kena

Fancy numbers with words

In lecture on December 10, 2012 at 12:00

People count things with words. Yet some internal features of number systems in human languages are fascinating.

Number systems come in a great variety of shape and form. English’s base-10 system is quite unusual in its regularity, as systems in human languages go. French mixes decimal and vigesimal forms. Huli in Papua New Guinea uses base 15, whereas Tongan in Polynesia simply enumerates the digits in base 10.

I found this page Number Systems of the world by the Japanese amateur linguist and Wikipedia activist Takasugi Shinji quite enlightening as an introduction to this vast diversity.

While browsing these listings, two aspects struck me.

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The first is that all number systems, without exceptions, are mixtures of bases which are either the first multiples of 5 (5, 10, 15, 20, rarely up to 100) or multiples of 12 (mostly 12 and 24). Why do we not see languages developing number systems out of base 3, 7, 13 or 17? Also, why 5 and 12 specifically?

The prevalence of base 5 and derivatives is to me obvious since I started counting, at around age four. It’s an inescapable consequence of people using their fingers to count: run out of fingers, and you have to keep count of how many hands using the finger on another hand. When doing so, the counting process is easier if each finger has its own name, hence the need for different words for each digit. And since we have twenty fingers (hands and feet), it’s all but expected that counting in base 5, 10, 15 and 20 have become most popular.

The origin of the popularity of base 12 became obvious to me some time later when I had to perform some simple arithmetic while holding a bag of groceries with one hand. With the thumb of one hand, I could mark and remember my count on the 12 phalanxes of the remaining four fingers. This trivial, almost unavoidable gesture is bound to have happened innumerable times during the history of mankind, and to have justified the appearance of singular words for the first 12 ordinals.

I can’t remember reading these particular reasons during my study of linguistics, so I would love to find a proper scientific argument in that direction. Nevertheless, this realization made it crystal clear that we should not expect any non-human intelligent entity to develop numbering in base 10 or 12 unless they happen to have five-fingered or twelve-phalanxed appendages.

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Besides the numbering bases, the aforementioned page reveals that each numbering system indexed there contains a mixtures of some features found in some other languages but not all.

For example, a system may be single-base (eg Huli, Tongan), multiple-base (eg 5+10, Nahuatl) or mostly single base with degenerate multiple-base forms for the lower ordinals. Most European languages feature the latter, using partial base-20 numbering for ordinals below 100, and from then on only regular forms in base 100.

They may or may not feature individual names for the first 12 ordinals, a vestigial inheritance of an earlier base-12 linguistic evolution. Most Northern European languages feature it, including English, Dutch, Swedish, German. In contrast, Latin, French, Spanish and Italian don’t; their words for eleven and twelve are usually a contraction of the word for one and two with the word (or an older form) for ten.

They may or may not feature substractive forms, where the ordinals before a boundary are counted as decreasing values substracted from the boundary. There are two variations on this theme.

In the elementary-substractive form, the value of the ordinal is computed from the words used using integer arithmetic. For example Roman numerals feature this, with eg IX representing nine (10-1); Ainu also names eight and nine as 10-2 and 10-1, respectively.

In the fractional-substractive form, non-integer arithmetic is involved. Danish says for example “half three times twenty” (halvtredsindstyvende) to mean 50 (3-½  ×20); Dutch has the single word anderhalf with etymology “half two” to mean 1,5 (2-½) and time in Dutch is also counted fractionally, with “half three” meaning 2:30 (3-½ hours).

Interestingly I have not found any language which has both elementary and fractional substractive forms in its number system.

Understanding these general features provides a glimpse at the general structure of number systems and the historical links between them over time, beyond and above the particulars of each individual language. An exciting topic in socio- or historico-linguistics could be, for example, to determine whether these features can be explained by behavioral or physiological traits of the human populations where they appeared originally. If anyone has heard of existing studies in this direction, please let me know!

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