Zero The Biography of a Dangerous Idea - Charles Seife Page 0,1
cardinals, and its use is only forced on us by the needs of cultivated modes of thought.
—ALFRED NORTH WHITEHEAD
It’s difficult for a modern person to imagine a life without zero, just as it’s hard to imagine life without the number seven or the number 31. However, there was a time where there was no zero—just as there was no seven and 31. It was before the beginning of history, so paleontologists have had to piece together the tale of the birth of mathematics from bits of stone and bone. From these fragments, researchers discovered that Stone Age mathematicians were a bit more rugged than modern ones. Instead of blackboards, they used wolves.
A key clue to the nature of Stone Age mathematics was unearthed in the late 1930s when archaeologist Karl Absolom, sifting through Czechoslovakian dirt, uncovered a 30,000-year-old wolf bone with a series of notches carved into it. Nobody knows whether Gog the caveman had used the bone to count the deer he killed, the paintings he drew, or the days he had gone without a bath, but it is pretty clear that early humans were counting something.
A wolf bone was the Stone Age equivalent of a supercomputer. Gog’s ancestors couldn’t even count up to two, and they certainly did not need zero. In the very beginning of mathematics, it seems that people could only distinguish between one and many. A caveman owned one spearhead or many spear-heads; he had eaten one crushed lizard or many crushed lizards. There was no way to express any quantities other than one and many. Over time, primitive languages evolved to distinguish between one, two, and many, and eventually one, two, three, many, but didn’t have terms for higher numbers. Some languages still have this shortcoming. The Siriona Indians of Bolivia and the Brazilian Yanoama people don’t have words for anything larger than three; instead, these two tribes use the words for “many” or “much.”
Thanks to the very nature of numbers—they can be added together to create new ones—the number system didn’t stop at three. After a while, clever tribesmen began to string number-words in a row to yield more numbers. The languages currently used by the Bacairi and the Bororo peoples of Brazil show this process in action; they have number systems that go “one,” “two,” “two and one,” “two and two,” “two and two and one,” and so forth. These people count by twos. Mathematicians call this a binary system.
Few people count by twos like the Bacairi and Bororo. The old wolf bone seems to be more typical of ancient counting systems. Gog’s wolf bone had 55 little notches in it, arranged into groups of five; there was a second notch after the first 25 marks. It looks suspiciously as if Gog was counting by fives, and then tallied groups in bunches of five. This makes a lot of sense. It is a lot faster to tally the number of marks in groups than it is to count them one by one. Modern mathematicians would say that Gog, the wolf carver, used a five-based or quinary counting system.
But why five? Deep down, it’s an arbitrary decision. If Gog put his tallies in groups of four, and counted in groups of four and 16, his number system would have worked just as well, as would groups of six and 36. The groupings don’t affect the number of marks on the bone; they only affect the way that Gog tallies them up in the end—and he will always get the same answer no matter how he counts them. However, Gog preferred to count in groups of five rather than four, and people all over the world shared Gog’s preference. It was an accident of nature that gave humans five fingers on each hand, and because of this accident, five seemed to be a favorite base system across many cultures. The early Greeks, for instance, used the word “fiving” to describe the process of tallying.
Even in the South American binary counting schemes, linguists see the beginnings of a quinary system. A different phrase in Bororo for “two and two and one” is “this is my hand all together.” Apparently, ancient peoples liked to count with their body parts, and five (a hand), ten (both hands), and twenty (both hands and both feet) were the favorites. In English, eleven and twelve seem to be derived from “one over [ten]” and “two over [ten],” while thirteen, fourteen, fifteen, and so on are contractions of “three and ten,”