The Linguistic Lives of Numbers

Ask a child to count and you discover something strange: the names of numbers are not arithmetic. One, two, three, four — and then a hard turn at eleven and twelve, words that pretend not to be eleventy-one and oneteen. Thirteen through nineteen reverse order. Twenty changes vowels. Hundred appears from nowhere, and a few centuries ago English speakers knew about a "long hundred" that meant 120. Every numeral system in the world has these scars. They are the fossils of how a culture thought about quantity before it had a theory of quantity.

The bodies that did the counting

Most languages base their number system on five, ten, or twenty — and the reason is sitting at the end of your arms. Five-fingered hand. Ten fingers. Twenty fingers and toes. The vigesimal systems of the Mayan calendar, of pre-revolutionary French (quatre-vingts for 80, "four twenties"), of Welsh and Basque and Yoruba, all settle on twenty as the natural ceiling because that is the count of one whole human. The decimal systems of most Indo-European languages settle on ten because shoes happened.

The exceptions are illuminating. Babylonian arithmetic was sexagesimal, base 60, which is why we still keep time in 60-second minutes and 60-minute hours. The choice of 60 was probably not bodily — the leading theory is that 60 has many divisors (2, 3, 4, 5, 6, 10, 12, 15, 20, 30), making it computationally convenient for a culture that did real arithmetic without decimal notation. Sumerian merchants needed to split inheritances and weights into halves, thirds, fourths, sixths; 60 makes all of that exact.

Languages that do not count past five

The Pirahã of the Brazilian Amazon famously do not have number words beyond approximations. Hoi means "small quantity" or "one"; hoi with rising tone means "a slightly larger quantity." There is no word for "exactly five." Daniel Everett's reporting on this in the 2000s set off a long argument in linguistics — partly because the Whorfian implications are uncomfortable, partly because subsequent fieldwork by Peter Gordon showed Pirahã speakers struggle on tasks that require exact discrimination of quantities greater than three.

The Pirahã are not unique. The Wári', the Mundürukú, and several Australian Aboriginal languages have similarly limited number vocabularies. The lesson is not that these speakers cannot reason about quantity — they obviously can, in everyday life — but that the cultural pressure to standardize quantity into discrete words depends on what the culture has to count. Without large-scale agriculture, taxation, or written records, exact integers above three rarely matter.

Grammatical number: more than singular and plural

English distinguishes one of a thing from more than one. Russian distinguishes one from a few from many: the noun ending changes for 1, for 2-4, and for 5+. Arabic and Sanskrit have a true dual, a separate number for exactly two — preserved in English only in vestiges like both and between (versus among). Slovenian has a fully active dual to this day; if a Slovenian speaker is talking about exactly two cats, the verb ending is different from the form for three cats.

Some languages go further. Lihir, an Austronesian language of Papua New Guinea, has five grammatical numbers: singular, dual, trial (exactly three), paucal (a few), and plural. The boundary between paucal and plural sits around seven. The grammar implies a culture for which "three brothers" is a different cognitive category from "four brothers," which is a different category from "many brothers."

Counting words that hide their math

French struggles with 70 and 90. Soixante-dix means "sixty-ten." Quatre-vingt-dix means "four-twenty-ten" — that is, 4×20+10. The Belgians and Swiss simplified centuries ago to septante and nonante, but standard French preserves the older vigesimal-decimal hybrid. School-aged children in Paris and Brussels learn arithmetic in slightly different cognitive frames.

Danish goes further. The Danish word for 50 is halvtreds, derived from halv-tred-sinds-tyve: "half-third-times-twenty," meaning 2½ × 20. Halvfjerds for 70 is "half-fourth-times-twenty" or 3½ × 20. The system is a frozen artifact of medieval Scandinavian arithmetic that no modern Dane reconstructs from first principles — they just memorize the numerals as opaque words. Foreign learners of Danish find this section of the language genuinely difficult.

Chinese and the regular shape

Mandarin numerals are unusually transparent. 11 is shi-yi ("ten-one"), 12 is shi-er ("ten-two"), 20 is er-shi ("two-ten"), 31 is san-shi-yi ("three-ten-one"). There are no irregular eleventy or twenty; the structure scales perfectly. Cognitive psychologists have argued for decades that Chinese-speaking children learn arithmetic faster on average partly because the number words themselves do the place-value teaching. Whether or not this hypothesis fully holds, it is striking how rare the Chinese pattern is. Most languages chose — or got stuck with — messier compromises.

The classifier languages

In Mandarin, you cannot say "three books." You must say "three [appropriate-classifier] book." For books, the classifier is ben. For long thin things (pencils, cigarettes), it is zhi. For flat things (sheets of paper, tickets), it is zhang. There are over a hundred classifiers in active use, and using the wrong one marks you immediately as a learner.

Japanese, Korean, Vietnamese, Thai, and most Southeast Asian languages share the structure. Linguists call these numeral classifier languages, and the count is roughly half the world's spoken languages. The pattern implies that for these speakers, raw quantity is incomplete information — you must also categorize the kind of thing being counted before counting becomes meaningful. English does this in a small way (two sheets of paper, three head of cattle, five grains of rice) but treats it as an exception rather than the rule.

What the diversity tells us

Mathematics is supposed to be culture-free — the universal language of physics, the lingua franca of science. The numerals are not. The systems we count with grew from bodies, markets, calendars, and accidents of language. The Chinese got place value; the Danes got fractional vigesimal embedded inside decimal; the French got 80 = four twenties; the Pirahã got "small" and "larger." English got eleven and twelve, and a long hundred we no longer use, and a "score" that survives mainly in Lincoln's address.

The reason this matters beyond linguistic curiosity: number words are the substrate on which mathematical thinking runs. The shape of the substrate biases what is easy to think and what is hard. The fact that mathematicians eventually agreed on Hindu-Arabic numerals and a small set of operators is one of history's great standardizations — but it sits on top of a deep, weird, irreducible diversity of how human languages decided to handle the strange task of saying exactly how many.