to a distinctive designation of the numbers.

Thus, in all languages there are deficits in the naming of numbers. To remedy this, distinctive number words would have to be introduced. But Locke did not want to impose this even for English, because, according to him, it was ultimately only the needs of the speaker that were decisive for the introduction of new number words.

3. Condillac’s theory of arithmetic as prototypical example of the necessity of signs for thinking

Following Locke, Étienne Bonnot de Condillac (1714-1780) also cited the example of those Indians who only had words for the numbers up to 20 and who could not imagine the number 1000. Condillac added that they could not even imagine numbers less than 1000 (Condillac 1947 [1746]: 41). People who do not have a method for building larger numbers from smaller units cannot go beyond the smaller numbers in their thoughts. The presence of such a method, in other words the necessary linguistic signs, is thus a prerequisite for the further development of thinking. If these linguistic signs were lacking, human knowledge would be constrained by a language-related limitation.

Condillac justified the necessity of number words with the limitations of our sensory imagination. The numbers two and three can be imagined in the form of two and three distinctive objects. The transition to four is simplified by imagining two objects respectively on two sides, for six he imagined two in three groups or three on two sides. To have an idea of larger numbers, it would be necessary to include several units in one group and to assign a sign to this group.

Pour moi, je n’apperçois les nombres deux ou trois, qu’autant que je me représente deux ou trois objets différents. Si je passe au nombre quatre, je suis obligé, pour plus de facilité, d’imaginer deux objets d’un côté et deux de l’autre : à celui de six, je ne puis me dispenser de les distribuer deux à deux, ou trois à trois ; et si je veux aller plus loin, il me faudra bientôt considérer plusieurs unités comme une seule, et les réunir pour cet effet à un seul objet.
(Condillac 1947 [1746]: 41)

Condillac attributed the cognitive deficit of the Indians who could not count to 1000 to the fact that their language did not have any method which would enable them to form new number words for larger numbers starting from the first numbers. If new number words could be formed by an analogous pattern, larger numbers could be easily named:

C’est que les premiers signes étant donnés, nous avons des règles pour en inventer d’autres. Ceux qui ignoreroient cette méthode, au point d’être obligés d’attacher chaque collection à des signes qui n’auroient point d’analogie entre eux, n’auroient aucun secours pour se guider dans l’invention des signes. Ils n’auroient donc pas la même facilité que nous pour se faire de nouvelles idées. Tel étoit vraisemblablement le cas de ces Américains. Ainsi seulement, ils n’avoient point d’idée du nombre mille, mais même il ne leur étoit pas aisé de s’en faire immédiatement au-dessus de vingt.
(Condillac 1947 [1746]: 41)

According to Condillac, progress in the acquisition of numbers thus depended solely on how accurately the initial numbers were named and how the subsequent numbers could be distinguished from them through corresponding names. There is nothing that holds several things in our thoughts together as the sign that has been assigned to it. For that reason one should not delude oneself by imagining that ideas of numbers, separate from their signs, would be something clear and determined:

Il ne faut pas se faire illusion, en s‘imaginant que les idées des nombres, séparées de leurs signes, soient quelque chose de clair et de déterminé.
(Condillac 1947 [1746]: 41)

As early as 1746, in his Essai sur l’origine des connoissances humaines, Condillac advocated transferring the reflections on numbers and the necessity for analogous number words to all of the sciences (Condillac 1947 [1746]: 42). In his unfinished work La langue des calculus, which was published posthumously in 1798, he extolled algebra as example of a language that is in itself analogous and therefore an ideal scientific language. For him, algebra was the only well-formed language (une langue bien faite), and it was the only language that could be well-formed because, unlike other languages, nothing about it is arbitrary. The analogy leads from expression to expression and the language leads as method to the right insights, which are not at all dependent on the authority of usage:

L’algèbre est une langue bien faite, et c’est la seule: rien n‘y paroît arbitraire. L’analogie qui n’échappe jamais, conduit sensiblement d’expression en expression. L’usage ici n’a aucune autorité. Il ne s’agit pas de parler comme les autres, il faut parler d’après la plus grande analogie pour arriver à la plus grande précision ; et ceux qui ont fait cette langue, ont senti que la simplicité du style en fait toute l’élégance : vérité peu connue dans nos langues vulgaires.
(Condillac 1948 [1798]: 420)

According to Condillac, however, it is not the language that determines the cognitive possibilities of the speaker and it is not the words that lead to the things, but rather only in the context with the development of science-immanent cognitive means can language be perfected and thus positively influence cognitive processes. Apart from that, the development of language and thinking is not merely a process that simply takes place in mutual dependence but rather is integrated into man’s needs to designate and their development. This relationship to numbers, which Locke already analyzed extensively, was apparently a generally accepted view at the end of the 18th century. For example, Joseph-Marie Degérando (1772-1842) asserted in his prize essay (Degérando 1800), using the example of numbers, that underdeveloped signs are no hindrance to thinking, but rather are an expression of the lack of any need to devote more attention to the thus designated knowledge areas.

