The Number Concept: Its Origin and Development. Levi L. Conant

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, Murdoch⁴⁶ 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 Abeokuta⁴⁹ 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 Quincey⁵⁰ 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