The counting principle makes number words unique

2.1 COCA-modified numerals

We collected all cardinal numerals from COCA and determined how many times a numeric expression occurred with a broadener or with a selector. The corpus, the process of data extraction, and the list of modifiers are detailed in the Supplementary Materials (SM3).

The total frequencies of numeric expressions and their frequencies when broadened and selected from in different segments of COCA are displayed in Table 1.

The numbers show that the most informative registers (news and academic texts) contain the highest number of numeric expressions (the top row, see also Biber [1995]; Woodin et al. [2023]). The second and third rows show that the vast majority of numeric expressions are neither broadened nor selected from by the modifiers here checked for. Yet, they are much more often broadened than they undergo selection by the operators here examined (133.4 times more across the four genres). As much as 12.1 % of the spoken segment numerals are overtly broadened by one of the broadeners we checked (as compared with 0.13 % selections, 90.3 times more). This is still quite a lower rate of broadening than that found by Ariel (2021) – close to 25 %, but first, Ariel’s count was manual and took into account any broadener encountered. Moreover, while the COCA spoken data represent diverse transcribed TV and radio broadcasts, many of these may in fact be read out loud. Ariel’s data, on the other hand, were restricted to spontaneous face-to-face conversations (the most natural habitat for language use and change). Numerals are broadened the least frequently in the academic texts (5 %). This is not surprising because precision is crucial in academic discourse.

As for selectors, they occur very infrequently (with less than 0.2 % of all numeric expressions) in all segments. Moreover, if we take a closer look at the examples, we can see that very many of them are in fact spurious hits. Some of them modify nouns, not numerals, as in (4a) and (4b), from the spoken segment:

To conclude, the large-scale data from COCA support our hypothesis: numeric expressions are much more often overtly broadened and virtually never undergo selection. This supports the idea that numerals do not comprise of multiple denotations.

2.2 HeTenTen-modified numerals

We used HeTenTen, a web-based corpus compiled using a web-crawler, for extracting the relevant Hebrew data.^[12] The details about the corpus and data extraction are provided in the Supplementary Materials (SM4). While the COCA searches were made on numeral expressions, which we then coded for whether they were modified by a number of category operators (broadeners and selectors), the Hebrew searches were made on modified numerals. In addition, we here also compare the modification of numerals with that of a typical lexeme. Three Hebrew denotation selectors were chosen: Mamash ‘real(ly)’, dey ‘pretty (much)’, and sug shel ‘kind of’.Each presupposes that multiple denotations are associated with the modified phrase and points to a different selection.^[13]

We contrasted these selectors with one approximator, be.erex ‘approximately/more or less’, which can modify both numerals and typical lexemes. For comparison with the numerals, we chose an open-class adjective rek ‘empty’, which is an absolute (precise) adjective. As can be seen below, the fact that it stands for an extreme value does not mean it needs to be overtly broadened. Quite the opposite. We predicted that if modified, this open-class lexeme would mostly be modified by one of the selectors, rather than by the approximator. The numerals, on the other hand, were predicted to mostly be modified by the approximator, rather than by any one of the selectors.

Our raw results are displayed in Table 2.

Table 2 shows that an overwhelming majority of the operators we examined were selectors (over 90 %). Be.erex ‘approximately’ tokens only account for under 10 % of our operators.^[14] Now, there were 5,503,344 expressions tagged as numerals in HeTenTen, and only 24,455 rek ‘empty’ tokens. In other words, there were 225 numerals per each token of rek. Nonetheless, there were only 1.4 times more selectors for the numerals than for the single typical lexeme. Although rek tokens were rarely modified by selectors (0.76 %, 1 in every 154 tokens), the parallel result for the numerals is significantly lower (0.005 %, 1 in every 20,767 tokens) – 157 times less.

