The history of numbers and counting is a fascinating one. 0 Read on to find out more. 1 Story 4 9 Cover 8 2 3 6 Number Sense It is not as if numbers have been there all though the human history. Munduruku, ALK in to a supermarket in China, indigenous hunter-gatherers in the Brazilian Monkey even though the hoardings would Amazon, have no words for numbers Counting? Wbe unintelligible, the numbers on beyond five in their language while Piraha, the currency and coins and the telephone another tribal community, do not have any numbers would be easily readable. Take name for numbers beyond three. But, they a stroll in the marketplace at Barcelona, have a sense of numbers – they can the market babble wafting around you discriminate between a heap having more might be incomprehensible, but the price than the other even though they may not tags are easy to read. Walk into any count one by one. remote region in Africa and you find that the written numbers – numerals – make sense, even though we cannot make head or tail of their scripts and languages. The world over, counting styles are different, languages are varied, scripts are distinct, but numeral symbols 0,1,2,3,4,5,6,7,8,9 are easily recognizable everywhere. The Indo-Arabic numerals have hegemonized the world. It is as if the world speaks in various languages and writes in a number of scripts, but works in only one kind of number symbols. Ubiquity of Indo-Arabic numerals is more than obvious. People of the Munduruku tribe have no words for numbers beyond five SCIENCE REPORTER, APRIL 2012 8 Cover Story T.V. VENKATESWARAN 5 UbiquitousUbiquitous 7 IndoIndo ArabicArabic NumeralsNumerals Not just humans, it appears that thought he could deceive the crow, but ever since the case of “Clever Hans”, a animals also have, albeit primitive, a the crow did not fall into this trap and horse that is said to have responded to number sense. An eighteenth century carefully waited for the second man to questions requiring mathematical anecdote aptly summarizes the number come out before returning. Neither did calculations by tapping his hoof, but later sense observed in the animal world. A three, four, or five men fool the clever bird. revealed to be a case of subtle nobleman wanted to shoot down a crow Each time, the crow would wait until all manipulation by the handler. As Ray Hyman that had built its nest atop a tower on his the hunters had departed. Eventually, the puts it, “Hans was responding to a simple, domain. However, whenever he hunters came as a party of six. When five involuntary postural adjustment by the approached the tower, the bird flew out of them had left the tower, the bird, not so questioner, which was his cue to start of gun range, and waited until the man numerate after all, confidently came tapping, and an unconscious, almost departed. As soon as he left, it returned to back, and was shot down by the sixth imperceptible head movement, which its nest. The man decided to ask a hunter! was his cue to stop.” neighbour for help. The two hunters Scientists have been sceptical of Nevertheless, subsequent entered the tower together and later only claims of mathematical abilities in animals experiments have revealed an one of them came out. The nobleman Scientists have been sceptical of claims of mathematical abilities in animals ever since the case Bone with of “Clever Hans”, a tally marks horse that is said to found in have responded to Ishango, a questions requiring village in Africa mathematical calculations by tapping his hoof. 9 SCIENCE REPORTER, APRIL 2012 Cover Story LEFT HAND 1 2 3 4 5 RIGHT HAND 15 10 25 20 5 unexpected capacity for ‘quantity discrimination’ in animals as varied as bees, salamanders, rats, dolphins and primates, suggesting that mathematical abilities could be more fundamental in biology than previously thought. Trained Counting on fingers monkeys can, it seems, perform rudimentary maths – they can compare Babylonian counting two heaps of dots and identify which one long notch of double length separated the has more. first twenty-five marks from the rest. It was Number sense, though not strictly in clear that the carving and grouping were the sense of counting, could be important deliberate human action of counting for animals to survive in the wild. If a something; perhaps the number of spears chimpanzee is unable to look up a tree they had or the mammoths that they had and quantify the amount of ripe fruit most hunted in the season, rather than mere likely it would go hungry. If a lion wants to scribble. attack another pack of lions it has to make This is not an isolated case. Soon tally a judgment how many are in each side. If marks such as Lebombo bone, a piece of a grazer is