Bases

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

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 more often than not this is all we are given. 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.

The Secondary Unit

 What there is no doubt we do need and have done from very remote times is a secondary unit. This must be clearly distinguished from a base  since the latter is an extendable sequence of ‘unit’ sizes, a ‘geometric series’ like 1, 10, 100, 1000 and so on. Why didn’t ‘early’ societies (with some exceptions) go straight for the base system? The answer is that they didn’t need it and that it doesn’t come naturally, at any rate to practical people. For most purposes two ‘significant amounts’ or the first and second powers of the base are quite sufficient. Even one will do if it is of reasonable size because you don’t actually have to stop at the ‘square’ of the base as if it were a brick wall cutting off all access to a numerical  beyond. Fifteen hundred, apart from being more succinct,  sounds a good deal more natural than the pedantic ‘one thousand and five hundred’ which is what we ought to say by rights (Note 1). Without even defining the hundred  we could still cope perfectly well with quantities up to 999 reckoning in so many tens e.g. by speaking of 810 as eighty-one tens and a five. Generally we do not need to go anything like so far and the language is littered with sets of number words which, though they show base potential, peter out into nothingness without ever even making it to the ‘cube’ stage.
The ‘natural’ way of cultivating the wilderness is to clear an area and then when you’ve planted that, to clear another. Natural at any rate if you have to do a fair amount of the work yourself. If you are a conqueror you may, of course, have an eye on infinity and eternity from the start but this is  folie de grandeur. The first man to introduce wholesale decimalisation was the Ch’in Emperor and he is thought to have hastened his death by imbibing elixirs of immortality.
Numbers were measures before they became numbers ¾ even ‘one’ itself, the ‘father (better mother) of all numbers’, is essentially a measure, one drop, one mouthful, one foot. Our standard weights and measures are really only numbers that remain tied to particular contexts and functions. The Imperial system of liquid measures is an application of base two with four left out since the numbers involved are 1, 2 and 8.

                   1 pint     =  1 pint

                   2 pints   =   1 quart

                   4 quarts =  1 gallon  =  8 pints

Verbally, the number system stops here : although there are obviously larger quantities than the gallon we have  no special words for them, it is someone else’s job to work them out.

In our number words and pre-decimal measures we find a surface  order with an underlying picturesque confusion where all sorts of sets of numbers leave their traces. In the Avoirdupois weights we seem to have two sets of numbers, one proceeding from the ‘small’ end and one from the ‘large’ end, most likely developed by persons performing different functions. It makes sense to have the pound finely divided for sales over the counter to individuals, thus the appearance of the  large, but not too large, secondary unit 16 in sixteen ounces to a pound. From the wholesaler’s point of view we want a large quantity defined straight off, the hundredweight (which is not a hundredweight). We now quarter the hundredweight as it is always useful to divide something into four equal parts and we nearly but not quite converge with the rising 16 system. But there are not 32 pounds to a quarter but the anomalous 28. Is a systematic base system preferable even supposing we had a more suitable base than 10? Not necessarily for the people doing the work. Their principal concern is not logical  consistency but the ready availability of convenient set amounts which the chosen number system or systems should favour and promote. Moreover, once they have what they want, various landmark fixed amounts, they leave the system to its own devices.
There is certainly, within the context of a pre-industrial economy, no need for a number sequence stretching out into infinity :  on the contrary this very feature would have in many cultures provoked a certain malaise as indeed it still does to persons like myself. The idea of really large magnitudes is frightening like the idea of really large intervals of time. Although it is now unfashionable, even politically incorrect, to speak of cultures having specific traits, it is surely no accident that it was the Hindu mathematicians who gave us the first fully positional indefinitely extendable written number system. Indian thinkers, both Hindu and Buddhist, were obsessed with large numbers and vast spans of time : the kalpa for example is a period which lasts 4320 million years. Armed with the decimal base number system the Indians built ‘number towers’ reaching unimaginable heights and not only could they write down these quantities but they had names for many of them. In one legend the Buddha, challenged to list the numbers (read ‘powers’) beyond 10⁷, answers with the names for all the powers up to the colossal tallaksana  or 10⁵³ ¾ i.e. 1 followed by fifty-three zeroes (Note 2) . The Indian approach to large numbers is quite different from that of Archimedes who is, surprisingly in some ways, much more in line with the earlier ‘clear an area’ approach. He wrote a treatise on large numbers but showed none of the delectation and religious awe that the Hindu and Buddhist mathematicians clearly felt. Archimedes was concerned to show that the finite Greek number system could be extended upwards and outwards to deal with colossal quantities like the amount of grains of sand in the, for him finite, universe. But his aim was to tame the beyond not to lose himself in admiration of it. To the amazement of modern commentators, he did not quite hit upon the artifice of full positional notation.
The acceptance (or imposition) of a single indefinitely extendable base system has taken a very long time and is of comparatively recent date. For centuries individual and local numbering systems co-existed with the State imposed one, Roman or Napoleonic, especially in country areas. Until very recently by far the greater part of the inhabitants of Europe were illiterate and many of them used their own numbering systems and ‘peasant numerals’ like the notched sticks of Swiss cowmen. Inns could scarcely have carried on at all without the one-base slate and chalk system where the reckoning was totted up at the end of the evening or month, and in Spain it was once common for the innkeeper to toss a pebble into the hood of a traveller’s cape for each drink consumed. Today rustic numbering systems like the ‘milk sticks’ of cowmen in the upper Alps or the use of knots or pebbles  are things of the past : everyone has at long last agreed to at least write numbers in the approved standard decimal fashion. But for all that we do not think or feel in the way we write. In effect we still use the Babylonian secondary unit, sixty, for the subdivision of the hour, although we  express the quantity in a ten-base. We think ‘sixty minutes’ as a ‘chunk of time’  divisible into so many units, not as six tens of time. We do indeed have the availability of intermediate amounts, five minutes, ten minutes, quarter of an hour, but they are subordinate to the hour and the minute. A day is experienced as a unity which is in the first place decomposable into the unequal ‘halves’ of daytime and nighttime. We do not experience or think the day as two tens of time plus four units.

