Popular Science Monthly/Volume 42/February 1893/Number Forms

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NUMBER FORMS.
By G. T. W. PATRICK,
PROFESSOR OF PHILOSOPHY IN THE STATE UNIVERSITY OF IOWA.

IN the Atlantic Monthly for February, 1873, Miss H. R. Hudson, writing on idiosyncrasies, says, "The nine digits will ascend in a straight line before my mind's eye, and the larger numbers will slant off at a queer angle" thus:

PSM V42 D522 Idiosyncratic arithmetic.jpg

About twelve years ago Francis Galton, in England, while engaged in an investigation into the visualizing peculiarities of different persons, discovered that the possession of "number forms" was not uncommon. Some of these "forms" were given by him to the public in Nature for January 15, 1880, and afterward a collection of about sixty-five of them was published in his book on Inquiries into Human Faculty. These were accompanied by many descriptive details, but Galton did not attempt any complete explanation of the number form in general.

Neither is it the purpose of this article to attempt such an explanation, but rather to add to Galton's list some thirty-five or forty forms, which I have incidentally collected during the last four years, together with some explanatory remarks and a few suggestions toward a future theory. It is hoped, too, that further attention may be called to the subject, and other contributions made to this curious chapter in psychology.

With about half a dozen exceptions, the accompanying forms have been gathered from college students of both sexes, varying in age from eighteen to twenty-five years. They are taken from drawings made in every case by the "seer" himself, in response to some such question as this: "When you think of the numbers

PSM V42 D523 Numbers as visualized forms.jpg

from 1 to 100, do you mentally see them in any form, or outline? If so, can you draw a representation of it?" At first about seventy-five students, of whom thirty were young women, were thus interrogated. In this examination it was probably understood that only well-defined and perhaps somewhat striking number forms were called for. As a result, only four forms were found, two from young women (Figs. 1 and 2) and two from young men (Figs. 3 and 4). This would correspond roughly with Galton's estimate that one out of every thirty adult males, and one out of every fifteen adult females, has a number form. My own later experience, however, has developed the fact that such a mode of investigation does not discover the full number of persons possessing forms, simple or complex. There are several reasons for this. The subject is not commonly understood when first presented. It would seem that a person having even a complicated number form might live and die without knowing it, or at least without once fixing his attention upon it or speaking of it to his nearest friends, although such a one might use his form in daily computation. It seems to him quite natural to see the numbers in that way, and the thought may never enter his mind that others should see them differently. Again, if one is consciousPSM V42 D524 Visualization of specific numbers.jpgFig. 5. of a peculiar form, he regards it as an idiosyncrasy and exhibits a certain shyness in revealing it. For this reason it is especially hard to get all the number forms from a company of children. They do not like to be laughed at, and will willingly keep silent about anything which they suspect may be another of those idiosyncrasies causing such mental torment to many children. Finally, those who do not have complicated forms are apt to think that the little curve, twist, or angle in which they see the numbers is quite too trifling a matter to mention. I am inclined to believe that one out of six adults would be a more accurate proportion, that the proportion among children would be still greater, and that it is perhaps a little more common among women than men.

The questions one would naturally ask a person having a number form are these: "How long have you seen the numbers in this way? Is the form fixed or is it changeable? What was its origin?" The answers to these questions are almost absolutely invariable. They would be as follows: "I have seen the numbers in this way ever since I can remember. The form is fixed and unchangeable. Its origin I do not know." In a very few cases when the nine digits always appear in mental vision as a mere straight line from left to right, the subject may conjecture that it originated with the printed forms from which they were learned. I have found that certain simple kinds of alphabet forms are very common. If a number of people be asked whether, when theyPSM V42 D524 Visualization of stepped numbers.jpgFig. 6. think of the alphabet from a to z, they see it in a visual picture, and if so in what particular form, it will be found that a considerable number will say that they see the letters in one, two, or three vertical columns reading downward. A simple illustration is seen in Fig. 7. When there are two or three columns, the same letter always appears to a given person at the top of each column, but I have not found two forms alike except when they consist of a single straight line. In these cases the suggestion is often made by the seer that the letters were so arranged in his primer. Obviously this explanation would not apply to any such alphabet forms as are shown in Figs. 13, 15, and 16. Nevertheless, it is probable that all strongly eye-minded people, if they do not visualize the alphabet in any other way, visualize it as they do other things, in the form in which they had usually seen it.

