Origins: One, Two…Three

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It seems to be the most basic human skill. We do it when submitting manuscripts to grouchy editors, shopping at the supermarket, and choosing sides for a ball game. It turns out, however, that this “simple” skill took thousands of years to develop. Just how did people learn to count?

It is a common misconception that we humans instinctively count to ten because we have ten fingers. Not all cultures even use a base-ten numerical system (or a numerical system of any base at all): Some Mesopotamians used a sexagesimal (base-60) system, a form of counting that survives on our watches; the French word for 80, quatre-vingts (or four twenties), may well be a survival from a base-20 system; and modern computers use a base-two system. There is no innate method of counting; one system is as good, and as artificial, as another.

Nor is developing a counting system, of any type, instinctual. Cognitive psychologists have demonstrated that pre-school children do not identify more than three sets: a set of one object, a set of two objects, and a set of three or more objects (also called “many”). Although toddlers do perform tasks involving numbers, such as setting the table, they use one-to-one correspondence: one plate for mommy, one plate for Mary, and so on—which is really just counting to one, over and over again. Human beings, it appears, are born with the ability to recognize only patterns of one thing, two things and many things and to count in one-to-one correspondence.

Additional evidence that counting is a sophisticated development comes from anthropologists and linguists—who report that, at least until some two centuries ago, numerous languages around the world had only two or three number words: generally words for “single,” “pair” and “many.” Of course, the lack of number words did not prevent people from going ably about their lives. When it was important to count, they used one-to-one correspondence. For example, a shepherd would match each of his sheep with a pebble, so that he could later tell whether the flock was still complete. Cultures that do not count beyond three do not develop a numerical system but simply use a token system to keep track of things. And all counting systems that continue beyond the number three are historical inventions; far from being “hard-wired” in our brains, they were conceived, refined and passed down from generation to generation.

Cultures have broken the “3 barrier” in various ways. For example, the Paiela of Papua New Guinea count by using parts of their body: The left-hand fingers starting with the little finger stand for one to five, the left wrist for six, the nose for 14, the right-hand fingers for 23–27. The highest number, 28, is shown by joining the two fists together.

Until the 1800s, the Tsimshians of British Columbia counted concretely. That is, they used different words to count different kinds of things: three men (gulgal), three canoes (galtskank), three long things (galskan), three flat things (guant) and three round things (gutle). The number of objects (three) and the kind of object (canoe) are fused in a single word. (We English-speakers also use concrete-counting words, as in “brace” [two partridges], “trio” [three musicians] and “quadruplets” [four children].) Like body counting, these so-called numerations, or parallel sequences of number words, rarely exceed 20, beyond which the Tsimshians used an object-specific term meaning “many”—much as we do with “gang,” “pile,” “bunch,” “flock,” “herd,” “school” and “fleet.”

Our own counting system, abstract counting, is very different. Our numbers one, two and three express the fact that groups of men, trees and canoes share the quality of oneness, twoness or threeness. Once numbers are split from (or abstracted from) the item counted, they can refer to anything. Abstract counting also makes it possible to break through the “20 barrier” and count as high as we wish.

From the evidence of counting systems around the world, mathematical historians speculate that counting has evolved over time in three main stages. The earliest and most rudimentary form is a simple token system in which things are counted in one-to-one correspondence. This stage may develop into an archaic counting system, such as the body counting of the 007Paiela or the concrete counting of the Tsimshians. Finally, there is abstract counting, which we use every day.

This is where archaeology enters the picture, for artifacts excavated in the Near East support the hypothesis that counting evolved in these three stages.

Archaeologists have found notched bones (above) in caves in modern Lebanon and Jordan that were once inhabited by Paleolithic (c. 15,000–10,000 B.C.) hunters and gatherers. The notches have been interpreted as tallies—perhaps recording animal kills, as anthropologists once thought. In recent years, however, scholars have begun to interpret the bones as lunar calendars, with each notch representing a sighting of the moon. In any case, if the notches are tallies, they illustrate perfectly the point of departure in the evolution of counting: one-to-one correspondence.

Even this simple device was a remarkable invention. It was the first tool used to record sets—in other words, the first counting machine. But this system was not perfect; since all the notches were alike, there was no way, out of context, to know what was counted.

Our earliest evidence of a specific counting system that could store data indefinitely comes from archaeological layers about 9,500 years old that also contain the first domesticated grains. This evidence consists of small, clay counting tokens (see photo of clay counting tokens), which have turned up by the hundreds at Near Eastern sites dating from 7500 to 3100 B.C. The shape of the counter determined what commodity it represented: A cone denoted a small measure of grain, a sphere denoted a large measure of grain, and an ovoid denoted a unit of oil. Although the tokens were used in one-to-one correspondence, with two jars of oil represented by two ovoids, this new system was more advanced than the notch system used by our Paleolithic ancestors. Here, the number of objects (one) is fused to the kind of object (oil). This system, then, is a form of concrete counting, even though number and object are fused in a sign rather than in a word.

In the Near East, people counted in one-to-one correspondence for thousands of years. Breaking this barrier was obviously a difficult step—and it came in the company of two other revolutionary developments at the end of the fourth millennium B.C.: the rise of the city-state and the advent of writing. In the administrative bureaucracies of the budding city-states, writing began to supersede tokens.

The first written signs were pictures of tokens. At first the counters were merely stamped onto a wet clay tablet (above). In two dimensions, the cone and the sphere tokens, representing small and large measures of grain, appear as a wedge and a circle.

Around 3100 B.C., images of certain tokens, such as the ovoid representing a jar of oil, began to be incised on clay tablets with a pointed stylus. At the same time, ancient accountants dispensed with the one-to-one correspondence. The written signs, or pictographs, do not express plurality by repeating the sign; instead, a sign representing a kind of object is preceded by abstract numbers. Remarkably, no new signs were created to express abstract numbers. Instead, the signs referring to measures of grain simply acquired another meaning: The wedge (a cone in two dimensions) standing for a small unit of grain became “1, ” and the circle (a sphere in two dimensions) representing a large unit of grain became “10.”

This creation of an abstract number system was a huge step. From “10, ” it was not far to “100” or “1000.” Now numbers could be added, subtracted, multiplied and divided. You could represent portions of things, as was soon done with taxes and interest rates. The sign for “10” led (with some leaps and bounds) to Archimedes and Newton.

Human beings learned to count at the same time that the Mesopotamian city-states were developing. Was it just a coincidence?