To the administrator, of course, the use of a consistent base-system is as necessary as the use of a ‘universal’ official language (Latin, English in the Commonwealth &c.). To him numbers have finally ceased to be tied to objects or activities, have become contextless,  in much the same way as, at a further level of abstraction, functions, to the contemporary mathematician, have ceased to be tied to numbers.

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  ½½½½½½ and ½½½½½½½ are  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 twenty.  Twenty 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.

However, on reflection I decide that introducing such a system would be premature. Within the bounds of a self-sufficient fishing, hunting or agricultural economy there would neither be any need for an indefinitely extendable number system nor would it have any special appeal. In the first place an inhabitant of such a society would not anticipate needing to assess really big quantities. Although a peasant needs more numbers than a hunter, in the past he probably rarely if ever needed numbers extending beyond about 400 , supposedly the upper numerical limit of a typical 19^th century Russian peasant. Large amounts of fruit, potatoes and so on would, of course, not be counted but be assessed by weight just like coins that we hand in to the bank. (Banknotes are still counted but in most banks the work is now done by a machine.)

The practical man, craftsmen, herdsman or farmer does not deal in ‘numbers’, he deals in fixed amounts that are significant in terms of his or her  daily work and/or perceptual apparatus. And such quantities do not usually  correspond to the transition points of an extendable base system like our own. To judge by the traces they have left on our language the two most popular ‘significant amounts’ beyond the unit in English speaking countries were, and to a certain extent still are, the dozen and the score. Although twelve as such is beyond our perceptual capacity, the image of two boxes each containing half a dozen eggs must by now have penetrated to the collective  unconscious, at any rate the English speaking one. Twelve, like ten, seems about the right size for making bundles or piles, but is better than ten in many ways because it can be halved, quartered and chopped into three equal portions. For dealing with larger amounts, the score which was originally a ‘score’ or notch a farmer made on a piece of wood as twenty animals passed through a gate, is about all you need so long as you know at least  twenty number words off by heart. The publisher of the red book travel guides to Europe is said to have counted the steps leading up to Milan Cathedral by transferring a pebble from one pocket to the other each time he mounted twenty steps.

