Paper presented at "The Origins of Writing – Genealogy of an Invention"
held by the Banca Popolare di Milano in October 2000, Milan
(OHP slides not included)

 

Numerical notation and abstraction of concepts

(Le notazioni numeriche e l’astrazione concettuale)

John Sören Pettersson, Karlstad University, 651 88 Karlstad, Sweden
john_soren.pettersson@kau.se

 

It appears as if one can trace the evolution of abstract concepts in the use of clay counters, Prof. Schmandt-Besserat’s tokens, and in the later development of numerical tablets and cuneiform tablets in the ancient Near East. These pebbles are found in different shapes and it seems reasonable to think that they were used for a sort of accounting. That is, each piece stood for a certain thing or a certain amount of a thing, whilst the shape of a token was related to the type of commodity. It was not possible to make a type, a shape, in the abstract; it was not possible to make a shape without making a counter.

In the fourth millennium BC we have the development of ball-shaped envelopes which the ancients used to secure the number of tokens. Eventually, token impressions on the outside of such envelopes lead to the invention of the numerical clay tablet on which token representations could be impressed with tokens or, more generally, with a stylus.

As one can see from the very writing implement – that is, a general ‘pen’ was used – the ancients realised that a general device could be used to make impressions rather than the specific tokens which were to be represented. Today, in the light of the computer, the tool may seem more important than the signs. Originally a counting device, the computer has now become an output device, an automatic ‘pen’. The pen in the form of the computer functions indirectly since a digital representation has to be made and the computer then generates the output from this representation. This has allowed for an extreme generality, far beyond the original number representation, and it makes written texts directly interplay with other modes of expression (Pettersson, 1998).

To what extent the use of stylus in ancient Sumer contributed to the abstraction of number representation from commodity representation is not obvious. But the following development where types of goods were incised beside the numerical annotation definitely saw the separation of number from what was counted.

When the notation no longer mixes number and item in the same sign, the numerical signs are true numerals, as they stand for abstract numbers. I will use the word ’numeral’ for written signs denoting numbers although linguists often use ’numeral’ for number words. And I will use the expression ’number word’ for words that denote numbers.

*

Simultaneously with the evolution of abstract numerals, ‘sheep’ and ‘cow’ and other things got signs of their own irrespective of in the quantity in which they appeared. As Prof. Schmandt-Besserat remarks in her Before Writing, "With the invention of numerals, pictography was no longer restricted to accounting but could open to other fields of human endeavor." (1992, p. 199, see also p. 194)

As for the tokens, Schmandt-Besserat has pointed out that we cannot jump to the conclusion that a bigger unit was regarded as, say, six times a smaller unit. Rather, it might have been primarily a representation of a larger measure, a larger quantity. It is, of course, hard to tell when internal relations between symbols became a primary feature of the systems; it is hard to tell whether this occurred already in ‘token times’ or only later, viz. in ‘tablet times’.

In any event, in tablet times we know definitely that systematic relations had developed: numerical tablets including a sum of individual entries prove that (e.g. Friberg, 1999).

*

Now, the abstraction could be taken further than only removing the commodity counted. I will use a few special words to describe different kinds of numerical notation.

An additive system repeats a numeral to express a number, like Roman III for ’three’, whilst a sign-value notation uses special signs to denote higher numbers, and again the Roman numerals constitute an example with V for ’five’ and X for ’ten’. Obviously, the Roman system is an additive sign-value notation. The ancient Egyptian hieroglyphic system provides another example. But one could imagine another kind of sign-value notation, where smaller numbers give the amount of bigger units. This is called a multiplicative system; thus the Chinese system is multiplicative. And then there are positional systems, which utilise the idea of place value, that is, there are no special signs for the higher ranks. Our decimal system is, of course, a positional system. (For this terminology, see Pettersson, 1996.)

The old Mesopotamian systems were additive sign-value notations, but the sign-values differed between systems. The reason why the ancients used several systems is of course that they were counting in different ways for different things, like original societies still do today as for instance Jack Goody has described (1977, p. 13; for a light introduction to these matters, see the recent children’s book by Schmandt-Besserat, 1999). Admittedly, it somewhat relatives the notion of ’true numerals’ and abstract numbers if each system is connected with specific types of measurement. Another fact to be noted is that the series of higher ranks in a Mesopotamian system was not powers of one and the same number. It was not like 1, 10, 100, 1000, or 1, 6, 36, and so on.

