Center Embedding as Cultural Imperative (not contact-induced innovation)—Michael Palin SpecGram Vol CLIV, No 1 Contents Cartoon Theories of Linguistics—Part 九—Lexicostatistics vs. Glottochronology—Phineas Q. Phlogiston, Ph.D.
Speculative Grammarian is proud to present yet another installment of indeterminate regularity in the Linguistic Anthropologic Monograph Endowment’s Bizarre Grammars of the World Series.

This is Not the Whole (Number) Story

An Anthropological Linguistic Study of Non-Integral Person in Åriðmatçəl Verbs0

Bizarre Grammars of the World, Vol. 63


Recently, while on assignment for the upcoming Mathematico-Linguistic Special Issue of Speculative Mathematician, I found myself in a remote corner of Central Africa among a tribe of brilliant number crunchers, the Åriðmatçəl, who all live in the small village of Jííáhm Ëtërìì, which means, quite appropriately, “Counting Wisdom”. By special arrangement with the publishers of SpecGram and SpecMath, this article will appear in the May 2008 issues of both journals.

Cultural Background

Despite their seemingly primitive living conditionsmud huts, grass skirts, hand-carved wooden spearsthe Åriðmatçəl lead an exceedingly sophisticated life of the mind. Their prowess with numerical concepts is unmistakable. They are quite conversant with a whole host of concepts that escape even advanced practitioners of linguistics: irrational numbers (“acyclic numbers”), prime numbers (“lonely numbers”), sparse, abundant, and perfect numbers (“ectomorphic, endomorphic, and mesomorphic numbers”), surreal numbers (“mildly difficult numbers”), and even Ramanujan numbers (“promiscuously stacking numbers”where “stacking” is the Åriðmatçəl term used for exponentiation). The average Åriðmatçəl child is also quite familiar with π, e, and imaginary numbers (such as i, the square root of -1; these are called “exalted numbers” by the Åriðmatçəl).

Their numerical dexterity is such that children as young as two or three can perform impressive computational feats, including fairly complex multiplication of complex numbers, without the use of pencil and paper (let alone calculators). These skills are necessary prerequisites, it seems, to properly inflect Åriðmatçəl verbs for grammatical number.

Linguistic Background

We should all be quite familiar with the concept of grammatical number. As a quick review, first person refers to the speaker, or a group that includes the speaker; second person refers to the addressee, or a group that includes the addressee; third person refers to some other or others. Often verbs, and sometimes nouns and pronouns, are marked for grammatical number in the Indo-European languages many readers will be familiar with. More complex inflections in less familiar languages can encode grammatical number of both subject and object. But nothing I have ever seen, or even heard of, remotely approaches the overloaded, multilayered complexity of grammatical number in Åriðmatçəl. To begin to understand it, we must first familiarize ourselves with certain Åriðmatçəl mathematical terminology.

The Åriðmatçəl are apparently unique in that they use a base-7 counting system. Their number expressions are very regular and transparently compositional:

Number roots:
0 = uʔu
1 = a
2 = u
3 = o
4 = e
5 = iŋiliŋi
6 = ə
       Multiplier prefixes:
70 = 1 = k-
71 = 7 = z-
72 = 49 = x-
73 = 343 = q-
74 = 2,401 = r-
75 = 16,807 = ʔ-
76 = 117,649 = w-
77 = kiw-
78 = ziw-
79 = xiw-
710 = qiw-
711 = riw-
712 = ʔiw-
713 = wiw-
714 = kiwiw-
715 = ziwiw-
716 = xiwiw-
717 = qiwiw-
718 = riwiw-
719 = ʔiwiw-
720 = wiwiw-
Numerical negation is indicated by the suffix -op, added to the term for most significant digit of the number. Inverse powers are indicated by the suffix -q, coupled with the appropriate prefix. For example, x-...-q indicates a multiplier of 7-2 = 1/49. Numbers are generally expressed from least significant to most significant digits. Exceptions to this unmarked order are allowed for word play and certain poetical forms. Thus, 34,604.5517 (approximately 8,873.81910) is qaq xiŋiliŋiq ziŋiliŋiq ke xə qe ro.

An example of numerical word play is provided in the following punning joke, made at my expense, after I attempted to help reconstruct a dwelling damaged in a freak rhino attack: How many linguists does it take to build a mud hut? 693! Of course, 69310 is 20017, or za qu, which sounds like zaq u, which is 2 1/7. My understanding of the humor (which is always lost when it has to be explained) is that it usually takes about four people to build a mud hut in a day, and I, the linguist, thought I was doing such a good job building the hut that I would estimate that only slightly more than two of me could do the job of four, when in fact I was so bad at it that in reality almost 700 of me could barely do the job properly. (Ha! That showed me! I think.)

