Popular Science Monthly/Volume 34/December 1888/Animal Arithmetic
ALL degrees of arithmetical aptitude may be found among the human races, from the genius of a Newton and a Laplace to an absolute inability to conceive the abstract notion of number aside from concrete facts furnished by direct perception. The savage is not deficient in the perception of the multiple. He never confounds one tree with two, or two with three, or four; and is well aware of the difference between two, three, or ten men, when he is going out to fight them. The thing that he can not do is to abstract the idea of a number from the things to which it is attached, and generalize it, without reference to the concrete objects with which he has seen it associated. He may comprehend two, because it is associated with his two hands and his two feet; three, by the aid of the triads with which he is acquainted, and of the triangle with which those objects may be arranged; four, from the four limbs of animals and the four corners of a square. But his ability to form such conceptions is very limited. The first steps in learning in this direction, in savages and children, are to distinguish the abstract notions of unity and plurality, and in plurality, of two and three from larger pluralities. The difficulty in the way of their reaching a concept of abstract numbers is their inability to form a mental representation: four trees not being identical in the savage's thought with four stones, he can not imagine that there is anything common between them. Nevertheless, he can distinguish clearly enough between four trees and three others, and the two groups will leave quite different impressions in his mind. Four trees in a row will also make a different impression from four trees in a square. He is most struck with differences of distribution in space, and derives from them his notions of differences in plurality. While he is a poor arithmetician he is a good geometrician.
It is by the exercise of this faculty that he finds his way so readily where he has once gone. He recognizes a wood he has been in by the relative distances apart of the trees, their heights, sizes, the inclination of their trunks to one another, the profile of their masses, and their kinds. He learns the landscape by the relief and accidents of the ground, the wave-lines of the horizon, and a thousand details which he fixes upon his memory by a single keen observation so clearly as to give imagination no chance to play tricks with him. He estimates distances by the weakening of tones and the convergence of the lines and planes of the perspective. But in all his comparisons of present and past sensations, definite notions of number rarely have any part.
The inequalities in the development of the power of counting and of arithmetical notions among the different human races should make us cautious about accepting the accounts of those who believe that some animals can be taught to count. It is extremely improbable, at least, that animals, however intelligent, without language, should acquire precise ideas which human beings succeed in apprehending only by the aid of language and education.
The animal can distinguish relative sizes, but the measure of quantities escapes him. Like the child and savage of inferior development, he can only distinguish between few and many, between unity, duality, and plurality, the various degrees of which must inevitably be more or less confused in his perceptions. He knows facts, but he distinguishes them chiefly by their order of coexistence in space, rather than of succession in time; and they are photographed in his brain in views of the whole, of which, not being able to separate them from the others, he can not distinguish the similar parts enough to count them. His recollection of places is a succession of kaleidoscopic pictures, in which, among the moving forms and successive sounds, he marks only what moves him, what gives him pleasure or pain, what flatters his instincts or answers to his wants, what arouses fear or desire in him. If he should go so far as to count, it would be only those objects which interest him with a view to his security.
The wolf and the fox, for instance, can distinguish whether a flock is guarded by one or two dogs, but can not tell any better than we can how many sheep there are in the flock; but if they see one or two separated from the rest, they will attack them. In the same way we can distinguish the forms of two, three, or four poplars on the bank of a stream, while we can not tell the number of trees in an avenue, but are always inclined to exaggerate it.
It is by this observation of lines, directions, and signs in the whole, rather than of numbers, that the animal acquires the faculty of recognizing roads it has passed over, and places where it has met its prey or escaped its enemies. It can orient itself to the horizon and measure distances, as the savage does; and, like him, it has the sense of the direction it ought to take to reach the object it is in search of; but it is reasonably certain that no calculation of any number of units enters into this intelligence. When a dog in hunting crosses a wood or a fallow ground, he is able, by a quick apperception, to describe all the curves and all the angles that permit him to avoid or turn the obstacles. He can adjust his leap to the width of the ditch which he has to jump, and find the point where he can pass a live hedge. This mechanical instinct is exercised spontaneously in him, by virtue of exact theorems of which he has no analytical consciousness, but which he applies with a precision that defies the ingenuity of mathematicians. Force, speed, mass, space, and time are calculated in an equation in which the means are closely adapted to the end, with a marvelous economy of efforts, and by the quickest if not always the shortest way.
