This commentary is in two sections. The first presents some specifically linguistic evidence regarding Dehaene's statement that, in animals and prelinguistic humans, ``only very small numbers (up to about 3) can be represented accurately''1. The second section contains some more general reflections on Dehaene's work.
Dehaene argues that adequately accurate analog mental representation of very small quantities, in the range 1-4, is ancient. Evidence from language corroborates this. Languages show a tendency to express the first two, three or four numbers differently from slightly higher ones. Some of this difference in behaviour makes the lowest-valued numerals more like adjectives. This suggests that these very low numerosities may be perceived and represented like the meanings of basic adjectives, such as red, hot and round, as sensible properties of (groups of) objects. Higher numerosities, from about 5 upward, are less directly, or less reliably, observable, and representation of them relies not just on perceptual properties, but probably (at least in part) on their being associated with a particular slot in a conventional recited counting sequence. It seems certain that in the historical development of languages, words for the lower numbers existed before words for higher numbers. The idiosyncrasies which tend to be shown by the very lowest numeral words preserve ancient linguistic patterns. Wittgenstein captures the idea nicely:
``Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses.'' (Philosophical Investigations)
Grammarians use the term `(grammatical) number' for the singular/plural distinction marked on nouns in most languages, and supplemented in some languages by categories such as dual, and even trial. Clearly, grammatical number in a language is distinct from its numeral system. Grammatical number tends to be indicated in the morphology of nouns (e.g. by an affix), whereas numerals are distinct words or phrases. Grammatical number in fact has only a somewhat loose relationship with numeral expressions in a language. Nouns marked for grammatical number need not be accompanied by numerals. In Arabic, for instance, kitaab means `(a) book' and kitabeen means `two books'. The important point here is that the numerical limit for systems of grammatical number is at around 3 . A few languages have distinct trial forms of nouns, for describing collections of 3; quite a few more languages have dual distinct forms, for describing pairs of things; and of course, almost all languages distinguish singular from plural.
In almost all languages, numbers up to the first base number (usually 10) are expressed by single words, and after the base number the system resorts to syntactic combinations. But very often the words for the values 1-10 are not uniform in their grammatical features. It is common for the very lowest numbers, from 1 to about 3 or 4, to be expressed by numeral words with distinctive characteristics and irregularities. The rest of this section documents such idiosyncrasies of the numerals in the range 1 - 4.
The numeral used attributively to modify a noun is not necessarily the same as the corresponding numeral in the conventional recited counting sequence. There are various degrees of idiosyncratic difference between a counting numeral and a quantifying (attributive) numeral. In some cases the counting numeral is completely dissimilar to the attributive numeral, and in other cases one form iis obviously a morphological variant of the other. Some examples are given below.
| LANGUAGE | NUMBER | ATTRIBUTIVE | COUNTING |
| NUMERAL | NUMERAL | ||
| German | 1 | ein, eine etc. | eins |
| German | 2 | zwei | zwo |
| Maltese | 2 | zewg | tnejn |
| Chinese | 2 | liang | erh |
| Hungarian | 2 | két | kettö |
| Basque | 2 | bi | biga |
The most common number to be expressed by a distinct counting form is 2. Others are extremely rarely. It might be expected that the occurrence of distinct counting forms would be greatest for the number 1, and decrease thereafter. Although the counting sequence begins at 1, it is only the utterance of a form for 2 after a form for 1 that confirms that the activity of counting is taking place.
Not all languages have a separate series of ordinal numerals. And some languages only have (or in practice their speakers only use) ordinals for a limited low-valued subset of numbers. Maltese only has distinct ordinals up to `4th'. In Spanish, although ordinals for values higher than 20 are available, e.g. centésimo, `100th', they are rarely used, and a construction with a cardinal is used instead.
Irregularity in the formation of ordinals occurs in various degrees. Some idiosyncratic modification of the cardinal stem may accompany an otherwise regular morphological process, as in English five/fifth. Or the ordinal may bear some unpredictable resemblance to the cardinal, as with English three/third. The extreme case is suppletion, which typically involves the very low numbers, 1, 2, and perhaps 3. Examples with 1/1st are: Greek ena/protos, Welsh un/cyntaf, Italian uno/primo, Finnish yksi/ensimmäinen, and English one/first. Examples with 2/2nd are Greek djo/devteros, Welsh dau/ail, Italian due/secondo and Finnish kaksi/toinen.
The following table summarizes the formation of ordinals in a sample of 17 mainly European languages, from many different language families. It shows clearly how various kinds of irregularity are clustered in the lower range of numbers, with most irregularity in the range 1-4.
| Number | Suppletive | Irregular | Regular | No distinct |
|---|---|---|---|---|
| ordinal form | ||||
| `1' | 14 | 3 | ||
| `2' | 6 | 3 | 8 | |
| `3' | 6 | 11 | ||
| `4' | 3 | 14 | ||
| `5' | 2 | 14 | 1 | |
| `6' | 1 | 15 | 1 | |
| `7' | 1 | 15 | 1 | |
| `8' | 1 | 15 | 1 | |
| `9' | 1 | 15 | 1 | |
| `10' | 16 | 1 |
Some languages have several sequences of numeral words, and choice of the appropriate numeral depends on the noun being modified. In some cases, such distinct numerals only exist for a small range of numbers, typically in the low range. For the numbers 2 - 6, Bulgarian has a series of numerals, applicable only to male humans. The similar forms for 7 - 10 are marginal or archaic.