However, numerals were to become more abstract. Out of the sexagesimal sign-value system evolved a positional system in the shift from (Neo) Sumerian to Old Babylonian times, around 2000 BC. This sexagesimal positional system was to be generally used in computations. Obviously, using only one system simplifies the use of counting boards and tables of pre-calculated expressions. The Mesopotamians used multiplication tables and inversion tables, the latter for division. A positional system allows for further economy when it comes to utilising such devices. The new system extended to fractions, so the tables with pre-calculated values could also be used for fractions. In fact, a positional system allows the re-employment of such tables in any range of numbers.

It should be realised that an ordinary additive notation, and especially an additive sign-value notation, is a very versatile instrument for computing additions and also subtractions. The Old Babylonian positional system did not have ciphered numerals, that is different symbols for consecutive numbers, but actually additively written numerals 1-9, and further a sign for ’ten’ to be repeated up to five times, so 59 could be written in additive sign-value notation with only two basic symbols. Then, for 60, a new position was used and the ’one’ symbol was re-employed. In this way, the Old Babylonian system made use of the simplicity of both types of system.

A prominent ‘60’ is, moreover, found not only in the earlier sexagesimal sign-value system but also in the two or three other most frequently employed sign-value notations. So the new system comprised the major structures of the major sign-value systems.

Now, if someone was intrigued by the 60-base of the Babylonian system, surely some will be intrigued by the ’six’ appearing in the earlier Sumerian systems. Five would be easier for us to accept, since we have all learnt to count by using our hands. Did the Sumerians have six fingers on each hand?

Or could the hand be counted as six? Well, that seems unmotivated when there are five unused fingers on the other hand. Admittedly, there are body counting systems in some parts of the world, where people count along the arm up through the face and then down again along the other arm (Comrie, 1999; Gvozdanovic, 1999). But then, obviously, when they have gone beyond the fingers, they have realised that they could equally well go beyond the hand.

Looking at various Sumerian and Elamite systems, one finds that the numerical relationship between sign-values that appear closest to each other is often 2 or 3, and not only 6 (see, e.g., Englund, 1996, Figure 14). That is, you make a pair or a triplet before you count these pairs or triplets. Limiting bundling to only ‘two’ or ‘three’ appears as most profoundly human: the ability to grasp sets with two or three objects is found in young children as well as among ‘primitives’ (i.a. Fischer, 1992; Neuman, 1989). Indeed, even if we are able to count to tens of thousands and beyond, we are not likely to recognise the number of a set greater than a few, if the objects are not arranged in an easily recognisable pattern.

If pairing and 3-grouping is what we would expect of the most ancient Near Eastern peoples, one can discern in the numerical tablets how these counting practices are formalised for each type of measurement. Exactly when a bigger measure is regarded as being exchangeable for smaller units is hard to tell – in token times or in tablet times. When this is possible, however, it is also possible to utilise relations that skip an intermediate quantity. That is, if A is a pair of B which in its turn is a bundle of three C, then six C could be replaced by one A. Thus, the numerical relation between A and C is ’six’.

So, there is no need for an extra finger on the Sumerian’s hand, and even more, the hand does not play such an essential role as might otherwise be thought. Rather, counting practices, including token manipulation, seem to be crucial. There is, admittedly, a relation of 10 or 5 in some instances, which suggests that they have used a hand in these instances, and possibly combined it with a doubling for the next or previous relation.

Parenthetically, one could ask why 4 or 9 do not appear among the relations, if it is possible to explain 6- and 10-relations as the result of early rationalisation. Perhaps the rule was that the same sort of bundling should not be employed to bundle a bundle. That is, you should not make pairs out of pairs, etc. Avoiding this clearly has some advantages when counting and when speaking about counting, as it avoids the risk of confusing magnitudes.

I could add that it is not a general feature of ‘early’ counting to avoid re-employing one type of bundling. Series of halving and doubling are frequent – we find it in early Sumerian numerals for weight (Friberg, 1999, p. 25) – and the Egyptian system for integers was based on powers of ten.

*

Bundling by fingers, by decades, may result in a sequence of number words as can be discerned in words like thirteen, fourteen, fifteen, and thirty, forty, fifty. In fact, for ordinary texts, Babylonians and Assyrians deviced a decimal system from the sexagesimally written numerals 1–99 by adding special signs for ‘hundred’ and ‘thousand’ (see, e.g., Pettersson, 1996, Table 69.2). They were thus able to write numbers in a way that conformed to their Semitic languages. In general, when people started to amass goods and control quantity, the way of bundling provided a means of naming the quantities. The concomitant establishment of a sequence of number words might make it possible to reel off a long series of numbers without having concrete objects before your eyes; children may learn it as a chant.

