Recently, while on assignment for the upcoming Mathematico-*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.

Despite their seemingly primitive living conditions*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.

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-

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

Numerical negation is indicated by the suffix

Number roots:

0 =uʔu

1 =a

2 =u

3 =o

4 =e

5 =iŋiliŋi

6 =ə

Multiplier prefixes:

7^{0}= 1 =k-

7^{1}= 7 =z-

7^{2}= 49 =x-

7^{3}= 343 =q-

7^{4}= 2,401 =r-

7^{5}= 16,807 =ʔ-

7^{6}= 117,649 =w-

7^{7}=kiw-

7^{8}=ziw-

7^{9}=xiw-

7^{10}=qiw-

7^{11}=riw-

7^{12}=ʔiw-

7^{13}=wiw-

7^{14}=kiwiw-

7^{15}=ziwiw-

7^{16}=xiwiw-

7^{17}=qiwiw-

7^{18}=riwiw-

7^{19}=ʔiwiw-

7^{20}=wiwiw-

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, 693_{10} is 2001_{7}, 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, 5*i* is *k rniŋiliŋi,* and 32

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-kutlu-kotlu-va-ketlu-go-kesay-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 12_{7} (9_{10}). 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-zuqtlu-kə zotlu-va-zeqtlu-go-ka ziŋiliŋisay-[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-ketlu-ka zatlu-ku zatlu-ko zəsay-[2 ^{2}]say-[2 ^{3}]say-[3 ^{2}]say-[3 ^{3}]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 (10_{7}).

we-dual-exclusive say we-plural-exclusive say you-dual-exclusive say tlu-xuqtlu-quqtlu-xoqsay-[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-quqtlu-xəq zoqtlu-zaq ku zasay-[2·10 ^{-2}/10]say-[3·10 ^{-2}·12]say-[4 ^{3}/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-kədn-kə zo tlu-kiŋiliŋi xadn-zaq ka tlu-go-zuq kiŋiliŋi xe qawish-[2] say-[3·(2)] wish-[3·12] say-[2·(3·12)] wish-[4 ^{2}/10] say-MASC-[4·12·(4^{2}/10)]wish-2 say-6 wish-36 say-105 wish-2.2 say-MASC-145.2

you-plural-exclusive will wish

we-dual saidshe wished they-plural will wish he said dn-qəq xoq tlu-roq qaq xuqdn-va-zeq dn-zaq ziŋiliŋi xə tlu-go-xeq kə

ziŋiliŋi xowish-[3·10 ^{-3}·12] say-[2^{2}/10·(3·10^{-3}·12)]wish-FEM-[4/10] wish-[4 ^{3}·12·(4/10)]

say-MASC-[4/10·(4^{3}·12·(4/10))]wish-0.036 say-0.0213 wish-FEM-0.4 wish-650.1 say-MASC-356.04

Evidentials of limited scope^{2} + y^{2})**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 zamutlu-ku tlu-umkə zamusay-[2] say-[√(2 ^{2}+ 2^{2})]say-[2] say-[√(3 ^{2}+ 2^{2})]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 qomusay-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 qomusay-MASC-[4·12] say-FEM-[√((4·12) ^{2}+ (4/10)^{2})] say-[2^{2}+ √((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-*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-ketlu-ka zasay-[2 ^{2}]say-[2 ^{3}]say-4 say-11 you-honor and I say they-honor dual-exclusive and we-dual-exclusive say tlu-krno kutlu-xrneq xuqsay-[3 i+ 2]say-[4 i·10^{-2}+ 2·10^{-2}]say-3 i+2say-0.04 i+0.02

Let us present one last, horribly detailed example.

you-honor and I wished

dn-zrnoq zuq

wish-[(3i+2)/10]

wish-[0.3i+0.2]

that she-honor and he will say

tlu-va-go-ziŋiliŋiq ke zrno zaq kiŋiliŋiopsay-FEM-MASC-[(4i+4)·12·((3i+2)/10)]

say-FEM-MASC-[34.5i-5.1]

that we-plural-exclusive said

tlu-umʔiwaq riwiŋiliŋiq kiŋiliŋi zu qə rrnoriwiŋiliŋiq qiwəq xiwəq ziwəq wəq ʔəq rəqqəq xəq zəq ka zə xe qə rəopmusay-[(3i+2)·√((4i+4)^{2}+ (2·10^{-3}/10)^{2})]

say-[√(36025.00000000051i-66461.6666666665)]

nothing.

uʔu

0

0

There is evidence of a no-_{7} years ago (approximately 350_{10} 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-*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.

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-

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!

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

- Connrie, Bernard, 1976.
*Nervous.* - Connrie, Bernard, 1985.
*Tense.* - Connrie, Bernard, 1997.
*Tired.* - Corbett, Greville, 2000.
*Number.* - Corbett, Greville, 2006.
*Agreement.* - Davis, Phillip J. and Reuben Hersh, 1999.
*The Mathematical Experience.* - Lakoff, George and Rafael Nuñez, 2000.
*Where Mathematics Comes From: How the embodied mind brings mathematics into being.* - Sten, Holgar, 1949.
*Le nombre grammatical.*(Travaux du Cercle Linguistique de Copenhague, 4.) - Wiese, Heike, 2003.
*Numbers, language, and the human mind.*

Claude Searsplainpockets |
Somewhere in Africa |