In the book A New Era of Thought, about which I have already had much to say, in the chapter Space the Scientific Basis of Altruism and Religion, Hinton says:
...When we come upon infinity in any mode of our thought, it is a sign that that mode of thought is dealing with a higher reality than it is adapted for, and in struggling to represent it can only do so by an infinite number of terms (of realities of a higher order).
Truly what is infinity, as the ordinary mind represents it to itself?
It is the only reality and at the same time it is the abyss, the bottomless pit into which the mind falls after having risen to heights to which it is not native.
Let us imagine for a moment that a man begins to feel infinity in everything: every thought, every idea leads him to the realisation of infinity.
This will inevitably happen to a man approaching an understanding of a higher order of reality.
But what will he feel under such circumstances?
He will sense a precipice, an abyss everywhere, no matter where he looks, and experience an incredible horror, fear, and sadness until this fear and sadness shall transform themselves into the joy of sensing a new reality.
... An intolerable sadness is the very first experience of the neophyte in occultism... says the author of Light on the Path.
First of all, it would feel astonishment and fright — fright approaching horror: because in order to find the new world it must lose the old one.
Let us imagine the predicament of an animal in which flashes of human understanding have begun to appear.
What will it sense to begin with? First of all, that its old world, the world of the animal, the world into which it was born, to which it has become accustomed, and which it imagines to be the only real world, is crumbling away and falling all around it. Everything that previously seemed real becomes false, delusive, fantastic, unreal. The impression of the unreality of all its environment will become very strong.
Until such a being shall learn to comprehend the reality of another, higher, order, until it shall understand that behind the crumbling old world a new and infinitely more beautiful one is opening up, considerable time will necessarily pass. During this time, a being in whom this new consciousness is in process of unfoldment must pass from one abyss of despair to another, from one negation to another. It must repudiate everything around itself. Only by the repudiation of everything will the possibility of entering into a new life be realised.
Formerly, when it was an animal, it reasoned:
How will it express this in its language of perceptions? Strictly speaking, it will not be able to express this at all, because it is impossible to express concepts in the language of an animal. The animal will simply mix up the sensations of the strange house and its own house. Confusedly, it will begin to feel some new properties in houses, and along with this it will feel less clearly those properties which made the strange house strange.
Simultaneously with this, the animal will begin to sense new properties which it did not know before. As a result it will undoubtedly experience the necessity for a system of generalisation of these new properties — the necessity for a new logic expressing the relations of the new order of things. But having no concepts, it will not be in a position to construe the axioms of Aristotelian logic and will express the impression of the new order in the form of the entirely absurd but more nearly true proposition:
This is that.
Or let us imagine that to the animal with the rudimentary logic expressing its sensations,
Then, sensing confusedly the idea of the likeness of two different objects but being without concepts, the animal will express this as something illogical from its own point of view. The idea, this and that are similar objects, the articulate two-dimensional being will translate into the language of its logic in the shape of the formula: this is that; and of course will pronounce it an absurdity, and that the sensation of the new order of things leads to logical absurdities. But it will be unable to express that which it senses in any other way.
We are in exactly the same position when we men come to the realisation of that other life, to the comprehension of higher things.
The same fright, the same loss of the real, the same impression of utter and never-ending illogicality, the same formula "this is that", will afflict us.
In order to realise the new world, we must understand the new logical order of things.
What is mathematics? The science of magnitudes.
What is logic? The science of concepts.
But these are not definitions; they are only the translation of the name. Mathematics, or the science of magnitudes, is that system which studies the quantitative relations between things; logic, or the science of concepts, is that system which studies the qualitative (categorical) relations between things.
Logic has been built up in quite the same way as mathematics. As with logic, so also with mathematics (at least generally known mathematics of "finite" and "constant" quantities). Both were deduced by us from the observation of the phenomena of our world. Generalising our observations, we gradually discovered those relations which we called the fundamental laws of the world.
In logic, these fundamental laws are included in the axioms of Aristotle (and Bacon).
