Chapter Xx
*
"Tertium Organum", by P.D. Ouspensky, [1922],
p. 243
Chapter Xx
The sense of infinity. The Neophyte's first ordeal. An intolerable sadness. The loss of everything real. What would an animal feel on becoming a man? The transition to the new logic. Our logic as founded on the observation of the laws of the phenomenal world. Its invalidity for the study of the world of noumena. The necessity for another logic. Analogy between the axioms of logic and of mathematics. Two Mathematics. The mathematics of real magnitudes (infinite and variable): and the mathematics of unreal, imaginary magnitudes (finite and constant). Transfinite numbers--numbers lying beyond Infinity. The possibility of different infinities.
There is in existence an idea which a man should always call to mind when too much subjugated by the illusions 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 the book "A New Era of Thought"--concerning which I have had already 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 realization 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?
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He will sense a precipice, an abyss everywhere, no matter where he looks; and experience indeed an incredible horror, fear and sadness, until this fear and sadness shall transform themselves into the joy of the sensing of a 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 already examined into the manner in which a two-dimensional being might approach to a comprehension of the third dimension. But we have never asked ourselves the question: what would it "feel", beginning to sense the third dimension, beginning to be conscious of "a new world" environing it?
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 "first of all?" First of all, that its old world, "the world of the animal", its comfortable, habitual world, the one in which it was born, to which it has become accustomed, and which it imagines to be the "only" real one, is crumbling away and falling all around it. Everything that before seemed real, becomes false, delusive, fantastic, unreal. The impression of the unreality of all its environment will be 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 one infinitely more beautiful and new is opening up, considerable time will necessarily pass. And during all 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 realized.
With the beginning of the gradual loss of the old world, the logic of the two-dimensional being--or that which stood for it for logic--will suffer continual violation, and its strongest impression
p. 245
will be that there is "no logic at all", that no laws of any sort even exist.
Formerly, when it was an animal, it reasoned:
"This is this".
\"This house is my own".
"That is that".
\"That house is strange".
"This is not that".
\"The strange house is not my own".
The strange house and its own house the animal regards as "different objects", having nothing in common. But now it will surprisedly understand that "the strange house" and "its own house" are EQUALLY houses.
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 generalization of these new proper. ties--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 its 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",
"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 similar, that they represent "one and"
p. 246
\"the same thing", that they are both "houses". The animal will never credit this "similarity". For it the two houses, its own, "where it is fed", and the strange one, "where it is beaten" if it enters, will remain "entirely different". There will be nothing in common in them for it, and the effort to prove to it the similarity of these two houses will lead to nothing "until it senses this itself". Then, sensing confusedly the idea of the likeness of two different objects, and 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 dead awaken"--i.e., when we "men", come to the realization 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 realize "the new world", we must understand "the new logical order of things".
"Our usual logic assists us in the investigation of the relations of the phenomenal world only. Many attempts have been made to define "what logic is". But logic is just as essentially undefinable as is mathematics.
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 quite in the same way as mathematics. As with logic, so also with mathematics (at least the generally known mathematics of "finite" and "constant" quantities), both were deduced
p. 247
by us from the observation of the phenomena of "our" world. Generalizing 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 of 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".)
"Everything is either A or Not-A".
"Everything will be either A or Not-A".
The logic of Aristotle and Bacon, developed and supplemented by their many followers, deals "with concepts only".
Logos, the word, is the object of logic. An idea, in order to become the object of logical reasoning, in order to be subjected to the laws of logic, must be expressed in a word. That which cannot be expressed in a word cannot enter into a logical system. More-over 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 between them complete similarity.
The axioms of logic,
"A is A".
"A is not Not-A". p. 248
\"Everything is either A or Not-A".
fully correspond to the fundamental axioms of mathematics, to the 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.
The similarity between the axioms of mathematics and those of logic extends very far, and this permits us to draw a conclusion about their similar origin.
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 with concepts only, and are related solely to them, so the axioms of mathematics apply to "finite" and "constant" magnitudes only, and are related solely to them.