The picture is reversed for the broadener we checked for. Virtually no tokens of rek ‘empty’ were overtly broadened (2, 0.008 %, or 1 in every 12,227 tokens), as compared with the numerals (4,466, 0.08 %, or 1 in every 1232 tokens). This means that proportionately, there are about 10 broadeners with numerals per one broadener for the typical lexeme. Examining the third and fourth rows of Table 2, we see that while the typical lexeme is overwhelmingly modified by the selectors, numerals were overwhelmingly modified by the broadener.

These findings are similar to the findings for English in Ariel (2021), as well as the COCA data presented in Section 2.1. But now, we must ask, how come a minority, albeit of a marginal size, of the numerals in English (4,115 cases in COCA) and Hebrew (265 cases) were seemingly modified by selectors. We manually examined all the Hebrew cases and found that these cases were for the most part only apparent exceptions. As many as 187/265 cases (70.6 %) were actually irrelevant for the question under discussion, for various reasons: search errors, cases where the number word does not in fact function as a numeral, cases where the “selector” has an altogether different function, or cases where the selector preceding the numeral actually modifies a phrase preceding it, or the selection pertains to the modified nominal rather than to the numeral (as in [4b] above).

Summing up, HeTenTen portrays the same picture as COCA, whereby numerals do not undergo selection. We are left with 78 cases of selectors modifying numerals (out of 4,731 modifiers), which represents a meager 1.6 % of modified numerals (the rest are broadeners).^[15] Section 3 presents our theoretical account for the differential distributions. But we must first argue against potential objections to our analysis (Section 2.3).

2.3 Countering potential objections

An apparent counter-example to our claim, which might be particularly salient to linguists, is the seeming naturalness of exactly as a numeral modifier. Exactly N is a very frequent collocation in linguistic analyses. But if, as we claim, numerals are precise by default, they should preferably collocate with the approximating about rather than with the narrowing exactly. Indeed, this is the case. A 200 random sample of exactly in BNC revealed that only 4 (2 %) are numerals. In comparison, an examination of a random sample of 500 about tokens revealed 79 cases where a numeral was modified by about (15.8 %). An exhaustive search for exactly three/3 revealed 23 tokens, 0.023 % of all three/3 tokens (100,279), but the 13,888 about three/3 tokens constitute 1.4 % of three/3 (600 times more). Similarly, exactly fifty/50 yielded 6 tokens, 0.04 % of the total fifty/50 tokens (15,851), but the 584 tokens of about fifty/50 constitute 3.7 % of fifty/50 (97 times more). Exactly rather collocates with open-class expressions, such as the same (1308/10,188, 12.8 % of the exactly tokens) and the opposite (66/10,188, 0.65 %). Indeed, 2.3 % of the same tokens (1308/57,487) are modified by exactly, as are 2.7 % of the opposite tokens (66/2459).

Another potential criticism might be that not all numerals are alike. Specifically, that while nonround numerals may indeed be interpreted precisely by default, this is not so for round numerals, nor for numerals within measurement phrases (e.g., seven inches) – the latter are assumed to be round too. Our claim is that round and measurement numerals too are precise by default, despite the fact that they are more easily taken as compatible with approximate values.^[16] This point is explicitly argued for in Katzir et al. (2024). We here note, however, that our findings above about the default preciseness of numerals do not result from the fact that we analyzed all numerals, where, one might think, nonround numerals are the majority. Based on a random sample of 120 number expressions in HeTenTen, we estimate that as many as 45.6 % of all numeral expressions are round and/or measurement numerals.

3.1 Defining dense and sparse domains

Ever since Trier (1934), it is commonly assumed that words belong to different conceptual fields, each lexical item naming some parceled out conceptual subdomain of the field. Analyzing words encoding types of knowledge in Old and Middle High German, Trier proposed that lexemes offer a continuous coverage of the whole relevant conceptual field. We agree, however, with later researchers who disputed the full coverage idea (Lehrer 1974; Lyons 1977; Ullmann 1967). Indeed, there can be nonlexicalized regions within many conceptual domains. In addition, we assume that functionally relevant domains are not restricted to predefined stable conceptual fields, each falling under a single superordinate category. Nice, attractive, and kind, for example, belong to a single lexical domain in some contexts, but of course, they each participate in other lexical domains as well.

