Levi Leonard Conant - The Number Concept

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5. amgnaitone = 1 hand complete.
6. itacono amgna pona tevinitpe = 1 on the other hand.
10. amgna aceponare = all of the 2 hands.
11. puitta pona tevinitpe = 1 on the foot.
16. itacono puitta pona tevinitpe = 1 on the other foot.
20. tevin itoto = 1 man.
21. itacono itoto jamgnar bona tevinitpe = 1 on the hands of another
man.

In the Guarani[75] language of Paraguay the same method is found, with a
different form of expression for 20. Here the numerals in question are

5. asepopetei = one hand.
10. asepomokoi = two hands.
20. asepo asepi abe = hands and feet.

Another slight variation is furnished by the Kiriri language,[76] which is
also one of the numerous South American Indian forms of speech, where we
find the words to be

5. mi biche misa = one hand.
10. mikriba misa sai = both hands.
20. mikriba misa idecho ibi sai = both hands together with the feet.

Illustrations of this kind might be multiplied almost indefinitely; and it
is well to note that they may be drawn from all parts of the world. South
America is peculiarly rich in native numeral words of this kind; and, as
the examples above cited show, it is the field to which one instinctively
turns when this subject is under discussion. The Zamuco numerals are, among
others, exceedingly interesting, giving us still a new variation in method.
They are[77]

1. tsomara.
2. gar.
3. gadiok.
4. gahagani.
5. tsuena yimana-ite = ended 1 hand.
6. tsomara-hi = 1 on the other.
7. gari-hi = 2 on the other.
8. gadiog-ihi = 3 on the other.
9. gahagani-hi = 4 on the other.
10. tsuena yimana-die = ended both hands.
11. tsomara yiri-tie = 1 on the foot.
12. gar yiritie = 2 on the foot.
20. tsuena yiri-die = ended both feet.

As is here indicated, the form of progression from 5 to 10, which we should
expect to be "hand-1," or "hand-and-1," or some kindred expression,
signifying that one hand had been completed, is simply "1 on the other."
Again, the expressions for 11, 12, etc., are merely "1 on the foot," "2 on
the foot," etc., while 20 is "both feet ended."

An equally interesting scale is furnished by the language of the
Maipures[78] of the Orinoco, who count

1. papita.
2. avanume.
3. apekiva.
4. apekipaki.
5. papitaerri capiti = 1 only hand.
6. papita yana pauria capiti purena = 1 of the other hand we take.
10. apanumerri capiti = 2 hands.
11. papita yana kiti purena = 1 of the toes we take.
20. papita camonee = 1 man.
40. avanume camonee = 2 men.
60. apekiva camonee = 3 men, etc.

In all the examples thus far given, 20 is expressed either by the
equivalent of "man" or by some formula introducing the word "feet." Both
these modes of expressing what our own ancestors termed a "score," are so
common that one hesitates to say which is of the more frequent use. The
following scale, from one of the Betoya dialects[79] of South America, is
quite remarkable among digital scales, making no use of either "man" or
"foot," but reckoning solely by fives, or hands, as the numerals indicate.

1. tey.
2. cayapa.
3. toazumba.
4. cajezea = 2 with plural termination.
5. teente = hand.
6. teyentetey = hand + 1.
7. teyente cayapa = hand + 2.
8. teyente toazumba = hand + 3.
9. teyente caesea = hand + 4.
10. caya ente, or caya huena = 2 hands.
11. caya ente-tey = 2 hands + 1.
15. toazumba-ente = 3 hands.
16. toazumba-ente-tey = 3 hands + 1.
20. caesea ente = 4 hands.

