A Preview of the Discussion

This thesis presents five interrelated proposals:

1. Numbers are sums composed of the arithmetical operation of addition (including multiplication, a shortened form of addition).

2. Numerical expressions are best interpreted syntactically as co-ordinate structures linking nouns to produce nominal compounds.

3. Numerical expressions exist as independent, context-free word strings that can be regarded as a singular form of paratactic and concatenated discourse.

4. Numerical expressions can be accounted for within the framework of X-bar theory by regarding them as additive or cumulative conjunction phrases (&P).

5. The &P analysis accounts for numerical expressions in both English and Chinese, incorporating both English “and” as a co-ordinator and Chinese ling (‘zero’) as a place holder.

I will first discuss the intrinsic qualities of numbers as arithmetical notations and syntactical nouns. This will be followed by a discussion and critique of Hurford’s classical interpretation of numerical expressions, focusing mainly on his phrasal structure. I will then discuss the characteristics of English and Chinese numerical expressions, paying special attention to the use of “and” in English numerical expressions and ling in Chinese numerical expressions. This will be followed by a discussion of the properties of co-ordination as a syntactical category, suggesting that it is co-ordination that best matches and expresses the arithmetical operation of addition that is central to the composition of numbers. I will then present my argument for regarding numerical expressions as additive conjunction phrases (&P) formed, in the manner of Munn (1993). The discussion concludes with an argument for

justifying the covert conjunction you as the head of &P in Chinese numerical expressions.

CHAPTER 2

THE NATURE OF NUMBERS

2.1 Numerals as Abstract Nouns

Hurford (1975, 1987, 2003) does not categorize numerals syntactically, and this creates a problem for any discussion of the phrasal construction of numerical expressions. Corbett (1978/2000) notes that number, as a categorical feature, is dominantly nominal, though verbs may also be marked for number. Moreover, linguists have often treated simple numbers as adjectives and multipliers and complex numbers as nouns. Nevertheless, Greenberg (1978) observes that numbers as numbers – in other words, abstract, context-free numbers – are always nouns. My thesis assumes that Greenberg is correct. All numerals, therefore, must be regarded as nouns. It follows, then, that when numerals combine to form a phrase, the result will be a nominal compound, a syntactic form that is common to both English and Chinese.

It is often taken for granted by linguists that low numbers – simple numbers, lexical numbers – especially those from 1 to 4 behave syntactically like adjectives, while higher numbers, 10 and all the complex numbers that follow, behave syntactically like nouns. But is this generalization true? Or does it represent an almost superstitious misunderstanding? I would like to propose that the grammatical nature of numbers needs to be examined more closely before we accept the common division of numbers into the categories of low adjectives and high nominals.

Greenberg (1978) states that numbers are either concrete or abstract, suggesting that it is possible to think of numbers in two distinct ways. Concrete numbers are derived from discourse.

In other words, concrete numbers quantify things. They refer to nouns. They combine with nouns to form phrases. The contextuality of concrete numbers is especially evident in numeral classifier languages such as Chinese where the appropriate classifier must be used with each noun enumerated. Abstract numbers, on the other hand, are created mentally as intellectual concepts. They exist independently and they are not constrained by discourse, contextuality, or reference to nouns. Abstract numbers are nouns. Their principal uses are counting by recitation and performing mathematical computations. Sometimes a simple number may have different

numeral expressions depending on whether it is concrete or abstract. In Chinese, for instance, 2 is expressed as er as the multiplier for 10 in the absolute system, but 2 is usually expressed as liang as the multiplier in the contextual system, especially for 100, 1000, and 10,000.

The difference between concrete numbers and abstract numbers has a long history of linguistic implications. The distinction between concrete numbers and abstract numbers is the same as the distinction between concrete nouns and abstract nouns. That is, concrete numbers are physical, because they are fused with nouns, but abstract numbers are not physical -- we cannot see, hear, taste, touch, or smell them -- because they are ideas and they are not fused with any nouns. There can be three books or three doors or three people. The number three is not limited to any one noun. To illustrate the difference between concrete nouns and abstract nouns Poncinie (1993) discusses Aristotle’s “double use” of the common term number. In Physics IV, 11, Aristotle defines number as both something that is counted and something that can be used to count something else. What can be counted is the concrete number, and what can be used to count is the abstract number. This may seem to be a simple distinction, but the more we think about it, the more difficult it is to understand. What happens for example, when we count sheep?

