Urban contact dialects emerged in urban settings among locally born young people and can serve as markers of a new, multiethnic urban identity. The chapter brings together instances of such dialects from Europe and Africa, two regions where these phenomena have received a lot of attention from contact-linguistic and sociolinguistic perspectives. In both settings, local contexts for urban contact dialects are characterised by an openness to multilingual practices. In African contexts, this multilingual perspective is usually also present at the macro level of the larger society; in Europe, the societal context is generally characterized by a more monolingual (and monoethnic) habitus. The comparative perspective adopted here shows that these differences in macro context support different structural and sociolinguistic outcomes, including contact-induced and contact-facilitated change; urban contact dialects taking the form of multilingual mixed languages or new vernaculars of a national majority language; the possible spread of these dialects to become general markers of youth or modernity; and negative public perceptions involving different language-ideological patterns.

In the previous chapter, we analysed the relationship between numbers and objects. We encountered numbers as efficient tools in different kinds of ‘measurements’, that is, tools that we assign to objects in a meaningful way in different contexts. Our discussion has identified the different purposes numbers fulfil in these contexts, and has shown us how we manage to employ them to tell us something about empirical properties. Now that we have an understanding of what we use numbers for, and how we do that, let us explore the philosophical background of our discussion and see what would be a good characterisation of numbers, in order to carve out the area in which our investigation will take place.

What does it mean to be a number? One of the earliest definitions of numbers that have been passed on to us is an analysis that the Greek mathematician Euclid gave in his ‘Elements’:

So, are numbers multitudes? What must a definition of numbers account for? Which characteristics distinguish numbers from other entities? At the end of the nineteenth and beginning of the twentieth centuries, these questions were discussed intensely in the philosophy of mathematics.

The criteria-based approach characterises numbers as elements of an infinite progression that is used as a tool in cardinal, ordinal, and nominal assignments. This account gives rise to a range of possible number sequences. We are not looking for the number sequence anymore, but for sequences that are employed as numbers, sequences that fulfil the function of numbers in our daily lives. In the present chapter we explore the consequences that can be derived from this view. The criteria-based view carves out the principal area of numbers, it gives us the criteria to recognise possible number sequences; with this characterisation in hand, we can now ask which sequences it is that we use for numerical purposes: what are the ‘numbers’ we employ in number assignments, the entities that constitute the core of our cognitive number domain?

So, what we are looking for now are sequences that fulfil the three ‘number’ requirements and that are actually used in the different types of number assignments: we want a designated number sequence that can serve as the basis for the cardinal, ordinal, and nominal number concepts we will discuss in chapters 5 and 6. Let us have a look at how we come to grasp numerical tools first: which sequence is it that children initially encounter when they are exposed to numbers which sequence do they employ in numerical routines like counting?

A striking feature of numbers is their enormous flexibility. A quality like colour, for instance, can only be conceived for visual objects, so that we have the notion of a red flower, but not the notion of a red thought. In contrast to that, there seem to be no restrictions on the objects numbers can apply to. In 1690 John Locke put it this way, in his ‘Essay Concerning Human Understanding’:

This refers to our usage of numbers as in ‘four men’ or ‘four angels’, where we identify a cardinality. This number assignment works for any objects, imagined or existent, no matter what qualities they might have otherwise; the only criterion here is that the objects must be distinct in order to be counted. In a seminal work on numbers from the nineteenth century, the mathematician and logician Gottlob Frege took this as an indication for the intimate relationship between numbers and thought, a relationship that will be a recurring topic throughout this book:

And this is only one respect in which numbers are flexible.

CARDINAL, ORDINAL, AND ‘#’-CONSTRUCTIONS

Cardinal, ordinal, and ‘#’-constructions express concepts of cardinality, numerical rank, and numerical label. Cardinal measure constructions (as opposed to cardinal counting constructions) include concepts of measure functions as part of their meaning. The different numerical concepts are integrated in the semantic representation of complex number word constructions. In the following representations, a is an empirical object (in the case of cardinal constructions the empirical object a is a set), p is a progression, s is a set, and /θriː/ is an element of the English counting sequence C (that is, /θriː/ is an element of a set of numerical tools).

Semantic representations for the different classes of number word constructions

• Cardinal counting construction: three pens: ɛa[PEN⊕(a) ∧ NQ(a, /θriː/)]

• Cardinal measure construction: a pumpkin of 3 kg: ɛa[PUMPKIN1(a) ∧ NQ(KG(a), /θriː/)]

• Ordinal construction: the third player: ιa ∃p[PLAYER1(a) ∧ NR(a, p, /θriː/)]

• ‘#’-construction (ordinal): player #3: ιa ∃p[PLAYER1(a) ∧ NR(a, p, /θriː/)]

• ‘#’-construction (nominal): player #3: ιa ∃s[PLAYER1(a) ∧ NL(a, s, /θriː/)]

Explanations of the elements used in the formulae

1. ɛ: The ‘ɛ’-operator is used to model nominal terms: F(ɛx(G(x))) =df. ∃x(G(x) ∧ F(x)).

2. ι: The ‘ι’-operator is used to model definite terms: ‘ιx(F(a))’ stands for the most salient F, that is, the most salient realisation of a concept F.

3. F1: ‘ɛa(F1(a))’ stands for ‘one F’, that is, F1(a) is true iff a is a singleton of realisations of F; a singleton of realisations of F is defined as a set of Fs with the numerical quantity ‘one’;

4. hence, employing our definition of NQ and of English words as an instance of numerical tools, we can state that F1(a) is true iff NQ(a, /w∧n/) is true.

