Desert Island Numbers

“He who examines 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.   After cogitating about number for some time , I set myself a mind-experiment. I imagined myself marooned like Robinson Crusoe on a desert island where the natives, though amiable and intelligent, did not seem to have a number system. After I had demonstrated what could be done with numbers like calculating when the next full moon would be (which they considered to be a kind of magic) they asked me what they had to do to make their own numbers. I laughed and said that numbers were not things that could be made like boats and mats. But then I stopped short and asked myself the question: What in fact do you need to construct a workable set of numbers? What are the minimal requirements?  And the answer that came to me was : 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’, probably not. Most likely the society he lives in does not have enough spoken or written words to represent such a quantity, at any rate if he has 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 great’, almost our ‘infinite’. Menninger cites the true story of the venerable South Sea Islander who, being asked how old he was, answered, “I am three”¹.
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 shells².

Number Objects and Object Numbers

 I imagine myself, then, Robinson Crusoe-like, looking for a suitable 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?
It is important that the objects chosen should be more or less identical since I have already decided in my solitary cogitations on numbering that the basic principle of number would seem to be that individual differences between objects do not matter numerically speaking. My ‘one-object’, whatever it is, is going to be used to represent indifferently a tree, a fish, a man, 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 need as many object-symbols as there are objects.
Secondly, since there are a lot of objects in the world, I want 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 object be durable, easy to see, especially when held in the hand 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 (prisoners of war, conscripts) have occasionally been used as numbers of a sort, for example to make a rough guess of 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 cheap today 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 abacus while marbles  were confined to grooves in the case of the Roman abacus. 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 inChinacenturies before ‘double-entry’ book-keeping became current inEuropethough the meaning was the reverse, black for negative, red for positive.

The Number Ball

On another occasion thinking of my island paradise awaiting its Archimedes, I hit upon the idea of a clay ‘Number Ball’. The advantage of this choice is that, apart from being portable, it allows one to get rid of a number once it is of no further interest and start again. A native might be sent, for example, to find out how many palm trees there were on a particular beach. Equipped with his Number Ball issued at the Central Data Hut he would arrive at the site and tear off as many little bits of clay as there were trees. He would report back to Central Office where the bits of clay would be recorded by an equivalent amount of scratches on a bone or knots in a cord, and would then squash everything together to recover the original ball.
This system has an interesting feature : it is two-way  in the sense that you can use the same apparatus for recording data but can then ‘de-record’ (wipe out) the data to recover the original set-up and start again. This means, firstly, that there is no wastage. There is also something aesthetically satisfying about such a simple apparatus having an  ‘inverse’ procedure built into it : once you have completed your task, the Number Ball is returned to what it was in the beginning like the Tao.
Most recording systems do not have this feature : if you make a scratch on a bone you cannot ‘de-record’ without damaging the recording device, and crossing out something written with pen and ink is both messy and inefficient (in films a crossed out line often gets deciphered and leads to the conviction of a criminal). Destroying data has in fact become a considerable problem in modern society, hence the sale of shredders and civil servants’ perpetual fear of e-mails being picked up.
Clay Number Balls would be too messy for modern interior use but Blu-Tack is an alternative I have experimented with a little. There is, however, a certain risk of the little bits of clay or Blu-Tack sticking together and thus falsifying the reckoning.
The Number Ball is something of an anomaly mathematically and even philosophically. The object-numbers produced, i.e. the little bits of clay, do not strictly fulfil the requirement that number objects should not merge on being brought into close proximity — they can be made to merge or kept apart at will, so we have an interesting intermediate case.
Also, and this is more significant, the Number Ball is not, properly speaking, a number but rather a source of numbers, a number generator. In this respect it resembles an algebraic formula since the latter is not in itself a number (in any sense) but can be made to spew out numbers, as many numbers as you require. (For example the formula f(n) = (2n –1)  gives you the odd numbers (counting 1) if you turn the handle by fitting in 1, 2, 3….. for n e.g. (2 ´ 1) – 1 =  1; (2 ´ 2) – 1 = 3; (2 ´ 3) – 1 = 5 and so on.)
Yet a Number Ball is not a formula or an idea : it remains an object. Of course, one could also call a box of matches or a set of draughtsmen  ‘number generators’ but there is a difference here : the object-numbers are present in the box as distinct items (matches) and are thus already numbers at least potentially, whilst bits of clay of Blu-Tack are not. A Clay Number Ball is actually a special type of generator since everything it produces comes from within itself and can be returned to itself. I have coined a term for this particular case : I call such an object an Aullunn. Although there are no complete Aullunn Generators in nature — not even, seemingly, the universe itself —   many natural phenomena approximate to this condition. The varied life in and around a pond to all intents and purposes emerges spontaneously ‘from inside’ and dies back into it; though we know that without some interaction with the environment, especially with sunlight, no generation would be possible.
Surprisingly I have not come across any accounts of tribes using clay Number Balls.