4. The broad effect of the numerical example for the dissemination of the view that thought is dependent on language

The statements of La Condamine about the lack of corresponding number words in native American languages and their epistemological evaluation by Locke and Condillac spread very quickly. Johann David Michaelis (1717–1791) also mentioned them in his award-winning answer to the question of the Berlin Royal Academy concerning the influence of the opinions of a people on its language, and of the language on its opinions, in which he cited the inability of the Indians to count above 20 and the metaphorical designation of quantity with reference to the hairs on the head. For them, all large numbers represented only a non-specific quantity that could not be determined and which was merely referred to as something large:

Einige Americanische Völcker können über zwantzig nicht zählen: was darüber gehet, vergleichen sie den Haaren ihres Haupts. Ein recht bequemer Ausdruck, die unordentliche Menge zu bezeichnen, an deren Bestimmung man verzweifelt. Sie müssen statt der größern Zahlen blos etwas großes dencken, ohne zu wissen wie groß es ist. Was vor Mühe muß es kosten, aus einem solchen Holtz einen Mathematicum zu bilden? und wie weit von der Mathesi unserer Bauern muß dort ein jeder zurück bleiben, der keinen göttlichen Geist, oder keinen fremden Unterricht hat? Dis kann nicht ohne weitern Einfluß bleiben: überall fehlen wir, wenn wir nicht rechnen können.
(Michaelis 1760: 34-35)

In contrast to La Condamine’s reference to Portuguese as a way out and Locke and Condillac’s reference to the possibility of constructing analogous number systems, Michaelis viewed the mathematical potential of these people as limited by their language, noting that no mathematician could be carved from such wood. Although Michaelis in his essay was basically concerned with the mutual influence of linguistic features and modes of thought of the peoples, in his opinion even uneducated peasants had a greater advantage when they had a language with an adequate vocabulary of number words. Much like Locke, Michaelis proposed that the ability to conceptualize very large numbers could be promoted by the corresponding words. We have nine ones and nine tens, hundred as the second power of ten and thousand as the third power. It would be useful for mathematical thinking if we had further non-compound words up to the tenth power. These would give concepts to every speaker which in the present condition of the language only scientists can acquire with much effort.

Der Vorzug einer Sprache könnte noch weiter getrieben werden, wenn sie sieben unzusammengesetzte Zahlwörter mehr hätte, als die unsrige. Wir haben neun Einer, sodann neun Zehner von 10 bis 90, endlich zehn in der zweiten und dritten Potenz, nehmlich Hundert und Tausend. Es wäre der Analogie des übrigen gemäß, wenn wir auch eigene Wörter für 10 bis in die zehnte Potenz hätten, das ist, bis auf zehntausend Millionen. Dadurch würden dem, der nur auf eine gemeine Art rechnen kann, zehntausend Millionen so faßlich werden, als jetzt tausend: und was uns jetzt tausendmahltausend ist, das wäre uns dann zehntausend Millionen in der zweiten Potenz, oder

100.000.000.000.000.000.000.

In einem solchen Volcke würde der gemeine Mann von den Größen, die der Astronomus misset oder dencket, Begriffe haben können, welche jetzt so manchem unmathematischen Gelehrten mangeln.
(Michaelis 1760: 37)

Towards the end of the 18th century the Jesuit Lorenzo Hervás y Panduro (1735-1809) addressed the issue of number words on a much broader basis. Volume 19 of his encyclopedia Idea dell’Universo, published in 1787, was entitled Aritmética delle nazioni e divisione del tempo fra vli orientali. Here Hervás described the system of numerals in American, European, Asian and African languages and remarked that many languages were based on the numbers 10, 100, 1000, which were constantly repeated in the formation of numerals. The arithmetic of Hervás is in reality a language-typological representation of the numerals in the languages known in the late 18th century (cf. Breva Claramonte 1993: 499). In the five language-related volumes of the Italian encyclopedia, Hervás presented the thus far largest collection about the languages of the world. Later Hervás published an even more extensive catalogue of languages in Spanish, the Catálogo de las lenguas de las naciones conocidas, y numeración, división, y clases destas, según la diversidad de sus idiomas y dialects (1800-1805). The most original achievement of Hervás is doubtless his classification of American languages. His purpose was to confirm statements made in the Bible using the factual evidence to be found in the world’s languages. Hervás asserted that an interpretation of Genesis, according to which God gave the first human beings one language from which all other languages were derived, could be proven through common roots and sounds in some words, even if the people who spoke the languages in question did not have any contact with each other. Thus, the words six and seven originated from one root common to more than 70 languages (Saggio pratico delle lingue, 1787: 9-11; cf. Breva Claramonte 1993: 500). Hervás also stated that the languages would help us to reconstruct secular history and to improve our knowledge about the peoples that speak them.

Since every language is richer in certain elements than other languages, and none reaches absolute perfection, language comparison appears as a contribution to the development of the entirety of human understanding. Several times during the 18th century it was proposed that a Collège des sciences étrangères should be established, in which members of indigenous tribes were to contribute insights gained on the basis of their languages (cf. Haßler/Neis 2009: 751-777). The ideologues were more or less able to realize this demand over the short term through expeditions made directly for scientific purposes. However, there is no written evidence of this interaction of various languages, nor of any attempts to train native speakers of indigenous languages without number words above twenty to be mathematicians.

Notes

[1] These include orientation in space (Bowerman 1996; Levinson 1996; Nuyts / Pederson 1997; Fuchs / Robert 1997), the names of colors (McLaury 1997), the linguistic-cultural processing of kinship terms (Kronenfeld 1998) and religious concepts (Boyer 1996 and the anthology of Niemeyer / Dirven 2000 and Pütz/Verspoor 2000). The entire third part of the anthology of Gumperz / Levinson (1996: 222-469) is devoted to the problem of relativity in language usage.

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How to cite this post

Haßler, Gerda. 2014. “Some Americans could not by any means count to 1000”. The cognitive effects of the lack of names for numbers in exotic languages from the perspective of linguistic theorists before Humboldt. History and Philosophy of the Language Sciences. https://hiphilangsci.net/2014/10/29/some-americans-could-not-by-any-means-count-to-1000-the-cognitive-effects-of-the-lack-of-names-for-numbers-in-exotic-languages-from-the-perspective-of-linguistic-theorists-before