5. […]

What is the status of non-verbal numerals like ‘5’ or ‘V’? Obviously they are not linked to a particular language, but used cross-linguistically; for instance, both ‘5’ and ‘V’ are used in English as well as in French contexts (although they are not used universally). And they are also not part of a particular alphabet: roman numerals are not restricted to contexts where we use, say, the Latin alphabet, but can be used together with the Cyrillic alphabet as well as with Chinese script, and the same is true for arabic or Chinese numerals. Which numeral system is used in the context of a particular script is a cultural phenomenon. The usage is not based on an inherent link between numeral systems and particular alphabets, but rather is due to historical coincidences or to the relative advantages of different numeral systems. And as the case of arabic and roman numerals illustrates, it is also quite common that two (or more) numeral systems are used within the same culture and together with the same script.

The independence of non-verbal numerals from script is particular obvious in Arabic. In contexts where Arabic script is used, one normally finds the East-Arabic numerals, which, like the West-Arabic ones that are used in Europe, originated in the Indian Brahmi-script. This script goes from left to right (hence like Latin, but unlike Arabic script).

As a result of our investigation thus far, we have developed a view of numbers as a set of species-specific mental tools, and we have characterised counting words as verbal manifestations of this toolkit and as the first numbers our species might have grasped. In the previous chapter, identified language as that human faculty which gave us the mental background to develop a systematic number concept. I suggested a route to number in human evolution which started from iconic representations of cardinality, went via finger tallies and iconic sequences of words, and led to cardinal number assignments as the first context in which counting words might have been used as genuine numerical tools. Let us now have a look at the emergence of number not in human history, but in the individual development of children. What does the picture look like here? What route do children take when they acquire a systematic concept of number?

The present chapter is dedicated to these questions. In particular I will discuss the acquisition of cardinal number assignments, relating our view of counting words and number contexts to the psychological evidence for their representation in individual development. I show what role the numerical status of counting words plays in their acquisition, and how children learn to apply them to empirical objects for the first time. Again, we will find initial stages of iconic cardinality representations, and will see how the development of the language faculty supports the emergence of genuine number assignments.

A note on the definitions of the functions NQ, NR, and NL, which account for our concepts of numerical quantity, numerical rank, and numerical label, respectively:

In what follows, I introduce an ontology for our conceptualisation of number contexts that is grounded in the representation of numerical tools. In order to keep things simple, I focus on counting sequences as our primary numerical tools. However, according to our criteria-based approach there are also other progressions that can serve as numerical tools. When we acquire these progressions, their representation is integrated into our concept of ‘number sequence’.

The definitions are hence generalised in order to cover all sequences that are introduced as number sequences: instead of ‘counting sequence’ I use the general term ‘numerical sequence’ for any sequence that fulfils the number criteria and is conventionally used as a set of numerical tools (where the above-defined sequence C is one instance for such a sequence).

As shown in chapter 7 non-verbal numerals can also play this role. In this case, the final argument of NQ, NR, and NL, respectively (the argument identified as ‘ɑ’ in the definitions) is not an element of a counting sequence C, but for instance an element of the sequence A of arabic numerals whose initial element in these mappings is then the element corresponding to ‘one’, namely ‘1’.

In the present chapter – the final chapter of this book – I want to discuss in more detail the expressions that denote number concepts. The reason why these expressions appeared in our discussion thus far was mainly that we had to make clear the distinction between non-referential counting words (our numerical tools) and referential number words (expressions for numerical concepts). In the first part of this chapter, I work out this distinction in some more detail. On this basis, the other sections will then be dedicated to the question of what referential number word constructions can tell us about the relationship between numbers and language.

In the second section we will investigate how the organisation of number word constructions relates to the organisation of numerical concepts. In this context we will see that some constructions can be used for ordinal and nominal reference alike. I show that our account of number contexts allows us to relate this overlap to structural similarities between ordinal and nominal number assignments.

In the third section of this chapter, I look into the grammatical status of referential number words. I show that cardinals (‘four pens’), ordinals (‘the fourth runner’) and ‘#’-constructions (constructions like ‘bus number four’) share grammatical features with different kinds of non-numerical expressions, namely with quantifiers like ‘few’ and ‘many’, with superlatives like ‘the fastest’, and with proper names like ‘Karen’, respectively.

A CRITERIA-BASED APPROACH TO NUMBERS

Criteria for possible number sequences N

1. All x ∈ N must be well distinguished.

2. N must be a progression.

3. N must be infinite.

Definition of numbers as numerical tools, based on these criteria

Any sequence that fulfils these criteria can fulfil numerical purposes, that is, it can be used as a sequence N of numerical tools. As long as this sequence is only used in number assignments for finite sets and does not serve as a basis for complex mathematics, the third feature (infiniteness) is optional.

A POSSIBLE SET N DRAWN FROM VERBAL ENTITIES: THE ENGLISH COUNTING SEQUENCE C

Generation of the elements of C by inductive definition

Elements of C are defined via their phonological representations. I define six classes of C: ‘Ones’ (one to nine), ‘Teens’ (ten to nineteen), ‘Tys’ (twenty to ninety-nine), ‘Hundreds’ (one hundred to nine hundred and ninety-nine), ‘Thousands’ (one thousand to nine hundred and ninety-ninethousand ninehundred andninety-nine), and ‘Millions’ (open-ended, from one million). Elements of the initial class of primitive counting words, Ones, are defined as a list, that is, by enumeration.

Apart from the two initial classes, each class consists of two subclasses: (1) the m-class of elements whose immediate constituents are combined by a rule of a multiplicative character (for example six-ty, two hundred), and (2) the a-class of elements whose immediate constituents are combined by a rule of an additive character (for example sixty-three, two hundred and ten).