Number Marks and Number Bearers

A very different method of producing a set of numbers is to have an object or substance which is not itself a number (or a number generator) but a ‘bearer of numbers’ : the numbers are marks on the surface of the number bearer or deformations of it. This system, which at first sight seems a lot closer to the written system we use today, is extremely ancient and possibly pre-dates the widespread use of distinct number objects. The markings on the Ishango Bone, which dates back to about 20 000 B.C., are thought by archaeologists to have numerical significance. Other bones have been found dating almost as far back with scratches on them that are thought by some to  indicate the number of kills to a hunter’s credit — one thinks at once of Billy the Kid, the “boy who had so many notches on his gun” (or was it Davy Crockett?).
The limitation of the notch system is that an incision is permanent which means that once the ‘number-bearer’ gets filled up it has to be stored somewhere or discarded like a diary. It thus tends to be used in rather special circumstances, either when one does not expect to be dealing with large quantities (rivals killed) or when one wants the information recorded to be permanent as, for example, in the case of inscriptions on State monuments.
Making charcoal marks on a wall, also an ancient practice, is ‘two-way’ in that one can rub out what one has written but the system would not be reliable for long-term recording of data because of effects of weather, flaking of surface &c. But numbers on a number bearer do not have to be marks : they can be reversible deformations, the prime example being knots in a cord. The great advantage of such systems is that, though very long lasting if the material is itself durable, the numerical data can easily be got rid of when no longer needed since knots can be untied. On the other hand because they take a lot of time to tie and untie, knots are unsuitable for rapid calculation and it would seem that the Inca State officials used quipus for storing data whilst they had some form of a counting-board system for calculations. Knots in a cord constitute a partial ‘two-way’ recording system — what is done can be undone — but they are at the same time quasi-permanent and so they combine some of the features of both types of system.
Knotted cords were in widespread use all over the world at one time and it is thought that mankind may even have gone though a ‘knotted cord’ era. The author of the Tao Te Ching (VIth century B.C.) who is a Luddite hostile to new-fangled inventions and to civilization generally speaks nostalgically of the days when mankind used knotted cords instead of written numbers.
In practice both systems are required, a ‘two-way’ number system which allows one to carry out calculations and then to efface them, plus a more permanent system which is used to record results if they are considered important enough. It has been agreed that the Incas only used quipus for permanent or semi-permanent records while they used stones and a counting board for calculation. The lack of a suitable ‘number-bearer’ to receive marks meant that inscribed number systems were a rarity until comparatively recently — baked clay tablets and papyrus were reserved for the bureaucratic elite and paper, a Chinese invention, only entered Europe in the latter Middle Ages and was expensive. Traders, even money-lenders and bankers, when they did  not use finger-reckoning of which more anon, used a two-way system, namely counters and counting boards, right into the Renaissance. The abacus, a two-way system, was never widespread in Europe for some reason except in Russia, but in the East has remained in use right through to modern times. The soroban or Japanese abacus is still used today and as late as the nineteen-fifties a Japanese clerk armed with a soroban competed successfully with an American naval rating using an early electronic  calculator. However, it must be pointed out that the Japanese achievement with the soroban depends on extensive practice in mental arithmetic which is still prized over there rather than any particular merits of the device itself.
The drawback of a ‘two-way’ system such as an abacus where you erase as you go is that you cannot check for mistakes and even the result itself, once reached, has to be erased when we perform our next calculation i.e. there is no inbuilt recording element, no memory. But when there is no easy way of erasing we oscillate wildly between conservation and destruction : we tend to accumulate a vast amount of stuff, then periodically have a sort out and throw it all away, the pearls with the dross. Like most authors and mathematicians from time to time I have to tip out the entire contents of a large dustbin to search for a scrap of paper with some idea or formula written on it.
The principal drawback of a one-way semi-permanent system such as ink on paper is that it is incredibly wasteful and was until recently so expensive that the bulk of the world’s population, the peasantry, practically never used it and employed a pocket knife and a flat piece of wood to record data. Even in the computer era we still use the chalk and blackboard two way  system though the chalked notice-board in the hall of buildings or private residences — to mark who is in or out — which was once commonplace is now virtually a thing of the past. I myself buy rolls of lining paper (which I clip down over a table) partly because I like to have plenty of room for drawing and calculation but also partly for reasons of economy — you get a lot of paper in a roll compared to an exercise book. It is a sobering thought that no less than a hundred years ago Ramanujan, one of the greatest names in Number Theory, like so many other Indian mathematicians of the time worked with slate and chalk because he found paper too expensive. Although to my knowledge no one has suggested this, I would guess that this is one of the main reasons why his early mathematical writings are so hard to follow — he left no tracks because he generally just copied out his conclusions, then literally wiped the slate clean. To many people the results seemed to come from nowhere and indeed he was often incapable of explaining how he got them.   (Ramanujan lived a century too early : today we have a relatively cheap improved ‘chalk and board’ system, the Whiteboard. At last marks can be easily erased without mess. I use large boards everyday and have somewhat moved on from lining paper to a more up to date recording system.)