The very lowest-valued numerals in a language sometimes have a different order from the rest. Basque, Arabic, Hebrew and Maltese place the numeral for 1, exceptionally, after the noun, like an adjective, and all other numerals precede the noun (in Vizcayan Basque the numeral for 2 also, exceptionally, follows the noun).
In Albanian, there are separate definite and indefinite numerals for the numbers 1-4. The paradigms for `1' are given below.
| INDEFINITE | DEFINITE | |||
|---|---|---|---|---|
| Masc. | Fem. | Masc. | Fem. | |
| NOM | një | një | njëri | njëra |
| GEN | njëri | njëre | njërit | njërës |
| DAT | njëri | njëre | njërit | njërës |
| ACC | një | një | njërin | njërën |
| ABL | njëri | njëre | njërit | njërës |
Lambertz' (1959) grammar, from which this information comes, also
mentions distinct definite and indefinite forms for the numerals 2-4,
but many, though not all, of his examples do not involve
morphological variants of the numerals themselves, but rather a definite
particle preposed to the numeral. My own informant, Eranda Kabashi, a
speaker of the Geg dialect of Kosovo, recognizes fewer morphological
distinctions among the numerals than Lambertz, but nevertheless has some
distinct forms for definites and indefinites, for example:
dy vajza
`two girls'
té dyja vajzat
`the two girls'
German has distinct indefinite genitive forms just for the numerals denoting 2 and 3, namely zweier and dreier.
The table below shows the maximum number of cases distinguished idiosyncratically on numerals, that is, instances where case-marking on numerals is not by a regular productive process which also applies to other parts of speech, such as adjectives or nouns. The numbers show, for each language, how many distinct rows there are in the case paradigm for each numeral.
| Numeral | Alb- | Ger- | Greek | Ice- | Rom- | Russ- | Zürich |
|---|---|---|---|---|---|---|---|
| anian | man | landic | any | ian | German | ||
| `1' | 3 | 4 | 3 | 4 | 2 | 6 | 2 |
| `2' | 3 | 2 | 1 | 4 | 2 | 4 | 1 |
| `3' | 1 | 2 | 2 | 4 | 2 | 4 | 1 |
| `4' | 3 | 1 | 2 | 4 | 2 | 4 | 1 |
| `5' | 1 | 1 | 1 | 1 | 2 | 3 | 1 |
| `6' | 1 | 1 | 1 | 1 | 2 | 3 | 1 |
| `7' | 1 | 1 | 1 | 1 | 1 | 3 | 1 |
| `8' | 1 | 1 | 1 | 1 | 1 | 3 | 1 |
| `9' | 1 | 1 | 1 | 1 | 1 | 3 | 1 |
| `10' | 1 | 1 | 1 | 1 | 2 | 3 | 1 |
In Russian, the numerals for 2, 3, and 4 assign genitive singular to a sister noun, whereas the numerals for 5 - 10 assign genitive plural.
Gender marking on attributive numerals is always in agreement with the
inherent gender of the sister noun.
Examples from Zurich German (Weber, 1964:132,133) are:
zwee Mane
two men
zwoo Fraue
two women
zwäi Chind
two children
drei Mane
three men
drüü Chind
three children
Idiosyncratic gender-marking is typically restricted to just the first few numerals, as the following table shows.
| N | AlbL | AlbN | Blg | Fr | Grm | Grk | Ice | Mlt | Rus | ScGl | Wel | ZD |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| `1' | 2 | 1 | 2 | 2 | 3 | 3 | 3 | 2 | 3 | 1 | 1 | 3 |
| `2' | 2 | 1 | 2 | 1 | 1 | 1 | 3 | 1 | 2 | 1 | 2 | 3 |
| `3' | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 2 |
| `4' | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 1 |
| `5' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| `6' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| `7' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| `8' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| `9' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| `10' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Dehaene's critique of Platonism is entirely harmonious with the view expounded in Hurford (1987, especially chapters 3 and 4). This work is missing the crucial neurological underpinning that Dehaene has begun to establish. But it outlines how the complete modern educated human grasp of whole numbers has evolved as a product of the interaction among simple linguistic signs, the disposition to recite ritual sequences of words, the utility of precise reference to collections of objects of various cardinalities, and recruitment of certain pre-existing syntactic processes (e.g. conjunction and pluralization). The pre-existing apparatus attributed to the child in respect of the very low numbers is very similar for both Dehaene and Hurford. In Hurford's account, concepts of numbers as abstract entities emerge from a process of abstraction from the more concrete meanings of linguistic expressions denoting collections of observable objects. It would be interesting to see to what extent Dehaene could agree with the details of this outline.