change, the finite number cannot be equal to itself. But here we see how, changing, the transfinite number remains equal to itself. After all, transfinite numbers are entirely real. We can find examples corresponding to the expression and even ∞∞ and

∞∞∞ in our world.

Let us take a line—any segment of a line. We know that the number of points on this line is equal to infinity, for a point has no dimension. If our segment is equal to one inch, and beside it we shall imagine a segment a mile long, then in the little segment each point will correspond to a point in the large one. The number of points in a segment one inch long is infinite. The number of points in a segment one mile long is also infinite. We get

∞ - ∞.

Let us now imagine a square, one side of which is a given segment, a. The number of lines in a square is infinite. The number of points in each line is infinite. Consequently, the number of points in a square is equal to infinity multiplied by itself an infinite number of times. This magnitude is undoubtedly infinitely greater than the first one: ∞, and at the same time they are equal, as all infinite magnitudes are equal, because, if there be an infinity, then it is one, and cannot change.

Upon the square a2, let us construct a cube. This cube consists of an infinite number of squares, just as a square consists of an infinite number of lines, and a line of an infinite number of points. Consequently, the number of points in the cube, a3 is equal to ∞∞∞, this expression is equal to the expression ∞∞ and ∞, i. e., this means that an infinity continues to grow, remaining at the same time unchanged.

Thus in transfinite numbers, we see that two magnitudes equal separately to a third, can be not equal to each other. Generally speaking, we see that the fundamental axioms of our mathematics do not work there, are not there valid. We have therefore a full right to establish the law, that the fundamental axioms of mathematics enumerated above are not applicable to transfinite numbers, but are applicable and valid only for finite numbers.

We may also say that the fundamental axioms of our mathe

THE NEW MATHEMATICS

251

Or in other words That is, each magBut if we take

matics are valid for constant magnitudes only. they demand unity of time and unity of place. nitude is equal to itself at a given moment. a magnitude which varies, and take it in different moments, then it will not be equal to itself. Of course, we may say that changing, it becomes another magnitude, that it is a given magnitude only so long as it does not change. But this is precisely the thing that I am talking about.

The axioms of our usual mathematics are applicable to finite and constant magnitudes only.

Thus quite in opposition to the usual view, we must admit that the mathematics of finite and constant magnitudes is unreal, i. e., that it deals with the unreal relations of unreal magnitudes; while the mathematics of infinite and fluent magnitudes is real, i. e., that it deals with the real relations of real magnitudes.

Truly the greatest magnitudes of the first mathematics has no dimension whatever, it is equal to zero, or a point, in comparison with any magnitude of the second mathematics, ALL MAGNITUDES OF WHICH, DESPITE THEIR DIVERSITY, ARE EQUAL AMONG THEMSELVES.

Thus both here, as in logic, the axioms of the new mathematics appear as absurdities:

A magnitude can be not equal to itself.

A part can be equal to the whole, or it can be greater than the whole.

One of two equal magnitudes can be infinitely greater than another. All Different magnitudes are equal among themselves.

A complete analogy is observed between the axioms of mathematics and those of logic. The logical unit—a concept—possesses all the properties of a finite and constant magnitude. The fundamental axioms of mathematics and logic are essentially one and the same. They are correct under the same conditions, and under the same conditions they cease to be correct.

Without any exaggeration we may say that the fundamental axioms of mathematics and of logic are correct only just as long as mathematics and logic deal with magnitudes which are artificial, conditional, and which do not exist in nature.

The truth is that in nature there are no finite, constant magnitudes, just as also there are no concepts. The finite, constant magnitude, and the concept are conditional abstractions, not reality, but merely the sections of reality, so to speak.

How shall we reconcile the idea of the absence of constant magnitudes with the idea of an immobile universe? At first sight one appears to contradict the other. But in reality this contradiction does not exist. Not this universe is immobile, but the greater universe, the world of many dimensions, of which we know that perpetually moving section called the three-dimensional infinite sphere. Moreover, the very concepts of motion and immobility need revision, because, as we usually understand them with the aid of our reason, they do not correspond to reality.