In his relations with man, attentive to the will of his master, whose language he has learned to understand, partially at least, the dog comprehends whether he must go or come, run or stop. But he attends less to the phonetics of the words than to the intonations of the voice, the direction of the glance, and the gestures.
The horse knows whether he is to go to the right or left by the energetically pronounced interjections "gee" and "haw," but he measures the extent to which he must veer in either direction by the pull on the bridle, so much as to turn quite round if the pull is at once energetic and quick. If theis equal upon both sides, he stops abruptly on his haunches. He learns the language of the stable in a very short time, much less than it takes to train the horseman. But there is no notion of numbers in it, although the horse is very sensitive to musical rhythm.
We can not, at a glance, estimate the number of men in a regiment, but we can calculate it rapidly, when we have noticed the number in a line, the number of lines in a company, and the number of companies. We are still more incapable of distinguishing one hundred points in rows from ninety or one hundred and twenty, without dividing them into tens. But an animal, which has no faculty of numerical abstraction, or of unities of different orders, or of multiplication, would be absolutely incapable of performing this operation. When small numbers of concrete things are in question, however, the aptitude of animals to distinguish these numbers is evident, and is, moreover, indispensable to their existence. A wolf or a boar, which could not perceive how many dogs were attacking it, would not know how to defend itself; but, if it counts them, it is by means of the tactics by which it opposes them, and in which it follows geometrical rules; for if four dogs attack it, two on the right and two on the left, it retreats, facing them, so as not to be between them, but to hold them as much as possible in front and within reach of its tusks. But if the pack is too numerous, the animal becomes wild, loses a clear notion of the number of its adversaries, and bites at hazard the one which presses closest upon it. A man in a similar situation would do much the same.
Birds have at least a vague idea of the number of eggs in their nests, for we can not take one out without causing in them a disquiet that becomes greater if we remove more. But they manifest a like distress when their eggs are only disarranged. Is this because the geometrical arrangement of the eggs is changed? Five eggs or four make a symmetrical arrangement as the bird disposes of them. But if some are taken out, and three, or two, or one are left, the disposition is very perceptibly changed. When the little birds have been hatched, the differences in their size, liveliness, figure, and voice, give the mother a means of distinguishing them individually. And even the eggs are perhaps not so indistinguishable to her as to us; for sexual and maternal instinct conveys special faculties in these matters. The domestic fowl seems to be less intelligent in such things than sparrows and other wild birds, but this is because domestication has modified her instincts.
Cats certainly know how many kittens they have, but they seem less affected by the loss of one of them according as more are left. If the mother loses one of four or five, she seeks for it a little while with considerable anxiety, and then becomes reconciled to the loss. If only one is left, she becomes greatly troubled, and, if that is taken from her, her distress appears extreme. This may be because of the pain she suffers in her teats when the milk ceases to be removed. When the young have become weaned, she can witness their disappearance with apparent indifference.
Dogs have been observed on various occasions to exhibit primary numerical perceptions in the concrete. When there are a number of them in a house, they quickly remark the absence of one of their companions. But they are rarely troubled by it, and make no effort to find the missing one, and they are still more ready to take notice of the absence of one of the members of their master's family. These traits are more easily explained by the clear knowledge which dogs have of individualities than by ascribing to them notions of unities as forming parts of numbers. The unequal degrees of attachment which they show for the different members of the family, and for the different persons who live in the house or visit it, proves that they make great differences between them. The idea of difference between several persons involves and supports a notion of their number. But it may lie there if wrapped up in their total perception.
Hounds pursuing a hare are troubled for an instant if they raise another one, and will sometimes stop, as if they were uncertain which one they ought to follow. Good dogs will not allow themselves to be diverted from the scent of the first animal, which they have already tired. When the setter pauses before a flock of partridges, the movements of his head and eyes follow the birds that stray to the right or left. If the flock is large, he can not estimate their number, nor can the hunter. The shepherd's dog watches his flock of sheep, and goes after the individual members that wander from it. But if one or two of them have been sold or carried off by a wolf during his absence, he does not miss them.
It is wholly improbable that the so-called learned dogs, which are said to have been taught to count, have really been lifted to the abstract notion of numbers. They have simply been taught to associate certain signs or words of their master with particular graphic signs, the geometrical figure of which has been impressed upon their memory. In the same way that the horse associates the words "gee!" and "haw!" with his right and his left, and dogs associate the sound of the horn or trumpet with the chase, they may associate with figures which are shown to them the names as pronounced of those figures, but without comprehension of their numerical relations, and without distinguishing them in any other way than by the difference in their shape. They may be taught to arrange them when commanded in any particular order, without it being necessary to suppose that they have any idea of their arithmetical significance.