Concerning the stability of number forms, any one may have his doubts removed by a few tests separated by months or years. In almost every case it will be found that, no matter how complicatedPSM V42 D525 Compound visualization of letters and numbers.jpgFig. 7. the form may be, the subject, after one, two, or three years, will draw from his mental picture of it a copy differing in no essential respect from the original copy. The number form represented in Fig. 3 was given to me in 1889. In October, 1892, I requested of the young man by letter a second copy, and in reply received one precisely like the first. Other tests gave similar results. Galton testifies to the unchangeable character of number forms in all cases where they are well defined. It is true, however, that they sometimes disappear entirely. They are found to be more common among children than adults. It is probable that in children who are not naturally vivid visualizers, or in cases where it does not serve any useful purpose, the form fails to survive. One case of such a lapse I have found in an adult.

The general character of number form is such that a person having one can not think of the related numbers without seeing them in a definite visual picture. A form or outline rises involuntarily before his mind. In some cases the seer can describe it as definitely located in space in relation to his own body. It is two feet long or six inches long. It stares him in the face or lies at his feet. It recedes to the right or left, or into the distance. Others can not answer the question as to the location. In most cases, though not in all, no individual number can be thought of without seeing it in its appropriate place in the usual outline. Sometimes the form seems to be useful to its possessor in computations, PSM V42 D525 Visualization of individual digits.jpgFig. 8. particularly in addition and subtraction. In other cases it seems to have no use at all further than that of all mental imagery, which will be considered below. It has been suggested that it is by means of a number form, or at least by a clear visualization of numbers, that the arithmetical prodigies accomplish their remarkable computations. Though it has been shown that many of them do visualize the numbers, and mentally see the different steps of their problem, yet this alone offers no adequate explanation of their mathematical agility. This hypothesis is further weakened by the recently developed fact that Inaudi, the ruling French mathematical wonder, is not a visionnaire at all, but a distinct auditaire who hears all his numbers.

Referring now to the accompanying forms, Figs. 1, 2, and 4 demand no further explanation. In Fig. 3 we have an interesting double form, the one to the left showing how the numbers from 1 to 15 appear when thought of by themselves or in connection with one another. But when any number below 15 is thought of in connection with any number above 15, it is seen as shown in the form to the right. Above 15 the numbers are unalterably fixed. The possessor of this form writes me as follows:

I do not believe I can think of a number apart from this outline. I refer all numbers to it, however large. One million is located where 1,000 is, and so of 1,000,000,000; 550 would be at 55 on the circle; 1,235 is at 35. You will notice that of the last two numbers I mention, the first is located at the point indicated by the first two figures, viz., 55; but the last number, 1,235, is located at 35, the last two figures. I can not explain this, but simply state it as a fact. I think possibly in large uneven numbers, I really, though almost unconsciously, separate the number into parts, in 1,235 the 1,200 either being ignored and my mind directed to 35, or else I in some manner connect the two locations but direct my attention more to one than the other. I stated above that I did not believe I could think of a number apart from this outline, and that is true when I think of some one number by itself and in adding and subtracting small numbers. If any one should ask me how many hours intervened from 3 to 11 o'clock, I would say 8, because I see that many on my number form, which immediately appears before my mind's eye, but I could not subtract 37 from 89 in that way. I would immediately locate the two numbers but I could not determine how many numbers intervened, and I find that in adding, subtracting, and multiplying odd numbers, and numbers beyond 15 say, I do it abstractly without referring to my form; but as I said, in thinking of any one number by itself, it is always connected with some point along that outline. This number form, by the way, stands upright and is about two feet in height—that is, the number 100 is two feet above 18 and about six inches to the right.

Among the seventy-five young men and women interrogated in the first experiment, was a rather diffident young woman who communicated to a classmate that while, she had no number form, there were certain associations that she always made with the nine digitis. Learning this, I questioned her, and she consented to write out the associations, which I reproduce here exactly as given:

1 = a child about two years old.
2 = a boy, ten or twelve years old, brown hair and eyes, frank, active, noisy, always ready to help.
3 = a girl, short hair, black, curly; sharp features, not pretty; slight; awful temper; shrill voice; bangs and slams around generally.

4 = a young lady, same characteristics as 2, but is calm, more quiet, studious, a home girl.

5 = a society girl, a policy girl, is always favored, has everything she wants; selfish; does not care how much trouble she makes other people; not always truthful.

6 = a young man, plain, matter-of-fact person, slow, good; will never amount to more than the average.

7 = a sort of villain; a schemer; dresses well, has polished manners, a good talker, bad habits; has a certain sense of honor; is able, but does not use his ability in the right direction; clearcut features, tall, dark.

8 = a lecturer or clergyman; good, solemn, careful, very pious.

9 = a lady, hair rather gray, tall, soft low voice, sweet face, very well educated, dresses in soft colors; a truly refined woman.