Already we have analyzed in detail how the idea of motion follows from our time-sense, i. e., from the imperfection of our space

sense.

Were our space-sense more perfect in relation to any given object, say to the body of a given man, we could embrace all his life in time, from birth to death. Then within the limits of this embrace that life would be for us a constant magnitude. But now, at every given moment of it, it is for us not a constant but a variable magnitude. That which we call a body does not exist in reality. It is only the section of that four-dimensional body that we never see. We ought always to remember that our entire three-dimensional world does not exist in reality. It is a creation of our imperfect senses, the result of their imperfection. This is not the world but merely that which we see of the world. The three-dimensional world—this is the four-dimensional world observed through the narrow slit of our senses. Therefore all magnitudes which we regard as such in the three-dimensional world are not real magnitudes, but merely artificially assumed.

They do not exist really, in the same way as the present does not exist really. This has been dwelt upon before. By the present we designate the transition from the future into the past. But this transition has no extension. Therefore the present does not exist. Only the future and the past exist.

Thus constant magnitudes in the three-dimensional world are only abstractions, just as motion in the three-dimensional world is, in substance, an abstraction. In the three-dimensional world

INFINITE MAGNITUDES

253 there is no change, no motion. In order to think motion, we already need the four-dimensional world. The three-dimensional world does not exist in reality, or it exists only during one ideal moment. In the next ideal moment there already exists another three-dimensional world. Therefore the magnitude A in the following moment is already not A, but B, in the next C, and so forth to infinity. It is equal to itself in one ideal moment only. In other words, within the limits of each ideal moment the axioms of mathematics are true; for the comparison of two ideal moments they are merely conditional, as the logic of Bacon is conditional in comparison with the logic of Aristotle. In time, i. e., in relation to variable magnitudes, from the standpoint of the ideal moment, they

are untrue.

The idea of constancy or variability emanates from the impotence of our limited reason to comprehend a thing otherwise than by its section. If we would comprehend a thing in four dimensions, let us say a human body from birth to death, then it will be the whole and constant body, the section of which we call a-changing-in-time human body. A moment of life, i. e., a body as we know it in the three-dimensional world, is a point on an infinite line. Could we comprehend this body as a whole, then we should know it as an absolutely constant magnitude, with all its multifariousness of forms, states and positions; but then to this constant magnitude the axioms of our mathematics and logic would be inapplicable, because it would be an infinite magnitude.

We cannot comprehend this infinite magnitude. We comprehend always its sections only. And our mathematics and logic are related to this imaginary section of the universe.

CHAPTER XXI

Man's transition to a higher logic. The necessity for rejecting everything "real." "Poverty of the spirit." The recognition of the infinite alone as real. Laws of the infinite. Logic of the finite—the Organon of Aristotle and the Novum Organum of Bacon. Logic of the infinite—Tertium Organum. The higher logic as an instrument of thought, as a key to the mysteries of nature, to the hidden side of life, to the world of noumena. A definition of the world of noumena on the basis of all the foregoing. The impression of the noumenal world on an unprepared consciousness. "The thrice unknown darkness in the contemplation of which all knowledge is resolved into ignorance."

E

VERYTHING that has been said about mathematical magnitudes is true also with regard to logical concepts. Finite mathematical magnitudes and logical concepts are subject to the same laws.

We have now established that the laws discovered by us in a space of three dimensions, and operating in that space, are inapplicable, incorrect and untrue in a space of a greater number of dimensions.

And as this is true of mathematics, so is it true of logic.

As soon as we begin to consider infinite and variable magnitudes instead of those which are finite and constant, we perceive that the fundamental axioms of our mathematics cannot be applied to the former class.

And as soon as we begin to think in other terms than those of concepts, we must be prepared to encounter an enormous number of absurdities from the standpoint of existing logic.

These absurdities seem to us such, because we approach the world of many dimensions with the logic of the three-dimensional world. It has been proven already that to an animal, i. e., to a twodimensional being, thinking not by concepts, but by perceptions, our logical ideas must seem absurd.