The series of numbers 1, 2, 3…in English correspond to the words one, two, three…used in counting. It is these number words which embed numbers in language, so as to provide the only historical context in which they could have originated as part of human cognition. Language means in the first place spoken language, which—from the neurophysiological evidence provided by palaeontology—has probably been part of human communication for some half a million years. Writing (which for the first time provided good historical evidence for the development of language) first appeared somewhere in the middle east about 10,000 years ago. It is highly significant that the first writing was developed so as to record quantities measured by numbers. This shows that at least some of the languages spoken before this time contained number words, although even today there are still languages in Australia, and perhaps elsewhere, which contain no numbers.1 In the case of Japanese, in which both the use of written numbers, or numerals, and the historical record begins with the first contact with Chinese culture some 1,600 years ago, there is clear evidence of the existence of number words in the language spoken before this time: such words are still part of Japanese, although their form has certainly been modified over the course of time.
If the series of natural numbers is unlimited (a proposition accepted in the western world since before the time of Plato), the same cannot be true of the number words in any language. There must be a limit to the process of counting, simply because no vocabulary can contain an unlimited number of words. Many languages containing such words go no further than some number less than 10,
but in all such cases the series of number words is continuous in the sense that there will be words for all numbers up to the limit imposed by the lexicon.2 If, for instance, this limit is 5, then there will always be a word for 4, and so on down to 1 (although the status of 1 as a number is often equivocal).3
In the formation of number words two general principles apply to almost all known languages. The first is that low numbers, up to a recognised limit, are represented by words, which, systematically, are quite unrelated. In English this is certainly true of the word for numbers up to 5, if not up to 10.4 The second principle is that almost all numbers, above another limit, if they exist at all, will be formed by some principle of composition. In English this limit is 10, so that ‘sixteen’, for instance, contains two components, meaning 6 and 10. In any language, this second limit cannot, a fortiori, be lower than the first, but there are certainly languages in which the two limits do not coincide, so that the numbers falling between them are sometimes composite and sometimes not.
The lexical principles applying in the realm of composite number may be quite chaotic,5 particularly for relatively low numbers. The numbers up to 100 in Indo-European languages provide such examples as quatre-vingt-douze for 92 in French, but Yoruba, one of the most important languages in West Africa, is far less ordered. No matter: at a certain point order must be imposed if larger numbers are to be represented. In French, once one has reached 100, there are no more than quite minor problems. For the middle range, generally between 10 and 100, where numbers are composite but subject to disorder, the countless different known languages can be ordered on a scale, with the simplest cases at one end and the most complex at the other. Chinese is a simple number language par excellence, where Yoruba is at the opposite extreme. The point is important since command over the middle range is important if numbers are to be used operationally, that is, in ways which make use of their essentially arithmetical properties.6 It is at this level also that visual representation—not necessarily by writing—of numbers enormously increases their operational utility.7
The hallmark of a simple system for composing number words is simply to be found in an economy in the rules governing the process.
The first step is to establish a base number, such as 10,8 which in successive powers, 100, 1,000, and so on, provides a frame in which to construct number words. This process plainly cannot go on for ever, and at some stage the normal counting process, although theoretically possible, no longer provides the ordinary means for forming numbers.9
16 The Japanese numbers game
In English the word ‘million’ is never used in communicating a telephone number: even the year, 1992, is seldom spoken out in full. At a certain level also, numbers in the frame will almost certainly be composite: in English this process begins with ‘ten thousand’, whereas in Japanese it does not begin until ju¯ man (hundred thousand).10 The series of low numbers below the base number, which on a system with base 10 will be the numbers 1 to 9, then provides a cyclic system, which, in combination with the frame numbers, allows all numbers likely to be used in spoken discourse to be expressed by a word without any ambiguity, although in some languages such as Japanese there are alternative words for some numbers.11 In a simple system, such as the Chinese, little more need be said, save that the number 1, as a member of the cyclic system, is generally suppressed when it governs a frame number, just as in English 10 is ‘ten’ and not some word derived from
‘one’ and ‘ten’. This process, known as one-deletion occurs in a great many languages. The general absence of a zero in such systems—at least when they originated—means that frame numbers, governed by a zero, are simply not part of the number word. For example, 207 is spoken as ‘two hundred and seven’, without any explicit indication of the missing tens.
Using the linguistic term morpheme to mean any non-reducible element in a language with its own recognised meaning, then the number words will be composed of a very limited number of morphemes:12 in English some twenty are sufficient to represent every number up to a million. These will comprise all the cyclic and non-composite frame numbers, which, being capable of standing alone, are unbound morphemes, together with such bound morphemes as ‘-ty’
and ‘thir-’, which only occur in a composite number word. In English number words, ‘and’, as in ‘two hundred and seven’, must be reckoned as a bound morpheme, whereas in this particular case the other components are unbound.13
Now, if the English case is simple it could be even simpler, as is shown by the case of Chinese, in which nine cyclic number words and four frame number words, all unbound morphemes, are sufficient for every number up to a hundred million. It contains no bound morphemes, with a limited number of unbound morphemes combining according to a single rule so as to represent all numbers capable of being expressed. This is the hallmark of a simple system.
