## NUMBER

What is number? By ‘number’ I mean whole number, positive integer such as 2, 34, 1457…  Conceptually, number is hard to pin down and the modern mathematical treatment which makes number depend on formal logical principles does not help us to understand what number ‘really’ is.  “He who sees things in their growth and first origins will obtain the clearest view of them” wrote Aristotle. So maybe we should start by asking why mankind ever bothered with numbers in the first place and what  mental/physical capacities are required to use them with confidence.

It would seem that numbers were invented principally to record data. Animals  get along perfectly well without a number system and innumerable ‘primitive’ peoples possessed a very rudimentary one, sometimes just one, two, three. The aborigine who often possessed no more than four tools did not need to count them and the pastoralist typically had a highly developed visual memory of his herd, an ability we have largely lost precisely because we don’t use it in our daily life (Note 1). It was the large, centrally controlled empires of the Middle East like Assyria and Babylon that developed both arithmetic and writing. The reasons are obvious: a hunter, goatherd or subsistence farmer in constant contact with his small store of worldly goods, does not need records, but a state official in charge of a large area does. Numbers were developed for the purpose of trade, stock-taking, assessment of military strength and above all taxation ― even today I would guess that numbers are employed more for bureaucratic purposes than scientific or pure mathematical ones (Note 2).

What about the required mental capacities? There are two, and seemingly only two, essential requirements. Firstly, one must be able to make a hard and fast distinction between what is ‘singular’ and ‘plural’, between a ‘one’ and a ‘more-than-one’. Secondly, one must be capable of ‘pairing off’ objects taken from two different sets, what mathematicians call carrying out a ’one-one correspondence’.

The difference between ‘one’ and ‘more-than-one’ is basic to human culture (Note 3). If we actually lived in a completely unified world, or felt that we did, there would be no need for numbers. Interestingly, in at least one ancient language, the word for ‘one’ is the same as the word for ‘alone’: without an initial sense of estrangement from the rest of the universe, number systems, and everything that is built upon them, would never have been invented.

‘Pairing off’ two sets, apples and pears for example, is the most basic procedure in the whole of mathematics and it was only relatively recently (end of 19th century) that it became clear that the basic arithmetic operations and numbers themselves        originate in messing about with collections of objects. Given any two sets, the first set A can either be paired off exactly member for member with the second set B, or it cannot be. In the negative case, the reason may be that A does not ‘stretch to’ B, i.e. is ‘less than’ (<), or alternatively it ‘goes beyond’ B (>), i.e. has at least one object to spare after a pairing off. Given two clearly defined sets of objects, one, and only one, of these three cases must apply.

But as Piaget points out, the ability to pair off, say, apples and pears is not sufficient. A child may be able to do this but baulk at pairing off shoes and chairs, or apples and people. To be fully numerate, one needs to be able, at least in principle, to ‘pair off’ any two collections of discrete objects. Not only children but whole societies hesitated at this point, considering that certain collections of things are ‘not to be compared’ because they are too different. One collection might be far away like the stars, the other near at hand, one collection comprised of  living things, the other of dead things and so on.

This gives us the cognitive baggage to be ‘numerate’ but a further step is necessary before we have a fully functioning number system. The society, tribe or social group  needs to decide on a set of more or less identical objects (later marks) which are to be the standard set against which all other sets are to be compared. So-called ‘primitive’ peoples used shells, beans or sticks as numbers for thousands of years and within living memory the Wedda of Ceylon carried out transactions with bundles of ‘number sticks’. Yoruba state officials of the Benin empire in Nigeria performed quite complicated additions and subtractions using only heaps of cowrie shells. Note that the use of a standard set is an enormous cultural and social advance  from simply pairing off specific sets of objects. The cowboy who had “so many notches on his gun” was (presumably) doing the latter, i.e. pairing off one dead man with one notch, and doubtless used other marks or words to refer to other objects. In many societies there were several sets of number words, marks or objects in use simultaneously, the choice depending on context or the objects being counted (Note 4).