We define a lexical domain as a set of contextually determined paradigmatic lexical alternatives, where the different lexemes encode distinct, yet similar enough and even interdependent meanings. This entails that they potentially compete with each other in the same context. On our account, Trier’s definition of a lexical field is actually inappropriate for typical lexemes: Typical lexemes, we claim, form part of sparse lexical domains. However, his mosaic-style definition is just right for the conceptual domain of counting: The natural number expressions belong to a densely packed lexical domain (see again Note 8). It is the +1 successor principle that is responsible for the numerals’ membership in a dense lexical domain. We note that the number line it creates is reinforced by the high cultural salience assigned to the counting routine, which is recited by very young (Western) children even before they understand the meaning of the numerals (Wynn 1992). In fact, Casati (2014) already proposed that numerals (and calendrical lexemes) are nonstandard lexemes just because they are inseparable from what he takes to be a tacit morphological, representation of the number line.

Table 3 summarizes the differences between dense and sparse lexical domains.

Consider six and its immediate neighbors.

1. While six denotes a distinct concept from both five and seven, the difference between them is minimal, as compared with the difference between, e.g., nice and attractive or table and chair. The difference between neighboring typical lexemes is neither minimal nor consistent.^[17] But the difference between adjacent numerals is defined over a single parameter (the +1 successor principle). Interestingly, the uniform (+1) difference between adjacent numerals is overlaid on top of a conserved cognitive perception of magnitude, where magnitude differences are not uniform. Thus, Dehaene’s (2011) participants perceived the difference between small numbers (e.g., 7 and 8) as larger than that between larger numbers (e.g., 27 and 28). The successor principle behind the natural number line renders these relative differences irrelevant, however, because they are not relevant to counting.

2. All words’ meanings (and interpretations) are constrained by their neighbors. But for typical words, there need not be full coverage of the conceptual domain, and there is a rather large nonlexicalized conceptual gap between neighbors (e.g., between nice and attractive), which a typical word can then “encroach” on. This accounts for the context-adaptability of typical lexemes (recall nice interpreted as ‘close to attractive’ in [3]). There is no conceptual gap between two neighboring (counting) numerals.^[18] This accounts for their rigid faithfulness to what we take to be their literal, precise meaning.

3. We have already seen that the numerals’ boundaries are rigid. This is why it takes an explicit broadener to create an approximate numeral concept, where it is then the operator, rather than the numeral, that is responsible for the broadening. Typical lexemes, on the other hand, broaden quite freely in the absence of overt broadeners (see again [3]). The result is that numerals’ interpretations are rigidly precise, regardless of the context they’re embedded in. Typical lexemes, on the other hand, adapt to the specific context they occur in and create different ad hoc concepts (see again Section 1). Finally, (4) the synchronic invariance of interpretation associated with numerals is not conducive to potential diachronic meaning changes. But the multiple-denotation typical lexemes give rise to contextually adapted interpretations and are then conducive to meaning change. We cite evidence for the extremely low rate of diachronic change for the numerals in Section 4.3.

A note is here in order about round and measurement numerals, which seem to refute our universal preciseness assumption. We address this issue in Ariel and Levshina (2021), where we argue that while the default number line for all numerals is the dense, +1 number line, some numerals (measurement ones, as well as what we call Personality numerals) can in addition shift to a sparse numeral line, where they too can “encroach” on their neighbors.^[19] At the same time, we argue that for these numerals too the default number line is the dense +1 domain.