These Axioms Are Untrue In Relation To Infinite And Variable Magnitudes, 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 of 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.
"As has been said before, we have in reality "two mathematics". One, "the mathematics of finite and constant numbers", represents a quite artificial construction for the solution of problems based on conditional data. The chief of these conditional data consists in the fact that in problems of "this mathematics" there is always taken "the t of the universe only", i.e., one section only of the universe is taken,
p. 249
which section is never taken in conjunction with another one. This mathematics of finite and constant magnitudes studies "an artificial universe", and is in itself something especially created on the basis of our observation of "phenomena", and serves for the simplification of these observations. "Beyond phenomena" the mathematics of finite and constant numbers cannot go. It is dealing with an imaginary world, with imaginary magnitudes. The practical results of those applied sciences which are built upon mathematical science should not confuse the observer, because these are merely the solutions of problems in definite artificial conditions.
The other, "the mathematics of infinite and variable magnitudes", represents something entirely real, built upon the reasonings in regard to "a real world".
The first is related to the world of phenomena, which represents in itself nothing other than "our incorrect" apprehension and perception of the world.
The second is related to the world of noumena, 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.
"The mathematics of transfinite numbers", so called, may serve as an example of "real mathematics," violating the fundamental axioms of our mathematics (and logic).
By "transfinite numbers", as their name implies, is meant numbers beyond infinity.
Infinity, as represented by the sign is the mathematical expression with which, as such, it is possible to perform all operations: divide, multiply, raise to powers. It is possible to raise infinity to the power of infinity--it will be. This magnitude is an infinite number of times greater than simple infinity. "And at the same time they are both equal": =. And this is the most remarkable property of transfinite numbers. You may perform with them any operations whatsoever, "they will change in a corresponding manner, remaining at the same time equal". This violates the fundamental laws of mathematics accepted for "finite" numbers. After a
p. 250
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 mathematics
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are valid for "constant" magnitudes only. Or in other words they demand "unity of time and unity of place". That is, each magnitude is equal to itself at a "given" moment. But if we take 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.
p. 252
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"
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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.
"Tertium Organum", by P.D. Ouspensky, [1922],
p. 243
Chapter Xx
The sense of infinity. The Neophyte's first ordeal. An intolerable sadness. The loss of everything real. What would an animal feel on becoming a man? The transition to the new logic. Our logic as founded on the observation of the laws of the phenomenal world. Its invalidity for the study of the world of noumena. The necessity for another logic. Analogy between the axioms of logic and of mathematics. Two Mathematics. The mathematics of real magnitudes (infinite and variable): and the mathematics of unreal, imaginary magnitudes (finite and constant). Transfinite numbers--numbers lying beyond Infinity. The possibility of different infinities.
There is in existence an idea which a man should always call to mind when too much subjugated by the illusions 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 the book "A New Era of Thought"--concerning which I have had already 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 realization 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?
p. 244
He will sense a precipice, an abyss everywhere, no matter where he looks; and experience indeed an incredible horror, fear and sadness, until this fear and sadness shall transform themselves into the joy of the sensing of a 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 already examined into the manner in which a two-dimensional being might approach to a comprehension of the third dimension. But we have never asked ourselves the question: what would it "feel", beginning to sense the third dimension, beginning to be conscious of "a new world" environing it?
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 "first of all?" First of all, that its old world, "the world of the animal", its comfortable, habitual world, the one in which it was born, to which it has become accustomed, and which it imagines to be the "only" real one, is crumbling away and falling all around it. Everything that before seemed real, becomes false, delusive, fantastic, unreal. The impression of the unreality of all its environment will be 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 one infinitely more beautiful and new is opening up, considerable time will necessarily pass. And during all 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 realized.
With the beginning of the gradual loss of the old world, the logic of the two-dimensional being--or that which stood for it for logic--will suffer continual violation, and its strongest impression
p. 245
will be that there is "no logic at all", that no laws of any sort even exist.
Formerly, when it was an animal, it reasoned:
"This is this".
\"This house is my own".
"That is that".
\"That house is strange".
"This is not that".
\"The strange house is not my own".
The strange house and its own house the animal regards as "different objects", having nothing in common. But now it will surprisedly understand that "the strange house" and "its own house" are EQUALLY houses.
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 generalization of these new proper. ties--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 its 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",
"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 similar, that they represent "one and"
p. 246
\"the same thing", that they are both "houses". The animal will never credit this "similarity". For it the two houses, its own, "where it is fed", and the strange one, "where it is beaten" if it enters, will remain "entirely different". There will be nothing in common in them for it, and the effort to prove to it the similarity of these two houses will lead to nothing "until it senses this itself". Then, sensing confusedly the idea of the likeness of two different objects, and 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 dead awaken"--i.e., when we "men", come to the realization 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 realize "the new world", we must understand "the new logical order of things".
"Our usual logic assists us in the investigation of the relations of the phenomenal world only. Many attempts have been made to define "what logic is". But logic is just as essentially undefinable as is mathematics.
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 quite in the same way as mathematics. As with logic, so also with mathematics (at least the generally known mathematics of "finite" and "constant" quantities), both were deduced
p. 247
by us from the observation of the phenomena of "our" world. Generalizing 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 of 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".)
"Everything is either A or Not-A".
"Everything will be either A or Not-A".
The logic of Aristotle and Bacon, developed and supplemented by their many followers, deals "with concepts only".
Logos, the word, is the object of logic. An idea, in order to become the object of logical reasoning, in order to be subjected to the laws of logic, must be expressed in a word. That which cannot be expressed in a word cannot enter into a logical system. More-over 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 between them complete similarity.
The axioms of logic,
"A is A".
"A is not Not-A". p. 248
\"Everything is either A or Not-A".
fully correspond to the fundamental axioms of mathematics, to the 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.
The similarity between the axioms of mathematics and those of logic extends very far, and this permits us to draw a conclusion about their similar origin.
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 with concepts only, and are related solely to them, so the axioms of mathematics apply to "finite" and "constant" magnitudes only, and are related solely to them.
These Axioms Are Untrue In Relation To Infinite And Variable Magnitudes, 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 of 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.
"As has been said before, we have in reality "two mathematics". One, "the mathematics of finite and constant numbers", represents a quite artificial construction for the solution of problems based on conditional data. The chief of these conditional data consists in the fact that in problems of "this mathematics" there is always taken "the t of the universe only", i.e., one section only of the universe is taken,
p. 249
which section is never taken in conjunction with another one. This mathematics of finite and constant magnitudes studies "an artificial universe", and is in itself something especially created on the basis of our observation of "phenomena", and serves for the simplification of these observations. "Beyond phenomena" the mathematics of finite and constant numbers cannot go. It is dealing with an imaginary world, with imaginary magnitudes. The practical results of those applied sciences which are built upon mathematical science should not confuse the observer, because these are merely the solutions of problems in definite artificial conditions.
The other, "the mathematics of infinite and variable magnitudes", represents something entirely real, built upon the reasonings in regard to "a real world".
The first is related to the world of phenomena, which represents in itself nothing other than "our incorrect" apprehension and perception of the world.
The second is related to the world of noumena, 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.
"The mathematics of transfinite numbers", so called, may serve as an example of "real mathematics," violating the fundamental axioms of our mathematics (and logic).
By "transfinite numbers", as their name implies, is meant numbers beyond infinity.
Infinity, as represented by the sign is the mathematical expression with which, as such, it is possible to perform all operations: divide, multiply, raise to powers. It is possible to raise infinity to the power of infinity--it will be. This magnitude is an infinite number of times greater than simple infinity. "And at the same time they are both equal": =. And this is the most remarkable property of transfinite numbers. You may perform with them any operations whatsoever, "they will change in a corresponding manner, remaining at the same time equal". This violates the fundamental laws of mathematics accepted for "finite" numbers. After a
p. 250
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 mathematics
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are valid for "constant" magnitudes only. Or in other words they demand "unity of time and unity of place". That is, each magnitude is equal to itself at a "given" moment. But if we take 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.
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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"
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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.
