Hominids began to walk and run, see, observe and plan. To plan there needs to be predictability, and that needs some sense of remembering pattern even before anything close to number symbolism. There is an environment of organic space and time (Cassirer, 1953, in Churchill, 1961, 23). Before numbers humankind would have recognised the regularity and predictability of natural phenomena. The first response would have been bodily: to work and eat in light and sleep in the dark (Churchill, 1961, 12). They would have seen what was rising and dying through the year. Animals would appear, and could be followed; walking to food and water took time and planning was important there.

Patterns need naming like numbers of animals born, charting visible stars, knowing about the numbers of berries and other food, and assessing the family and tribal group. Yet, even at the start of civilisation located in farming, cluster numbers would have been enough, and these are not unit numbers as such (Churchill, 1961, 13).

A lot of this archaeology is imaginative. Perhaps the first numbers were using stones either in positions or marked, and trees could have been notched (Churchill, 1961, 13).

This sort of numbering is reckoning. This means finding measures for things and facilitating predictability. A progression is following, predicting and using the waxing and waning of the moon as a marker of time. 4236 BCE is the first historical record of marking days into thirties according to the movements of the moon, and attaching the moon to the calendar function, producing blocks of 360 days (Churchill, 1961, 12). This is why 60 and 360 are important numbers in the earliest number systems. They directly suggest that numbers began in order to assist following the calendar (Churchill, 1961, 13). Numbering in thirties would have generated a need for a symbol system to supersede a basic visual system (Churchill, 1961, 13). This is how reckoning becomes numbering, where words replace or add to reliance on pictures. Nevertheless people are always more sophisticated than the limited evidence may suggest, though some evidence suggests a complex ability to observe the stars and to build constructions using the sun and stars.

However, early reckoning was object specific. Universal numbering was not understood, and so translations of numbering activities were made. Thought processes were descriptive of the objects rather than getting into an understanding of magnitude as a universal measure (Churchill, 1961, 14); collective names were therefore qualitative as well as quantitative. Therefore early languages did not have universal numbers, but applied numbering for different classes of objects (Churchill, 1961, 14). This was persistent even when universal numbering became better understood.

Some primitive societies still differ their names for the same numbering according to application. The Moanu Islanders number one to ten differently for spirits, people, animals, trees, canoes, houses, villages, coconuts and plantations. The Tsimshian language of British Columbia has numerals for flat objects, round objects, time intervals, animals, people, boats, long objects and measurements.

As well as objects a such, early numbering advanced to become descriptive of classifying objects. The equivalent is between a crowd of people and sheep in a flock. Again primitive societies indicate this. Malayo-Polynesian languages work around classifications. Numbering into tails is applied, say, to horses as these have tails. Numbering into round bodies is applied, say, to stones as these are rounded (from Cassirer, 1953, in Churchill, 1961, 15).

An advancement still is model groupings with size constancy against which other groups can be compared (Churchill, 1961, 16). It also suggests the importance of picture symbols in setting up these primary reference points. So there are two wings, the three leafed clover, four legged animals and five fingers (Churchill, 1961, 16). Other objects are compared with these and so there is some classification by number, although again comparison is by objects and not universal numbering. These are visual object comparisons of natural groupings and despite advancement these may not in themselves have developed abstracted numbers (Wertheimer, 1938, quoted in Churchill, 1961, 16-17).

People settled and there was a need to divide large clusters into stages (Churchill, 1961, 17). More names were needed for quantities, but quantities had to be split and joined to other quantities, and across types. The key impact was trade, exchanging quantities of one thing for something completely different, which needs both a measure of counting and a measure of value. The problem became how to count (reckon) when there is no logical object connection. There was a dynamic towards a homogeneous system that bridged the increasingly unconnected or little connected plurality of things (Dantzig, 1947, quoted in Churchill, 1961, 17). There was also the need to count well beyond the small quantities of every day hunter gatherer experience.

The movement to counting and abstraction was arguably dangerous because early societies, even those setting up towns, had to remain in touch with their object environment. An important need in moving on was to retain attachment to the body and sensory experience, and this is why finger counting and reference to the body, or to what was felt, understood and close, remained important as the numerical process advanced. So as counting became abstracted, people needed object reassurance (see Churchill, 1961, 18).