Commodity counting has its limits, as has purely idle counting, even if children can be made to enjoy reeling off numbers up to one hundred. More extensive elaboration of numbers would be furthered by other purposes, such as estimating future profits, or mathematics; or astronomy and other kinds of religion. Objects and notations of accounting systems can provide the means for this elaboration by providing the means for the computations related to such flights of fantasy.

As these numbers are employed more and more without a concrete relationship, it would be natural to claim that they become more firmly established as abstract numbers – even if there could be different degrees or kinds of abstract numbers, as I hinted earlier. But I think it would be fair to say that the more abstract they become in this respect, the more self-sufficient they will appear to their users. When they are mastered, they will appear as things in themselves. One will become accustomed to their properties, and they will not appear as abstractions. Rattling off any sequence of numbers will seem like mentioning the names of old acquaintances.

Manipulating and enumerating the numbers firmly establishes the numbers as entities of their own. There is a certain seductiveness in this, however, which threatens to undo the advancements in thought made possible through the evolution of numerals and abstract numbers. Before going into this, it might be worth pointing out the conceptual productivity of all this manipulating and sequencing.

*

Zero and negative numbers have had a hard time emerging. I shall dwell only briefly on this topic even if it is a popular topic – indeed, as late as this year, a ‘biography’ of Zero was published, which includes negative and other strange numbers, written by a New Scientist journalist, Charles Seife (2000).

One might be led to think that the Old Babylonian positional system necessitated the introduction of a zero – how else could a numeral ‘1’ for 60 be interpreted as 60 and not as ‘one’? The ancient scribes, for their part, took various other precautions. They could state the size of a number in words (Friberg, 1990, p. 536) or use a larger space between numerals to indicate the absense of an intermediate power of sixty (Pettersson, 1996, Table 69.2). Sometimes they employed columns. And the very fact that the base was sixty rather than merely ten, made it less urgent to mark empty positions than we might presume. The risk to confuse magnitudes was not great. But after a long while, in the third century BC, special void-marking signs became frequent.

However, these signs do not really allow for counting zero as one of the numbers. They are simply void-markers in a positional system. The Indians had devised a decimal positional system in the latter half of the first millennium and there are Indian texts speaking about zero as well as negative numbers. Perhaps this was due to influence from Chinese decimal counting boards. Negative numbers promote the concept of zero as a number: if –1 plus 2 is 1, then there has to be a number which equals –1 plus 1.

It took some while before the Indian numerals, via the Arabs, reached Europe. The industrious people of Northern Italy, especially the bankers, embraced it (Seife, 2000). A symbol for voids that has the same status as the digits surely implies a concept of the number zero which places it among the other numbers.

Again, it took some while for negative numbers to be re-invented in the West, but ever since there has been no end to imagination. For instance, with the set of negative numbers came also another set of numbers: the so-called imaginary numbers which equal the ‘forbidden’ square roots of negative numbers. But no new set of numerals have been devised; rather, the Indian numerals have been employed to count these fictious numbers as well.

As I once wrote, ending an encyclopaedic section about "Numerical Notation", the European forms of the Indian numerals have spread world-wide, and their sole global rival is … the bar code! (Pettersson, 1996, p. 804) Admittedly, the bar code system, like other codes used in connection with electronic media, is just a general representational device applicable to any commodity. In fact, the bar code is used to represent what was left over when true numerals once were invented and started to represent numbers in the abstract. And the ordinary numerals appearing on bar code tags are just a prosthetic for us to be able to formally identify bar code codes. They are, as it were, not used as ‘numerical’ numerals.

It sounds like the final abstraction: moving numerals away even from abstract numbers. But it isn’t, because it is merely a return to the token system. A specific bar code is a token type, whilst the occurence of a bar code on a commodity is a token.

*

To round off this discussion of abstractness, it is time to note that it is harder to discuss the precise meaning of ‘abstract number’ when it comes to developmental psychology – to pedagogy – than it is when discussing the whole human race. For a culture at large, we can easily discern the separation of numerical and commodity signs, as in the Sumerian case. And for number words, one could investigate their degree of general applicability and thus find measurements for their degree of abstractness. But what do numerals and number words mean to a child? In pedagogy, we have to ask whether the individual child has a notion of abstract numbers. …or should it be: whether the child has an abstract notion of numbers?

Presently, I am discussing with some researchers in pedagogics what to model in interactive graphic computer interfaces for non-readers, mainly young children. While the aim is language-developing devices, we focus initially on the notion of numbers, and the question will not only be what teachers teach – like written numerals, the sequence of numbers, the concept of quantity, etc. – but also how to measure children’s apprehension and proficiency (especially, in this research, through an interface aimed at non-readers, or beginning readers).

