Authors: Robert Musil
“All that about imaginary numbers.”
“Yes. It's not particularly difficult, is it? All you have to do is remember that the square root of minus one is the basic unit you work with.”
“But that's just it. I mean, there's no such thing. The square of every number, whether it's positive or negative, produces a positive quantity. So there
can't
be any real number that could be the square root of a minus quantity.”
“Quite so. But why shouldn't one try to perform the operation of working out the square root of a minus quantity, all the same? Of course it can't produce any real value, and so that's why one calls the result an imaginary one. It's as though one were to say:
someone always used to sit here, so let's put a chair ready for him
today too, and even if he has died in the meantime, we shall go on behaving as if he were coming.”
“But how can you when you know with certainty, with mathematical certainty, that it's impossible?”
“Well, you lust go on behaving as if it weren't so, in spite of everything. It'll probably produce some sort of result. And after all, where is this so different from irrational numbers-division that is never finished, a fraction of which the value will never, never, never be finally arrived at, no matter how long you may go on calculating away at it? And what can you imagine from being told that parallel lines intersect at infinity? It seems to me if one were to be over-conscientious there wouldn't be any such thing as mathematics at all.”
“You're quite right about that. If one pictures it that way, it's queer enough. But what is actually so odd is that you
can really
go through quite ordinary operations with imaginary or other impossible quantities, all the same, and come out at the end with a tangible result!”
“Well, yes, the imaginary factors must cancel each other out in the course of the operation just so that does happen.”
“Yes, yes, I know all that just as well as you do. But isn't there still something very odd indeed about the whole thing? I don't quite know how to put it. Look, think of it like this: in a calculation like that you begin with ordinary solid numbers, representing measures of length or weight or something else that's quite tangible-at any rate, they're real numbers. And at the end you have real numbers. But these two lots of real numbers are connected
by something that simply doesn’
t exist. Isn't that like a bridge where the piles are there only at the beginning and at the end, with none in the middle, and yet one crosses it lust as surely and safely as if the whole of it were there? That sort of operation makes me feel a bit giddy, as if it led part of the way God knows where. But what I really feel is so uncanny is the force that lies in a problem like that, which keeps such a firm hold on you that in the end you land safely on the other side.”
Beineberg grinned. “You're starting to talk almost like the chaplain, aren't you? You see an apple- that's light-waves and the eye and so forth-and you stretch out your hand to steal it-that's the muscles and the nerves that set them in action-but between these two there lies something else that produces one out of the other, and that is the immortal soul, which in doing so has committed a sin... ah yes, indeed, none of your actions can be explained with-out the soul, which plays upon you as upon the keys of a piano... And he imitated the cadences in which the chaplain was in the habit of producing this old simile. “Not that I find all that stuff particularly interesting.”
“I thought you were the very person who would find it interesting. Anyway, it made me think of you at once because-if it's really impossible to explain it-it almost amounts to a piece of evidence for what you believe.”
“Why shouldn't it be impossible to explain? I'm inclined to think it's quite likely that in this case the inventors of mathematics have tripped over their own feet. Why, after all, shouldn't something that lies beyond the limits of our intellect have played a little joke on the intellect? But I'm not going to rack my brains about it: these things never get anyone anywhere.”
That same day Törless asked the mathematics master for permission to call on him, in order to discuss some points in the last lesson.
The next day, during the noon break, he went upstairs to the master's little apartment.
He had gained an entirely new respect for mathematics, for now it seemed all of a sudden to have ceased to be a dead school subject and to have turned into something very much alive. And arising out of this respect he felt something like envy of the master, who must be on familiar terms with all these processes and relationships and who carried the knowledge of them about with him always, like the key to a locked garden. But above and beyond this Törless was also impelled by curiosity, though it was, to be sure, rather diffident curiosity. He had never before been in the room of a grownup young man, and there was a certain titillation in wondering what things looked like in the life of such a person, a different person, one who knew things and yet was composed and calm, 'it least so far as one could tell from the external objects surrounding him.
He had always been shy and withdrawn in his relations with both teachers and believed that as a result he was not particularly well liked by them. Hence his request, as he now paused in agitation outside the door, seemed to him an act of daring in which the main object was less to get some further light on his difficulties-for at the back of his mind he had already begun to doubt that he would get any-than to cast a glance, as it were, past the master and into this man's daily cohabitation with mathematics.
He was shown into the study. It was a long narrow room with a single window; near the window was a desk spattered with ink-blots, and against the wall was a sofa covered in some scratchy green ribbed material, with tassels. Over this sofa a faded student's cap hung on the wall, together with a number of photographs, the size of visiting-cards, brown and now grown dark with age, dating from the master's university days. On the oval table with the knock-kneed legs, which were of a would-be grace and prettiness that had somehow gone wrong, there lay a pipe and some leafy, crude-cut tobacco. The whole room was permeated with the smell of cheap tobacco-smoke.
Törless had scarcely had time to make these observations and note a trace of discomfort in himself, as on contact with something unsavoury, when the master came in.
He was a fair, nervous young man of no more than thirty, and quite a sound mathematician, who had already submitted several important papers to the academy.
He at once sat down at his desk, rummaged about a little among the papers strewn upon it (later it struck Törless that he had positively taken refuge there), then, crossing his legs, he began to polish his
pince-nez
with his handkerchief, and fixed an expectant gaze on Törless.
Meanwhile Törless had been scrutinising him too. He observed a pair of thick white
woolen
socks and saw that over them the bands of the underpants had been rubbed black by the blacking on the boots.