"When Sir John Lubbock speculated about teaching his dog to read, he played upon this faculty of associating vocal signs with certain forms or figures, and even with graphic signs, figures, or letters. The dog certainly had no comprehension of the ideographic value of the figures which were shown him drawn upon the pasteboard; but their shape, stamped in his memory, was associated with the sounds that were spoken to him when they were shown him, and with certain acts that he was to perform to obtain caresses and rewards from his master. He thus soon learned to pick out the cards which he had to bring to ask for drink or food or to go out in order to have his desire satisfied. The quickest way of teaching children to read is to show them at the same time the image of the object and the word that designates it, so that the two shall be associated in their minds, and they are tempted to speak in the same way. But to appreciate the abstract sense of the noun or the verb requires a degree of intelligence and faculties of comparison which none of our domestic animals has as yet attained.
It would, moreover, be very extraordinary to find in an animal so far removed in organization from man as the dog a quasi-identity of mental faculties, and an educability which is wanting in entire human races. It is enough to show that between the intellectual state of dogs and of Bushmen, Tasmanians, and Veddahs, who can count only two, and then say "many," the difference is as slight as possible, and the passage insensible.
It is, moreover, evident that horses and dogs know as well as a savage that they have four limbs. Foxes caught in trails will use the most ingenious devices to extricate themselves, and will even gnaw off one of their paws rather than be prisoners. This requires an effort of the will contrary to the instincts, surpassing the degree of moral energy of which most men are capable; and, further than this, the act demonstrates a power of applying means to the end, which is an act of intelligence not less complicated than the effort required for counting twenty on the fingers and toes, as is the manner with most savages.
The faculty of abstraction and generalization is developed exclusively by the aid of descriptive and ideological language, which classifies things and acts under different words or auditive images. With inferior races this faculty is weaker in proportion as their language is less analytical and less rich in abstract terms. It is impossible to excite it in an animal, because, in the absence of a language common to man and him, he is destitute of every means of acquiring it. There is really no bridge between animal and human intelligence. While our language, being descriptive and objective, associates a sound with each visual image, the language of the animal only expresses emotions and passions. As a rule, it is as untranslatable to us as our language is to them. It is only when we try to paint, describe, relate, and express ideas, that they can not understand us, for nothing is easier than to cause them to share our emotions, tenderness, anger, or hatred. They understand our mimicry better than we can understand theirs, and by mimicry we can make them understand the causes of our emotions of a certain kind. The only condition is, that we be dealing with species of a social nature.
When a hunting-dog sees his master with his gaiters, carrying his game-bag and gun, he understands that he is to go with him. He may even have acquired the habit of associating the recollection of a sound with these objects, and thus know the names by which we designate them in the language which he hears us speak. He may also be taught to fetch the gaiters, shoes, and game-bag when told to do so. If, when he has brought one shoe, he is told to fetch the other, he understands that there are two. To this point he certainly has the notion of duality. He can not be ignorant, after he has executed this order several times, that the words "the other" mean the second shoe. If, after having been trained by an English master, he passes over to a Frenchman, he learns that his order l'autre means the same thing. He takes no notice of the difference in the sound of the words, because they are both uttered with the same accentuation and intonation, and under the impulse of the same feelings. To him human speech is a yelping, which he interprets by the same rule as he does his own ejaculations.
A second unit, added to the first, is in reality the beginning of all numeration and the foundation of arithmetic. As there are human races that have never gone further, we need not be surprised if animals stop there. But they do not stop there. They go on from this by successive additions, while we have reached the stage of multiplication, and have framed arbitrary systems of numeration, and have thus made of calculation an art founded on ideal notions. Animals, on the other hand, have a concrete notion of numbers more highly developed than we suppose, and perhaps more highly developed than it is with us, in proportion as abstraction is less easy with them. We need not suppose that the animal is destitute of all abstract notions and incapable of all generalization. Far from it; but the general notions being made of resemblances and the individual notions of differences, it is more struck than we are by individual characteristics. In this it again approaches the child and the savage, neither of whom has the generic notion of man. The child has the individual notion of its mother or its nurse, whom it distinguishes from all other persons around; but the generic notion, composed of all the common traits of the persons around, is of slow growth. The languages of savages are, for the most part, wanting in words for tree or animal, to comprehend the class, but have definite names for all the trees and animals that are useful to the tribe, or which they fear. We, therefore, may affirm that the dog has no generic idea of man, animal, or plant, but only ideas of particular men, particular women or children; and that every species, whether of animal or plant, is thought of by it as a representation of its individual figure, with all the differences that distinguish it from the others that it has seen. Our imagination by itself can not bring up the idea of an animal or plant which is not a particular animal or plant; and any effort we may make in this direction will end in there passing through the mind a succession of images of different animals and plants. If the use of generic names is taken from us, the general notion will go with it.