No explanation of these peculiar associations could be offered. Each person arose in a distinct mental image whenever the corresponding digit was thought of. One notices, of course, that we have here the prevailing types of mankind as seen by a young girl. I have recently found another case quite similar. Here, also, the subject is a young woman, and she can give no explanation of the origin of her associations. In hearing or reading long numbers rapidly, she says that she does not have time to see the mental pictures, but single numbers, especially if written by herself, instantly call them up. The associations are as follows:

1 is without definite character, as is also 8, with the exception that the former reminds me of a short person, and the latter of a very stout person, but neither has sex or other characteristics.

2 is always a graceful woman, beautifully dressed. She is slender, with a beautiful delicate face.

3 is a chubby little girl, with dark eyes and bright quiet ways.

4 is a plain woman, rather tall, with pale hair brushed tightly back from a severe face. She is dressed very plainly, and the lines of her figure are angular. She is abstinent, intolerant, and hard to get along with.

5 is a man, dark, medium height, dressed in gray clothes. He is a business or professional man, successful and not particularly intellectual. 5 is always associated with the color gray.

6 is a pleasant-faced woman, medium height and stature, with hair brushed back plainly, and with quick, quiet ways. She is dressed plainly and neatly, and always looks pretty. She is an excellent housekeeper, and I think of her as engaged in household duties. I do not know the color of her hair.

7 is a man of quite opposite type from 5. He is very tall and dark, of musical or poetic temperament. I don't know how he is dressed, except that his whole figure is dark as I imagine him.

9 is another man, more like 7 than 5, also dark and dressed in black clothes. He is fine looking and a professional man.

To a class of twenty-nine students, of whom eight were young women, the following questions were recently given:


1. When you think of the numbers from 1 to 100, do you see them in any particular form? If so, will you write or draw it on paper?

2. When you think of the alphabet from a to z, do you see the letters in any particular form?

3. Have you any associations of color with the numbers or letters?

 

To these questions twenty-nine written answers were received, disclosing four number forms and a few simple alphabet forms. Immediately afterward, however, two others of the class told me privately that they thought they did have forms, although they had not reported them in writing. These were found, indeed, to be perfectly well defined, and are shown in Figs. 6 and 7. The other four are shown in Figs. 5, 8, 9, and 10 (a). One curious alphabet form was found (Fig. 10, a), but no color associations. This method of inquiry revealed in this case, at least, a much larger percentage of number forms than that given by Galton. These six forms present also some new types. Fig. 5 is from a young man, who sees only the numbers 5, 6, 7, 8, and 9. Of these, 7 is by far the most conspicuous, and is described as a black figure, fine and perfectly formed, standing on a reddish background. 6 and 8 are less distinct; 5 and 9 still less. Fig. 7 shows the number form of a young man, who sees the numbers from 1 to 9 in Italics on a horizontal line. The others are straight and form a right angle with the first. All the odd numbers appear to him as weak, affording in counting unsatisfactory places to stop. The

1 2 3 4 5 6 7 8 9 10
10 20 30 40 50 60 70 80 90 100
11 21 31 41 51 61 71 81 91 1000
12 22 32 42 52 62 72 82 92 10000
13 23 33 43 53 63 73 83 93 100000
14 24 34 44 54 64 74 84 94 1000000
15 25 35 45 55 65 75 85 95 10000000
16 26 36 46 56 66 76 86 96 100000000
17 27 37 47 57 67 77 87 97 1000000000
18 28 38 48 58 68 78 88 98 10000000000
19 29 39 49 59 69 79 89 99 100000000000

Fig. 9.

even numbers are firm and strong, while 10 and its multiples are much larger and more prominent.

Fig. 8 is from a young woman who sees the numbers in a straight line; 1, 5, 10, 15, etc., appearing more distinct than the others, with wider spaces after 5 and its multiples.

One of the most interesting forms in this collection is that shown in Fig. 10 (a). This young man sees only 1 and 0 distinctly; 2 and 9 stand in their proper places, but are less distinct;

PSM V42 D529 Visualizations of alphabets and colors.png
Fig. 10. Fig. 11. 

while 3 and 8 are seen but faintly in a shadowy form. The intervening figures are not seen at all, but the appropriate space for them is there. His alphabet form accompanying discloses the

PSM V42 D529 Visualizations of calendar dates and days.jpg
Fig. 12. Fig. 13. 

same principle. A, m, n, and s are very distinct; b and y are fainter; c and x are shadowy; a blank space intervenes, sufficient for the other letters. This young man has a brother and two sisters, whom he severally asked concerning their views of the letters and figures, without mentioning his own. The forms shown in Fig. 10 (b), (c), and (d) are the results. The brothers and sisters differ widely in age, did not learn their letters or figures from the

PSM V42 D530 Visualizations of a calendar.jpg
Fig. 14. Fig. 15.

same books, and had never, until this time, spoken of their forms to one another. The very striking similarity, together with the odd character of the forms, shows strong hereditary tendencies in this case. There are other instances of family likeness in number forms. It is not, however, invariable. Figs. 2, 12, 14, 15, and 16 are forms from members of one family. There are some similarities, but they are not striking.

