Introduction

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 conditions—mud huts, grass skirts, hand-carved wooden spears—the 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-

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 12₇^th root of 32₇ is ko zumku zomko zu (incidentally, this phrase is—at least in part because of its phonological attributes—the 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)]    say-MASC-[4/10·(43·12·(4/10))]
wish-0.036 say-0.0213		wish-FEM-0.4 wish-650.1 say-MASC-356.04

Evidentials of limited scope—for reported speech—are 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 wish-[(3i+2)/10] wish-[0.3i+0.2]		that she-honor and he will say tlu-va-go-ziŋiliŋiq ke zrno zaq kiŋiliŋiop say-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 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)] say-[√(36025.00000000051i-66461.6666666665)]		nothing. uʔu 0 0

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.

	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.
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