So what are the criteria for the choice of a standard set? It is essential that the objects (or marks) chosen should be more or less identical since the whole principle of numbering is that individual differences such as colour, weight, shape and so on are irrelevant numerically speaking. Number is a sort of informational minimum: of all the information available we throw away practically everything since all that matters is how the objects concerned pair off with those of our standard set. Number, which is based on distinction by quantity, required a cultural and social revolution since it had to replace distinction by type which was far more important to the hunter/foodgatherer ― comestible or poisonous, friend or foe, male or female.

Secondly, we want a plentiful supply of object numbers so the chosen ‘one-object’ must be abundant or easy to make, thus the use of shells, sticks and beans. Thirdly, the chosen ‘one-object’ must be portable and thus fairly small and light. Fourthly, it is essential that the number objects do not fuse or adhere to each other when brought into close proximity.

All these requirements make the choice of a basic number object (or object-number) by no means as simple as it might appear and eventually led to the use of marks on a background such as charcoal strokes on plaster, or knots in a cord, rather than objects as such.

Numbering has come a long way since the use of shells or scratches on bones but the ingenious improvements leading up to our current Arab/Hindu place value number system have largely obscured the underlying principles of numbering.      The choice of a ‘one-object’, or mark, plus the ability to replicate this object or mark more or less indefinitely is the basis of a number system. The principal improvements subsequent to the replacement of number-objects by ‘number-marks’, have been ‘cipherisation’ and the use of bases.

In the case of cipherisation we allow a single word or mark to represent what is in fact a ‘more than one’, thus contradicting the basic distinction on which numbering depends. If we take 1 as our ‘one-mark’, 11111 ought by rights to represent what we call five and write as 5. Though this step was long to come,  the motivation is obvious: simple repetition is cumbersome and leads to error ― one can with difficulty distinguish 1111111 from 111111. Verbal number systems seem to have led the way in this: no  languages I know of say ‘one-one-one’ for 3 and very few simply repeat a ‘one-word’ and a ‘two-word’ (though there are examples of this).

The use of bases such as our base 10, depends on the idea of a ‘greater one’, i.e. an object that is at once ‘one’ and ‘more-than-one’ such as a tight bundle of similar sticks. And if we now extend the principle and make an even bigger bundle out of the previous bundles while keeping the ‘scaling’ the same, we have a fully fledged base number system. The choice of ten for a base is most likely a historical accident since we have exactly five fingers and thumbs on each hand. The hand was the first computer and finger counting was widely practiced until quite recent times: the Venerable Bede wrote a treatise on the subject.

The final advance was the use of ‘place value’: by shifting the mark to the left (or right in some cases) you make it ‘bigger’ by the same factor as your chosen base. Although we don’t see it like this, 4567, is a concise way of writing four thousands, five hundreds, six tens and seven ones.

Human beings, especially in the modern world, spend a vast amount of time and effort moving objects from one location to another, from one country to another or from supermarket to kitchen. One set of possessions increases in size, another decreases, giving rise to the arithmetic operations of ‘adding’ and ‘subtraction’. And to make the vast array of material things manageable we need to divide them up neatly into subsets. And  for stock-taking and related purposes, we need agreed numerical symbols for the objects and people being shifted about. A tribal society can afford to ignore numbers (but not shape), an empire cannot.     SH

Note 1 A missionary in South America noted with amnazement that some tribes like the Abipone had only three number words but during migration could see at a glance from their saddles whether a single one of their dogs was missing out of the ‘immense horde’. From Menninger, Number Words and Number Symbols p. 10

Note 2  To judge by the sort of problems they tackled, the Babylonian and Egyptian scribes were obviously interested in numbers for their own sake as well, i.e. were already pure mathematicians, but the primary motivation was undoubtedly socio-economic. Even geometry, which comes from the Greek word for ‘land-measurement’, was originally developed by the Egyptians in order to tax peasants with irregular shaped plots bordering the Nile.

Note 3  Some historians and ethnologists argue that the tripartite distinction ‘one-two-more than two’, rather than ‘one-many’, is the basic distinction. Thus the cases singular, dual and plural of certain ancient languages such as Greek.

Note 4  The Nootkans, for example, had different terms for counting or speaking of (a) people or salmon; (b) anything round in shape (c) anything long and narrow. And modern Japanese retains ‘numerical classifiers’.