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November 23, 2017

Mathematics seems to be not only an exclusively human activity but also a very recent one (in evolutionary terms). Animals, with one or two possible exceptions (bees, whales), do not use symbolic systems for communicating information or as behavioural aids. This is not to imply that our way of doing things is necessarily preferable or even ‘more advanced’: it would be hard to better the extraordinary feats of migrating birds returning to the selfsame nesting spots year after year or the accuracy of predators pursuing rapidly moving prey. Quite how animals achieve such feats is still not completely understood, at any rate with respect to migratory birds, but, certainly, they do not consult ephemerides or solve differential equations. Much the same goes for early societies: they did not develop complex numerical systems because, most of the time,  they did not need them. Why did they not need them? For two reasons, firstly because the quantities involved, e.g. number of personal possessions or number of members of the tribe, were small and, secondly, because the sensory apparatus of early humans was extremely acute, more acute than ours. Being able to review objects in their precise locations often does away with the need to count them. Do you know how many chairs there are in your house or flat? How many rooms even? You don’t need to know the numbers involved because you can mentally review such a familiar landscape and even approach it from different sides, go into it, behind it and so on. Moreover, since in earlier times a good visual and auditory memory was necessary for survival, it was intensely cultivated alongside physical fitness. Predominantly oral societies routinely produced individuals whose memory capacities seem scarcely credible to us today: sacred books as long as the Rig-Veda or the Koran were learned off by heart and reputedly still are in some parts of India and Pakistan. Mathematics itself (as we understand the term) only took off with the advent of large, centralised, bureaucratic societies such as Assyria, Babylon and Egypt. In such cases, a good visual memory was inadequate since the scribe or official would not be personally familiar with what he was supposed to be assessing. It was the necessity to record and process data efficiently that gave rise to mathematics in these imperial societies and, conversely, an advanced numbering system only becomes essential when what you are dealing with exceeds the range of your personal experience. “The main condition under which arithmetical operations  become useful is economic action at a distance and such conditions do not arise for hunters or for the simpler forms of agricultural society” (Denny, Cultural Ecology of Mathematics) (Note 1). Unromantic though it sounds, arithmetic seems to have been developed mainly for the purpose of stocktaking, taxation and large-scale warfare while geometry (literally ‘land-measurement’) was, according to Herodotus, invented by the Egyptians in order to survey accurately (and subsequently tax) the irregular plots of peasants bordering the Nile.

Early arithmetic and numbering generally was concerned (1) with recording what was already known (at least approximately) and (2) finding out and recording what was not known — but which could, hopefully, be extracted from the relevant data. A census carried out in a series of villages would tell a regional official how densely populated the area was,  and such a piece of data needed to be recorded in a form that other officials would be able to comprehend. This is (1), recording what is already known — at any rate locally . If we  want to work out the food supplies necessary to keep all these people alive in a time of famine, or how many young men the region is likely to be able to provide for the army, we have a primitive kind of equation. This is a case of (2), finding out and then recording what is, prior to the census or other data collection, is not known locally. There is, however, no hard and fast line separating (1) and (2) since simply combining the separate data about each village does provide new information, i.e. the total number of inhabitants in the region which, doubtless, no single villager knew.
An efficient number system is necessary both for assessing and recording important quantities and in practice this  means that two systems, or two versions of the same system, are required, a temporary  system and a more permanent system. If quantities are small, we can assess a given quantity (how many pigs? how many coconut trees?) using our hands as the temporary recording system but, since we need our hands for other purposes, we also need a separate, much more durable, recording system which could be clusters of shells (Benin Empire in Nigeria), knots in a string (Inca Empire in Peru) or marks on some long-lasting material such as bone, bark or papyrus (Egypt). Even today, numbers are still primarily used simply for recording data — rather than for pure-mathematical purposes. Coping with numerical data has, in fact, been a perennial problem for advanced societies from ancient Egypt right down to the present day.

