It’s all over at five
Humans can estimate quantities of up to four objects accurately. When there are more than four, they have to start counting. (Foto: Uwe-Jens Kahl/pixelio.de)
Humans can estimate quantities of up to four objects accurately. When there are more than four, they have to start counting. This innate limit is also reflected in the counting systems of ancient civilizations.
“One – two – three – four – many!” Anyone with little children knows that people can only determine a limited number of objects if they are unable to count. Since 1871, it has also been proven scientifically that man can estimate a quantity of no more than four objects accurately every time. It was the English economist William Stanley Jevons who conducted the corresponding experiments and published the results in the science journal Nature.
For a test on himself, Jevons used the following setup: he tossed black beans into a white box and immediately closed his eyes to prevent himself from counting them normally. After more than 1000 such experiments, he concluded that only in the case of one to four beans was he able to estimate their number accurately. When there were five beans or more, the estimates became increasingly erroneous, meaning that from five upwards the exact number of beans could only be determined by counting them normally.
“This human limitation influenced the counting systems of ancient advanced civilizations, even way back in time. It played a key role in the development of new symbols for numbers beyond 4,” says now Professor Hans Gross, Chairman Emeritus of the Department of Biochemistry at the University of Würzburg. Under the heading “Give me 5 …” Gross presents his findings in the latest edition of the journal Communicative & Integrative Biology.
Ancient advanced civilizations: discontinuity between four and five
The fact is that in many ancient advanced civilizations there is noticeable discontinuity in the transition from the number 4 to the number 5 when written down. In the earliest days of Ancient Rome, for example, 1 to 5 were written like this: I, II, III, IIII, V. It was only later, during the classical period, that the IIII changed to a IV (5 minus 1). In Ancient South Arabia, people wrote I, II, III, IIII, U. The Maya in Central America saw the numbers from 1 to 5 like so:*, **, ***, ****, I. Early Chinese wrote I, II, IIII, IIII, and X.
“In these advanced civilizations with a well-developed calendar and accounting system people consciously or subconsciously felt or understood that up to 4 objects could be identified correctly every time without counting them and that from as few as five dots or strokes counting was required. Hence, they gave the number 5 its own new symbol,” says Hans Gross.
However, it is not just advanced civilizations that used this technique: “Even the Vikings, who spent very little time on astronomy or accounting, wrote numbers in their runic calendars in a similar manner,” says Gross. There, one to four dots corresponded to the respective numbers, while a > was used to symbolize the number 5.
Regular patterns make identification easier
But what is the situation with dice players, for example? After all, they are also able to identify at a glance whether they have thrown a 5 or a 6. “Another effect comes into play here: pattern recognition,” says Gross. Unlike in the box of beans, where the objects arrange themselves in a random pattern each time, the dots on the dice are always in the same place. They form a regular pattern that tells any player how many dots there are without him having to go to the trouble of counting them first.
Ancient Egyptians chose a similar path for their written counting system. “They did not introduce a new symbol for 5 onwards. Instead, they arranged the strokes after 4 into certain patterns,” says Gross. Three strokes above and two below, for example, stood for 5. Three blocks of three strokes each gave 9. It was not until 10 that a new symbol was introduced: an inverted U. “It is clear that the Egyptians also recognized that they had to find a way of presenting numbers such that they could be identified at a glance without having to count them,” explains the scientist.
Arabic numbers enable progress
It was not until many centuries later that the problems with 5 and higher numbers disappeared – with the invention of zero in India in the 8th century and the introduction of the Arabic numerals we still use to this day between the 13th and the 15th centuries. Both are developments that “enabled tremendous growth in trade and science,” as Hans Gross says.
And yet, it is still possible to find remnants of the ancient approach to counting. When people count objects using tally sheets, they draw a stroke for each number up to 4 (I, II, III, IIII). But instead of writing IIIII for 5, they simply draw a diagonal line through the IIII, thereby creating a new numeral that saves them having to count off five strokes.
“Give me 5 … The invention of number five in ancient civilizations”, Hans J. Gross, Communicative & Integrative Biology 4:1, 62-63, doi: 10.4161/cib.4.1.13762
Contact: Prof. Dr. Hans J. Gross, T: +49 (0)931 31-84027, e-mail: 
hj.gross@biozentrum.uni-wuerzburg.de
18.03.2011, 11:34 Uhr