Annotating and Inferring Compositional Structures in Numeral Systems Across Languages (2025)

Abstract: Numeral systems across the world’s languages vary in fascinating ways, both regarding their synchronic structure and the diachronic processes that determined how they evolved in their current shape. For a proper comparison of numeral systems across different languages, however, it is important to code them in a standardized form that allows for the comparison of basic properties. Here, we present a simple but effective coding scheme for numeral annotation, along with a workflow that helps to code numeral systems in a computer-assisted manner, providing sample data for numerals from 1 to 40 in 25 typologically diverse languages. We perform a thorough analysis of the sample, focusing on the systematic comparison between the underlying and the surface morphological structure. We further experiment with automated models for morpheme segmentation, where we find allomorphy as the major reason for segmentation errors. Finally, we show that subword tokenization algorithms are not viable for discovering morphemes in low-resource scenarios.

## introduction numeral systems represented by the words for cardinal numbers used in counting are an interesting kind of linguistic data: they code a part of the lexicon of human languages that is potentially large and often exhibits a regularity that increases with higher numbers. regularity is reflected in the recycling of linguistic material used to create higher numbers, where morphemes for smaller number words are often reused to motivate the formation of larger numerals. in addition, numeral systems are also maximally distinctive. being used to distinguish ordinal numbers, we rarely find cases in which two distinct numbers are expressed by the same word form, even if numeral words themselves can have multiple meanings outside of the number domain (as can be easily seen when browsing number words in the database of cross-linguistic colexifications, @xcite . another important aspect of numeral systems is that they are not created in an ad-hoc fashion but have instead often evolved over hundreds of years. the evolution can leave traces in numeral systems that counter-act former regularity, leading to allophonic variation in the morphemes that compose numeral words. language contact can also feature as an important aspect of evolution, resulting in extreme cases where languages use two or more numeral systems in combination, reflecting different stages of their history. the fact that most numeral systems are compositional, while at the same time being distinctive and discrete in their denotation, makes them an interesting test object for linguistic analyses that deal with lexical compositionality in the context of language change. while one would otherwise have to cope with problems resulting from various kinds of morphological and semantic variation, numeral systems can be seen as an ideal test ground for the annotation and inference of compositional structures in the lexicon of human languages. in the following, we will try to illustrate this point in more detail. after a short overview on numeral systems in the context of descriptive and computational linguistics ( § 2), we present a small collection of numeral systems along with methods that can be used to annotate numeral systems manually or to segment numeral words automatically into morphemes ( § 3). after testing these methods and reporting the results on our small cross-linguistic sample of numeral systems ( § 4) we discuss our findings and point to ideas for future work ( § 5). ## background the cross-linguistic diversity of numeral systems has attracted the interest of scholars since hervás y panduro's comparative work (1786), which pre-sented data from missionaries on many then littleknown languages. today, the most comprehensive database on numerals is @xcite , who collected data on more than 5,000 varieties, often provided by linguists with first-hand experience of the respective languages. the constant increase in data has allowed for the study of numeral systems from a formal (see e.g. brandt @xcite and a typological perspective. the latter approach reached a turning point with @xcite 54 generalizations, most of which stood the test of time @xcite . even though their synchronic structure may be opaque, numeral systems are diachronically motivated and are built through a limited number of cross-linguistic strategies @xcite . they typically combine a small set of morphemes (mainly numbers, but also linking elements) according to three parameters, including (1) the choice of the base(s), (2) the operations applied to the base(s), following the implicational hierarchy: @xmath0; and the most common bases are 'five', 'ten', and 'twenty', whose conceptual sources are, respectively, the fingers of the hand, of both hands, and of all hands and toes @xcite ; on finger counting and its cultural variability, see @xcite . decimal systems are the most frequent worldwide, followed by vigesimal and quinary systems @xcite . languages can employ more than one base, resulting in hybrid numeral systems. while languages with no numerals or only the number 'one' are rare @xcite , the numeral systems of many languages, particularly in south america, new guinea and australia, are restricted to a few numerals @xcite . according to @xcite , this indicates that the speakers did not count and enumeration was not the primary use of these number words. @xcite observed that pidgins and creoles tend to have more complex numeral systems than the global average, with their frequent origin as trade languages being a potential contributing factor. numeral systems often developed out of contact, which usually comes with societal change, and borrowing may also involve the lowest numbers @xcite . while numeral systems all over the world have been quite intensively investigated in the past, very few computational studies @xcite formalize approaches to model compositionality and reveal motivation patterns underlying individual number words. recent advances in the annotation of lexical motivation patterns @xcite and the automated segmentation of words into morphemes @xcite open new possibilities for a computational investigation of numeral systems that we will discuss in more detail in the following. ## discussion and conclusion in this study, we have demonstrated an efficient, transparent, and robust workflow for the annotation and analysis of numeral systems. the workflow features a detailed annotation scheme for shared morphemes across word forms, accounts for potential allomorphy, and can be carried out in a computerassisted manner, using a web-based annotation tool. as a result, we presented a small sample of annotated numeral systems from 25 typologically diverse languages from eurasia and south america. we used this sample to evaluate how well unsupervised methods for automated morpheme segmentation work in extremely low-resource scenarios with an extraordinarily strong morphological signal. the results suggest that the major error source of these models is allomorphy. when this factor is accounted for, rather satisfactory morphological analyses can be inferred automatically. for future research on morpheme segmentation in lowresource scenarios, the handling of allomorphy will therefore be crucial. several statistical measures of numeral systems introduced here confirm intuitive correlations, such that smaller morpheme inventories necessarily entail a higher expressivity of the individual morphemes. it remains unclear, however, if a measure of morphological complexity can be inferred from our measures, since information-theoretic approaches that have been proposed to measure morphological complexity on corpus data do not convey any useful information about the morphological structure of numeral systems. curiously, it seems that the performance of morfessor aligns with (impressionistic) human judgement of how transparent a numeral system is. since morfessor is based on the minimum description length (mdl, @xcite principle, which has been proposed as a framework for measuring complexity in numeral systems @xcite , it might serve as a useful indicator for complexity when applied on the underlying data representation. we conclude that due to their high degree of compositionality, numerals serve as an ideal controlled sample for developing and testing the annotation and inference of morphological structures in multilingual wordlists. in the future, we hope to further expand our sample of numeral systems and test more methods for automated morpheme segmentation.