In the last chapter mention was made of the scanty numeral systems of the
Australian tribes, but a single scale was alluded to as reaching the
comparatively high limit of 20. This system is that belonging to the
Pikumbuls,[80] and the count runs thus:

1. mal.
2. bular.
3. guliba.
4. bularbular = 2-2.
5. mulanbu.
6. malmulanbu mummi = 1 and 5 added on.
7. bularmulanbu mummi = 2 and 5 added on.
8. gulibamulanbu mummi = 3 and 5 added on.
9. bularbularmulanbu mummi = 4 and 5 added on.
10. bularin murra = belonging to the 2 hands.
11. maldinna mummi = 1 of the toes added on (to the 10 fingers).
12. bular dinna mummi = 2 of the toes added on.
13. guliba dinna mummi = 3 of the toes added on.
14. bular bular dinna mummi = 4 of the toes added on.
15. mulanba dinna = 5 of the toes added on.
16. mal dinna mulanbu = 1 and 5 toes.
17. bular dinna mulanbu = 2 and 5 toes.
18. guliba dinna mulanbu = 3 and 5 toes.
19. bular bular dinna mulanbu = 4 and 5 toes.
20. bularin dinna = belonging to the 2 feet.

As has already been stated, there is good ground for believing that this
system was originally as limited as those obtained from other Australian
tribes, and that its extension from 4, or perhaps from 5 onward, is of
comparatively recent date.

A somewhat peculiar numeral nomenclature is found in the language of the
Klamath Indians of Oregon. The first ten words in the Klamath scale
are:[81]

1. nash, or nas.
2. lap = hand.
3. ndan.
4. vunep = hand up.
5. tunep = hand away.
6. nadshkshapta = 1 I have bent over.
7. lapkshapta = 2 I have bent over.
8. ndankshapta = 3 I have bent over.
9. nadshskeksh = 1 left over.
10. taunep = hand hand?

In describing this system Mr. Gatschet says: "If the origin of the Klamath
numerals is thus correctly traced, their inventors must have counted only
the four long fingers without the thumb, and 5 was counted while saying
_hand away! hand off!_ The 'four,' or _hand high! hand up!_ intimates that
the hand was held up high after counting its four digits; and some term
expressing this gesture was, in the case of _nine_, substituted by 'one
left over' ... which means to say, 'only one is left until all the fingers
are counted.'" It will be observed that the Klamath introduces not only the
ordinary finger manipulation, but a gesture of the entire hand as well. It
is a common thing to find something of the kind to indicate the completion
of 5 or 10, and in one or two instances it has already been alluded to.
Sometimes one or both of the closed fists are held up; sometimes the open
hand, with all the fingers extended, is used; and sometimes an entirely
independent gesture is introduced. These are, in general, of no special
importance; but one custom in vogue among some of the prairie tribes of
Indians, to which my attention was called by Dr. J. Owen Dorsey,[82] should
be mentioned. It is a gesture which signifies multiplication, and is
performed by throwing the hand to the left. Thus, after counting 5, a wave
of the hand to the left means 50. As multiplication is rather unusual among
savage tribes, this is noteworthy, and would seem to indicate on the part
of the Indian a higher degree of intelligence than is ordinarily possessed
by uncivilized races.

In the numeral scale as we possess it in English, we find it necessary to
retain the name of the last unit of each kind used, in order to describe
definitely any numeral employed. Thus, fifteen, one hundred forty-two, six
thousand seven hundred twenty-seven, give in full detail the numbers they
are intended to describe. In primitive scales this is not always considered
necessary; thus, the Zamucos express their teens without using their word
for 10 at all. They say simply, 1 on the foot, 2 on the foot, etc.
Corresponding abbreviations are often met; so often, indeed, that no
further mention of them is needed. They mark one extreme, the extreme of
brevity, found in the savage method of building up hand, foot, and finger
names for numerals; while the Zuni scale marks the extreme of prolixity in
the formation of such words. A somewhat ruder composition than any yet
noticed is shown in the numerals of the Vilelo scale,[83] which are:

1. agit, or yaagit.
2. uke.
3. nipetuei.
4. yepkatalet.
5. isig-nisle-yaagit = hand fingers 1.
6. isig-teet-yaagit = hand with 1.
7. isig-teet-uke = hand with 2.
8. isig-teet-nipetuei = hand with 3.
9. isig-teet-yepkatalet = hand with 4.
10. isig-uke-nisle = second hand fingers (lit. hand-two-fingers).
11. isig-uke-nisle-teet-yaagit = second hand fingers with 1.
20. isig-ape-nisle-lauel = hand foot fingers all.

In the examples thus far given, it will be noticed that the actual names of
individual fingers do not appear. In general, such words as thumb,
forefinger, little finger, are not found, but rather the hand-1, 1 on the
next, or 1 over and above, which we have already seen, are the type forms
for which we are to look. Individual finger names do occur, however, as in
the scale of the Hudson's Bay Eskimos,[84] where the three following words
are used both as numerals and as finger names:

8. kittukleemoot = middle finger.
9. mikkeelukkamoot = fourth finger.
10. eerkitkoka = little finger.

Words of similar origin are found in the original Jiviro scale,[85] where
the native numerals are:

1. ala.
2. catu.
3. cala.
4. encatu.
5. alacoetegladu = 1 hand.
6. intimutu = thumb (of second hand).
7. tannituna = index finger.
8. tannituna cabiasu = the finger next the index finger.
9. bitin oetegla cabiasu = hand next to complete.
10. catoegladu = 2 hands.

As if to emphasize the rarity of this method of forming numerals, the
Jiviros afterward discarded the last five of the above scale, replacing
them by words borrowed from the Quichuas, or ancient Peruvians. The same
process may have been followed by other tribes, and in this way numerals
which were originally digital may have disappeared. But we have no evidence
that this has ever happened in any extensive manner. We are, rather,
impelled to accept the occasional numerals of this class as exceptions to
the general rule, until we have at our disposal further evidence of an
exact and critical nature, which would cause us to modify this opinion. An
elaborate philological study by Dr. J.H. Trumbull[86] of the numerals used
by many of the North American Indian tribes reveals the presence in the
languages of these tribes of a few, but only a few, finger names which are
used without change as numeral expressions also. Sometimes the finger gives
a name not its own to the numeral with which it is associated in
counting--as in the Chippeway dialect, which has _nawi-nindj_, middle of
the hand, and _nisswi_, 3; and the Cheyenne, where _notoyos_, middle
finger, and _na-nohhtu_, 8, are closely related. In other parts of the
world isolated examples of the transference of finger names to numerals are
also found. Of these a well-known example is furnished by the Zulu
numerals, where "_tatisitupa_, taking the thumb, becomes a numeral for six.
Then the verb _komba_, to point, indicating the forefinger, or 'pointer,'
makes the next numeral, seven. Thus, answering the question, 'How much did
your master give you?' a Zulu would say, '_U kombile_,' 'He pointed with
his forefinger,' _i.e._ 'He gave me seven'; and this curious way of using
the numeral verb is also shown in such an example as '_amahasi akombile_,'
'the horses have pointed,' _i.e._ 'there were seven of them.' In like
manner, _Kijangalobili_, 'keep back two fingers,' _i.e._ eight, and
_Kijangalolunje_, 'keep back one finger,' _i.e._ nine, lead on to _kumi_,
ten."[87]

Returning for a moment to the consideration of number systems in the
formation of which the influence of the hand has been paramount, we find
still further variations of the method already noticed of constructing
names for the fives, tens, and twenties, as well as for the intermediate
numbers. Instead of the simple words "hand," "foot," etc., we not
infrequently meet with some paraphrase for one or for all these terms, the
derivation of which is unmistakable. The Nengones,[88] an island tribe of
the Indian Ocean, though using the word "man" for 20, do not employ
explicit hand or foot words, but count

1. sa.
2. rewe.
3. tini.
4. etse.
5. se dono = the end (of the first hand).
6. dono ne sa = end and 1.
7. dono ne rewe = end and 2.
8. dono ne tini = end and 3.
9. dono ne etse = end and 4.
10. rewe tubenine = 2 series (of fingers).
11. rewe tubenine ne sa re tsemene = 2 series and 1 on the next?
20. sa re nome = 1 man.
30. sa re nome ne rewe tubenine = 1 man and 2 series.
40. rewe ne nome = 2 men.

Examples like the above are not infrequent. The Aztecs used for 10 the word
_matlactli_, hand-half, _i.e._ the hand half of a man, and for 20
_cempoalli_, one counting.[89] The Point Barrow Eskimos call 10 _kodlin_,
the upper part, _i.e._ of a man. One of the Ewe dialects of Western
Africa[90] has _ewo_, done, for 10; while, curiously enough, 9, _asieke_,
is a digital word, meaning "to part (from) the hand."

In numerous instances also some characteristic word not of hand derivation
is found, like the Yoruba _ogodzi_, string, which becomes a numeral for 40,
because 40 cowries made a "string"; and the Maori _tekau_, bunch, which
signifies 10. The origin of this seems to have been the custom of counting
yams and fish by "bunches" of ten each.[91]

Another method of forming numeral words above 5 or 10 is found in the
presence of such expressions as second 1, second 2, etc. In languages of
rude construction and incomplete development the simple numeral scale is
often found to end with 5, and all succeeding numerals to be formed from
the first 5. The progression from that point may be 5-1, 5-2, etc., as in
the numerous quinary scales to be noticed later, or it may be second 1,
second 2, etc., as in the Niam Niam dialect of Central Africa, where the
scale is[92]

1. sa.
2. uwi.
3. biata.
4. biama.
5. biswi.
6. batissa = 2d 1.
7. batiwwi = 2d 2.
8. batti-biata = 2d 3.
9. batti-biama = 2d 4.
10. bauwe = 2d 5.

That this method of progression is not confined to the least developed
languages, however, is shown by a most cursory examination of the numerals
of our American Indian tribes, where numeral formation like that exhibited
above is exceedingly common. In the Kootenay dialect,[93] of British
Columbia, _qaetsa_, 4, and _wo-qaetsa,_ 8, are obviously related, the
latter word probably meaning a second 4. Most of the native languages of
British Columbia form their words for 7 and 8 from those which signify 2
and 3; as, for example, the Heiltsuk,[94] which shows in the following
words a most obvious correspondence:

2. matl. 7. matlaaus.
3. yutq. 8. yutquaus.

In the Choctaw language[95] the relation between 2 and 7, and 3 and 8, is
no less clear. Here the words are:

2. tuklo. 7. untuklo.
3. tuchina. 8. untuchina.

The Nez Perces[96] repeat the first three words of their scale in their 6,
7, and 8 respectively, as a comparison of these numerals will show.

1. naks. 6. oilaks.
2. lapit. 7. oinapt.
3. mitat. 8. oimatat.

In all these cases the essential point of the method is contained in the
repetition, in one way or another, of the numerals of the second quinate,
without the use with each one of the word for 5. This may make 6, 7, 8, and
9 appear as second 1, second 2, etc., or another 1, another 2, etc.; or,
more simply still, as 1 more, 2 more, etc. It is the method which was
briefly discussed in the early part of the present chapter, and is by no
means uncommon. In a decimal scale this repetition would begin with 11
instead of 6; as in the system found in use in Tagala and Pampanaga, two of
the Philippine Islands, where, for example, 11, 12, and 13 are:[97]

11. labi-n-isa = over 1.
12. labi-n-dalaua = over 2.
13. labi-n-tatlo = over 3.

A precisely similar method of numeral building is used by some of our
Western Indian tribes. Selecting a few of the Assiniboine numerals[98] as
an illustration, we have

11. ak kai washe = more 1.
12. ak kai noom pah = more 2.
13. ak kai yam me nee = more 3.
14. ak kai to pah = more 4.
15. ak kai zap tah = more 5.
16. ak kai shak pah = more 6, etc.

A still more primitive structure is shown in the numerals of the
Mboushas[99] of Equatorial Africa. Instead of using 5-1, 5-2, 5-3, 5-4, or
2d 1, 2d 2, 2d 3, 2d 4, in forming their numerals from 6 to 9, they proceed
in the following remarkable and, at first thought, inexplicable manner to
form their compound numerals:

1. ivoco.
2. beba.
3. belalo.
4. benai.
5. betano.
6. ivoco beba = 1-2.
7. ivoco belalo = 1-3.
8. ivoco benai = 1-4.
9. ivoco betano = 1-5.
10. dioum.

No explanation is given by Mr. du Chaillu for such an apparently
incomprehensible form of expression as, for example, 1-3, for 7. Some
peculiar finger pantomime may accompany the counting, which, were it known,
would enlighten us on the Mbousha's method of arriving at so anomalous a
scale. Mere repetition in the second quinate of the words used in the first
might readily be explained by supposing the use of fingers absolutely
indispensable as an aid to counting, and that a certain word would have one
meaning when associated with a certain finger of the left hand, and another
meaning when associated with one of the fingers of the right. Such scales
are, if the following are correct, actually in existence among the islands
of the Pacific.

BALAD.[100] UEA.[100]

1. parai. 1. tahi.
2. paroo. 2. lua.
3. pargen. 3. tolu.
4. parbai. 4. fa.
5. panim. 5. lima.
6. parai. 6. tahi.
7. paroo. 7. lua.
8. pargen. 8. tolu.
9. parbai. 9. fa.
10. panim. 10. lima.

Such examples are, I believe, entirely unique among primitive number
systems.

In numeral scales where the formative process has been of the general
nature just exhibited, irregularities of various kinds are of frequent
occurrence. Hand numerals may appear, and then suddenly disappear, just
where we should look for them with the greatest degree of certainty. In the
Ende,[101] a dialect of the Flores Islands, 5, 6, and 7 are of hand
formation, while 8 and 9 are of entirely different origin, as the scale
shows.

1. sa.
2. zua.
3. telu.
4. wutu.
5. lima
6. lima sa = hand 1.
7. lima zua = hand 2.
8. rua butu = 2 x 4.
9. trasa = 10 - 1?
10. sabulu.

One special point to be noticed in this scale is the irregularity that
prevails between 7, 8, 9. The formation of 7 is of the most ordinary kind;
8 is 2 fours--common enough duplication; while 9 appears to be 10 - 1. All
of these modes of compounding are, in their own way, regular; but the
irregularity consists in using all three of them in connective numerals in
the same system. But, odd as this jumble seems, it is more than matched by
that found in the scale of the Karankawa Indians,[102] an extinct tribe
formerly inhabiting the coast region of Texas. The first ten numerals of
this singular array are:

1. natsa.
2. haikia.
3. kachayi.
4. hayo hakn = 2 x 2.
5. natsa behema = 1 father, _i.e._ of the fingers.
6. hayo haikia = 3 x 2?
7. haikia natsa = 2 + 5?
8. haikia behema = 2 fathers?
9. haikia doatn = 2d from 10?
10. doatn habe.

Systems like the above, where chaos instead of order seems to be the ruling
principle, are of occasional occurrence, but they are decidedly the
exception.

In some of the cases that have been adduced for illustration it is to be
noticed that the process of combination begins with 7 instead of with 6.
Among others, the scale of the Pigmies of Central Africa[103] and that of
the Mosquitos[104] of Central America show this tendency. In the Pigmy
scale the words for 1 and 6 are so closely akin that one cannot resist the
impression that 6 was to them a new 1, and was thus named.

MOSQUITO. PIGMY.

1. kumi. ujju.
2. wal. ibari.
3. niupa. ikaro.
4. wal-wal = 2-2. ikwanganya.
5. mata-sip = fingers of 1 hand. bumuti.
6. matlalkabe. ijju.
7. matlalkabe pura kumi = 6 and 1. bumutti-na-ibali = 5 and 2.
8. matlalkabe pura wal = 6 and 2. bumutti-na-ikaro = 5 and 3.
9. matlalkabe pura niupa = 6 and 3. bumutti-na-ikwanganya = 5 and 4.
10. mata wal sip = fingers of 2 hands. mabo = half man.

The Mosquito scale is quite exceptional in forming 7, 8, and 9 from 6,
instead of from 5. The usual method, where combinations appear between 6
and 10, is exhibited by the Pigmy scale. Still another species of numeral
form, quite different from any that have already been noticed, is found in
the Yoruba[105] scale, which is in many respects one of the most peculiar
in existence. Here the words for 11, 12, etc., are formed by adding the
suffix _-la_, great, to the words for 1, 2, etc., thus:

1. eni, or okan.
2. edzi.
3. eta.
4. erin.
5. arun.
6. efa.
7. edze.
8. edzo.
9. esan.
10. ewa.
11. okanla = great 1.
12. edzila = great 2.
13. etala = great 3.
14. erinla = great 4, etc.
40. ogodzi = string.
200. igba = heap.

The word for 40 was adopted because cowrie shells, which are used for
counting, were strung by forties; and _igba_, 200, because a heap of 200
shells was five strings, and thus formed a convenient higher unit for
reckoning. Proceeding in this curious manner,[106] they called 50 strings 1
_afo_ or head; and to illustrate their singular mode of reckoning--the king
of the Dahomans, having made war on the Yorubans, and attacked their army,
was repulsed and defeated with a loss of "two heads, twenty strings, and
twenty cowries" of men, or 4820.

The number scale of the Abipones,[107] one of the low tribes of the
Paraguay region, contains two genuine curiosities, and by reason of those
it deserves a place among any collection of numeral scales designed to
exhibit the formation of this class of words. It is:

1. initara = 1 alone.
2. inoaka.
3. inoaka yekaini = 2 and 1.
4. geyenknate = toes of an ostrich.
5. neenhalek = a five coloured, spotted hide,
or hanambegen = fingers of 1 hand.
10. lanamrihegem = fingers of both hands.
20. lanamrihegem cat gracherhaka anamichirihegem = fingers of both
hands together with toes of both feet.

That the number sense of the Abipones is but little, if at all, above that
of the native Australian tribes, is shown by their expressing 3 by the
combination 2 and 1. This limitation, as we have already seen, is shared by
the Botocudos, the Chiquitos, and many of the other native races of South
America. But the Abipones, in seeking for words with which to enable
themselves to pass beyond the limit 3, invented the singular terms just
given for 4 and 5. The ostrich, having three toes in front and one behind
on each foot presented them with a living example of 3 + 1; hence "toes of
an ostrich" became their numeral for 4. Similarly, the number of colours in
a certain hide being five, the name for that hide was adopted as their next
numeral. At this point they began to resort to digital numeration also; and
any higher number is expressed by that method.

In the sense in which the word is defined by mathematicians, _number_ is a
pure, abstract concept. But a moment's reflection will show that, as it
originates among savage races, number is, and from the limitations of their
intellect must be, entirely concrete. An abstract conception is something
quite foreign to the essentially primitive mind, as missionaries and
explorers have found to their chagrin. The savage can form no mental
concept of what civilized man means by such a word as "soul"; nor would his
idea of the abstract number 5 be much clearer. When he says _five_, he
uses, in many cases at least, the same word that serves him when he wishes
to say _hand_; and his mental concept when he says _five_ is of a hand. The
concrete idea of a closed fist or an open hand with outstretched fingers,
is what is upper-most in his mind. He knows no more and cares no more about
the pure number 5 than he does about the law of the conservation of energy.
He sees in his mental picture only the real, material image, and his only
comprehension of the number is, "these objects are as many as the fingers
on my hand." Then, in the lapse of the long interval of centuries which
intervene between lowest barbarism and highest civilization, the abstract
and the concrete become slowly dissociated, the one from the other. First
the actual hand picture fades away, and the number is recognized without
the original assistance furnished by the derivation of the word. But the
number is still for a long time a certain number _of objects_, and not an
independent concept. It is only when the savage ceases to be wholly an
animal, and becomes a thinking human being, that number in the abstract can
come within the grasp of his mind. It is at this point that mere reckoning
ceases, and arithmetic begins.

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