Starting with reckoning as before, use of fingers and marks is useless when coming to the larger numbers that trade inevitably involves. Even then numbering was limited to symbolic naming transfers regarding objects. Malay and Aztec count in stones, the Niuès of the Southern Pacific count in fruits and the Javans count in grains. Zulus say taking a thumb for six and say he pointed for seven. (Smith, 1923, quoted in Churchill, 1961, 18) Sotha say complete the hand for five and jump is six (Churchill, 1961, 19). The British use the foot that became a foot in length and the inch comes from the French pouce which came from the Latin pollex and originated as a thumb unit (Churchill, 1961, 18-19). These object symbolic transfers became universalised.

The need for strict ordering was being established prior to any abstracted numerical system. This ordering at first only suggested going beyond the qualities of the objects themselves (see Churchill, 1961, 19). There is however a crucial movement from the object to an act around the object (Cassirer, 1953, quoted in Churchill, 1961, 20). That act became universal and separated: homogeneous to cover heterogeneity. This means building convention. The object world was still a comfort and attachment, but abstraction was taking place in imposing order. Even today people build for themselves comforting rituals of how to get up from bed to breakfast and to go out, or the method to go to sleep. These attachments and memory devices for efficient method are the same means to universal numerical detachment.

So the history of numbering is awareness and dependence on the environment, then a beginning to record, then an ability to match with collective quantities and eventually categories. Numbering was object specific, but was forced to universalise when trade meant an exchange of the otherwise little or unconnected. To count means to order, and this came from the act around and then slowly abstracted from the object world, whilst keeping reference points to our object environment and to behaviours.

What this abstraction facilitates is theory. Once the concrete attachments are removed from the task and method (other than for historical reference points - something about who we are as a people), then the numbers have the potential for mathematics. There is still language and depths of meaning, but the mathematical symbols become in a sense their own language with their own forms of task. The symbols must be agreed, and may have deep biological, social and cultural origins, but the task is now not those reference points but systematic handling of number.

One crucial area of mathematics is the use of powers in the radix or base of counting. Queensland natives counted "one, two, two and one, two twos, much. Other numbers around three, four and five were likely. Twelve, known as the duodecimal scale, had the advantage of divisibility by two, three and four. (Smith, 1923, quoted and used in Churchill, 1961, 24-25) The crucial influence was the moon, however, because the moon showed some regularity and cycle within the annual cycle of the sun.

The Maya System was a moon system of 360 days with a complex uneven base system of 1, 20, 360, 7200. It was probably not used for counting due to its inconsistency but still shows the need in symbolism for an order of magnitude indicated by place. The use of pebble and rod in the symbolism suggests an abacus could have been used for dating. (Churchill, 1961, 25-26)

The Sumerians and Egyptians did use numbers to compute from around 3500 BCE. Both of these used an inconsistent base of ten in the symbolisms. (Churchill, 1961, 27)

Sumerians had a form of yearly calendar as early as 4700 BCE. By the 1900s BCE they had the monetary elements of trade and business (bills, receipts, accounts and measurement) (Churchill, 1961, 27). The Babylonians counted in units from one to ten, in tens to sixty, and then 60 by 10, and then 600 by 6. Their symbols were a rotated numbers of a wedge-shaped reed on damp clay where 60 looks like 1 and 60 is that and a 10 symbol and four circle at 3600 and one goes within for 36000. A development of the reed symbol also accounted for minus 1. (Churchill, 1961, 28)

Inconsistency of symbolism is most obvious and least of a problem where counting is little used, as in the Maya system, but awkwardness did not stop increasingly high levels of computation as with the Babylonians using multiplication and division, squares and square roots, geometric progressions and unit fractions. (Churchill, 1961, 29)

Before 2000 BCE the Egyptians turned maths into engineering with the building (and astronomy) of the Great Pyramid and extensive irrigation. The mathematical treatise the Rhind papyrus was written in this era with its resolving fractions, multiplying and dividing fractions, working out volume and cubic measures, division of areas and land, working out angles of slope, and other mathematics. This was despite symbolic clumsiness and the method of recording. (Churchill, 1961, 29) They had hieroglyphic (picture based and each highly specific and variable) numerals on stone from Nile Valley quarries and hieratic numerals (variable symbols) on papyrus (a paper from strips of water reed pulp), wood and pottery. Demotic numerals were used in the Rhind papyrus. Egyptians repeated established symbols to achieve specific numbers. (Churchill, 1961, 30-31)

