Desert Island Numbers : Introduction

“He who sees things in their growth and first origins will obtain the clearest view of them” (Aristotle).

What exactly are numbers? This is by no means an easy question to answer despite the fact that we use numbers every day of our lives in some form or other. Do we not, then, know what we are doing?
One way of answering this question is to launch into an investigation of ‘number’ as a concept or basic principle. This approach quickly leads on to some deep and perplexing issues that go back at least to Pythagoras and are still with us today (though mathematicians try to avoid tackling them if they possibly can).
At the opposite extreme, we have the Set-theoretic approach according to which ‘numbers’ (integers, rational numbers, irrationals &c. &c.) are just ‘things’ that emerge as just one application of the six or seven basic Axioms (preliminary assumptions) of Zermelo-Fraenkel Set Theory. This way of proceeding avoids metaphysical speculation altogether but at considerable  cost : the end products are not recognizable as numbers. Moreover, no one ever learned mathematics this way and most likely never will.
I favour a more practical approach. Let us ask ourselves why mankind ever bothered with numbers in the first place? Do we really need them, and if so what for? Essentially, we need them to ‘represent’ or ‘stand in for’ certain objects, i.e. numbers are ‘symbols’ in the straightforward sense that they ‘stand in’ for something else, something they are not. So why bother with symbols? Why not use the real thing? The answer is that the real thing may be far away, may be too heavy to carry about, too small to see without a microscope and so on. It is frequently not practicable to deal with  the real thing, especially if you want to manipulate it in various ways, make it larger, smaller, join it to something else and so forth. So we employ a substitute which represents it.

     Historically, numbers did not evolve as the result of philosophic speculation or as an intellectual pastime. Arithmetic was developed for mundane practical reasons : numbers and operations with numbers were required for trade, stock-taking, taxation, carrying out censuses, assessing military strength and a host of other unromantic administrative tasks. Innumerable tribes got along pretty well without much of a number system at all — sometimes nothing more than ‘one, two, three’. It was the large, centrally controlled empires of the Middle East like Assyria and Babylon who developed both writing and arithmetic. The reasons are pretty 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 vast area with varied resources does.
After cogitating about number for some time and not getting very far, I set myself a mind-experiment. I imagined myself marooned like Gauguin in a pre-mathematical society and asked myself the question: What exactly do you need to make a workable set of numbers? What are the minimal requirements?  And the answer is : all you need is a set of more or less identical portable objects that do not merge or stick to each other when brought close together. Before being ink marks on paper or dots on a computer screen

mankind’s numbers were objects — pebbles, shells, twigs, knots in a cord, things you can touch and handle.

Parable of the Goatherd

  Consider an illiterate goatherd such as existed in many parts of the globe until a few decades ago, and     possibly still does in very remote parts. He brings in his goats each night and leads them out through a gate each morning. On the right hand side of the gate is a pile of stones and as he lets each goat through the gate he shifts a stone from the right hand pile to make a similar pile on the left. One goat, one stone.
At the end of the day he lets the goats back in and shifts a stone back to the right hand side of the gate (right hand from the inside but on the goatherd’s left side if he is coming back.) If there are any stones remaining when the last goat has been ushered in, he knows there are goats missing — or at least one goat missing.
Does our goatherd know how many goats he has in his herd? In our sense of ‘how many’, perhaps not. In the past the society he lived inmay not even have had enough spoken or written words to represent such a quantity, at any rate if he had a sizeable herd. Innumerable tribal languages had no words to express quantities beyond our ‘forty’ and in many cases, incredible though this seems, the vocabulary of number was limited to the equivalent of our ‘one, two, three’ where ‘three’ had the meaning ‘inconceivably large’, ‘not numberable’  —. Karl Menninger, in his wonderful book, Number Words and Number Symbols, cites the true story of a venerable South Sea Islander who, being asked how old he was, answered, “I am three”.
But in fact our goatherd is not innumerate, he does have a set of numbers, his pile of stones. The stones are his numbers. If asked how many goats he had  in his flock, he would probably indicate with his hand the pile of stones on the right side of the gate. And if asked at the end of the day how many goats were missing, supposing some were missing, once again he would indicate the stones left. It is as ridiculous to suppose that numbers must be marks on a piece of paper or pictels on a screen as it is to suppose that poetry has to be words on a page.
 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’. Although the development of a centralised imperial state apparatus generally gave rise to written  number systems, this was by no means invariably the case. The state officials of the Inca of Peru managed a vast empire without any form of written records : they used the quipu system where knots in coloured cords served as numbers. And the Yoruba officials of the equally extensive Benin empire in Nigeria performed quite complicated additions and multiplications using only heaps of cowrie shells.