In the absence of articulate and descriptive language, and there being no object competent to serve us as a phonetic and auditive representation, we would think directly of things by a kind of precise interior view that permits no error or verbal sophism, and not as by a kind of internal audition that tends to replace things by their names, that makes us speak our thought within ourselves before speaking it aloud, and which we mistake as well as deceive others, when the interior definition which we give to the words does not correspond with the thing defined. It is especially difficult, in the absence of a common language between man and the animal, to make the latter comprehend what we require from it, and the object of the acts which we solicit it to perform. The dog has no desire except to obey and please us; the trouble is in explaining to him what we want of him.
A dog had been taught to go, when commanded, to the shed for wood for the fireplace. The exercise amused him; and, when he had brought one stick, he liked nothing better than to return for another, so that he had to be told to stop. But one time, when he was alone and lonesome, he pulled down all the wood, stick by stick. He had not comprehended the purpose of the act which they made him perform, supposing it to be a sport, like the ordinary carrying of a stick. Could this dog have been taught to count by sending him for two sticks and then for three, and so on to larger numbers? We doubt it, because he had not even disengaged from the act which he was ordered to do the general idea that all the pieces of wood which he brought were to be burned in the fireplace, that he was never sent for them except for that purpose, and that he should only fetch as many as were needed.
If efforts to educate animals have been even more fruitless in the hands of scientific investigators than of workingmen proceeding without theoretical views, it is because great errors have been committed in the analysis of human faculties, in making such suppositions, for instance, as that arithmetical notions are more elementary than geometrical ones. Having done this, they have sought to teach animals, whose capability is for measuring, to count. Having become habituated by our industrial civilization and the economical laws of exchange to the intervention of the idea of number in all our wants, acts, and works, we have lost perception of the insignificant part which it has in animal life as compared with that of the idea of size. Animals have a very exact sense of size. They can measure time and distance better than we can. The sparrows in our parks, when affecting the highest degree of confidence in us, know how to keep just enough distance from us to be able to evade us. It also seems to be demonstrated that all animals have more or less of the faculty of estimating the number of objects coexisting in space; that is, in a varying degree, of analyzing the similar or identical elements in their visual or auditive perceptions, so long as the number is small enough.
Have they also the faculty of estimating numbers as successive repetitions of the same facts in time, or of counting the reiteration of the same perceptions? I was once told of a workman who was in the habit of giving sugar every day to a dog which he met in going to his work. The dog counted on his daily return. He gave three pieces of sugar, one after the other, and the dog would wait and look till it had got the third piece, when it seemed satisfied and did not ask for any more. It had, therefore, the notion of these three successive facts, and could count them. I learn from good authority that a tame sparrow was accustomed to go out daily from the house where it lived in freedom just before it was time for the children to come out from school. It would wait at the school-house door for a child of the family with which it lived, and return perched upon its shoulder. One day it went out but did not return, having probably fallen a prey to a cat.
Nothing is more frequent among animals than daily acts at fixed hours; but we have proof also that animals can measure longer periods. A dog was used to go every Saturday evening for his master, who came to spend Sunday at home, and went away again on Monday. But the dog, instead of following his master away, showed his displeasure at the parting by sulking in a corner. Could this dog count the six days of the week during which his master was absent? It is more likely that his return was foreshadowed by certain things going on in the house that only occurred on that occasion.