Other inflectionally important mathematical terms include the following: Imaginary numbers: The Åriðmatçəl word for the number i is rn, which is inserted into the term for the most significant digit of a number as an infix (or, perhaps, through tmesis). For example, 5i is krniŋiliŋi, and 32i (base 7, naturally) is ku zrno. Square roots (and other roots): Roots are indicated with the curious circumfix -m-...-m-. The power of the root is indicated by prefixing and suffixing the number around the circumfix. The k- prefix is omitted for square, cube, fourth, fifth, and sixth roots. Thus, the square root of 4 is umkemu. The fourth root of 1447 is emke ze xame. The 127th root of 327 is ko zumku zomko zu (incidentally, this phrase isat least in part because of its phonological attributesthe punchline to a complicated joke I do not understand). The fifth root of five (iŋiliŋimkiŋiliŋimiŋiliŋi) is, for similar reasons, part of a popular nursery rhyme. We will mostly be concerned with the square root, um-...-mu.

Linguistic Data

A fair number of terrifyingly complex glosses do not suffice to demonstrate the relevant phenomena, but it is all I can stand. N.B.: All numbers in the glosses below are in base 7. For reasons that will become numerically apparent, what is normally termed “first person” is represented by the number two in Åriðmatçəl, “second person” by the number three, and “third person” by the number four. In the simplest case, that of the first, second, or third person singular, the grammatical number is suffixed to the verb:

I say       you say       she says       he says
tlu-ku tlu-ko tlu-va-ke tlu-go-ke
say-2 say-3 say-FEM-4 say-MASC-4

Åriðmatçəl has three simple tenses: past, present, and future. Past tense is indicated by dividing the grammatical number by 7 (that is, shifting the decimal, or septimal, point to the left). The future tense is indicated by multiplying the grammatical number by 127 (910). This is said to represent the myriad number of ways the future can unfold, and, conversely, the severely diminished possibilities to be found exploring the past.

I said       you will say       she said       he will say
tlu-zuq tlu-kə zo tlu-va-zeq tlu-go-ka ziŋiliŋi
say-[2/10] say-[3·12] say-FEM-[4/10] say-MASC-[4·12]
say-0.2 say-36 say-FEM-0.4 say-MASC-51

Singular, dual, and plural are distinguished by exponentiating (or “stacking”) the grammatical person. Singular is unmarked, dual is squared, plural is cubed.

we-dual say       we-plural say       you-dual say       you-plural say
tlu-ke tlu-ka za tlu-ku za tlu-ko zə
say-[22] say-[23] say-[32] say-[33]
say-4 say-11 say-12 say-36

Clusivity can also be indicated, with inclusivity unmarked, and exclusivity marked with negative exponentiation, but with the exponent shifted onto the base (107).

we-dual-exclusive say       we-plural-exclusive say       you-dual-exclusive say
tlu-xuq tlu-quq tlu-xoq
say-[2·10-2] say-[2·10-3] say-[3·10-2]
say-0.02 say-0.002 say-0.03

Below are some examples combining the features we have seen so far. Note the ambiguity of tluquq above and below.

we-dual-exclusive said       you-dual-exclusive will say       they-plural said
tlu-quq tlu-xəq zoq tlu-zaq ku za
say-[2·10-2/10] say-[3·10-2·12] say-[43/10]
say-0.002 say-0.36 say-12.1

The subjunctive is used in Åriðmatçəl in subordinate clauses with verbs of desire or belief. The grammatical number of the subordinate verb is multiplied by the grammatical number of the main verb to indicate the subjunctive.

I wish you say       you will wish I say       they-dual wished he will say
dn-ka tlu-kə dn-kə zo tlu-kiŋiliŋi xa dn-zaq ka tlu-go-zuq kiŋiliŋi xe qa
wish-[2] say-[3·(2)] wish-[3·12] say-[2·(3·12)] wish-[42/10] say-MASC-[4·12·(42/10)]
wish-2 say-6 wish-36 say-105 wish-2.2 say-MASC-145.2

you-plural-exclusive will wish
    we-dual said
     she wished they-plural will wish he said
dn-qəq xoq tlu-roq qaq xuq dn-va-zeq dn-zaq ziŋiliŋi xə tlu-go-xeq kə
    ziŋiliŋi xo
wish-[3·10-3·12] say-[22/10·(3·10-3·12)] wish-FEM-[4/10] wish-[43·12·(4/10)]
wish-0.036 say-0.0213 wish-FEM-0.4 wish-650.1 say-MASC-356.04

Evidentials of limited scopefor reported speechare also used in Åriðmatçəl. The speaker and the other whose words the speaker reports are considered to exist, for evidentiality purposes, in separate, orthogonal dimensions. Thus the grammatical number of the reported speech is the Euclidean length of the vector (computed as √(x2 + y2) for vector <x,y>) composed of the speaker’s grammatical number in one dimension and the grammatical number of the speaker being reported on in the other dimension.