3.2 A word2vec analysis

In this subsection, we use a word2vec model trained on COCA for comparison of numerals with a variety of typical lexemes. Our data show remarkably different distributional patterns for the two types of expressions. The model indeed shows that the number domain is highly densely populated, so that lexemes’ meanings are rather similar to each other, whereas typical lexemes form part of sparse domains, where meanings are relatively different from one another.^[20]

In order to measure how dense different lexical fields are, and compare them with numerals, we used a vector space model. The model creates a vector space with hundreds of dimensions, as a rule. Every vector corresponds to a word. If two words are used in similar contexts, they will have comparable values on the same dimensions and will be located close to each other in the multidimensional semantic space (see an introduction of vector space models for usage-based linguists in Levshina and Heylen [2014]).

Specifically, we use a popular algorithm word2vec (Mikolov et al. 2013). This is a neural network that learns to predict words from their neighbors based on a large corpus. For example, how likely is the word crown to appear if some word in its neighborhood is queen? In order to find out, the algorithm builds two layers. One is the hidden layer, in which the rows are the context words, and the columns are hundreds of dimensions. Every word is represented by a vector with different values on those dimensions. The other one is the output layer, which is used to generate the probabilities of different words in the neighborhood of the context words. If two words are used in similar contexts, like king and queen, then word2vec should have similar probabilities for other words in their neighborhood. For example, words like crown or monarchy would have similarly high probabilities given king or queen, while words like microscope and proton would probably have similarly low probabilities. An important idea is that in order to have similar output probabilities, the word vectors in the hidden layer should also be similar. So the words that predict other words in a similar way should have similar semantics and similar distributional vectors. For technical details, see SM6 in the Supplementary Materials.

The numerals we compare represent numbers from two to ninety-nine, which we compared with typical lexemes belonging to three different lexical domains: adjectives describing intelligence, precision adjectives, and verbs of cooking from Lehrer (1974). The complete lists of lexemes are available in SM6.

After we created the semantic vectors for all these words, we computed cosine measures, which represent semantic distances between the words. We calculated the average distances between the members of the fields, which helped us estimate how dense or sparse a lexical field is on average. We also used the distances to visualize the data in ordinal Multidimensional Scaling, as shown below.

Figure 2 displays the distances between the numerals and between the adjectives representing intelligence. Figure 3 shows the numerals and the adjectives related to precision. Figure 4 displays the numerals and the cooking verbs that were found in the corpus. All three plots demonstrate that the numerals are clustered very densely, while the other domains are sparse. The semantic areas occupied by the sparse-domain lexemes are larger than that of the numbers. This difference is all the more striking given the much larger number of lexemes in the numeral field. In order to control for the number of items in each domain, we computed the average cosine-based similarity between all pairs of lexemes. The average cosine-based similarity between the numbers is 0.78, whereas for the others, the number is much lower: for the verbs of cooking, it is 0.11; for the adjectives describing precision, it is 0.05; and for the adjectives representing intelligence, it is only 0.02.

Figure 2:

Multidimensional scaling based on word2vec similarity measures: numbers and adjectives representing intelligence.

Figure 3:

Multidimensional scaling based on word2vec similarity measures: numbers and adjectives representing precision.

Figure 4:

Multidimensional scaling based on word2vec similarity measures: numbers and verbs of cooking.

In sum, Figures 2 –4 show remarkably different distributional patterns for the two types of expressions. The model shows that the number domain is highly densely populated, so that lexemes’ meanings are rather similar to each other, whereas typical lexemes form part of sparse domains, where meanings are relatively different from each other.

We note that the results of this case study have interesting implications for the recent discussions of Large Language Models (LLMs) and their capabilities. Although state-of-the-art LLMs display impressively good results in text generation, their abilities to perform basic arithmetic operations, especially multiplication and division of large numbers, are more modest. Even OpenAI’s most powerful model GPT-4.0 often makes mistakes (Yuan et al. 2023). We believe that the dense semantics of numbers could be one factor responsible for this shortcoming. The contexts in which numbers are used are too alike to allow the algorithm to learn the differences between them and to generate accurate responses.

4.1 The sparse shall narrow and the dense shall broaden

We have seen that category operators typically manifest a complementary distribution between numerals and typical lexemes. Thus, selectors kind of/sort of overwhelmingly modify typical lexemes (see Sections 2.1 and 2.2 for both English and Hebrew).^[21] So do real(ly) (e.g., real audiophile, SBC: 016) and typical,^[22] and the same is true for selector typical in SBC.^[23] Moreover, broadeners around, about, and approximately overwhelmingly modify numerals (Ariel 2021). But a complementary distribution account faces two types of counter-examples: (i) the quantitatively marginal, but still exceptional cases noted in Sections 2.1 and 2.2, where numerals were modified by selectors; (ii) the existence of modifiers used for both numerals and typical lexemes.

We here suggest that this almost complementary distribution is not due to selectional restrictions or biases. Rather, it’s the fundamental difference between dense and sparse systems that explains the distribution and interpretation of the category modifiers. Indeed, should a selector modify a numeral, it actually broadens the numeral’s meaning, and when a broadener modifies a typical lexeme, it imposes some selection on it.

Consider selectors first. As expected, selectors kind of and sort of modifying typical lexemes (e.g., kinda sore [SBC: 043]) select nonprototypical interpretations (‘close to sore, but not prototypically sore’). But in the rare cases where sort of did modify a (measurement) numeral (e.g., sort of five minutes [BNC]), its effect was actually to broaden, rather than to select a subset of the denotations. This is why sort of five does not exclude ‘exactly 5’. Rather, it creates a broadened range, say ‘4–6’ around the explicit ‘5’. In other words, even if a conventional denotation selector modifies a numeral, it is typically coerced into broadening.

Even the ballpark constructions (e.g., thousands), which lie outside the dense number line, are hardly ever narrowed by selectors. There are 56,012 tokens of this construction in the spoken + magazine sections of COCA. Despite the fact that they are definitely associated with multiple denotations, only 12 of them (0.02 %) were modified by a selector. Importantly, in none of these cases did the numeral undergo selection. All 7 reallys served as ‘truly’, 3/5 sort/kind of were not used as numeral modifiers, and most interestingly, two tokens functioned as broadeners rather than selectors:

We maintain that when a numeral is (rarely) the target of a selector, the selector turns into a broadener, interpreted as ‘approximately’ (whether the numeral is a round measurement one [6a] or a bona fide nonround count numeral [6b]):

Second, most broadeners are restricted to modifying numerals, as in (7), where the denotation is some value within a range constructed around ‘16’:

But more or less also modifies typical lexemes. This is a counter-example for a simple complementary distribution account for the modifiers. Note, however, that while the speaker-intended concept conveyed by sixteen percent more or less is not in addition selective (it does not exclude the pivot ‘precisely 16’), decide in (8a) does exclude a prototypical ‘decision’. The same is true for the ‘knowledge’ involved in (8b):

Indeed, the co-occurring broadener more or less and selector kind of in 8(a) give rise to the very same ‘nonprototypical decide’ interpretation. In other words, just like selectors, broadeners too involve selection when typical lexemes are involved. Contra Bauer and Huddleston (2002]: 1677), who claim that the suffix –ish similarly approximates numerals and typical lexemes, we propose that it only approximates numerals. When modifying typical lexemes –ish functions as a selector: Reddish (SBC) and youngish (LSAC) denote nonprototypical ‘red’/‘young’. But modifying time measuring numerals, such as oneish and thirtyish (LSAC), it is interpreted as ‘approximately’ (Ariel 2021).

We ran a mini questionnaire in order to support these possibly delicate judgments. We picked the Hebrew ‘more or less’, which is acceptable with both typical lexemes and numerals, and checked whether it functions differently when modifying a typical versus a numeral expression. Thirteen native Hebrew speakers were asked to judge the relative acceptability of the Hebrew counterparts of the following two sentences:

We chose a contrastive construction with but, and especially with instead, in order to show that whereas more or less functions as a selector when modifying pink, thus excluding ‘central pink’, it functions as a broadener in more or less thirty, hence not excluding ‘thirty’. If so, “pink” can constitute an alternative to ‘more or less pink’, but ‘thirty’ cannot constitute an alternative to ‘more or less thirty’. Results were quite clear. While many participants commented that both sentences were not perfect,^[24] a clear majority (11/13) confirmed that (9a) is more acceptable than (9b). Only one participant preferred (9b) (one had no preference). Following his judgment that (9a) is better, one of our participants added the following: “The first one [a] is a bit strange, but one can understand it with a reading that someone had a specific color in mind which is not exactly pink and was disappointed to get “exactly pink” instead. In contrast, in the second sentence [b], there is an actual contradiction in my view, between the two clauses – “30” cannot come instead of “more or less thirty” here in my opinion” (R.G. p.c., Dec. 13, 2022, original emphasis). This is exactly our point. It is the difference between dense and sparse systems that accounts for the modifiers’ differential functions. The dense-system numerals can only broaden, whereas sparse-system lexemes necessarily undergo selection.

4.2 Numerals in a noncounting community

On our approach, it would seem that numerals should always be construed as members of a dense domain and show no prototype effects. While we claim that this is the overwhelmingly dominant pattern of use, exceptions have been noted in the literature. Interestingly, such cases are restricted to noncounting communities, where specifically the absence of a counting routine has been pointed as the source of the treatment of numerals as approximate (Pica et al. 2004). We take this as further evidence for the inherent connection we propose between a dense domain and precise numerals. The absence of a salient +1 number line in these communities hinders a construal of the numeral domain as dense.^[25]

Flaherty and Senghas (2011), Spaepen et al. (2013), and Spaepen et al. (2011) studied Nicaraguan unschooled homesigners who never learned to recite a counting system, although they live in a numerate culture, and they themselves engage with numbers in connection with money, for example. Here is an example of a participant who was asked to write the Arabic numerals for 1–20 (Flaherty and Senghas 2011):

Note that quite a few numbers are missing (5, 9, 14, 17, 20), and others are out of order (7, 11, 16, 15). Age-matched hearing participants, equally unschooled, performed this task perfectly. Since they hardly have experience in counting, these signers showed only 27–55 % success rates in ephemeral matching tasks (e.g., where the experimenter taps their knee N times and they have to do the same). Now, if what entrenches the dense domain of the numerals is a practiced counting routine, it is not surprising that these signers treat numerals (other than subitizing ones) as approximate: “They may hold up either 5, 6, or 7 fingers to represent six items” (Spaepen et al. 2013: 497). Their numeral representations are one-to-one mappings between quantities of objects and signs, the latter participating in a typical, sparse lexical domain. This is why, we propose, their number signs do not have rigid boundaries and may be interpreted approximately.

4.3 The diachronic stability of numerals

“Most words have about 50 % chance of being replaced by a new noncognate word roughly every 2,000–4,000_y (= language years–M.A. and N.L.)” (Pagel et al. 2013: 1). Since words routinely adjust their interpretations in accordance with the context they occur in should some contextual adaptation become highly salient (due to high frequency), it may actually come to serve as the encoded meaning of the expression. Renewal is then necessary in order to encode the original meaning. English nice corresponds to Latin “not knowing.” Indeed, once nice evolved different meanings, a renewed lexeme ignorant was needed.

Numerals, however, “are replaced far more slowly … once every 10,000, 20,000, or even more years” (Pagel et al. 2013: 1). Consider Figure 5, which shows the rate of renewal of Indo-European numerals 1–5, when compared to a basic Swadesh list of 200 words from Pagel and Meade (2017).

Figure 5:

Rates of lexical replacement per annum for Indo-European 1–5 within 200 Swadesh list words. The numerals are in the shaded area. Pagel and Meade (2017) show a similar picture for Bantu and for Austronesian languages, except that specifically ‘one’ is renewed more often than 2–5. Indeed, ‘one’ has other uses than numeral 1.

We propose that the dense system they are part of explains the absence of synchronic polysemy for numerals, which in turn accounts for their remarkable diachronic stability.^[26]