Number Objects and Object Numbers

 I imagine myself, then, Robinson Crusoe-like, looking for a set of objects which are to be the basis of a workable number system which I can use myself and, possibly, introduce to the inhabitants of the island supposing there are some and that they are as yet innocent of numbers. What criteria are going to influence my choice?
Firstly, it is important that the objects chosen should be more or less identical since I have already decided that the basic principle of number is that individual differences between objects do not matter. My ‘one-object’, whatever it is, is going to be used to represent indifferently a tree, a fish, a man, a god, indeed anything at all provided the ‘thing’ I want to represent is singular, is a ‘one’. If I used different number-objects to represent different objects there would be no net gain — I would soon need as many object-symbols as there are objects.
Secondly, since there are a lot of objects in the world, I need a plentiful supply of numbers to represent them, so my chosen ‘one-object’ must be abundant. Alternatively, if I am going to make my own numbers I require the raw material to be abundant — wood for example — and the manufacturing process to be relatively rapid and easy so that I feel I can always make more numbers if I run out of them.
Thirdly, the chosen ‘one-object’ must be portable and to be portable must be fairly small and light. For, once again, if I have a stationary set of numbers there is little net gain: one of my main goals in developing numbers is so I can move around to assess numerically a distant clump of trees or a distant village. Although in special cases such as censuses and elections  the (human) objects do actually come to the numbers — come to a place where they are numbered — it is generally necessary to take the numbering apparatus to the objects and, in the very important case of spying, this is essential (in the days before long-range surveillance was feasible).
Fourthly, it is essential that the number-objects (or object-numbers, the terms are perfectly equivalent) do not merge or adhere to each other when brought into close proximity. It must be possible to make the numbers into a group while the objects remain distinguishable whilst in this group. Why is this important? Because this is the commonest set-up  we shall be modelling numerically. If we were regularly confronted with entities that flowed into each other, fused, only to separate a little later,  we would need  to introduce this feature into our mathematics But we spend most of our lives amassing objects, removing them from one group’s ownership to another’s (commerce), shifting them from house to house or port to port and so on. And the majority of these objects do not merge when brought together — even liquids are transported in containers and so function numerically as solid objects.
Other requirements are that the ‘one-object’ be durable, easy to see and can be easily held or placed on the palm of the hand. Also, we do not want the number-object to have the power of locomotion or it might move off before we have finished the counting! This more or less rules out living things or at any rate mammals as being suitable numbers though human beings in very special circumstances (soldiers, prisoners of war) have been used as numbers of a sort, for example to make a rough guess at the enemy’s strength on the basis of terrain occupied and suchlike cases.
All these requirements make the choice of a number object by no means so simple as it might seem at first sight. Grains of sand are abundant  but not easy to see, grains of salt stick to the hand. Beads are about the right size and are today cheap but they would have been luxury articles on a desert island in the past. Also they tend to roll around as do marbles which is why beads were threaded onto wires in the usual ordinary abacus while marbles  were confined to grooves in the case of the Roman abacus (see right). The most suitable objects, at any rate for a  rural society, turn out to be exactly the ones actually used by tribes : shells, beans and sticks. Shells are abundant, light, portable, and can be neatly stacked into piles. Sticks have the additional advantage that they can be split in two and so, if one runs out of numbers on a field trip, one can make more numbers on the spot.
               The question is not for me entirely an academic one as I have been looking for suitable object-numbers for some time now (in order to practise concrete arithmetic) and haven’t found an ideal choice yet.                        Draughtsmen  stay flat and stack up beautifully if they are of the old type with rings on top, thus allowing for the representation of ‘powers’. They also have the advantage of coming in two colours though it would be even better if they were a different colour underneath so that in order to ‘change sign’ you could just turn a draughtsman upside down. But  they are not abundant : one set of draughtsman is not enough and if you buy different makes  you find they don’t stack up properly.
Coins are feasible numbers, especially ones with holes in them like old Chinese ones since they can be threaded together. The counters used on counting boards in markets and even banks right through to the seventeenth century were virtually indistinguishable from coins. Roman calculi, of glass or stone, were unmarked but from the Renaissance onwards counters were elaborately decorated.
Matches are not at all bad, especially outsize ones like Brymay Long Matches and you can even represent positive and negative quantities if so inclined by using ones with brown and red heads — though the latter are becoming scarce now for health and safety reasons. Number sticks, painted red and black, were used in China centuries before ‘double-entry’ book-keeping became current in Europe though the meaning was the reverse, black for negative, red for positive.            To be continued

Note 1