While it is possible to teach young children to pair the members of two sets to see whether one set outnumbers the other, it is not that easy to implant the notion of distinct quantities. And that is not surprising. Looking at the history of numerical notation as well as at ‘primitives’ in various corners of the world, we find that counting above three is not an obvious task. (Fischer, 1992; Neuman 1989; Ifrah 2000, pp. 4ff.)

If pairing, on the other hand, is not very difficult to teach, then the children seemingly need only to ‘know the numbers’ in the sense of getting to know the names of the numbers, or the system of numerals. It would seem as if this is enough to make it possible for them to express distinct, and discrete, quantities, and also to understand the relation between numbers, since both the spoken sequence of number words and the system of written numerals imply an order among the numbers.

Admittedly, 16+1 would not be 17, if it were not for the existence of the established sequence ‘…14, 15, 16, 17’. Otherwise ‘one added to sixteen’ would simply be ‘sixteen and one more’. So the sequence of numbers, as possible to line up in the abstract, is important, as are the production rules for constructing numerical expressions in our positional system. At the same time, knowing either or both for a fair amount of numbers, say up to twenty in words and up to one hundred with numerals, would not prove an understanding of discrete quantities. Counting ‘one, two, three’ or scribbling digits ‘1’, ‘2’, and ‘3’ beside objects on a paper, is easily confused by young children with temporarily giving names to the items counted. ‘Three’ then is not the name of the set as a whole but only for the last member. Grasping how we adults use the last mentioned number in different senses – for the ‘third’ and for the set of three (Piaget, 1969) – is not easy.

When designing interactive teaching material which measure children’s apprehension we therefore have to consider investigations like the one performed a decade ago by the Swedish researcher Dagmar Neuman (1989). When she investigated the conception of number as a measure of quantity among young school children, there were some who performed very poorly in tests; for instance, when asked how many pens they would have if they dropped 7 out of 10, they answered 6. However, she performed the tests orally and gave the children a chance to explain how they thought when they solved the problems.

It turned out that the poorest children were enumerating the objects in the problem by naming them with the words of the number sequence. Getting rid of seven was then understood as excluding the pens named ‘10’ down to ‘7’. That is, the last mentioned number, ‘7’, functioned as the name of the set removed. The remainder was then ‘6’, or for some children actually ‘1’ or even ‘0’, because they continued to count backwards to find the name of the remaining set.

These children clearly have an abstract sequence of numbers to rely on – the sequence even starts at zero for some of them. But the numbers are not our numbers! Formally, the very sequence in itself looks totally correct. But the idea of quantity, or at least our idea of quantity, has not entered their conceptual sphere. Lots of people have pondered upon what has given us the sense of numbers we have today; in particular, the use of fingers has been presented as a way of getting from a sequential sense to a quantitative sense. Etymological proofs may enter the discussion here, but rather than revealing my ignorance in such matters, may it suffice to note that etymology is not of any particular interest when it comes to children. Rather, it is the numerical notation – based on ten numerals and a principle of position, as well as a never-failing presence of the hands, that make it advisable to base young children’s numeracy on the fingers.

Actually, Neuman has reached the conclusion that "Mastery of the number relations within the number range 1–10 seems to be a necessary condition for performing arithmetic operations outside this range." (Marton & Neuman, 1990, pp. 8–14.) In the light of the rather small bundles made by Sumerians and others, and our general inability to see the number of items in a set of more than a few items, one might wish we had had fewer fingers on our hands so we could have had a number system based on 3 rather than 10.

However, Neuman points out that when splitting the numbers six to ten into two parts for training basic addition or subtraction, the larger part is in most cases 5 or bigger. That is, the larger part contains "the undivided full hand". That leaves the division to the second hand where one is dealing with one or two very small sets of fingers: e.g. 9 could split into 6 & 3 = (5+1) & 3; that is, the second hand splits into ‘one and three’. This in effect recreates ‘early’ bundling and avoids counting in spite of the decimal structure of our fingers.

*

In sum, then, it is a great evolution of imagination that has taken place through the utilisation of pebbles, and body parts, and written numerals. This evolution surely belongs to the genealogy of writing. Unfortunately, it also belongs to the genealogy of counting difficulties. The present profusion of abstract numbers – in the number sequence and in written numerals – could conceal numeracy from the ignorant. Likewise, the ignorant person’s scribbling of numerals and smattering of number words could make us unaware of his or her ignorance of our number concept. It is hard to know when a number, written or uttered in the abstract, refers to an abstract number or merely to a misunderstanding.

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