I say that I say       I say that you say
tlu-ku tlu-umka zamu tlu-ku tlu-umkə zamu
say-[2] say-[√(22 + 22)] say-[2] say-[√(32 + 22)]
say-2 say-√(11) say-2 say-√(16)

he will say that she said
tlu-go-ka ziŋiliŋi tlu-va-umxuq zuq ka zo xiŋiliŋi qomu
say-MASC-[4·12] say-FEM-[√((4·12)2 + (4/10)2)]
say-MASC-51 say-FEM-√(3531.22)

More than two levels of indirection are possible.

he will say that she said I say
tlu-go-ka ziŋiliŋi tlu-va-umxuq zuq ka zo
    xiŋiliŋi qomu tlu-va-umxuq zuq kiŋiliŋi zo xiŋiliŋi qomu
say-MASC-[4·12] say-FEM-[√((4·12)2 + (4/10)2)] say-[22 + √((4·12)2 + (4/10)2)]
say-MASC-51 say-FEM-√(3531.22) say-√(3535.22)

To further complicate matters, it is possible to use a certain honorific form which separates and “exalts” one or more individuals from within a larger group. Rather than using the “we-dual” inflection with “you and I”, it is possible to use a more complex form that indicates “you-honorific and I”. The honorific is indicated by separating the honorific and non-honorific grammatical numbers, and multiplying the honorific value by the imaginary (“exalted”) number, i. The honored entity and the matching imaginary numerical part of the grammatical number come first.

you and I (we-dual) say       they-dual and we-dual (we-plural) say
tlu-ke tlu-ka za
say-[22] say-[23]
say-4 say-11
you-honor and I say they-honor dual-exclusive and we-dual-exclusive say
tlu-krno ku tlu-xrneq xuq
say-[3i + 2] say-[4i·10-2 + 2·10-2]
say-3i+2 say-0.04i+0.02

Let us present one last, horribly detailed example.

you-honor and I wished
dn-zrnoq zuq

     that she-honor and he will say
tlu-va-go-ziŋiliŋiq ke zrno zaq kiŋiliŋiop

that we-plural-exclusive said
tlu-umʔiwaq riwiŋiliŋiq kiŋiliŋi zu qə rrno
      riwiŋiliŋiq qiwəq xiwəq ziwəq wəq ʔəq rəq
      qəq xəq zəq ka zə xe qə rəopmu
say-[(3i+2)·√((4i+4)2 + (2·10-3/10)2)]


Additional Linguistic Tidbits

There is evidence of a no-longer-used “literary past” tense which was computed as the harmonic mean of the grammatical number of the actor in the story and the grammatical number of the narrative voice. It was expressed, naturally, as a fraction. However, this tense seems to have fallen out of use about 1,0007 years ago (approximately 35010 years ago), when the Åriðmatçəl abandoned fractional grammatical number in favor of using decimal (or, more properly, septimal) grammatical numbers.

There is also a “zeroth person”, represented by the number one (ka), which can be used for the personification of nature or of the universe as a whole. This is mathematically and culturally significant because the number 1 is invariant under the operations used to indicate singular, dual, plural, inclusive, and exclusive. In Åriðmatçəl lore, the universe is a similarly unchanging background against which the human drama unfolds.

There are tales, likely untrue but intriguing nevertheless, of wise elders who hide in caves, wander the vast desert, or banish themselves to mountain tops, where they attempt to construct ever-lengthening sentences that require a verb with an inflection for an ever-increasingly accurate approximation of a transcendental number (interestingly, also “transcendental number” in Åriðmatçəl), such as π or e. To achieve perfection, such a sentence would obviously (so my mathematician colleagues assure me) have to be infinitely long, so there is no reason to assume that any of the Åriðmatçəl would be foolish enough to attempt such a task.

Linguistic Analysis

As is obvious from the data above, there is a maddening amount of complexity and ambiguity in this system. Not only are speakers required to perform truly exceptional mathematical feats in order to conjugate a moderately complicated verb, hearers have the even tougher task of analyzing and reverse-engineering the appropriate circumstance that would generate the value uttered by the speaker. This seems impossibly difficult, but Åriðmatçəl speakers manage to do it, thousands of times a day.

One can try to imagine the grammatical numbers used in a phrase like I believe that you-dual wish that we-dual-exclusive and you-honorific will say that she and they-plural exclusive said ... but to do so surely invites madness!

Tentative Conclusions

More research is necessary to unravel the intricacies of this system. Said research will require more and abundant funding.


Claude Searsplainpockets Somewhere in Africa

0 This paper was made possible by LAME grant #13, and the phoneme /5/.

Center Embedding as Cultural Imperative (not contact-induced innovation)—Michael Palin
Cartoon Theories of Linguistics—Part 九—Lexicostatistics vs. Glottochronology—Phineas Q. Phlogiston, Ph.D.
SpecGram Vol CLIV, No 1 Contents