Houzeau de la Haie tells of a pelican living in a fisherman's family at Santo Domingo that was fed upon the refuse of the fish-cleaning. Looking for its food, it went to the shore every day and waited for the boats to come back. The fishermen rested on Sunday, and the bird acquired so clear a notion of the return of that day, when it had to fast, that it would not stir from the tree on which it was accustomed to spend its time. It is not necessary to suppose that the pelican had learned to count the six days at the end of which its masters would not go fishing; but, while it really estimated daily the time when it must make its excursion to the shore, it was informed of the return of Sunday by observation of what was going on in the house, as, for instance, by the fishermen putting on their Sunday clothes; in the same way as the dog knew when its master was going to hunt by seeing him with his gun and game-bag. In such instances, animals show that they have the faculty of associating ideas, of observing consecutive facts, and establishing a correlative connection between them—things which have been proved by abundance of other evidence, and which demonstrate not less intelligence than acquaintance with the ten signs exposing the first ten numbers, or the use of a system of numeration to express larger numbers.
Broderip tells of an English Protestant minister's dog which escaped every Sunday and followed its master to church. It was shut up on one Saturday evening, but on the next week when they went to shut it up it could not be found, and hid itself till the service-hour on Sunday, when it appeared again at the church. In acting thus, it had evidently reasoned out all its conduct, displaying memory, foresight, and calculation. It is not likely, however, that it acted upon a count of the days, but rather on the knowledge that it had to turn the spit on Saturday, or the day before its master went to church.
If this period of the seven days of the week does not exceed the intelligence of a dog, the dog should be able easily to measure periods of two or three days. Houzeau says that he tried for three consecutive weeks to repeat the same walk with his dogs, every two days at exactly the same hour. It would have been enough for them to count two to determine the period. On the twentieth day, or the tenth periodical repetition of the excursion, although the dogs enjoyed the excursion exceedingly, he never remarked that they anticipated it spontaneously, or had a thought of it before witnessing his preparations to go. From this, Houzeau concluded that dogs could not count the days. But when actions repeated daily at fixed hours were in question, the dogs knew when the time came. Broderip's dog and the Santo Domingo pelican had learned, in the course of years, that the same succession of events took place every Sunday. It was not, therefore, by an isolated fact, but by an aggregation of facts, that they became aware of the return of that day; for not only did certain things take place regularly in the family, but Sunday noises, like the ringing of the bells, and unusual comings and goings, occurred in the place. After continued experience, the animals acquired knowledge of the succession of the events, and governed their conduct accordingly.
Houzeau also learned that some animals are capable of measuring lapses of time that particularly interest them. He says that female crocodiles abandon their eggs in the sand for ten or fifteen days, according to the species, and return to the spot at the exact time when they are to be hatched. It is easily conceivable that animals have, in general, a more precise measure of periods which concern the needs of their organic or specific life, than of the more artificial periods to which they have become habituated in the domesticated state or in consequence of their relations with man, because an hereditary habit has always more force than habits acquired by education.
Houzeau cites facts showing that some animals can count the number of similar objects or acts, provided the numbers are not too high. When a magpie is watched by a company of hunters, it will not move till they go away. If they go one after another, it can not be deceived by one of them staying behind unless there are more than four of them. Another story of similar bearing is that of the tramway mules at New Orleans, which are relieved and fed after making five trips. They make their trips patiently and quietly till the end of the fifth, when they give evident signs that they expect their usual refreshment. The horses in the coal-mines of Hainault make thirty trips a day, taking their places again, after every trip, at the head of the train, in readiness for the next trip; but at the end of the thirtieth trip they turn their heads in the opposite direction, or toward the stable.
Facts of this kind ought to be tested by most precise experiments bearing upon the conditions under which they are produced, and upon different subjects. Are not the horses warned of the end of their stint by some exterior sign, such as a change of conductors, the departure of a squad of workmen, or the arrival of the horses that are to take their places, or by the meal-hour? Is not the conclusion that they count the number of their trips arrived at too quickly?
It would predicate a very high degree of development to suppose that a horse could count up to thirty in any given number of hours. A man in such case would nearly always make mistakes, unless he had some means of registering the trips as they were completed.
It is nevertheless established that some birds and quadrupeds are capable of counting up to four or five, and perhaps more. It can not be disputed that the higher limit of this faculty may vary according to species, and also to individual traits, since the mathematical faculties of men are very great in their variations. But we have reasons for believing that the geometrical faculty in animals supersedes the arithmetical faculty, and that the latter has been developed in man under the influences of industrial civilization and commercial exchanges, which have, in nearly all cases, caused the notion of numbers to be substituted for that of measure. — Translated for the Popular Science Monthly from the Revue Scientifique.