Tertium Organum (P.D. Ouspensky) · 160 of 225 · tr. Nicholas Bessaraboff and Claude Bragdon
Chapter XX (part 4)
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 a 2 , 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.
The doctrine of the universe as boundless and without center or circumference, populated by innumerable worlds — Bruno's rejection of the closed Aristotelian sphere; the divine and the cosmos as ontologically infinite.
The text posits that the real world is represented by infinite and variable magnitudes, contrasting with the artificial finite universe of ordinary observation.
Gurdjieff and Ouspensky's claim that ordinary human beings live in a state of waking sleep, reacting mechanically without unified will or consciousness, and that the spiritual path begins with recognising this condition.
As an Ouspensky text, it describes ordinary perception as 'incorrect apprehension' creating an 'artificial universe,' aligning with the concept of mechanical consciousness.
Numbers, proportions, and mathematical relationships as the underlying structure of divine reality.
[auto] The passage centrally argues that mathematics of infinite magnitudes represents the real world while finite mathematics is artificial and unreal.
The dissolution of binary oppositions (male/female, inner/outer, above/below) as a mark of spiritual completion.
[auto] It describes how transfinite numbers change yet remain equal to themselves, transcending the binary opposition between change and identity found in finite logic.
Using apparent contradiction, reversal, or riddle to destabilise ordinary understanding and induce insight.
[auto] The text utilizes mathematical paradoxes, such as infinity equaling its own power, to illustrate properties of the real world that violate finite logic.