Much later Greeks and Romans used a base of ten with special symbols. (Churchill, 1961, 31)

These systems then used the principle of repetition (Churchill, 1961, 31) with higher value symbols to reduce length. This is not the same as a system of position. Integers ought to be varied in one position but the same choice of them in different base positions. Their value depends on the integer rising and the position of magnitude. There is a need for one less than the base value for the number of variable symbols, and then something else. The key to this is the zero, which stands in position but has no number value. Its mathematical quality is positional only, and facilitates the whole system. So those with a base ten have nine number symbols and one zero symbol.

Maya and the Sumerians had a zero symbol for no number but they did not use it in a way leading to arithmetic and measurement (Churchill, 1961, 33-34). The zero was part of the Hindu-Arabic system of numerals in the 500s CE (Churchill, 1961, 33-34). So the origin of the modern positional number system with a zero we know is Hindu, brought to the West by Arab copyists and developers while Europe was in the Dark Ages (but Europe read about it).

The Hindus' system works because it is visual. By position is seen value. The symbols themselves do not need counting along. The Hindus called zero sunyabindi, or sunya for short, meaning empty or void and is directly linked to the symbolism of the trimurti godhead. Arabs called this sifr or as-sifr (from which cipher comes) (Churchill, 1961, 37). Zephirum followed as-sifr and zero is found in Calandri's Arithmetic in 1491 (Churchill, 1961, 37-39). The zero's route to Europe was probably via trading but by the 1100s the Hindu system was in European books and indeed likely seen as superior (Churchill, 1961, 39). However, mathematicians remained divided when the systems competed by the end of the 1500s. There were the abacists of old and the new style algorists of the Hindu-Arabic system. Resistance was probably because of the claimed superiority of Christianity and Christendom. (Churchill, 1961, 39)

Also zero was a dangerous number: the idea that there can be a void. It was dangerous in India; it is dangerous being linked with the symbol of the trimurti (godhead). More practically, numerals were not often written down (no easily-available paper and pencils) and the contemporary methods of multiplying and dividing were not yet invented (Churchill, 1961, 39). The superior system did however win through, and England adopted the system of tens in the 1600s as indeed the Hindu-Arabic system was common in Europe (Churchill, 1961, 34).

From this was facilitated science and the ages of European civilisation no less. Maths became very powerful, and so were statistics. Often it seems anything can be understood using statistics. If matters of human understanding want to seem scientific, they attempt to use mathematical formulae. However, those who are seduced by number should realise that for a long time the human and environmental use of reckoning linked it to sense experience and the natural world and it was a long time abstracting. Human behaviour rarely is purely rational, never mind regularly numerical, and is surely qualitative rather than quantitative in its deeper meanings. (See the discussion in Churchill, 1961, 41-42)

New knowledge needs mathematics, yet at extremely large and notably extremely small quantities cease to equate with experimentation - or the rules of mathematics begin to warp. Computers with huge power reach into these challenging areas of maths where the decimal number system reverts to powers both positive and negative. Predictability becomes probability. Virtual number iterations produce patterns that are, or are not quite, symmetrical, but reproduce themselves at both ever increasing and ever reducing levels, and seem to equate with patterns in the natural world. There may be a fractal reality at levels of the world we encounter as real.

Churchill, E. M. (1961), Counting and Measuring: An Approach to Number Education in the Infant School, London: Routledge and Kegan Paul, 11-44.

Refers to:

Cassirer, E. (1953), The Philosophy of Symbolic Forms, vol. 1, Yale University Press.

Dantzig, T. (1947), Number, the Language of Science, third edition, Allen and Unwin.

Smith, D. E. (1923), The History of Mathematics, vol. 1, Ginn.

Wertheimer, M. (1938), 'Numbers and Number Concepts in Primitive Peoples' in Ellis (ed.) (1938), Sourcebook of Gestalt Psychology, Hartcourt Brace.

Adrian Worsfold