If we count three sheep, there is no doubt that the abstract number is three. But what is the term three sheep? Is it the quantity of the group? Or is it a numbered group? Is it three? Or is it three sheep? This is the concrete number, and it seems to suggest two interpretations.

Poncinie claims that the totality of the group, having contingent relations to other numbers, is an extensional use of the abstract number, while the numbered group is a concrete term akin to aggregate or class. What this means, then, is that there are really three kinds of numerical expressions. First, there is the abstract number. But the concrete number can be seen in two different ways: either as a totality referring to a group, or as a totaled group complete with existential features. To understand this split usage of the concrete number, let us return to the sheep. Can we apprehend and comprehend the difference between three sheep and three sheep?

It is not an easy thing to do.

To complicate the matter, we probably intuit that three sheep is a different proposition than seventy-eight sheep. It is easier to regard three sheep as a numbered group of sheep than it is to regard seventy-eight sheep in the same manner, and it is easier to regard seventy-eight sheep as the totality of the group of sheep than it is to regard three sheep in that way. Why is this? To answer we must return to the common linguistic assumption that simple numbers are adjectival

and complex numbers are nominal. Menninger (1969/1992) offers an explanation of how this belief probably began. Since only the first four numbers are usually considered to be adjectival, Menninger argues that these numbers correspond to the primitive habit of counting on one’s fingers. In other words, low numbers are intimately connected to the physical reality of the entities they quantify. It is easier to see, touch, and imagine three sheep than it is to see, touch, and imagine seventy-eight sheep. This also means that it is easier to regard three as a word that describes sheep than it is to regard seventy-eight as a word that describes sheep. Thus three seems like an adjective, while seventy-eight seems like a noun.

Menninger provides a compelling reason why numbers – even low numbers – should not be regarded as adjectives. We might be tempted to claim that “three-ness” or even “seventy-eight-ness” are attributes of the counted sheep, the same as whiteness might be an attribute of these sheep. But there is a vast difference between numbers as attributes and colours as attributes. One sheep can be white, but one sheep cannot be three or seventy-eight. Numbers always refer to totalities, and as such they are always conceptual. In other words, numbers are always abstract, always detachable from the nouns they count. Contrary to popular linguistic belief, this is true of simple numbers as much as it is true of complex numbers. Menninger (p. 11) says a number word – a numerical expression – is “a special kind of word.” He means it is an abstract noun that is, by its nature, somewhat mysterious.

Menninger illustrates the importance of abstract numbers by telling how Archimedes proposed measuring the universe by computing the number of grains of sand it would take to fill it. “The whole point of Archimedes’ discussion is that even so inconceivably large a number as that of the grains of sand contained in the universe can not only be clearly understood and verbally expressed, it can even be easily exceeded: the limitless progression of the number sequence has finally been recognized!” (p. 141) One of humanity’s greatest achievements, equaled only by the invention of writing and reading, was the discovery of the abstract nature of numbers. In grammatical terms, this means that numbers are actually nouns that name ideas or concepts that exist far beyond their connections to particular physical entities.

2.2 Numerals as Nominal Compounds

Ultimately, complex numeral expressions are nominal compounds. This means that in the phrasal structure of numerals two nouns – either simple or multiplier or even complex in form – are combined to form a larger complex noun. As Li and Thompson (1981) point out, both English and Chinese make frequent use of nominal compounds. Examples of English numerical nominal compounds are given in (1).

(1) a. 14: nine + five → fourteen

b. 78: seventy + eight → seventy-eight

c. 130: one hundred + thirty → one hundred and thirty d. 60: six x ten → sixty

e. 1700: 17 x 100 → seventeen hundred (or, one thousand, seven hundred)

f. 1,000,000: one thousand x one thousand→ one million

When two numerals are combined in English, either through addition as in (1a - c) or through multiplication as in (1d - f), mathematics determines the resulting number, but the syntactical projection of nouns determines the resulting numeral expression. This is very obvious in (1b) where the two numerals are combined in such a way that the only difference between saying the operation of adding the two numerals and saying the resulting sum is the substitution of a hyphen for ‘and’. In (1c) the meaning of the operational phrase ‘plus’ appears as ‘and’ in the numerical expression. In (1a) and (1d) transformations for -teen and -ty are involved. I will discuss these transformations presently. In (1f) the underlying operation for the product, is transformed by mathematics into a completely different lexical expression for the nominal compound. In (1e) the nominal compound can be pronounced in either of two ways. Hurford (1975) would prefer the second alternative because it conforms to his “packing strategy” concept whereby numerical expressions should incorporate the highest possible multiplier – in this case, one thousand, not one hundred.

Examples of Chinese numerical nominal compounds are given in (2).

(2) a. 16: shi + liu → shi liu b. 20: er x shi → er shi

c. 35: (san x shi) + wu → san shi wu

d. 855: (ba x bai) + (wu x shi) + wu → ba bai wu shi wu

e. 1, 0370: (yi x wan) + ling + (san x bai) + (qi x shi) → yi wan ling san bai qi shi f. 21, 3220 (er x shi x yi x wan) + (san x qian) + (er x bai) + (er x shi) →

er shi yi wan san qian er bai er shi

As in English, numeral expressions in Chinese are composed as nominal compounds through the mathematical operations of addition and multiplication. These two basic operations are illustrated clearly in (2a) and (2b). Moreover, the way the operations of multiplication and addition can be further combined is evident in (2c) and (2d). In (2e) and (2f) we notice two important features of Chinese nominal compounds. First, Chinese numerals are arranged in strings of four, unlike English numerals which are arranged in strings of three. For this reason 10, 000 or wan becomes an important Chinese multiplier. The second distinctive feature of Chinese numeral expressions is the appearance of ling, meaning ‘zero’, as a place-holder to mark the beginning of the last string of numbers if it begins with a 0. I will discuss the importance of ling in much detail as this thesis unfolds.

There are also other differences between Chinese and English numerals resulting from transformations. For instance, the number 16 is expressed with normal word order in Chinese as shi liu – literally, 10 + 6. In English, however, the numeral expression for 16 involves switching the normal word order and changing the morpheme ‘ten’ through inflection to ‘teen’, resulting in

‘sixteen’. These transformations are true of all numeral expressions in English for the numbers 13 through 19, with distinct lexical morphemes for 11 and 12, while in Chinese all the numbers between 10 and 20 are expressed by maintaining both the lexical morpheme shi for 10 and the normal word order of 10 + 1 . . . 9. Moreover, in Chinese the numeral expression for 20 maintains both normal word order and morphological form, resulting in the nominal compound

er shi, representing the multiplicative operation of 2 x 10. Once again, this pattern holds true for 30, 40, and so on, up to the next important lexical morpheme bai representing 100. In English, on the other hand, transformation morphologically changes ‘ten’ to ‘ty’ in ‘twenty’, ‘thirty’, and so on, up to ‘ninety’. Finally, a number such as 35 is expressed in Chinese as san shi wu, literally 3-10-5, whereas in English the numeral expression is ‘thirty-five’, combining the 3 x 10 operation into ‘thirty’, resulting in two morphemes for 35, whereas in Chinese all three basic morphemes are preserved. This is to say that Chinese retains both the base multiplier 10 and normal word order, while English transforms the base multiplier and deletes it in expression.

It is significant to note here that the nominal compound form of numeral expressions occurs because they are constructed in the syntax. In other words, complex numeral expressions are unique. They do not, like simple numerals and multiplier numerals, exist in the lexicon, although, as I will soon argue, the complex numerals 11 – 99 behave, especially in English, as if they were independent lexemes. Is there a limit to the construction of complex numerals as nominal compounds? We have already seen that the ancient Greek philosopher Archimedes believed that it is possible to construct a numeral expression large enough to describe the universe. Given the abstract nature of numbers, this would seem to be entirely possible.

Nevertheless, Hurford (1975) points out that numerical expressions need to be well-formed or sanctioned by natural language usage, and extremely large “theoretical” numbers may have no usable expression. Radzinski (1991) agrees and observes that it is the interface of arithmetical operations and grammatical operations that determine which numeral expressions are well-formed or even plausible.

