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Bases

June 4, 2013

Bases

The first thing to realize about bases is that Nature does not bother with them. Nature does not group objects into tens, hundreds, thousands and so on, not even into twos, fours, eights. Bases in the mathematical sense are entirely a matter of human convenience — Nature only uses base one. We are so used to thinking decimally and writing numbers in columns that we tend to consider that an amount, expressed in our modern Hindu-Arabic  positional numerals, is somehow truer than if it were expressed in, say, Chinese Stick Numbers. Even, there is a tendency to think that the modern representation of a quantity  is somehow numerically truer or more real  than the actual quantity  — an example of the delusionary thinking that doing mathematics all too readily gives rise to. Asked how many stones there are in a certain pile, an unhelpful but perfectly correct answer would be to transfer all the stones into a wheelbarrow and empty them out at the enquirer’s feet.

          So why do we bother with bases? Partly for reasons of space : an amount expressed in base one requires a lot of paper or wood or whatever material we are employing and this was and is an important consideration. But the main reason is that our perception of number is so defective that we find it very hard indeed  to distinguish between amounts of similar objects or marks beyond a certain very small quantity (seven at most). So what do we do? What we need is a second fixed quantity, a second ‘unit’, in terms of which we can assess larger quantities. What do we choose?

‘Ones’ are given us by Nature and the sort of objects we shall want to assess either are collections of ‘ones’, like trees or sheep, or can be treated as if they were, e.g. mountains, villages and so on. We speak of ‘whole’ numbers thus implying that they are in some sense ‘entire’, ‘indivisible’. If our standard ‘one-object’ is a pebble it really is indivisible in a pre-industrial society and a cowrie shell, though it can be broken in two, ceases to be a proper shell if this is done. There is, in concrete number systems,  no question of dividing a ‘one’ and ‘fractions’ (‘broken numbers’), if defined at all, are represented by smaller objects or marks, not by splitting up the ‘one-object’. Whole number arithmetic was by Greek times firmly associated with the physical theory of atomism as put forward by Democritus who himself wrote works on arithmetic now lost.

A mass of ‘ones’ is perceptually unmanageable unless related to certain standard amounts we are familiar with. But Nature is extremely unhelpful in providing us with exact standard amounts. Litters of kittens and puppies are by no means standard, and apart from a slight prejudice in favour of the quantity five, the amount of petals in a flower or branches on a tree varies amazingly.
Familiar fixed amounts such as the ‘number’ of hills on the skyline could only be of strictly local relevance and at the end of the day about the only available standard amounts given to man by Nature are the fingers which, counting the thumbs, come in fives. It is thus no surprise that number systems are dominated by the amounts five, ten and twenty.

Alternatively, of course, we could start from the other end and opt for a ‘man-made’ secondary unit whose size would depend on our perceptual needs and what exactly it is we want to assess or measure. These three criteria 1.) availability of a standard amount; 2.) human perceptual  limitations and 3.) appropriateness for assessment purposes, conflict and one of the main problems of early numbering was how to reach an acceptable compromise between them.

Two is the first possibility for a ‘secondary unit’ but, although it has come into its  own in the computer era (because of the two states On and Off), it is clearly too small to be of much use for ordinary  purposes. A language spoken in the Torres Straits had a word for our ‘one’, namely urapan and a word for our ‘two’ okasa and that was about it. Their numerals went

1.       urapan                   4.       okasa okasa

2.       okasa                     5.       okasa okasa urapan

3.       okasa urapan         

Understandably, since even a number as small as 11 would require six words, the natives referred to anything above 6 as ras — ‘a lot’ (Conant, p. 105).

A few things, or rather events, are viewed in threes witness phrases like “third time lucky” and we group quite a lot of things in fours (seasons, points of the compass &c.) but  the obvious first choice for a ‘secondary unit’ is five. Beyond five we really feel the need for a ‘secondary unit’ since  collections like    l l l l l l l l and   l l l l l l re  practically indistinguishable. Also, as it happens we have the five fingers to be able to check (by pairing off) whether we are separating out the items into groups correctly.

The Old Man of the Sea in Homer ‘fives’ his seals but for most herdsmen five would still have been rather too small as a secondary unit. So where do we go next? If we remain guided by the fingers the next possibilities are ‘both hands’ and what many primitive languages referred to as ‘the whole man’ i.e. ten and twentyTwenty is in some ways a better choice since, if we keep the option open of reverting to five for trifling amounts we can cope with very sizeable collections using batches of twenty. The Yoruba used twenty cowrie shells as their principal counting amount after the unit. Some modern European  languages which have long since become decimal show traces of an earlier vigesimal (twenty-based) system which probably suited farmers better. Hence Biblical terms like ‘three score years and ten’ in English and the French soixante dix-huit (sixty-eighteen).