PSM V42 D530 Visualizations of coordinates and time.jpg
Fig. 16.

Fig. 11 shows the number form of a girl of nine years. This is a colored form, its peculiarity consisting in the fact that the numbers containing two or more figures maintain their individuality and appear in a color formed by mixing the colors of its constituent digits. Thus 13 is not white and pink but pale pink. Other colored or partly colored forms appear in Figs. 17 and 18. In the former, 5 is scarlet, while the other numbers are not seen in colors. This form, like that in Fig. 4, lies in space of three dimensions, and like that of Fig. 10 has some faint and some missing numbers. Other peculiarities of this form are best presented in the words of the seer herself:

In all these forms the figures and letters appear to me in my own handwriting, except in the divisions of the day. In these, I have the abstract idea of morning, etc., in mind, but with the distinct divisions as in the diagram. All the plans are very much larger than here represented. The figures begin at my left and cross to my right before me, curving at 8; and 1,000 seems about my fingertips when my arm is extended straight before me. The circle of the seasons is about as large as would lie between my arms extended straight before me. The days of the week occupy a line at my left, about a yard long. The divisions of the day are perpendicular, as though hung on a wall, and morning begins at a level with my hand.

The number form in Fig. 18 is peculiar in this respect, that it reads from right to left. The seer is an artist, and it may be worthPSM V42 D531 Color related calendar visualizations.jpgFig. 17. mentioning that she is not left-handed. Other features of her form she describes as follows:

The line of the figures runs down to 8, 18, 28, 38, 48. They turn and ascend to the multiples of 10, but after 40 I see only the numbers found in the multiplication table—42, 48, etc. The numbers 18 and 19 form a very dark corner; 20 is quite light; 24 again is dark, the darkness continuing to 30; 30 is again light, and the numbers following are all quite light. The source of light seems to be 60, which is much higher than the rest, the light touching those on either side; 75 is very distinct.

Number forms being all unlike, adequate explanation of them becomes practically impossible. Speaking very generally, however, their origin may be traced to one great cause—namely, the attempt or necessities of children to give a concrete form to the abstract. Now, numbers are among the first abstractions that children have to wrestle with. Our earliest abstract ideas, perhaps also our later ones, are, as it is now well known, either mere samples of individual things, or else a kind of composite picture of them. The child's concept of boy, girl, dog, horse, are nothing more than visual pictures of some particular boy, girl, dog, horse, or else a composite picture of a limited number of individuals. Now, numbers do not admit of such composite pictures. They are bald abstractions that the poor child must manage in some way. In most cases, if he be an eye-minded child, he merely visualizes the Arabic numeral. He may give it individuality further by clothing it in a particular color, or even personifying it outright as in the two cases given. Color audition and association of color with written characters are explained by many as due to physiological conditions, especially to the contiguity of the cortical center involved. Dr. Krohn, in a recent review of the subject in the American Journal of Psychology, adopts this explanation in part. It is doubtful whether this hypothesis is necessary to explain the comparatively rare cases of color associations; it certainly is not necessary for explaining number forms. In any case, physiological association would be due, either in its origin or as a justification of its survival, to useful psychic associations. Now, in the matter of number forms, suppose

PSM V42 D532 Calendar and numerical visualizations.jpg
Fig. 18.

that the child is required, as is very early the case in counting, to think of the numbers not separately, but in relation to each other. He has then the problem of arranging abstractions in a series, and, if he is naturally an ear-minded child, will arrange them as a mere series of associated sounds. If, however, he is eye-minded, he may consciously or unconsciously hit upon the device of a visual spatial image, and thus enable himself to comprehend and remember the numbers as he does other things by a mental picture. A number form thus becomes a little system of topical mnemonics. Its continuance, either in the individual or, in cases of inherited forms, in the family, is of course due to physiological conditions. In every case, however, its origin is probably to be traced to useful psychic associations.

 


 
M. Stanislas Heunier has been able to produce artificially, by a process of reflection, an appearance like that of the gemination or doubling of the canals of Mars, and suggests hypothetically that the phenomenon in question may be one of that character.