A brief list of ‘significant quantities’ on a world-wide scale with reasons for their significance would perhaps be:

5                         hand, fingers,  right size perceptually

6                         half of dozen

10                        both hands, right size for base

12                        right size for base, many factors

 20                        hands and feet, multiple of 5 and 10

60                        many factors, multiple of all previous

100                    square of 10, many factors

Some of the above numbers are significant perceptually, notably 5 since this is around the stage when we cease to be able to assess objects numerically without counting them individually. Thus 5 combines significance because of its use in one-one correspondences (by way of the fingers) with significance as a perceptual ‘unit’. But it has the serious drawback that it cannot be divided up at all (has no factors). It is thus significant but not convenient as a ‘secondary amount’.
Having many factors is really more a matter of convenience than  significance as such but since previous significant amounts are amongst the factors of 60 and 100 these numbers acquire significance acquire significance by proxy. 60 is particularly rich in factors  and has the remarkable property of being a multiple of all previous significant amounts.
100 means nothing to us perceptually though it undoubtedly did to a Roman centurion who would have had in his mind’s eye the terrain covered by his infantry when lined up ready to give battle. 100 is around the ‘acquaintance’ mark, i.e. near the maximum number of persons one is able to relate to personally ¾ I believe I have read somewhere (Desmond Morris?) that 128 is about the limit and that this generally corresponds to the maximum number of persons one has in one’s address book.
But of course 60 and 100 are above all significant because of their divisibility ¾ the main use of 100 is in percentages though it still has the defect that one cannot divide it into three properly.  It must be stressed that a number’s divisibility is not just a matter of interest to modern number theorists : wholesalers or state suppliers receive commodities in bulk which they must subsequently sell or distribute to individuals and it is important that standard quantities should be easy to divide up. This is the reason why so many of the old Imperial measures are built around 20 or the powers of 2 ¾ as it is one of the main reasons why there is such hostility to metric weights and measures.
   One of the troubles with the transition points of a base-system, the ‘powers’ of the base, is that they are significant and convenient not in a practical but purely mathematical sense. Technically speaking, the unit, the base and its powers are the successive terms of a geometric sequence  1, b, b², b³, b⁴ ……. with common ratio b. My choice of twelve for the bundles that are to go into the second alleyway means setting b at 12. We would thus have 1, 12, 144, 1728… (since 144 = 12², 1728 = 12³). Now 144 and 1728 apart from being too large are not meaningful amounts in our day to day experience. The same goes for smaller choices of base.  5 is perhaps the most ‘significant amount’ of all in real-life terms but 5² = 25 is nothing special and 5³ = 5 ´ 5 ´ 5 = 125  even less. 6 certainly has some valid claims on our attention as a significant quantity but 36 has none.
One suspects that the success of a hundred as a ‘significant amount’ is due to its being a multiple of the significant amounts 5 and 20 ¾ it is actually 5 ´ 20 ¾ rather than it being the square of ten. A thousand is just a word meaning ‘large quantity’ and the ambiguous meaning of billion (a million millions or a thousand millions?) shows how vague such large quantities are. A million only has meaning with respect to wealth — and even this sense has been eroded by devaluation so that we find it necessary to replace the word millionaire by multi-millionaire which is even vaguer.

 Non-base extendable systems

Most people assume that once you have defined your ‘secondary unit’ you are somehow obliged to turn it into a true base (and I tended to think along these lines myself before writing this book). But of course you aren’t. The ‘tertiary unit’ or next standard amount we choose to define can be anything at all in principle. One of the few numerically advanced peoples that still used object numbers, the Yoruba,  took a pile of twenty cowrie shells as their ‘secondary unit’. They then combined five such piles of cowrie shells to make 100 in our reckoning, and combined two such piles to define their second most important amount after the unit, 200 in modern numbers. If they had operated a true base system the next halting point would have been 20² or 400 which presumably they considered too large. (Algebraically the Yoruba sequence goes 1, b, 10b and not 1, b, b² ). For a somewhat different, but nonetheless pragmatic reason, the Mayans, who also took 20 as their ‘secondary unit’, then moved on to 360 (instead of 400) for the next transition point in order to get close to the number of days in a year — or so it has been conjectured.