A secondary  unit is, unlike the unit, not actually indivisible — since it is still made up of standard ones — so how do we keep it together if we are using objects as numbers? This depends on the choice of standard object and in practice is one of the motivations for the choice of object in the first place (or second place at least). Heaps of pebbles are heavy enough not to blow away but can all too easily be disturbed by people bumping into them, while piles of flat objects unless they are paper thin readily tip over and in any case really flat objects are hard to come by in nature. This is where shells are advantageous since if of the cowrie variety they stack up neatly and, even better, can be pierced and threaded on strings to make number rosaries. Beads make good numbers but since they are manufactured items they would not have been amongst the very earliest examples of object numbers.

The clay Number Ball I have already mentioned would not be suitable for secondary (or tertiary) standard amounts precisely because the bits tend to adhere together : its use would be for assessing limited quantities in the field which, if required, could be recorded back at the Number Hut using a different system, knotting or incising.

On my island I opt for a stick of standard length as my ‘one-object’ and I instruct the natives to tie sticks together into a bundle when we reach the fixed secondary amount which tentatively I fix at our twelve. Lacking a  sign on this computer for a bundle of sticks I represent this amount by . At the back of the Number Hut I set up partitions to make alleyways for concrete numbers and I suspend from the roof an example of the bundle the alleyway is to contain and its decomposition into smaller bundles or individual sticks as a sort of Système Internationale prototype. For the moment I only propose to use the extreme right two alleyways, the first for individual sticks and the second for bundles of the specified size.  Whenever we have  ⁄ sticks in the extreme right alley, they are tied together and transferred (literally ‘carried’) to the next alleyway. The system can be used for the temporary recording of data but it is best to restrict its use to calculation, simple additions and subtractions, while using a different system for recording purposes. I can, for example, paint three vertical lines on a piece of bark to make it into a Number Board and paint in particular configurations of the sticks and bundles.  When painting I do not use short cuts, I just represent the sticks and bundles as well as I can turning stick numbers into stroke numbers.  As yet I do not proceed any further : all quantities are to be represented by sticks and/or Number Bundles of a single fixed amount and, for the time being, I do not set an upper limit to the amount of bundles. Thus the components of our number system so far are only  l  and     where

       =        l l l l l l l l l l l l  

A  feature of this still very rudimentary system is that at any moment a bundle     can be reduced to so many sticks simply by untying the cord and transferring them into the appropriate alley. Even, it is possible to have second thoughts about the ‘secondary unit’ and change it for another, since all you have to do is untie the bundles and tie the sticks up again using a different set amount. We might, for example, want to revert to five if the quantity to be assessed turns out not to be so great, or, conversely, jump ahead to twenty for a really large herd of goats or clump of trees. With systems that depend on threading objects on a string or wire, changing the secondary unit is either impossible or time-consuming and so would tend not to be done.

If the first ‘greater unit’ is set at ten and we are dealing with sticks, the problem of distinguishing different numbers less than ten  remains — we have met requirements 1.) and 3.) but not 2.) . The earliest Egyptian written numbers, perhaps based on still earlier number sticks, got round this problem, or tried to, by arranging the sticks in set patterns. But the patterns are not very distinctive or memorable. Far more striking are the excellent domino or dice dot numbers. Domino patterned numbers stop at six and the arrangements for the playing-card seven, eight and nine are not so striking — I have sometimes I caught myself having to look at the corner to distinguish between a seven and a nine.

The stratagem of arranging near identical marks in a pattern is an attempt to enlist shape in the service of number : if you recognize the shape you don’t need to count the dots. Shape recognition is distinction by type which the principle of distinction by number must displace in the cultural development of the species. For all that , even today, we feel at ease with shape and respond to it ‘naturally’ (perhaps because of the sexual instinct and childhood memories) while number is at first unappealing, it appears cold and  inhuman. The supposedly artificial distinction between the arts and the sciences is rooted in this primeval struggle between distinction by shape and distinction by number, a struggle which the latter is obviously  winning. In the pre-industrial past it was quite the reverse  : most ‘primitive’ tribes considered that distinctions of shape and thickness were so much more significant than numerical distinctions that they developed a large  and sophisticiated vocabulary to deal with the former while contenting themselves with half a dozen number words. And even in the domain of number itself shape cast its shadow: many societies used different number words  depending on the overall shape of the objects being counted. The Nootkans, for example, used special terms for counting round objects and traces of this practice persist in the ‘numerical classifiers’ of modern Japanese and Chinese1.

A drawback of numerical distinction by patterning is that every new number requires its own special arrangement which must then be committed to memory. This certainly limits the range but the same patterns could be re-used with slight differences or could be combined in various ways. One would not have thought the effort involved was that great, not that much more than is involved in learning the alphabet. Also, the idea of familiarising people with numerals by way of card and board games is delightful (though presumably not done  deliberately). For some reason this promising system never got extended beyond six (otherwise we would have in our heads patterns for higher numbers) and taken in itself constitutes something of a dead-end in the history of numbering.

Domino numbers are a curious and attractive relic of days long gone. 

Q1. If we use dominoes as numbers, what base are they in? And what is the largest amount that can be represented by a single set of dominoes ?  

(To be continued)

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