 A Standard Set of Ones

 Prior to my arrival on the desert island in question the inhabitants did not bother with numbers at all and had no number objects, not even the bundles of number sticks used by the Wedda of Ceylon. How exactly they got on without any form of mathematics I shall discuss in a moment (not as badly as you might think). Almost certainly they had a handful of number words and were able to point and perhaps make intelligible signs with the hands. I decide on my chosen set of objects which are henceforth to serve as numbers, cowrie shells for example, and instruct the natives in their use.
The point is that I am taking this as an all-round numerical watershed : prior to having a set of standard objects or marks set aside for numbering I call such a society innumerate, while if it has such a set it is numerate. The line must be drawn somewhere. Most people today consider that, with one or two rare exceptions (bees, for instance), animals do not have number sense as we understand it. They are excluded by my definition because none of them use sets of number objects or signs. (I can hardly stress too strongly that this is not ‘putting down animals’ — animals, like many early societies  do not use numbers because they do not feel the need for them.) At the other extreme it would, I think, be ludicrous to take as a definition of numeracy familiarity with the Axioms for Fields as laid out in a modern algebra book. Such a criterion would exclude Newton, not only because he did not know of the Axioms for Fields which are a twentieth century concoction, but because, even had they been formulated in his  time, he would almost certainly have disapproved of them.¹ My simple criterion lets in the Incas and the Yoruba who administered huge areas without any form of written or incised records, and even the Wedda.
More specifically I declare a society to be numerate if it operates a set of standard objects or signs which represent any set of discrete objects that can be put in one-one correspondence with it, or a portion of it.²
          The decisive step, then, is when a society accepts that a particular ‘one’, cowrie shell, bean, vertical stroke ½ or symbol 1, can validly represent any individual object, any ‘one’.
The step taken is really momentous — so momentous that most societies that reached such a stage baulked at a complete generalisation and made at the very least one or two exceptions. In very many societies certain sets of marks were permitted to represent specific objects alone, or certain categories of objects. Until modern times, craftsmen, tradesmen and the like often used special sets of numerals (generally marks on tally-sticks) for specific merchandise such as milk, wheat and so on despite having some familiarity with ‘ordinary’ numerals. Even in a society as advanced as Athensin the Vth century BC there was as yet no completely universal symbol corresponding to our ‘2’. In state documents dealing with the city’s income and expediture we find that different symbols are used, depending on whether the official is referring to “2 talents”, “2 staters” or “2 obols” (Menninger, p. 269).³
The step involved is not only momentous but shocking and indeed has still not been accepted psychologically by the vast majority of the population. Why does one feel an instinctive repugnance about being treated ‘as a number’? Why is there such deep-rooted though inarticulate hostility towards administrators and bureaucrats, the users of numbers? Because numbering wipes out every feature of individuality — that is what numbering is about.

“Where is mankind? These ciphers do not speak
          Of what is gay or sad, wistful or intimate,
          All persons and all feelings are equivalent;”

The two developments which have probably done the most to change the conditions in which we live, mathematics and physics, are not only non-human sciences but in a very real sense actually anti-human. This is the reason for the extraordinarily persistent popular antipathy towards the mathematical sciences — after all we are humans and cannot be expected to welcome with open arms something which does not recognize the fact! Numerically no distinction is made between a gnat, a star, a loved one or a murderer — they are all in some sense single objects and so the same mark can  be and is used for all of them. Hume wrote trenchantly, “From the point of view of the universe, there is no difference between the death of a man and the death of an oyster”. He might just as well have said, “from the point of view of arithmetic, there is no difference between a man and an oyster”. And as for physics, the numerical science, the same applies :Newton’s Laws predict that in a vacuum your mother or child would fall to the earth at the same rate as a piece of cardboard or a slug. ~
Prior to the twentieth century practically all societies were religious. For many it was not so much the implied threat to human beings that made people wary of numbers as the implied threat to the gods. For a god or goddess was also a ‘one’ and the immortals could be (and were) numbered in the same way as cattle were numbered. Practically, this led to several societies having two or three completely different sets of numerals, one kept for religious matters, one that dealt with mundane matters and perhaps a third for matters of state. Thus the Mayans had three sets in use. The Chinese had a special set of ordinal numbers for noting the length of the reigns of Emperors.  Even today I would hazard the guess that we still use the inefficient and not very picturesque Roman numerals for dates (because we somehow feel the passing of the years is ‘different’, is  semi-sacred and so needs special symbolic treatment. Certainly, it was the astronomer/priests who gave us the calendar in the first place and the last widespread reform, the Gregorian calendar, was instigated by the Papacy. Only quite recently  has the measurement of time finally fallen into the hands of entirely secular officials at Sèvres, Paris. The location (Paris) is fitting since it was the French Revolution which did away with the myriad local measuring systems that existed during the ancient regime and which deliberately invented (and imposed on most of Europe) a new ‘scientific’ system of measurement, the metric system. Whereas previous basic measurements had generally been based on human dimensions (the cubit, for example) or related to human activities — an acre was supposedly the amount of land a good worker could plough in a day — the metre was a sub-multiple of the circumference of the Earth.

Notes :