Logos, the word, is the object of logic. In order to become the object of logical reasoning, in order to be subjected to the laws of logic, an idea must be expressed in a word. That which cannot be expressed in a word cannot enter into a logical system. Moreover, a word can enter into a logical system, can be subjected to logical laws, only as a concept.
At the same time we know very well that not everything can be expressed in words. In our life and in our feelings there is much that cannot be expressed in concepts. Thus it is clear that even at the present moment, at the present stage of our development, not everything can be entirely logical for us. There are many things which in their substance are outside of logic altogether. This includes the entire region of feelings, emotions, religion. All art is just one entire illogicality; and as we shall presently see, mathematics, the most exact of sciences, is entirely illogical.
If we compare the axioms of the logic of Aristotle and of Bacon with the axioms of mathematics as it is commonly known, we find them to be completely similar.
The laws of mathematics and of logic are the laws of the reflection of the phenomenal world in our receptivity and in our reasoning faculty.
Just as the axioms of logic can deal only with concepts, and are related solely to concepts, so the axioms of mathematics apply only to finite and constant magnitudes, and are related solely to such magnitudes.
In relation to infinite and variable magnitudes, these axioms are untrue, just as the axioms of logic are untrue even in relation to emotions, to symbols, to the musicality and the hidden meaning of words, to say nothing of those ideas which cannot be expressed in words.
What does this mean?
It means that the axioms of logic and mathematics are deduced by us from the observation of phenomena, i.e., of the phenomenal world, and represent in themselves a certain conditional incorrectness which is necessary for the knowledge of the unreal "subjective" world — in the true meaning of that word.
The other, the mathematics of infinite and variable magnitudes, represents something entirely real, built upon reasonings in regard to a real world.
The first is related to the world of phenomena [see Chapter XII — Ed.], which in itself represents nothing other than our incorrect apprehension and perception of the world.
The second is related to the world of noumena [see Chapter XIII — Ed.], which represents in itself the world as it is.
The first is unreal: it exists in our consciousness, in our imagination.
The second is real: it expresses the relations of a real world.
As their name implies, by transfinite numbers is meant numbers beyond infinity.
Infinity (the sign for which is a figure 8 turned on its side) is the mathematical expression with which, as such, it is possible to perform the operations of multiplication, division, and raising to powers. It is possible to raise infinity to the power of infinity to obtain a magnitude that is an infinite number of times greater than simple infinity. And at the same time they are both equal: infinity to the power of infinity is equal to infinity. This is the most remarkable property of transfinite numbers. Any operations may be performed with them, and they will change in corresponding manner, yet remaining equal. This violates the fundamental laws of mathematics accepted for finite numbers. After a 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 in our world examples corresponding to the expression infinity, infinity to the power of infinity, and even infinity times infinity to the power of infinity.
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 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 the segment one inch long is infinite. The number of points in the segment one mile long is also infinite. We get infinity equals infinity.
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 is 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 infinity times infinity to the power of infinity. This means that infinity continues to grow, remaining at the same time unchanged.
We may also say that the fundamental axioms of our mathematics are valid for constant magnitudes only. In other words, they demand unity of time as well as unity of space. That is, each magnitude is equal to itself at a given moment. But if we take a magnitude which varies, and take it at different moments, then it will not be equal to itself. Of course, we may say that by changing, it becomes another magnitude; that it is a given magnitude only so long as it does not change. But this is precisely what I am talking about.
The axioms of our usual mathematics are applicable only to finite and constant quantities.
Thus, quite in opposition to the usual view, we must admit that the mathematics of finite and constant quantities 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 magnitude 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:
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.
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.
We have already analysed in detail [see Chapter IV — Ed.] 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 really exist, just as the present does not really exist. This has already been dwelt upon [see Past, Present, Future — Ed.]. 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, 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 on 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 — 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 always comprehend only sections through it. And our mathematics and logic are related to this imaginary section of the Universe.