In contrast the handkerchief peeping out of the breast pocket was all white and dainty, and though the tie was a made-up one, it counterbalanced this by being as magnificently gaudy as a painter's palette.
Törless could not help feeling further repelled by these little observations; he scarcely found it in him to go on hoping that this man was really in possession of significant knowledge, when there was nothing whatsoever about either his person or his surroundings to suggest that it might be so. He had been secretly imagining a mathematician's study quite differently and as somehow expressive of the awe-inspiring matters that were excogitated there. The ordinariness of what he saw affronted him; he projected this on to mathematics, and his respect began to give way before a mistrustful reluctance.
And since the master was now shifting impatiently on his chair, not knowing what to make of this long silence and this scrutinising gaze- even at this stage there was already an atmosphere of misunderstanding between the two people in the room.
“And now let us. . . now you . . .I shall be pleased to tell you whatever you want to know,” the master began at last.
Törless then came out with his difficulties, exerting himself to make clear what they meant to him. But he felt as though he were talking through a dense and gloomy fog, and his best words died away in his throat.
The master smiled, now and then gave a little fidgety cough, said:
“If y
ou don't mind,” and lit a cigarette, at which he took hasty puffs. The cigarette-paper-and this was yet another thing that Törless noticed and found incredibly sordid-at each puff became greasy and crumpled up, crackling a little. The master took off his
pince-nez,
put it on again, nod
ded . . And finally he cut Törle
ss short. “I am delighted, my dear Törless, yes, lam indeed delighted-' he said, interrupting him, “your qualms are indications of a seriousness and a readiness to think for yourself, of a . . . h'm . . . but it is not at all easy to give you the explanation you want. . . . you must not misunderstand what I am going to say.
“It is like this, you see-you have been speaking of the intervention of transcendent, h'm, yes-of what are called
transcendent
factors.
“Now of course I don't know what you feel about this. It's always a very delicate matter dealing with the suprasensual and all that lies beyond the strict limits of reason. I am not really qualified to intervene there in any way. It doesn't come into my field. One may hold this view or that, and I should greatly wish to avoid entering into any sort of controversy with an
yone . . . But as regards math
ematics,” and he stressed the word 'mathematics' as though he were slamming some fateful door once and for all, “yes, as regards mathematics, we can be quite definite that here the relationships work out in a natural and purely mathematical way.
“Only I should really-in order to be strictly scientific-I should really have to begin by posing certain preliminary hypotheses that you would scarcely grasp, at your stage. And apart from that, we have not the time.
“You know, I am quite prepared to admit that, for instance, these imaginary numbers, these quantities that have no real existence whatsoever, ha-ha, are no easy nut for a young student to crack. You must accept the fact that such mathematical concepts are nothing more or less than concepts inherent in the nature of purely mathematical thought. You must bear in mind that to anyone at the elementary stage at which you still are it is very difficult to give the right explanation of many things that have to be touched upon. Fortunately, very few boys at your stage feel this, but if one does really come along, as you have today-and of course, as I said before, I am delighted-really all one can say is: My dear young friend, you must simply take it on trust. Some day, when you know ten times as much mathematics as you do today, you will understand-but for the present: believe!
“There is nothing else for it, my dear Törless. Mathematics is a whole world in itself and one has to have lived in it for quite a while in order to feel all that essentially pertains to it.”
Törless was glad when the master stopped talking. Since he had heard that door slam it had seemed to him that the words were moving farther and farther away from him . . . towards that other, indifferent realm where all correct and yet utterly irrelevant explanations lie.
But he was dazed by the torrent of words and the failure, and did not instantly grasp the fact that now he should get up and go.
So, in order to put an end to it once and for all, the master looked for one last, convincing argument.
On a little table lay a volume of Kant, the sort of volume that lies about for the sake of appearances. This the m
aster took up and held out to T
örless.
“You see this book. Here is philosophy. It treats of the grounds determining our actions. And if you could fathom this, if you could feel your way into the depths of this, you would come up against nothing but just such principles, which are inherent in the nature of thought and do in fact determine everything, although they themselves cannot be understood immediately and without more ado. It is very similar to the case with mathematics. And nevertheless we continually act on these principles. There you have the proof of how important these things are. But,” he said, smiling, as he saw Törless actually opening the book and turning the pages, “that is something you may well leave on one side for the present. I only wanted to give you an example which you may remember some day, later on. For the present I think it would still be a little beyond you.”
All the rest of that day Törless was in a state of inward upheaval. The fact that he had had the volume of Kant in his hand-this quite haphazard circumstance, to which he had paid little attention at the time-now worked mightily within him. The name of Kant was familiar enough to him, though only as a name, and its currency value for him was that which it had generally among those who even remotely occupied themselves with things of the mind-it was the last word in philosophy. And this authority it had was indeed part of the reason why Törless had hitherto spent so little time on serious reading.
For very young people, once they have got over the stage of wanting to be cab-drivers, gardeners or confectioners when they grow up, in their imaginings are inclined to set their ambitions for life in whatever field seems to hold out most chance for them to distinguish themselves. If they say they want to be doctors, it is sure to be because some time, somewhere, they have seen a well-furnished waiting-room crowded with patients, or a glass case containing mysterious and alarming surgical instruments, or the like; if they dream of a diplomatic career, it is because they are thinking of the urbane glamour of cosmopolitan drawing-rooms; in short, they choose their occupation according to the milieu in which they would most like to see themselves, and according to the pose in which they like themselves best.