**Mathematical Structures
of Early Indo-European Numeral Systems**

In honor of Joseph H. Greenberg

Carol F. Justus

Indo-European (IE) language grammars have been organized in terms
of inflectional and derivational categories. Such categories often reflect the
system that Latin grammarians, building on their Greek forebears, established.
Greenberg's work in defining categories of language on the basis of the extent
to which human languages vary offers a way to capture systematic differences
in terms of universal categories of language.

Interests of Greenberg, the anthropologist, however, also extended
to linguistic systems that encoded cultural systems as when he (1978) pointed
out that the names of number words may change because a language has adopted
a new numeral base. Such an understanding presupposes that mathematical functions
are encoded in the linguistics of a numeral system. Such mathematical functions
underlying a "pure" numeral system are that of numeral base and exponentiation
on that base (Greenberg 2000:373-374). So exponential bases of the decimal system,
besides 10 (one times the base) include 100 (10 the power of one), 1000 (10
to the power of three), and so on, powers that result from the mathematical
principle, exponentiation.

Since Szemerényi (1960), the Proto-Indo-European (PIE) numeral system
has generally been thought to have been a decimal system. This is based primarily
on Szemerényi's phonological analysis of the decades, the multiples of
10 before 100 (Greenberg's 'augends'). Among other problems (Justus 1996:66-77),
the reconstruction of a decimal system for PIE does not account for the Germanic
long hundred of 120, a number which is neither a power of 10 nor of 12 (Justus
1999:138-140). It is not a "pure" numeral system nor does it belong
to a "mixed base" category (e.g., French quatre vingt 'four-20(s)'
or '80': Greenberg 2000:774). Earlier studies have related it to a duodecimal
or sexagesimal system, systems which Greenberg (2000:773) says remain unexplained.

Following Greenberg's anthropological concerns, one turns to the cultural context
of early counting. Studies of the Sumerian sexagesimal system have shown that
it was not based on 60 but on alternating multiples of 10 and 6 (Damerow &
Englund 1989; Damerow 1996). Such early systems differed from modern numeral
systems in that they lacked a base and with it the function, exponentiation.
Justus (1999:145) termed alternating units of multiplication such as 10 and
6 'collection units', while Luján (1999) termed them 'improper bases'.
Both terms reflect their pre-exponential, non-base status.

Greenberg (1978) established that changes in the linguistics of numeral systems reflected changes in a base. This study argues that the change from pre-exponential to exponential system built on a base also affects the linguistic form for expression of numerals. It further illustrates how the form of data within specific linguistic domains is related to variation in the human cultural systems that they encode.

**Sources Cited:**

Damerow, Peter. 1996. Abstraction and Representation: Essays on the Cultural
Evolution of Thinking. Dordrecht.

_______. & Robert Englund. 1989. The Proto-Elamite Texts from Tepe Yahya.
(Bulletin of the American School of Prehistoric Research, 39.) Cambridge, MA:
Harvard University, Peabody Museum.

Greenberg, Joseph H. 1978. Generalizations about Numeral Systems. In: Universals
of Human Language, ed. by J. H. Greenberg, Charles A. Ferguson, & Edith
A. Moravcsik,249-295. Stanford: Stanford University Press.

______. 2000. 75. Numeral. In: Morphology. Vol. I. An International Handbook
on Inflection and Word-Formation, ed. by Geert Booij, Christian Lehmann, &
Joachim Mugdan with the collaboration of Wolfgang Kesselheim & Stavros Skopeteas,
770-783. Berlin & NY: Walter de Gruyter.

Justus, Carol F. 1996. Numeracy and the Germanic Upper Decades. Journal of Indo-European
Studies 24.45-80 (=http://www.utexas.edu/cola/depts/lrc/numerals/cfj-jies/cfj1-section1.html)

______. 1999. Indo-European Numerals Since Szemerényi. The Emergence
of the Modern Language Sciences. Studies on the Transition from Historical-Comparative
to Structural Linguistics in Honour of E. F. K. Koerner, ed. by Sheila Embleton,
John E. Joseph, & Hans-Josef Niederehe. Vol. 2: Methodological Perspectives
and Applications, Chapter 30 (=131-152). Amsterdam & Philadelphia: John
Benjamins.

Luján, Eugenio R. 1999. Towards a Typology of Change in Numeral Systems.
Language Change and Typological Variation: In Honor of Winfred P. Lehmann. Vol.
I: Language Change and Phonology, ed. by Edgar C. Polomé & Carol
F. Justus, 183-200. (Journal of Indo-European Studies Monograph, 30.) Washington,
DC: Institute for the Study of Man.

Szemerényi, Oswald. 1960. Studies in the Indo-European System of Numerals.
Heidelberg: Carl Winter.