Give me 5...The invention of number five in ancient civilizations

Abstract

Precise recognition of small numbers of objects without counting is an archaic, inborn ability of humans. Since almost 140 years it is known that we can recognize precisely only up to four objects if sequential counting is prevented. Vertebrates and invertebrates such as honeybees can recognize and remember three and up to four objects, respectively. A synopsis of counting systems in ancient civilizations reveals that our limited ability to recognize only four objects without counting influenced our counting and numbering systems and enforced the need for new symbols for numbers beyond four.

This article refers to:

For humans it is long known that we can recognize only up to four objects accurately without sequential counting. In 1871, an English economist reported on his experiments in which he presented a number of black beans in a white box for a tiny fraction of a second so rapidly that counting was impossible. He determined that the maximum number of objects to be correctly identified was “ four” under these conditions.1 Sequential counting, a cultural achievement of mankind,2 and the rapid, inborn, archaic recognition of small object numbers, later named “subitizing”,3 are obviously completely different mental abilities which have nothing to do with each other. There is a wide agreement that the ability of animals to remember about three objects (reviewed in ref. 4) and of humans to recognize four objects without explicit counting is an inborn property which reflects the difference between animals and humans and which can neither be taught nor improved by training. Interestingly, there is some discussion about the numerical abilities of trained grey parrots (Psittacus erithacus) that seem to really “count” up to six items.5

We have recently reported4 that even an invertebrate, the honeybee, with its very small brain of one million neurons, has at least the same numerical ability as the chimpanzee with its ∼20 billion brain cells, thus indicating that the size of the brain is not important for numerical competence.6

The ability of honeybees to identify object numbers up to three and partially four means that object numbers beyond 3 or 4 have the meaning of “many”. Surprisingly, the recognition of object numbers by honeybees is correct even if images of completely different objects are offered, e.g., mixtures of blue dots, yellow lemons, purple flowers and green leaves, respectively.4

What about the biological role of the inborn ability to recognize small numbers of objects without sequential counting? Our own capacity of sequential counting appears to have outdated this ability, however, for certain animals it may be useful to have a feeling for the number of offspring or eggs in their nest, for a chimpanzee to estimate the number of fruits on different branches and for a honeybee to recognize the number of blossoms on a branch or the number of competing foragers on a blossom, thus allowing the decision to join or to quit.4

This notion has an intriguing consequence for our own way of counting:

It is a well established fact that |||| is rapidly identified as “four” without sequential counting, whereas ||||| and |||||| are not so easily realized as “5” and “6”, respectively, since it requires real counting. This ability to recognize object numbers up to “four” without counting is the reason why in case of checklists we draw up to four vertical lines and cross them out in order to complete “five”. We do not draw more than 4 vertical lines (e.g., ||||| to |||||||||), obviously because of our inborn limitation to identify more than four objects without sequential counting. We instinctively avoid the time consuming, laborious counting of “five” or more whenever possible.

A look at the counting systems of advanced ancient civilizations with an interest in astronomy, trade or tax administration reveals a phenomenon which supports the idea that, for numbers higher than four, a new symbol had to be invented in order to avoid sequential counting. The following examples (reviewed in ref. 2) support this notion.

In ancient, pre-classical Roman times, numbers from “one” to “five” were written as |, ||, |||, |||| and “V” as a new symbol, followed by VI, VII for 6 and 7, respectively. Later, in the early classical period, number “four” was depicted as IV (= “five minus one”).

Accordingly, in archaic China, the symbols for “one” to “five” were |, ||, |||, |||| and “X”.

During South Arabian antiquity they were |, ||, |||, |||| and “U”, respectively.

Another example are the Maya in ancient Middle America who had also invented a new symbol for number 5, using dots for “one” to “four” and the symbol “|” for “five”.

Even the northern European Vikings who had only very limited astronomic interests, except in the position of the sun, the moon and the polar star, and who preferred robbing without the trouble of commercial bookkeeping, had numbers for their rune calendars: dots for “one” to “four” and “>” as a new symbol for “five”.

However, there are obviously exceptions, since the dice-player recognizes 5 or even 6 dots immediately without real counting. The reason is that we have here a clear case of pattern recognition. This also indicates that it matters whether 5 or more objects are arranged by chance like beans in the box1 or in a regular arrangement like |||||| (both cases require counting), or form a pattern like 5 and 6 in the game of dice.

Surprisingly, the ancient Egyptians solved the problem of dealing with numbers between 5 and 9 in the following way. They obviously realized the need to avoid counting object numbers of 5 or more. However, they did not introduce a new symbol for 5, but used an alternative presentation as shown in Figure 1: 5 (= 3 + 2) and 7 (= 4 + 3) or 9 (= 3 + 3 + 3). This can easily be identified via “subitizing” and pattern recognition without sequential counting.

In conclusion, it appears that most ancient civilizations invented new symbols for the number “five” or used pattern recognition in order to avoid sequential counting of five or more objects because of our limited inborn numerical competence. All these problems disappeared with the early invention of “zero” in the 8^th century in India and later with the introduction of the Arabic numbers around the 13^th to 15^th century, thus facilitating an enormous, worldwide growth of science and commerce.

Figures and Tables

Figure 1 Counting in ancient Egypt. Symbols like for

for “10” or

for “100” serve the same purpose as symbols for “5,” namely to facilitate counting by a combination of “subitizing” and pattern recognition

Acknowledgements

I thank Profs. H. Beier, H. Hoehn, D. Schindler, J. Tautz, HG. Weigand (Würzburg University, Germany), D. Söll (Yale University, CT) and S. Zhang (Australian National University, Canberra) for advice and for critical reading of this manuscript.

Addendum to: