NUMBER SYSTEM LIMITS.
With respect to the limits to which the number systems of the various
uncivilized races of the earth extend, recent anthropological research has
developed many interesting facts. In the case of the Chiquitos and a few
other native races of Bolivia we found no distinct number sense at all, as
far as could be judged from the absence, in their language, of numerals in
the proper sense of the word. How they indicated any number greater than
_one_ is a point still requiring investigation. In all other known
instances we find actual number systems, or what may for the sake of
uniformity be dignified by that name. In many cases, however, the numerals
existing are so few, and the ability to count is so limited, that the term
_number system_ is really an entire misnomer.
Among the rudest tribes, those whose mode of living approaches most nearly
to utter savagery, we find a certain uniformity of method. The entire
number system may consist of but two words, _one_ and _many_; or of three
words, _one_, _two_, _many_. Or, the count may proceed to 3, 4, 5, 10, 20,
or 100; passing always, or almost always, from the distinct numeral limit
to the indefinite _many_ or several, which serves for the expression of any
number not readily grasped by the mind. As a matter of fact, most races
count as high as 10; but to this statement the exceptions are so numerous
that they deserve examination in some detail. In certain parts of the
world, notably among the native races of South America, Australia, and many
of the islands of Polynesia and Melanesia, a surprising paucity of numeral
words has been observed. The Encabellada of the Rio Napo have but two
distinct numerals; _tey_, 1, and _cayapa_, 2.[20] The Chaco languages[21]
of the Guaycuru stock are also notably poor in this respect. In the Mbocobi
dialect of this language the only native numerals are _yña tvak_, 1, and
_yfioaca_, 2. The Puris[22] count _omi_, 1, _curiri_, 2, _prica_, many; and
the Botocudos[23] _mokenam_, 1, _uruhu_, many. The Fuegans,[24] supposed to
have been able at one time to count to 10, have but three
numerals,--_kaoueli_, 1, _compaipi_, 2, _maten_, 3. The Campas of Peru[25]
possess only three separate words for the expression of number,--_patrio_,
1, _pitteni_, 2, _mahuani_, 3. Above 3 they proceed by combinations, as 1
and 3 for 4, 1 and 1 and 3 for 5. Counting above 10 is, however, entirely
inconceivable to them, and any number beyond that limit they indicate by
_tohaine_, many. The Conibos,[26] of the same region, had, before their
contact with the Spanish, only _atchoupre_, 1, and _rrabui_, 2; though they
made some slight progress above 2 by means of reduplication. The Orejones,
one of the low, degraded tribes of the Upper Amazon,[27] have no names for
number except _nayhay_, 1, _nenacome_, 2, _feninichacome_, 3,
_ononoeomere_, 4. In the extensive vocabularies given by Von Martins,[28]
many similar examples are found. For the Bororos he gives only _couai_, 1,
_maeouai_, 2, _ouai_, 3. The last word, with the proper finger pantomime,
serves also for any higher number which falls within the grasp of their
comprehension. The Guachi manage to reach 5, but their numeration is of the
rudest kind, as the following scale shows: _tamak_, 1, _eu-echo,_ 2,
_eu-echo-kailau,_ 3, _eu-echo-way,_ 4, _localau_, 5. The Carajas counted by
a scale equally rude, and their conception of number seemed equally vague,
until contact with the neighbouring tribes furnished them with the means of
going beyond their original limit. Their scale shows clearly the uncertain,
feeble number sense which is so marked in the interior of South America. It
contains _wadewo_, 1, _wadebothoa_, 2, _wadeboaheodo_, 3, _wadebojeodo_,
4, _wadewajouclay_, 5, _wadewasori_, 6, or many.
Turning to the languages of the extinct, or fast vanishing, tribes of
Australia, we find a still more noteworthy absence of numeral expressions.
In the Gudang dialect[29] but two numerals are found--_pirman_, 1, and
_ilabiu_, 2; in the Weedookarry, _ekkamurda_, 1, and _kootera_, 2; and in
the Queanbeyan, _midjemban_, 1, and _bollan_, 2. In a score or more of
instances the numerals stop at 3. The natives of Keppel Bay count _webben_,
1, _booli_, 2, _koorel_, 3; of the Boyne River, _karroon_, 1, _boodla_, 2,
_numma_, 3; of the Flinders River, _kooroin_, 1, _kurto_, 2, _kurto
kooroin_, 3; at the mouth of the Norman River, _lum_, 1, _buggar_, 2,
_orinch_, 3; the Eaw tribe, _koothea_, 1, _woother_, 2, _marronoo_, 3; the
Moree, _mal_, 1, _boolar_, 2, _kooliba_, 3; the Port Essington,[30] _erad_,
1, _nargarick_, 2, _nargarickelerad_, 3; the Darnly Islanders,[31] _netat_,
1, _naes_, 2, _naesa netat_, 3; and so on through a long list of tribes
whose numeral scales are equally scanty. A still larger number of tribes
show an ability to count one step further, to 4; but beyond this limit the
majority of Australian and Tasmanian tribes do not go. It seems most
remarkable that any human being should possess the ability to count to 4,
and not to 5. The number of fingers on one hand furnishes so obvious a
limit to any of these rudimentary systems, that positive evidence is needed
before one can accept the statement. A careful examination of the numerals
in upwards of a hundred Australian dialects leaves no doubt, however, that
such is the fact. The Australians in almost all cases count by pairs; and
so pronounced is this tendency that they pay but little attention to the
fingers. Some tribes do not appear ever to count beyond 2--a single pair.
Many more go one step further; but if they do, they are as likely as not to
designate their next numeral as two-one, or possibly, one-two. If this step
is taken, we may or may not find one more added to it, thus completing the
second pair. Still, the Australian's capacity for understanding anything
which pertains to number is so painfully limited that even here there is
sometimes an indefinite expression formed, as many, heap, or plenty,
instead of any distinct numeral; and it is probably true that no Australian
language contains a pure, simple numeral for 4. Curr, the best authority on
this subject, believes that, where a distinct word for 4 is given,
investigators have been deceived in every case.[32] If counting is carried
beyond 4, it is always by means of reduplication. A few tribes gave
expressions for 5, fewer still for 6, and a very small number appeared able
to reach 7. Possibly the ability to count extended still further; but if
so, it consisted undoubtedly in reckoning one pair after another, without
any consciousness whatever of the sum total save as a larger number.
The numerals of a few additional tribes will show clearly that all distinct
perception of number is lost as soon as these races attempt to count above
3, or at most, 4. The Yuckaburra[33] natives can go no further than
_wigsin_, 1, _bullaroo_, 2, _goolbora_, 3. Above here all is referred to as
_moorgha_, many. The Marachowies[34] have but three distinct
numerals,--_cooma_, 1, _cootera_, 2, _murra_, 3. For 4 they say _minna_,
many. At Streaky Bay we find a similar list, with the same words, _kooma_
and _kootera_, for 1 and 2, but entirely different terms, _karboo_ and
_yalkata_ for 3 and many. The same method obtains in the Minnal Yungar
tribe, where the only numerals are _kain_, 1, _kujal_, 2, _moa_, 3, and
_bulla_, plenty. In the Pinjarra dialect we find _doombart_, 1, _gugal_, 2,
_murdine_, 3, _boola_, plenty; and in the dialect described as belonging to
"Eyre's Sand Patch," three definite terms are given--_kean_, 1, _koojal_,
2, _yalgatta_, 3, while a fourth, _murna_, served to describe anything
greater. In all these examples the fourth numeral is indefinite; and the
same statement is true of many other Australian languages. But more
commonly still we find 4, and perhaps 3 also, expressed by reduplication.
In the Port Mackay dialect[35] the latter numeral is compound, the count
being _warpur_, 1, _boolera_, 2, _boolera warpur_, 3. For 4 the term is not
given. In the dialect which prevailed between the Albert and Tweed
rivers[36] the scale appears as _yaburu_, 1, _boolaroo_, 2, _boolaroo
yaburu_, 3, and _gurul_ for 4 or anything beyond. The Wiraduroi[37] have
_numbai_, 1, _bula_, 2, _bula numbai_, 3, _bungu_, 4, or many, and _bungu
galan_ or _bian galan_, 5, or very many. The Kamilaroi[38] scale is still
more irregular, compounding above 4 with little apparent method. The
numerals are _mal_, 1, _bular_, 2, _guliba_, 3, _bular bular_, 4, _bular
guliba_, 5, _guliba guliba_, 6. The last two numerals show that 5 is to
these natives simply 2-3, and 6 is 3-3. For additional examples of a
similar nature the extended list of Australian scales given in Chapter V.
may be consulted.
Taken as a whole, the Australian and Tasmanian tribes seem to have been
distinctly inferior to those of South America in their ability to use and
to comprehend numerals. In all but two or three cases the Tasmanians[39]
were found to be unable to proceed beyond 2; and as the foregoing examples
have indicated, their Australian neighbours were but little better off. In
one or two instances we do find Australian numeral scales which reach 10,
and perhaps we may safely say 20. One of these is given in full in a
subsequent chapter, and its structure gives rise to the suspicion that it
was originally as limited as those of kindred tribes, and that it underwent
a considerable development after the natives had come in contact with the
Europeans. There is good reason to believe that no Australian in his wild
state could ever count intelligently to 7.[40]
In certain portions of Asia, Africa, Melanesia, Polynesia, and North
America, are to be found races whose number systems are almost and
sometimes quite as limited as are those of the South. American and
Australian tribes already cited, but nowhere else do we find these so
abundant as in the two continents just mentioned, where example after
example might be cited of tribes whose ability to count is circumscribed
within the narrowest limits. The Veddas[41] of Ceylon have but two
numerals, _ekkame[=i]_, 1, _dekkameï_, 2. Beyond this they count
_otameekaï, otameekaï, otameekaï_, etc.; _i.e._ "and one more, and one
more, and one more," and so on indefinitely. The Andamans,[42] inhabitants
of a group of islands in the Bay of Bengal, are equally limited in their
power of counting. They have _ubatulda_, 1, and _ikporda_, 2; but they can
go no further, except in a manner similar to that of the Veddas. Above two
they proceed wholly by means of the fingers, saying as they tap the nose
with each successive finger, _anka_, "and this." Only the more intelligent
of the Andamans can count at all, many of them seeming to be as nearly
destitute of the number sense as it is possible for a human being to be.
The Bushmen[43] of South Africa have but two numerals, the pronunciation of
which can hardly be indicated without other resources than those of the
English alphabet. Their word for 3 means, simply, many, as in the case of
some of the Australian tribes. The Watchandies[44] have but two simple
numerals, and their entire number system is _cooteon_, 1, _utaura_, 2,
_utarra cooteoo_, 3, _atarra utarra_, 4. Beyond this they can only say,
_booltha_, many, and _booltha bat_, very many. Although they have the
expressions here given for 3 and 4, they are reluctant to use them, and
only do so when absolutely required. The natives of Lower California[45]
cannot count above 5. A few of the more intelligent among them understand
the meaning of 2 fives, but this number seems entirely beyond the
comprehension of the ordinary native. The Comanches, curiously enough, are
so reluctant to employ their number words that they appear to prefer finger
pantomime instead, thus giving rise to the impression which at one time
became current, that they had no numerals at all for ordinary counting.
Aside from the specific examples already given, a considerable number of
sweeping generalizations may be made, tending to show how rudimentary the
number sense may be in aboriginal life. Scores of the native dialects of
Australia and South America have been found containing number systems but
little more extensive than those alluded to above. The negro tribes of
Africa give the same testimony, as do many of the native races of Central
America, Mexico, and the Pacific coast of the United States and Canada, the
northern part of Siberia, Greenland, Labrador, and the arctic archipelago.
In speaking of the Eskimos of Point Barrow, Murdoch[46] says: "It was not
easy to obtain any accurate information about the numeral system of these
people, since in ordinary conversation they are not in the habit of
specifying any numbers above five." Counting is often carried higher than
this among certain of these northern tribes, but, save for occasional
examples, it is limited at best. Dr. Franz Boas, who has travelled
extensively among the Eskimos, and whose observations are always of the
most accurate nature, once told the author that he never met an Eskimo who
could count above 15. Their numerals actually do extend much higher; and a
stray numeral of Danish origin is now and then met with, showing that the
more intelligent among them are able to comprehend numbers of much greater
magnitude than this. But as Dr. Boas was engaged in active work among them
for three years, we may conclude that the Eskimo has an arithmetic but
little more extended than that which sufficed for the Australians and the
forest tribes of Brazil. Early Russian explorers among the northern tribes
of Siberia noticed the same difficulty in ordinary, every-day reckoning
among the natives. At first thought we might, then, state it as a general
law that those races which are lowest in the scale of civilization, have
the feeblest number sense also; or in other words, the least possible power
of grasping the abstract idea of number.
But to this law there are many and important exceptions. The concurrent
testimony of explorers seems to be that savage races possess, in the great
majority of cases, the ability to count at least as high as 10. This limit
is often extended to 20, and not infrequently to 100. Again, we find 1000
as the limit; or perhaps 10,000; and sometimes the savage carries his
number system on into the hundreds of thousands or millions. Indeed, the
high limit to which some savage races carry their numeration is far more
worthy of remark than the entire absence of the number sense exhibited by
others of apparently equal intelligence. If the life of any tribe is such
as to induce trade and barter with their neighbours, a considerable
quickness in reckoning will be developed among them. Otherwise this power
will remain dormant because there is but little in the ordinary life of
primitive man to call for its exercise.
In giving 1, 2, 3, 5, 10, or any other small number as a system limit, it
must not be overlooked that this limit mentioned is in all cases the limit
of the spoken numerals at the savage's command. The actual ability to count
is almost always, and one is tempted to say always, somewhat greater than
their vocabularies would indicate. The Bushman has no number word that will
express for him anything higher than 2; but with the assistance of his
fingers he gropes his way on as far as 10. The Veddas, the Andamans, the
Guachi, the Botocudos, the Eskimos, and the thousand and one other tribes
which furnish such scanty numeral systems, almost all proceed with more or
less readiness as far as their fingers will carry them. As a matter of
fact, this limit is frequently extended to 20; the toes, the fingers of a
second man, or a recount of the savage's own fingers, serving as a tale for
the second 10. Allusion is again made to this in a later chapter, where the
subject of counting on the fingers and toes is examined more in detail.
In saying that a savage can count to 10, to 20, or to 100, but little idea
is given of his real mental conception of any except the smallest numbers.
Want of familiarity with the use of numbers, and lack of convenient means
of comparison, must result in extreme indefiniteness of mental conception
and almost entire absence of exactness. The experience of Captain
Parry,[47] who found that the Eskimos made mistakes before they reached 7,
and of Humboldt,[48] who says that a Chayma might be made to say that his
age was either 18 or 60, has been duplicated by all investigators who have
had actual experience among savage races. Nor, on the other hand, is the
development of a numeral system an infallible index of mental power, or of
any real approach toward civilization. A continued use of the trading and
bargaining faculties must and does result in a familiarity with numbers
sufficient to enable savages to perform unexpected feats in reckoning.
