The acquisition of a counting sequence Pre-numerical props for number assignments Gateway to number Language and the emergence of counting and cardinality 6 The organisation of our cognitive number domain Representation of counting words as numerical tools Concepts of numerical quantity Abstract cardinalities Concepts of measure Concepts of numerical rank Concepts of numerical label The architecture of the number domain 7 Non-verbal number systems Arabic numerals as a non-verbal number sequence The development of non-verbal numerals: From iconic representations to numerical tools Two alternative number sequences: The correlation of counting words and Arabic numerals 8 Numbers in language: The grammatical integration of numerical tools

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Is language the key to number? This article argues that the human language faculty provides the cognitive equipment that enables humans to develop a systematic number concept.Crucially, this concept is based on non-iconic representations that involve relations be tween relations: relations between numbers are linked with relations between objects. In contrast to this, language-independent numerosity concepts provide only iconic representations. The pattern of forming relations between relations lies at the heart of our language faculty, suggesting that it is language that enables h umans to make the step from these iconic representations, which we share with other species, to a generalised concept of number.

We argue for a perspective on bilingual heritage speakers as native speakers of both their languages and present results from a large-scale, cross-linguistic study that took such a perspective and approached bilinguals and monolinguals on equal grounds. We targeted comparable language use in bilingual and monolingual speakers, crucially covering broader repertoires than just formal language. A main database was the open-access RUEG corpus, which covers comparable informal vs. formal and spoken vs. written productions by adolescent and adult bilinguals with heritage-Greek, -Russian, and -Turkish in Germany and the United States and with heritage-German in the United States, and matching data from monolinguals in Germany, the United States, Greece, Russia, and Turkey. Our main results lie in three areas. (1) We found non-canonical patterns not only in bilingual, but also in monolingual speakers, including patterns that have so far been considered absent from native grammars, in domains of morphology, syntax, intonation, and pragmatics. (2) We found a degree of lexical and morphosyntactic inter-speaker variability in monolinguals that was sometimes higher than that of bilinguals, further challenging the model of the streamlined native speaker. (3) In majority language use, non-canonical patterns were dominant in spoken and/or informal registers, and this was true for monolinguals and bilinguals. In some cases, bilingual speakers were leading quantitatively. In heritage settings where the language was not part of formal schooling, we found tendencies of register leveling, presumably due to the fact that speakers had limited access to formal registers of the heritage language. Our findings thus indicate possible quantitative differences and different register distributions rather than distinct grammatical patterns in bilingual and monolingual speakers. This supports the integration of heritage speakers into the native-speaker continuum. Approaching heritage speakers from this perspective helps us to better understand the empirical data and can shed light on language variation and change in native grammars. Furthermore, our findings for monolinguals lead us to reconsider the state-of-the art on majority languages, given recurring evidence for non-canonical patterns that deviate from what has been assumed in the literature so far, and might have been attributed to bilingualism had we not included informal and spoken registers in monolinguals and bilinguals alike.

Humans possess a number concept that differs from its predecessors in animal cognition in two crucial respects: (1) it is based on a numerical sequence whose elements are not confined to quantitative contexts, but can indicate cardinal/quantitative as well as ordinal and even nominal properties of empirical objects (e.g. 'five buses': cardinal; 'the fifth bus': ordinal; 'the #5 bus': nominal), and (2) it can involve recursion and, via recursion, discrete infinity. In contrast to that, the predecessors of numerical cognition that we find in animals and human infants rely on finite and iconic representations that are limited to cardinality and do not support a unified concept of number.In this paper, I argue that the way such a unified number concept could evolve in humans is via verbal sequences that are employed as numerical tools, that is, sequences of words whose elements are associated with empirical objects in number assignments. In particular, I show that a certain kind of number words, namely the counting sequences of natural languages, can be characterised as a central instance of verbal numerical tools.I describe a possible scenario for the emergence of such verbal numerical tools in human history that starts from iconic roots and that suggests that in a process of co-evolution, the gradual emergence of counting sequences and the development of an increasingly comprehensive number concept supported each other.On this account, it is language that opened the way for numerical cognition, suggesting that it is no accident that the same species that possesses the language faculty as a unique trait, should also be the one that developed a systematic concept of number.