When it comes to the written forms for number words, the position is altered quite radically. In the first place, counting by means of written signs is possible simply by repeating the same mark, so that 1, 2, 3…can be represented by /, //, ///…a process without any parallel in spoken
language.14 Such representation may be made more manageable by using a base, say 5, to establish recognised larger units.15 This can be achieved by using a fifth stroke to round off the first four, so that one writes not but . The Japanese use the character , whose five strokes are written in the following order: , so that 9, for instance, would be represented by .
Right at the very beginning, therefore, the simplest written form loses contact with its spoken equivalent. It is moreover extremely difficult to re-establish contact, quite simply because conventions of the spoken language, such as one-deletion and zero-suppression, make any literal written representation extremely confusing. The reason for this is to be found in the characteristic uses of written numbers. So long as such use is confined to purely documentary transactions, such as payment by means of a bank-note or the keeping of archives such as census returns, the actual written form is not crucial. As soon, however, as numbers must be used operationally, as in the elementary arithmetic used in accounting, the written number system must be amenable to simple arithmetical rules, or algorithms, such as children now learn in primary school for addition, subtraction, multiplication and division.
Given the demands made of numbers in their written form, it is not surprising that we only know of one original case of a language which represents every numerical morpheme occurring by its own distinctive character, so that there is a perfect one-to-one correspondence between the written and the spoken versions: this language is Chinese.16 Even then the original written numbers did not pass this test;17 historically it is of decisive importance that at a certain stage the written forms were adapted so that they did so, and it is these forms which are still in use today, not only in China but in Japan. The transformation, which took place well over two thousand years ago, was only possible because of the utterly simple syntax of number words in the spoken language.
Whatever the advantage of having such transparent written forms, the price still to be paid was that these were still not adequate for any but the simplest arithmetical operations.
To this problem there is only one solution, which is the visual representation of numbers by a place-value system. The best-known example of this is provided by Arabic18 numerals, from 0 to 9, whose value in any numeral is determined by their place in it. This means that 2 in 207 is 200, simply because it occupies the third place from the right end, and hundred is the third component in the frame sequence one, ten, hundred, thousand and so on.19 Once this convention is adopted, the divorce between written and spoken form
18 The Japanese numbers game
is near complete, and simply does not matter. The fact that the French quatre-vingt-douze for 92 is perhaps frustrating if one is learning to speak French, but a French child has no more difficulty in dealing with 92 in a sum than an English child, for whom the spoken form ‘ninety-two’ is much more transparent. But what about a Chinese child, for whom the written form will correspond precisely with the spoken form ji7 shì èr. Can the written form be used in arithmetic. The practical answer is no, so that at the present day Chinese children are also taught to work with Arabic numerals, which of course fit the spoken language no better than they do French or English. At the same time the Chinese, for arithmetical operations, continue to use the abacus, a calculating instrument which represents the numbers occurring according to a place-value system isomorphic to that of the Arabic numerals. The actual form of numerals on an abacus is more elementary, since every number, from 0 to 9, is a combination of beads, with those below the bar representing units, and those above multiples of five. This is illustrated in figure 9.1 (see p. 128).20 The lesson is that to deal with numbers at all efficiently, in arithmetical operations, there is no alternative to a place-value system. This is even true with electronic computers, with the difference only that they work on a base 2 and not 10.21
However the problem of representing numbers, whether in spoken or written form, is solved, the question still remains as to how linguistic factors affect their actual use. This of course is but one aspect of the subject of this book, the use and understanding of numbers, but it is a decisively important one, particularly in the extremely idiosyncratic context of modern Japan. Numbers, in any culture in which their operational use is at all advanced, must face in two directions, that of language on one side and that of arithmetic on the other. The two directions relate, if somewhat imperfectly, to the difference between the spoken and written forms and, in a rather different cognitive domain, to that between ordinal and cardinal numbers. In the English-speaking world convention dictates that a number is written out in full when its use is descriptive rather than operational: contrast the ‘nineteenth century’ with a ‘19p stamp’. In the case of Chinese, and derivatively that of Japanese also, this distinction is only possible if Arabic numerals are used, but such use was unknown before the late nineteenth century. In the English case it corresponds roughly to the distinction between language and arithmetic, although the case of the 19p stamp, as many others, faces in both directions. Those in the sorting office handling the letter can
see from the face of the stamp whether the right postage has been paid, so that in this case the numerical information is descriptive. On the other hand, those working behind the counter, who sell the stamp, perhaps with others of the same or a different amount, and then have to work out the change for a bank-note offered in payment, are involved in an operation depending upon the arithmetical properties of 19. It is not for nothing that counter-clerks in Japanese post-offices have their own abacus to help them with such calculations. None of this is possible with the 19 in ‘19th century’ except in a very contrived sort of way: 4×19, for instance, is not conceivable as an operation applicable in such a case.