The early Egyptian ‘hieroglyphic’ number system is perhaps the clearest and simplest number system ever invented. A single item, a datum, was originally represented by a picture of a papyrus leaf which soon just became a stroke. The Egyptians, like most (but by no means all) societies used a base-ten system, i.e. once you have a given collection of strokes, you make it into a ‘first base’ (our ten), when you have the same quantity of ‘first bases’ you make it a second base (our hundred) and so on. In principle the different bases  could be distinguished by size ― if unity is a stroke, ‘ten’ is a longer stroke, ‘hundred’ a longer stroke still &c. &c.  The inconvenience of such a number system is that it requires a lot of space if you are dealing with large quantities, which the Egyptian officials often were (it is thought that some Egyptian cities at their height had nearly a million inhabitants). Considerations of space have in fact played a very large part in the development of number systems and recording technology generally. The Egyptians did not distinguish the ‘one-symbol’ from the symbol for first base, the symbol for the first base from the second and so on by comparative size: they had separate pictograms for ‘one’, ‘first base’, ‘second base’ and so on. Our ten was a bent leaf, our hundred a coiled rope, our thousand a lotus flower, our ten thousand a snake, our hundred thousand a tadpole or frog and our million a “seated scribe holding up his hands in astonishment”.  In this system you only had to learn the meaning of seven hieroglyphs whichis not a very great task. But with these seven symbols repeated when necessary any quantity less than a ‘million million’ (original meaning of ‘billion’)  could be  represented. “They [the Egyptian officials] could record the number of captives available for slave labour and share them out for public works. They could estimate how much food and drink, how many blocks of stone of different shapes and sizes, how many slaves and overseers would be needed from day to day to build the pyramids” (McLeish, Number).

Note that in the Egyptian system, as opposed to the ‘increasing size’ system which hardly any society ever used, a new single symbol is needed for each larger base; any given symbol is never repeated more than a certain number of times (nine times in a ten-base system). Each new symbol is thus not just a bigger and better version of the basic ‘one-symbol’ but something quite different. Some of the new symbols seem somewhat arbitrary since one sees no obvious connection between a quantity we call a hundred  and a coiled rope for example. On the other hand, the Egyptian symbol for our 100, 000, either a frog or a tadpole, may well have been chosen because frog spawn contains a vast number of eggs, as someone recently suggested to me. Since, even today, our brain finds it much easier to store images of real things rather than abstract signs, the Egyptian system was extremely easy to memorise.
This is not really what we mean by a ‘cyphered’ number system, however, since, in the Egyptian system all quantities less than our ten are still represented by the one-symbol repeated the appropriate number of times. The Greeks took the ‘different symbol’ principle much further by introducing single symbols for all quantities greater than one and less than first base, as we ourselves do. Thus our ‘four‘ is not represented by a plurality of one-symbols such as l l l l   but by a single symbol, δ. We are so used to this principle that we do not realize what a significant departure it really was. The Greeks managed this by making their alphabetic letters double up as numerals and so β, the second letter, became ‘one-one’ or our ‘two’.The 27 letters of the Greek alphabet, which they took over from the Phoenicians,  were divided up into three sets of nine, the first set for quantities up to, and including, our 9, the second set for our 10, 20… 90 and the third for the hundreds. Various artifices such as having a bar above the letter-number enabled one to extend the system beyond 900 but the Greek alphabetic system could not be extended indefinitely in the way that ours can — because the Greeks did not hit on the idea of place value and positional notation. Archimedes himself, the greatest mathematician of the ancient world, felt obliged to write a treatise, The Sand Reckoner, to argue that, in principle at least, all the grains of sand in the world could be numbered ― but even he never hit upon the stratagem of place value.
The great advantage of the Greek alphabetic system was its conciseness. The economy of the Greek, and later the Hindu-Arabic number system was, in its day, as important as the miniaturisation of the components of contemporary computers that has revolutionised the world of communication technology: saving space for the recording of data has been and remains one of the most important of all human concerns.

SH 23/11/17



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