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LRC Blog

July 14, 2010

Todd B. Krause

Indo-European Number Bases

N.B. The email exchange below may have been edited, e.g. to remove content not essential to the main point(s) or to standardize English spelling/grammar.


Dear LRC,

Apologies for such an unsolicited email, but I was wondering if you might be able to help me. I am currently writing a short history of mathematics for young people, and in the course of my research on the origins of number have come across the interesting fact that in the case of Native North American Indian tribes, the words for their numbers apparently have an interesting etymology, with the word for "five" often being closely related to the word for "hand," and the words "ten" or "twenty" often being related to "man" or "all" or something similar, reflecting the likely origin of decimal and vigesimal systems in the counting of digits (see e.g. Closs, M.F., Native American Mathematics, UTP 1986).

I would be most interested to know if there exist similar relationships between number words in English (which I assume have their origins in PIE?) and digits, or if there are traces of other number bases than ten (e.g. the traces of a vigesimal numbering system that persists in French). I took the liberty of writing to you as I have been unable to find much documentation on the subject. If you felt able to offer any help or advice at all, or could point me in the direction of some suitable material, I would be most grateful.

Yours, with thanks, T. M.


Howdy Mr. M.,

Thank you very much for your inquiry. Such "unsolicited" emails are what keep life here at the LRC interesting and, with questions like yours, downright fun. I apologize for the delayed response, but we've actually received quite a number of email inquiries of late, and we have been trying to respond to each in turn. I note you have also contacted Drs. Bauer and Slocum; if by chance my response parts ways with theirs, then please take their advice, as they are scholars with much more experience in this field.

I'd like first and foremost to point you to three references that might be of use in answering your question:

  1. The Oxford Introduction to Proto-Indo-European and the Proto-Indo-European World, Mallory and Adams, Oxford U. Press, 2006;
  2. Theoretical Bases of Indo-European Linguistics, W.P. Lehmann, Routledge, 1993;
  3. New Comparative Grammar of Greek and Latin, Sihler, Oxford U. Press, 1995.

The first presumes no particular knowledge of linguistics and speaks to a general audience in terms clear, entertaining, and authoritative. The second reference assumes some background in linguistics, but not a lot, and describes both the foundations and history of PIE studies better than most any other work I've found. The last reference is of a more technical nature, assuming at least a knowledge of Greek or Latin. But many sections open with more general discourses which invariably contain profound insights.

I gather from the way you describe your project that perhaps the following statement by Lehmann applies to you as well: "I am chiefly concerned here with the [historical linguistic] approach applied with the aim of providing interpretations and explanations rather than with furnishing etymologies." (Lehmann, p. 254)

That is, I'll try to refrain from giving a list of etymologies as we Indo-Europeanists are prone to do. This, it turns out, is just as well, since Mallory and Adams note: "To sum up the etymological discussion, it would seem that two of the basic numbers, one of the words for 'one' (*h1oinos [etc.]) and the word for 'hundred', have excellent etymologies while two more, 'eight' and 'nine', have plausible ones. The rest remain mysterious." (Mallory and Adams, p. 316)

If we try to focus on the explanations, rather than the particular etymologies, it turns out there is little definite we can say. In some sense, however, this is a good thing: as a scholarly community we have data old enough, and have studied it long enough, both to entertain many suggestions as to origins and at the same time to recognize that the data does not support the suggestions with the accuracy to which we have grown accustomed. That is, PIE studies are mature enough to know when we're just wishing an explanation were true.

Concerning numeral reference to the human body, we do find a link between, e.g., Greek pente 'five' and Hittite pankus 'all, whole'. It takes no great leap of imagination to suggest that the referent in question is the human hand, and that five fingers form a complete set in some sense. But the degree to which this can be seen as a standard for the numeric system as a whole is not clear. As Lehmann points out, in Sumerian (a non-IE language) for example, numbers beyond 5 are represented as 5+1, 5+2, etc., and so we can plausibly see 5 as a standard for the system. We do not find such transparency in the structure of numerals beyond 5 in PIE.