unable to judge the relative baboon fibula, found in the caves of abundance of food in two patches, Lebombo Mountains in Swaziland, dated perhaps it would die of hunger. to be 35,000 years old, with 29 notches Evolutionary psychologists posit that such were found. A similar bone with 29 notches, evolutionary pressures, perhaps, have led and about 18000 years old, has been to evolution of sophisticated procedures discovered in Nicobar in India. for number sense in animals. Not Archaeologists interpret it as the counting surprisingly, primates and human toddlers of the cycle of waxing and waning of the too exhibit this primitive number sense. moon or perhaps menstrual period. At a fundamental level, these Birth of Counting archaeological artefacts attest that early When Karl Absolon, a Czech archaeologist, humans were counting rather than merely discovered a wolf-bone in the dust and making an estimate and the pattern debris of a 30,000-year-old Stone Age recognition had evolved into counting. settlement, it became an instant sensation, Further, whatever that they were counting The Mayan so much so that the London Illustrated was important enough for them to warrant Not all such tally sticks excavated so system of News, a weekly tabloid printed in London, the keeping of records. Every notch, far had just a few dreary lines carved on counting carried its picture in its 2 October 1937 called tally mark, precisely represented them. Ishango bone, a tiny 10 cm curved issue. This bone fragment discovered at something that they counted, implying a bone excavated at the Semliki River in Zaire Dolni Vestonice, a Palaeolithic site, is touted rudimentary version of an important dated to about 20,000 years old, is a to be one of the earliest evidences that mathematical cognition – the one-to-one significant find. The three rows of notched Stone Age mammoth hunters counted correspondence between elements of columns engraved on it are clearly something – said to be the dawn of two different sets of objects, or cardinality patterned and two of them add up to 60. mathematics. of numbers, in this case between the set Also one of these two rows is grouped in The bone had fifty-five little ‘tally’ of notches on the bone and the set of to 11,13,17,19 – all prime numbers notches carefully carved and they were whatever the prehistoric humans were between ten and twenty. The third row has arranged into groups of five. An additional counting. a particular pattern: 3 followed by its SCIENCE REPORTER, APRIL 2012 10 CoverCoverCoverCover Story Story Counting tokens in stone as “five ten seven”. On the other hand, and X. Ten also has a unique symbol. In many European systems of number words like manner, fifty is L and hundred is C, five are irregular up to 100. For example in hundred is D and thousand is M. French, 92 is said as “four twenty twelve,” Of particular interest is the counting corresponding to 4*20 + 12. system of Siriona Indians of Bolivia and the But quinary counting is not universal, Brazilian Yanoama. Their counting goes like nor counting upon ten fingers. “one,” “two,” “two and one,” “two and With the scientific Comparative studies have shown that two,” “two and two and one,” and so forth, revolution setting in counting of fingers, excluding the thumb, very similar to what mathematicians would is also an established practice among call binary counting. Similarly, in Australian Europe, fetters fell and some North American aboriginal tribes and Aboriginal language, Kala Lagaw Ya, the soon Indo-Arabic numerals thus numbers are grouped into fours numbers one through six are urapon, instead of five – that is, quaternary counting. ukasar, ukasar-urapon, ukasar-ukasar, became the only form to be Archaeologists contend that one of the ukasar-ukasar-urapon, ukasar-ukasar- used. ancient Indian numerals, Gandharan ukasar. But why two? We are yet to fathom. Kharosthi numerals, are partially Hundreds and thousands of quaternary – base 4 – counting. cuneiform records have been excavated The Ndom language of Papua New from the ancient Sumer region, presently double, then 4, followed by its double, then Guinea is reported to be senary – that is in Iraq. In these written records preserved 10, followed by its half, and indicates more base 6. The Yuki Tribes in Americas count in clay, there are symbols only for 1, 10, 60 sophisticated arithmetical reasoning. the space between the fingers and have and 3600, suggesting that the ancient evolved an octal-base 8-number system. Sumerians used sexagesimal – base sixty – Variety of Counting Systems Consider the four fingers of your left hand: number system. But why 60, and not 10 The bunching of lines into fives by the Dolni ignoring the thumb.