TERTIUM ORGANUM

THE THIRD CANON OF THOUGHT
A KEY TO THE ENIGMAS OF THE WORLD

CHAPTER 20

The sensation of infinity. The first test of a Neophyte. Intolerable sadness. Loss of everything real. What would an animal experience on becoming a man? Transition to a new logic. Our logic as based on the observation of laws of the phenomenal world. Its unsuitability for the study of the noumenal world. The need of a new logic. Analogous axioms in logic and mathematics. TWO MATHEMATICS. The mathematics of real magnitudes (infinite and variable); and mathematics of unreal imaginary magnitudes (finite and constant). Transfinite numbers -numbers lying BEYOND INFINITY. The possibility of different infinities.
There exists an idea which a man should always try to remember when he finds himself too engrossed in the sense of the reality of the unreal visible world in which everything has a beginning and an end. It is the idea of infinity, the fact of infinity.
In his book A New Era of Thought, 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 [realities of a higher order],
And, indeed, what is infinity as an ordinary man pictures it?
It is the only reality, and at the same time it is the abyss, the bottomless pit into which our mind falls after having risen to a height where it cannot keep a foothold.
Now, let us imagine for a moment that a man begins to sense infinity in everything, every thought, every idea leads him to the sensation of infinity.
This is bound to happen to a man who passes to the understanding of a higher order of reality.
What will he feel then?
He is bound to feel an abyss and a bottomless pit wherever he looks. And this feeling is bound to bring with it a sense of incredible fear, terror and sadness, until this terror and sadness become transformed into the joy of feeling new reality. 'An intolerable sadness is the very first experience of the Neophyte in occultism,' says the author of
Light on the Path.
We have previously examined the way in which a two-dimensional being might come to the understanding of the third dimension. But we have not asked ourselves what such a being would feel when it begins to sense the third dimension, begins to be conscious of the 'new world' around itself.
The first feeling is bound to be surprise and fear -a fear approaching terror,for before it finds the new world it must lose the old.
Let us imagine an animal in which flashes of human understanding begin to appear.
What will be its first sensation? The first sensation will be that its old world, the world of the animal, a comfortable habitual world, the world into which it was born, to which it has grown accustomed, the only world it represents to itself as real, is crumbling away and falling into ruins all around. Everything that before appeared real becomes false, deceptive, fantastic, unreal. The sensation of the unreality of everything around must be very strong.
Until such a being learns to perceive realities of another, a higher order, until it realizes that beyond the crumbling old world there opens up a new world, infinitely more beautiful -a long time must pass. Meanwhile the beingin whom new consciousness is being born must pass from one gulf of despair to another, from one negation to another. It must repudiate everything around it. And only then, having repudiated everything, will the possibility of passing into a new life be his.
With the gradual loss of the old world the logic of the two-dimensional being, or what in its case took the place of logic, will suffer constant violation, and its strongest sensation will be that there is no logic, no laws of any kind.
Formerly, when it was an animal, it reasoned thus:
This is this This house is mine That is that That house is strange This is not that The strange house is not mine.
Thus a strange house and its own house an animal regards as different objects having nothing in common. And now it will suddenly understand that both the strange house and its own house are equally - houses. How will it express this in its language of representations? In all probability it will be unable to express it at all, because it is impossible to express concepts in the language of an animal. The animal will simply confuse the sensations of the strange house and its own house. It will begin to sense dimly some new properties in houses, and at the same time the properties which had made the strange house strange it will begin to sense less clearly. Simultaneously it will begin to sense new properties it did not know before. As a result it will necessarily feel in need of some system for the generalization of these new properties - the need of a new logic expressing the relations of the new order of things. But having no concepts, it will be unable to construe the axioms of Aristotelean logic and will express its sense of the new order in the form of a perfectly absurd proposition which, nevertheless, is much nearer truth.

This is that.

Or else let us imagine that to an animal in whom rudiments of logic find expression in the sensations

This is this That is that This is not that

somebody tries to prove that two different objects, two houses -its own and a strange one -are the same, that they represent the same thing, that both are -houses. The animal will never credit their sameness. For it the two houses - its own where it is fed, and the strange one where it is beaten if it comes in, will remain totally different. For it they will have nothing in common. No attempt to prove that these houses are the same thing will lead to anything until the animal senses this itself. Then, sensing dimly the idea of the sameness of two different objects and having no concepts, the animal will express this as something illogical from its point of view. An articulate twodimensional being will translate the idea -this and that are the same object, into the language of its own logic in the form of the formula: This is that. Of course, it will say that it is nonsense, that the sense of a new order of things leads to logical absurdities. But it will be unable to express its sensations in any different way.
We are exactly in the same position when we, the dead, awaken i.e. when we, men, arrive at the sensation of a different life, the understanding of higher entities.
The same fear, the same loss of the real, the same sensation of an all-round illogicality, the same formula: This is that.
To realize the new world we must understand the new logicality.
Our ordinary logic helps us to gauge only the relations existing in the phenomenal world. A great many attempts have been made to define what logic is. But in its essence logic is just as undefinable as mathematics.
What is mathematics? The science of magnitudes.
What is logic? The science of concepts.
But these are not definitions, they are merely a translation of the name. Mathematics, or the science of magnitudes, is a system studying quantitative relations between things; logic or the science of concepts is a system studying qualitative (categorical) relations between things.
Logic is constructed on exactly the same plan as mathematics. Both logic and mathematics (at least the mathematics of 'finite' and 'constant' numbers) are deduced by us from observing the phenomena of our world. By means of generalizing our observations we gradually found relations which we called the fundamental laws of the world.
In logic these fundamental laws are contained in the axioms of Aristotle and Bacon.

A is A (That which was A will be A) A is not not A (That which was not A will be not A) Each thing is either A or not A (Each thing must be A or not A)

The logic of Aristotle and Bacon, elaborated and supplemented by their numerous followers, operates solely with concepts.
Logos, the word, is the subject of logic. To become the subject of logical reasoning, to be governed by the laws of logic an idea must be expressed in a word. What cannot be expressed in a word cannot enter into a logical system. Moreover, the word can enter a logical system, be subject to logical laws only as a concept.
At the same time we know perfectly well that not everything can be expressed in words. In our life and in our feelings there is a great deal that cannot be fitted into concepts. So it is clear that even at this moment, at the present stage of our development, by no means everything can be logical for us. A great many things are essentially outside logic. Such is the entire domain of feelings, emotions, religion. All art is a complete illogicality. And we shall see presently that mathematics, the most exact of all sciences, is also completely illogical.*
If we compare the logical axioms of Aristotle and Bacon with the axioms of the generally known mathematics, we shall see that they are entirely identical.
The axioms of logic * Strictly speaking, the science parallel to logic is not mathematics, but geometry.

A is A
A is not not A

Each thing is either A or not A
completely correspond to the fundamental axioms of mathematics, axioms of identity and difference.
Every magnitude is equal to itself. The part is
less than the whole.
Two magnitudes, equal separately to a third, are equal to each other, etc.
This similarity between the axioms of mathematics and logic goes very deep, and this allows us to draw the conclusion that they have the same origin.
The laws of mathematics and the laws of logic are the laws of the reflection of the phenomenal world in our perception and thinking.
Just as logical axioms can operate only with concepts and refer only to concepts, so mathematical axioms can operate only with finite and constant magnitudes and refer only to them.
In relation to infinite and variable magnitudes these axioms are incorrect, just as logical axioms are incorrect even in relation to emotions, to symbols, to music and to the hidden meaning of the word, to say nothing of that content of ideas which cannot be put into words.
What does it mean?
It means that axioms of logic and mathematics are deduced by us from the observation of phenomena, i.e. the phenomenal world, and represent a certain conditional incorrectness, necessary for the cognition of the unreal, 'subjective' world in the true meaning of the word.
It has been pointed out earlier that in fact we have two mathematics. One — the mathematics of finite and constant numbers, represents an entirely artificial construction for solving problems on the basis of conditional data. The chief of these conditional data consists in the fact that in problems of this mathematics there is always taken only the 't' of the universe, i.e. only one section of the universe which is never mixed with another section. Thus the mathematics of finite and constant magnitudes studies an artificial universe and is itself something specially created on the basis of our observations of phenomena and as a means of facilitating these observations. The mathematics of finite and constant numbers is unable to go beyond phenomena. It deals with an imaginary world, with imaginary magnitudes. (The practical results of those applied sciences which are based on mathematical sciences need not disturb the observer, because they are merely the solutions of problems in definite artificial conditions.)
The other, the mathematics of infinite and variable magnitude is something entirely real, constructed on the basis of mental deductions about the real world.
The first refers to the world of phenomena, which is nothing else than our incorrect perception and representation of the world.
The second refers to the world of noumena, which is the world as it is.
The first is unreal and exists only in our consciousness, in our imagination.
The second is real and expresses the relations of the real world.

An example of 'real mathematics', violating the fundamental axioms of our mathematics (and logic) is the so-called mathematics of transfinite numbers.
Transfinite numbers, as their name implies, are numbers beyond infinity.
Infinity, as represented by the sign         is a mathematical expression with which, as such, it is possible to carry out all operations: divide, multiply, raise to powers. It is possible to raise infinity to the power of infinity - the result will be          . This magnitude is an infinite number of times greater than a simple infinity. And at the same time they are equal                    . Precisely this is the most remarkable thing in transfinite numbers. You can carry out with them any operation you like, and they will change correspondingly, remaining at the same time equal. This violates the fundamental laws of mathematics, accepted for finite numbers. Having changed, a finite number can no longer be equal to itself. And yet we see here that, in changing, a transfinite number remains equal to itself.
Moreover, transfinite numbers are entirely real. We can find examples in the real world corresponding to expressions      and even       and        .
Let us take a line, any segment of a line. We know that the number of points in this line is equal to infinity, because a point has no dimensions. If our segment equals an inch, and side by side with it we imagine a segment which equals a mile, then each point in the small segment will have a corresponding point in the large segment. The number of points in the segment an inch long is infinite. The number of points in a mile is also infinite. The result is                    .
Now let us imagine a square of which the given line a constitutes one side. The number of lines in a square is infinite. The number of points in every line is infinite. Consequently the number of points in a square equals infinity multiplied by itself an infinite number of times        . This magnitude is undoubtedly infinitely greater than the first ∞. And at the same time they are equal, as all infinite magnitudes are equal, because if there is infinity, it is one and it cannot change.
On the square        which we have obtained, let us construct a cube. This cube consists of an infinite number of squares, just as the square consists of an infinite number of lines, and the line -of an infinite number of points. Consequently the number of points in the cube        equals         . This expression is equal to the expressions         and       , which means that infinity continues to grow, remaining at the same time unchanged.

Thus we see in transfinite numbers that two magnitudes, each of which separately equals a third, may be not equal to each other. Altogether we seethat the fundamental axioms of our mathematics do not operate there, are not applicable there. And we have every right to establish the law that the fundamental axioms of mathematics, cited above, are not applicable there but are valid and applicable only for finite numbers.
Moreover, we can say that the fundamental axioms of our mathematics are valid only for constant magnitudes. In other words, they require unity of time and place, namely, each magnitude is equal to itself at a given moment. But if we take a variable magnitude, and take it at different moments, it will not be equal to itself. Of course one may say that, in changing, it becomes another magnitude, that it is a given magnitude only so long as it does not change. But this is exactly what I mean.
Axioms of our mathematics are applicable only to finite and constant magnitudes.
So, in direct opposition to the usual view, we have to admit that mathematics of finite and constant magnitudes is unreal, i.e. it deals with unreal relations of unreal magnitudes, whereas the mathematics of infinite and fluent magnitudes is real, i.e. it deals with the real relations of real magnitudes.
Indeed, the greatest magnitude of the first mathematics has no dimension whatever, is equal to nought or to a point in comparison with any magnitude of the second mathematics ALL THE MAGNITUDES OF WHICH, IN ALL THEIR VARIETY, ARE EQUAL AMONG THEMSELVES.
Thus here, as well as in logic, the axioms of the new mathematics appear as absurdities.
A magnitude can be not equal to itself.
The part can be equal to the whole or can be greater.
One of two equal magnitudes can be infinitely greater than
the other. All DIFFERENT magnitudes are equal to each other.
We observe a complete analogy between the axioms of mathematics and those of logic. The logical unit -the concept -possesses all the properties of a finite and constant magnitude. The fundamental axioms of mathematics and logic are essentially the same. And they are correct in similar conditions and cease to be correct in similar conditions.
We may say without the slightest exaggeration that the fundamental axioms of logic and mathematics are correct only so long as logic and mathematics operate with artificial, conditional units which do not exist in nature.
The truth is that there are no finite, constant magnitudes in nature, just as there are no concepts. A finite, constant magnitude and a concept are conditional abstractions; they are not reality but, so to speak, sections of reality.
How to connect the idea of the absence of constant magnitudes with the idea of a static universe? At the first glance, the one contradicts the other. But in actual fact this contradiction does not exist. Not this one, but the greater universe is static, the world of many dimensions of which we know the eternally moving section called the threedimensional infinite sphere. In addition, the very concepts of motion and immobility need to be reconsidered, because in the way our mind usually understands them, they do not correspond to reality.
We have already examined in detail how the idea of motion results from our sense of time, i.e. from the imperfection of our sense of space.
If our space-sense were more perfect, then, in relation to any given object, say to a given human body, we would perceive the whole of its life in time, from birth to death. Then within the limits of this compass it would be for us a constant magnitude. But now, at every moment of its life it is for us not a constant, but a variable magnitude. And what we call the body does not actually exist. It is only a section of a fourdimensional body which we never see. We must remember that all our threedimensional world actually does not exist. It is the creation of our imperfect senses, the result of their imperfection. It is not the world; it is only what we see of the world. The three-dimensional world is the four-dimensional world observed through the narrow slit of our senses. Therefore all the magnitudes we accept as such in the three-dimensional world, are not real magnitudes, but are only artificially assumed.
They have no real existence, just as the present has no real existence. This has already been said. What we call the present is the transition from the future into the past. But this transition has no extension. Consequently, the present does not exist. Only the future and the past exist.
Thus constant magnitudes in the three-dimensional world are abstractions; just as motion in three-dimensional world is in actual fact an abstraction. In the three-dimensional world there is no change, no motion. For the conception of motion we need a four-dimensional world. The threedimensional world does not exist in reality, or it exists only during one ideal moment. In another ideal moment there is already another three-dimensional world. Consequently, magnitude A is no longer A the next moment, but becomes B; the next moment it is C and so on, ad infinitum. It is equal toitself only during one ideal moment. In other words, within the limits of one ideal moment the axioms of mathematics are valid; for the comparison of two ideal moments they are only conditional, just as Bacon's logic is conditional compared to the logic of Aristotle. In time, i.e. in relation to magnitudes which are variable from the point of view of the ideal moment, they are incorrect.
The idea of constancy or variability is the outcome of the incapacity of our limited mind to know a thing otherwise than in the form of its section. But if we achieve the knowledge of a thing in four dimensions, say a human bodyfrom birth to death, it will be a whole and constant magnitude, a section of which we call the human body changing in time. A moment of life, i.e. the body as we know it in the three-dimensional world, is a point on an infinite line. If we could know this body as a whole, we would know it as an absolutely constant magnitude with all its variety of forms, states and positions. But in that case the axioms of our mathematics would not be applicable to this constant magnitude, because it would be an infinite magnitude.
This infinite magnitude we cannot know. We always know only its section. And to this imaginary section of the universe belong our mathematics and logic.




Tertium Organum

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