It does not in practice matter too much for addition and subtraction if the transition points are not in proper sequence (‘proper’ as we see it today) though it is desirable that they should be multiples of the first ‘significant amount’. Thus, anticipating a further fixed amount in my stick system I have already decided to opt for 60 since it is a multiple of 12 while 20 or 100 are not. Odd though it sounds, there is much to be said for defining a large ‘secondary unit’ and then defining  ‘units’ rather smaller instead of larger than it. This is in effect what the Babylonians did by taking 60 as principal amount after the unit which they noted as    . Such an enormous secondary unit makes it absolutely essential to have one or more halting points, or sub-bases,  in the intermediary territory which the Babylonians provided at the five and ten points. They defined the five transition by grouping the one-symbols and introduced a special mark for ten      but otherwise they used only the ‘one-symbol’ right up to 60 itself. For quantities > 60  the Babylonians proceeded by using 60 as a  true base, i.e. the next halt was at 60² = 3,600. In their case they seemed to have no misgivings about using such a huge  amount as tertiary unit which seems to contradict what I said earlier. But the Babylonian scribes who developed and used the sexagesimal number system were not hunters or herdsmen but officials helping to run a vast empire. They needed large numbers and spent their lives dealing in them as did the Egyptian scribes.

Practically speaking we require very different fixed standard amounts depending on the context. To divide a pound weight into sixty ounces would appear slightly crazy but we find it most convenient to divide up a fairly short interval of time, the hour, into sixty minutes, while we divide up a somewhat larger interval, the day, much less finely. The numbers 60, 12 and 24 are not imposed on us in the way the number of days in the year is : we could divide up the day into 60 or 72  or any (even) number of ‘hours’ and divide each ‘hour’ into 12 or for that matter 17 ‘minutes’. The unsystematic way in which we divide up the day seems right : there is, as far as I know, no SI project to decimalise time (though the ancient Egyptians did just this) and the very idea fills me with horror.

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Note 1

“Yet it was the Indians who reckoned the age of the Earth as 4.3 billion years, when even in the 19th century many scientists were convinced it was at most 100,000 years old ( the current estimate being 4.6 billion).” The apparent source for this is: Pingree, David, Astronomy in India, in Astronomy Before the Telescope, p.123-42.  Quoted Chasing the Sun by Richard Cohen, p. 132

¹  I recently came across an interesting example of how restricting the idea of always keeping to a base is. I noticed, or had read somewhere, that the Binomial Coefficients were powers of 11 and this made sense since they can be defined by starting with 1 and getting the next term by shifting what you’ve got across one column and adding. Thus

          1                             1  =   11⁰

                                      1        0 

                                      1        1                            11  =  11¹

                             1        1        0  

                             1        2        1                           121  =  11²

                   1        2        1        0

                   1        3        3        1                         1331  =  11³

          1        3        3        1        0

          1        4        6        4        1                       14641  =  11⁴

         

 However, what happens now?  The next line of Pascal’s Triangle is supposed to be

                   1        5        10      10      5        1 

 This isn’t a power of 11 surely.  But who says you can’t overstep the base if you want to?

 (1 x 10⁵) + (5 x 10⁴) +  (10  x 10³) +  (10 x 10²) +  (5 x 10) +  1 

  =   100,000 + 50,000 + 10,000 + 1,000 + 51   = 161,051   = 11⁵

   

Classifiers      In other cultures different bases were used depending on the different objects being counted. Flat objects like cloths were counted by the Aztecs in twenties, while round objects like oranges were counted in tens.  The use of classifiers obviously marks an intermediary stage between the era when numbers were completely tied to objects and the era when they became contextless as now. We still retain words like ‘twin’ and ‘duet’ to emphasize special cases of ‘twoness’, note also ‘sextet’, ‘octet’ &c.     The complete dissociation of verbal and written numerals from shape and substance is today universally seen as ‘a good thing’ especially by mathematicians. But classifiers were doubtless once extremely useful because they emphasized what people at the time felt to be important about certain everyday objects and activities, and they remain both a picturesque reminder of the origins of mathematics in the world of objects and our sense-perceptions. The removal of all such features from mathematics proper seems to be a necessary evil but at least let us recognize that it is in part an evil : the banning of contextual meaning from mathematics, the language of science and administration, is typical of the depoeticization of the modern world.

Base sixty
It is not known why the Babylonians chose 60 as their most significant amount after the unit. The fact that 60² = 360  is close to the number of days in the year may have something to do with it. Certainly, 60 would not have been the original choice. It has been suggested that 60 evolved as a compromise solution to the separate claims of 5 and 12  which were already well established as ‘secondary units’ within the territories conquered by the Babylonians. Since 60 has as factors all the main bases and significant amounts smaller than it, including 10 and 20, it had something for everyone : it was a numerical Pax Romana.