2.3 Numerical Expressions as a Form of Discourse

Isomorphism between syntax and phonology has long been considered dubious, and yet

consideration of numerical expressions raises the possibility that it might be necessary to search the interface of these two linguistic forms to fully comprehend how numerals are composed. We have already seen that numerals are abstract, context-free nouns. Verbs (and therefore adverbs) never exist in numerical expressions. Neither do prepositions. Conjunctions are the only other possible syntactic category to be found in numerical expressions. It should be noted, however,

that the presence of a conjunction – always cumulative, represented by “&” – is regarded by Hurford (1975. p. 50) as an optional transformation. The conjunction “and” is certainly present in English numerical expressions, but a comparable conjunction, such as you, is not overtly present in Chinese numerical expressions. I shall argue, nonetheless, that the conjunction you covertly heads an &P in my binary-branching phrasal analysis of numerical expressions. In order to justify this analysis I shall appeal, to a large extent, to phonological data to support my syntactic interpretation.

Once again, Hurford’s common sense approach to the construction of numerical expressions cannot be ignored. He says, in describing his analysis (ibid.): “I have not made an attempt to integrate the grammar of numerals into a grammar for the rest of the language.” We have already seen that the grammar of numerals involves, more than anything else, the creation of sums through the operation of addition (supplemented by multiplication). We have also seen that the creation of sums results in numerals being, syntactically speaking, nominal compounds.

But is now time to observe that numerical expressions may also be regarded as independent forms of discourse whose principal characteristic is that they are composed with an absolutely rigid word order arranged paratactically as unique concatenations. Although they are basically nouns or noun phrases, these word strings do not assign case internally, and they do not exhibit any movement whatsoever – making the Empty Category Principle irrelevant. Hurford also mentions that not all lexical items denote objects. In fact, some lexical items denote

relationships. This is where & enters the grammar of numerals, as overt “and” in English numerical expressions and as covert you in Chinese numerical expressions. Either way, it is important to remember that there is always a distinction to be made between the grammar of numerals and grammar as a whole. As my argument progresses, I attempt to integrate the grammar of numerals into the general grammar, but in order to do this I will have to refer to the fact that the grammar of numerals represents a special form of discourse.

2.4 English Numerical Expressions

In English numerical expressions are divided into strings of three. For example, the number 357 contains a units’ place, a tens’ place, and a hundreds’ place. This first string may be called the

hundreds’ level. This basic ordering of places is repeated in the next higher level or string of three digits, which may be called the thousands’ level. This means that at the thousands’ level there are units, tens, and hundreds (of thousands). Moreover, the same pattern holds true for the next higher level, that is, the millions’ level. Once again, there may be units, tens, and hundreds (of millions). This system may be expanded to subsequently ever higher levels, practically ad infinitum. These facts are illustrated by the table in (3).

(3) Level of Hundreds Level of thousands Level of millions

Units 0-9 of one thousand 0-9 thousands 0-9 millions

Tens 10-99 of one thousand 10-99 thousands 10-99 millions

Hundreds 100 -999 of one thousand 100-999 thousands 100-999 millions

This system may be illustrated, for example, by the number 716, 429, 357. The strings of three digits are conventionally marked by commas in arithmetical notation. At the hundreds’ level we see 357; at the thousands’ level, 429, and at the millions’ level, 716. In the string of three digits at each of these levels there is a units’ place, a tens’ place, and a hundreds’ place.

When these numbers are given numeral expression, each of the levels contains the conjunction “and.” Thus, the number in our example is pronounced (without the emphasis suggested by the italics, which are provided for purely lexical demonstration) “seven hundred and sixteen million, four hundred and twenty nine thousand, three hundred and fifty-seven.”

Furthermore, something else is immediately noticeable from this numeral expression. Not only is “and” repeated at each level or in each string of three digits, but “hundred” is also repeated immediately preceding each “and.” This means that each string of string of digits is divided into two parts: the hundreds’ place on the left and a combination of the tens’ place and the units’

place on the right. The conjunction “and” stands in the middle of the division. This arrangement

place on the right. The conjunction “and” stands in the middle of the division. This arrangement