In language the use of number words is primarily descriptive, so that, in English grammar at least, they may be seen as adjectives that qualify a noun.22 The young child, as it learns to use numbers, must somehow marry the numbers it has learnt by counting with different categories of countable objects. Logically ‘seven apples’ is prior to an understanding of ‘seven’, however counter-intuitive this statement may be. What ‘seven apples’ represents is the intersection of the class of aggregates containing apples and that of collections with seven members. If we look at this latter class from the perspective of our own culture, we see that it is extremely heterogeneous. It includes not only the seven apples just counted, but the ‘seven stars in the sky’, the ‘seven deadly sins’, the ‘seven days of the week’ and the ‘seven hills of Rome’, together with countless other collections which one could conceive of or constitute ad hoc.
The cardinal number 7 is the only thing these collections have in common: linguistically ‘seven’ is the only modifier which applies to all of them, but it is still not an inherent property of any of them. Take away one apple from the pile, and neither that apple nor those remaining are changed in any way, but the property of being 7 is completely eliminated, and seven, as a modifier, no longer describes (except in the past tense) anything connected with the apples. If the property of being 7 in this one case is eliminated, the concept of 7 remains, like the smile on the Cheshire Cat, which, as Alice found, was the last part of it to disappear from sight (giving rise to all sorts of epistemological problems).
It is precisely this concept of 7 which is operational. It is no accident that the numerical quantities occurring in everyday arithmetic relate for the most part to abstract quantities, such as units of measure or money (which may be taken to measure value).
Whatever concrete results may be reached in contexts in which arithmetical operations are used, the latter are abstract, autonomous and context free. Arithmetically there is no difference between
20 The Japanese numbers game
calculating the price of 7 19p postage stamps and calculating the weight of 7 19lb sacks of potatoes.
This fact is particularly important in the context of Japanese, which does not allow numbers the adjectival property of applying to all nouns. In Japanese, the words for seven (nana) and apple (ringo) cannot directly combine to express ‘seven apples’, and Japanese is certainly not the only language subject to this restriction.23 The problem is solved by the use of counters, or in Japanese, jos*shi,24 which can be governed by numbers, and so provide a sort of interface between any given number and any noun which that number must govern. This means the constant use of phrases such as the English ‘seven head of cattle’. Thus ‘seven apples’ can only be expressed in Japanese with the help of the counter ko, so that the complete form is nana ko no ringo. In some cases the counter can also stand alone, so that sannin is simply ‘three people’, whereas sannin no kodomo is ‘three children’, or sannin no sensei, ‘three teachers’—nin being simply the general counter for human beings,25 leaving aside such special cases as Shinto gods, who have their own counter. In the case of the measurement of physical dimensions, time and money, the appropriate units generally need no counter, so that fifty (goj*) Yen (-en) is simply goj*en. Having said all this, there are still special rules for using the autochthonous Japanese numerals (which have not been introduced) in forms which count up to four people and up to ten days, or sometimes more.
Japanese, therefore, makes explicit the logical priority of ‘seven apples’ (or seven of anything else) over the number 7. This part of Japanese cognition is reinforced by the use of the abacus, since the numbers occurring in any calculation count the beads used in it.26 The abacus therefore represents the sum 7+11 as 7 beads added to 11 beads, and indeed abacus calculation is simply known as shu-zan, meaning
‘bead calculation’. The secret of success is to be found in the use of a place-value system for ordering the beads so as to represent any number occurring, for the written form of numerals, being ordered according to a different system, cannot be used directly for calculation.
The price paid for this success is that a number, say 7, can only be abstracted from ‘seven apples’ by being moved to seven beads. This is a matter of cognition, not of arithmetical skill, which in practice does not suffer from the limitations of the Japanese language. Indeed in everyday counting and arithmetic numerals are used perfectly correctly without counters.
Finally, there is the matter of the distinction between cardinal and ordinal numbers. In terms of word formation, the Japanese process for deriving ordinal from cardinal numbers is very simple, as it is in many other languages.27 The simplest rule is to add the suffix -bamme to any number word, so that san-bamme (third) is formed from san (three). In
Finally, there is the matter of the distinction between cardinal and ordinal numbers. In terms of word formation, the Japanese process for deriving ordinal from cardinal numbers is very simple, as it is in many other languages.27 The simplest rule is to add the suffix -bamme to any number word, so that san-bamme (third) is formed from san (three). In