Concerning the question of mathematical base, we do indeed find a systematic use of PIE *dekmt 'ten' in the construction of higher numerals. In particular, the numerals in the teens tend to have the additive form 1(+)10, 2(+)10, 3+10, 4+10, etc. That is, Greek for example shows dodeka 'twelve', literally 'two-ten'; but then tres kai deka 'three and ten' is 'thirteen'. The decades use 'ten' in multiplicative constructions, e.g. PIE *dwi-(d)kmtih1 'two-tens', i.e. 'twenty', hence Latin viginti. So it seems safe to say that the original PIE numeric system was decimal.

We must however take care in making such assertions. For example, Old Norse often employs hundrath 'hundred' with the numeric value 120. And as Lehmann points out, Homeric Greek hekatombe should literally be a 'sacrifice of 100 oxen'; but the number used is rarely 100, and we even find one instance in the Iliad where this sacrifice is performed with only 12 oxen (Lehmann, p. 254). So perhaps the original PIE meant something like 'a whole lot' (though in this instance the occurrence of 12 is interesting).

At the same time, we must recognize the possibility of innovation and not be too hasty to project into archaic history what only derives from the recent past. For example, English dozen was only imported in the Middle English period from French. Moreover, English score as representing 20 is an innovation: the word itself was borrowed into English from Old Norse skor 'a cut, tally' (an etymological cousin of the Old Norse word in English, incidentally, is shear). So neither of these, nor the French innovation soixante-douze 'sixty-twelve' = 72, are particular clues to a possible earlier base-12 system (Sihler, p. 402).

Being an Indo-Europeanist, I could go on for ages with interesting trivia about the PIE numeral system. Personally I find it fascinating because, much like yourself, I have a keen interest in the history of science in general, and mathematics in particular. But I will try to show some restraint. Instead I think it might be worth including an extended passage from Sihler's work:

"From the beginning of IE studies attempts have been made to discover metaphors and metonymies behind the words for numbers. So for example *newm 'nine' was held to be based on *new(o)- 'new', on the theory that before the term was coined the highest number was 'eight'. PIE *dekmt 'ten' was taken as *d(w)e- 'two' and the zero grade of *kemt- 'hand', whose o-grade is seen in the pan-Gmc. hand words (which is the only attestation among the IE language of the element in question with the meaning 'hand'). Hands and fingers in terms for numbers are certainly found in languages, so this interpretation is more convincing than the 'nine' = 'new number' business, which is fantastical. Among the most persuasive of these suggestions is the connection of *penkwe 'five' with *ponkw-to-, *ponkwu- 'all, whole' (Lat. cunctus, Hitt. pa-an-ku-us), that is 'five' = 'the whole [hand]'.

"Although they are entertaining to ponder, such etymologies assume a degree of naivete on the part of the speakers of PIE which was more imaginable when PIE was taken to be something like Original Human Language, whose speakers would be driven to epistemology when faced with a high-tech concept such as 'nine'. Besides, given the time-depths, we have only vague resemblances not testable hypotheses, which makes it practically impossible to decide between chance similarities and genuine history." (Sihler, p. 402)

Thus, even for very plausible explanations such as 'ten' = 'two hands', we must take care to consider how much evidence we find supporting the assertion. And perhaps a much more important point from a modern perspective is that we not treat the scientific and mathematical knowledge of ancient cultures as 'primitive'; in many instances it was highly developed, but tuned to the needs of the culture at hand. For example, the Latin term septemtriones, literally 'seven oxen', came to mean 'north' in common usage. Such usage can only take hold in a speech community that notices that the Big Dipper is circumpolar and never dips fully below the horizon (at that latitude). Similarly in the Maya calendar, we find that the winal is not a base-20 unit like the rest, since this brings the numeric system better in line with the annual motions of the heavens.

I hope this helps in some way to answer your question. Please let me know if I can be of any further assistance, and best of luck on your history of mathematics.

Sincerely, T. K.