Among some of the West African tribes this has actually been found to be
the case; and among the Yorubas of Abeokuta[49] the extraordinary saying,
"You may seem very clever, but you can't tell nine times nine," shows how
surprisingly this faculty has been developed, considering the general
condition of savagery in which the tribe lived. There can be no doubt that,
in general, the growth of the number sense keeps pace with the growth of
the intelligence in other respects. But when it is remembered that the
Tonga Islanders have numerals up to 100,000, and the Tembus, the Fingoes,
the Pondos, and a dozen other South African tribes go as high as 1,000,000;
and that Leigh Hunt never could learn the multiplication table, one must
confess that this law occasionally presents to our consideration remarkable
exceptions.
While considering the extent of the savage's arithmetical knowledge, of his
ability to count and to grasp the meaning of number, it may not be amiss to
ask ourselves the question, what is the extent of the development of our
own number sense? To what limit can we absorb the idea of number, with a
complete appreciation of the idea of the number of units involved in any
written or spoken quantity? Our perfect system of numeration enables us to
express without difficulty any desired number, no matter how great or how
small it be. But how much of actually clear comprehension does the number
thus expressed convey to the mind? We say that one place is 100 miles from
another; that A paid B 1000 dollars for a certain piece of property; that a
given city contains 10,000 inhabitants; that 100,000 bushels of wheat were
shipped from Duluth or Odessa on such a day; that 1,000,000 feet of lumber
were destroyed by the fire of yesterday,--and as we pass from the smallest
to the largest of the numbers thus instanced, and from the largest on to
those still larger, we repeat the question just asked; and we repeat it
with a new sense of our own mental limitation. The number 100
unquestionably stands for a distinct conception. Perhaps the same may be
said for 1000, though this could not be postulated with equal certainty.
But what of 10,000? If that number of persons were gathered together into a
single hall or amphitheatre, could an estimate be made by the average
onlooker which would approximate with any degree of accuracy the size of
the assembly? Or if an observer were stationed at a certain point, and
10,000 persons were to pass him in single file without his counting them as
they passed, what sort of an estimate would he make of their number? The
truth seems to be that our mental conception of number is much more limited
than is commonly thought, and that we unconsciously adopt some new unit as
a standard of comparison when we wish to render intelligible to our minds
any number of considerable magnitude. For example, we say that A has a
fortune of $1,000,000. The impression is at once conveyed of a considerable
degree of wealth, but it is rather from the fact that that fortune
represents an annual income of $40,000 than, from the actual magnitude of
the fortune itself. The number 1,000,000 is, in itself, so greatly in
excess of anything that enters into our daily experience that we have but a
vague conception of it, except as something very great. We are not, after
all, so very much better off than the child who, with his arms about his
mother's neck, informs her with perfect gravity and sincerity that he
"loves her a million bushels." His idea is merely of some very great
amount, and our own is often but little clearer when we use the expressions
which are so easily represented by a few digits. Among the uneducated
portions of civilized communities the limit of clear comprehension of
number is not only relatively, but absolutely, very low. Travellers in
Russia have informed the writer that the peasants of that country have no
distinct idea of a number consisting of but a few hundred even. There is no
reason to doubt this testimony. The entire life of a peasant might be
passed without his ever having occasion to use a number as great as 500,
and as a result he might have respecting that number an idea less distinct
than a trained mathematician would have of the distance from the earth to
the sun. De Quincey[50] incidentally mentions this characteristic in
narrating a conversation which occurred while he was at Carnarvon, a little
town in Wales. "It was on this occasion," he says, "that I learned how
vague are the ideas of number in unpractised minds. 'What number of people
do you think,' I said to an elderly person, 'will be assembled this day at
Carnarvon?' 'What number?' rejoined the person addressed; 'what number?
Well, really, now, I should reckon--perhaps a matter of four million.' Four
millions of _extra_ people in little Carnarvon, that could barely find
accommodation (I should calculate) for an extra four hundred!" So the
Eskimo and the South American Indian are, after all, not so very far behind
the "elderly person" of Carnarvon, in the distinct perception of a number
which familiarity renders to us absurdly small.