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.
Themes
◆Infinite Cosmoscosmology
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.
All passages →◆Mechanical Humanityanthropology
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.
All passages →◆Unity Of Beingtheology
The doctrine that all existence is ultimately one substance or one divine reality; non-duality as a theological claim.
The passage suggests that infinity is ultimately one and cannot change, hinting at a unified reality beyond multiplicity.
All passages →◇Cosmic Dualismtheology
Ontological opposition between two primal principles (light/darkness, truth/lie, spirit/matter) as the fundamental structure of reality.
[auto] It establishes a fundamental ontological opposition between the unreal world of phenomena (consciousness) and the real world of noumena.
All passages →◇Numerical Mysticismtheology
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.
All passages →◇Opposites Transcendedanthropology
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.
All passages →◇Paradox As Teachingethics
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.
All passages →Related passages · 7 parallels
◆neoplatonism|Select Works of Plotinus|Select Works, Section 622
“Section 2 2. What, then, of the "Number of the Infinite"? To begin with, how is Number consistent with infinity? Objects of sense are not unlimited and therefore the Number applying to them cannot be so. Nor is an enumerator able to number…”
Both passages explore the concept of infinity and its relationship with numbers and reality. Passage A discusses the inconsistency of infinity with finite numbers and the Intellectual sphere, while Passage B delves into the mathematics of infinite and variable magnitudes, highlighting the real nature of transfinite numbers.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 623
“Section 3 3. And there is the question How can the infinite have existence and remain unlimited: whatever is in actual existence is by that very fact determined numerically. But, first, if multiplicity holds a true place among Beings, how c…”
Both passages explore the concept of infinity and its relationship to reality and perception. They discuss the limitations of finite mathematics and the nature of infinite magnitudes, though from different philosophical and mystical perspectives.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 632
“Section 12 12. We may be told that unity and monad have no real existence, that the only unity is some definite object that is one thing, so that all comes to an attitude of the mind towards things considered singly. But, to begin with, why…”
Both passages discuss the concept of infinity and its relationship to numbers and mathematics. They explore the idea of transfinite numbers and the limitations of finite mathematics, showing a genuine conceptual parallel across different mystical traditions.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 633 (part 1)
“Section 13 13.
It cannot reasonably be thought that the notion of unity is derived from the object since this is physical- man, animal, even stone, a presentation of that order is something very different from unity [which must be a thing…”
Both passages explore the concept of infinity and its relationship to reality and perception. They discuss the limitations of finite mathematics and the nature of infinite magnitudes, albeit from different philosophical and mystical perspectives.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 637
“Section 17 17. But what of the Infinite Number we hear of; does not all this reasoning set it under limit? And rightly so if the thing is to be a number; limitlessness and number are in contradiction. How, then, do we come to use the term?…”
Both passages discuss the concept of infinity and its relationship to mathematics and reality. They explore the idea of transfinite numbers and the distinction between finite and infinite magnitudes, showing a genuine conceptual parallel across different mystical traditions.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 638
“Section 18 18. It appears then that Number in that realm is definite; it is we that can conceive the "More than is present"; the infinity lies in our counting: in the Real is no conceiving more than has been conceived; all stands entire; no…”
Both passages explore the concept of infinity and the nature of reality beyond finite perception. Passage A discusses the distinction between the mathematics of finite and infinite magnitudes, relating them to the worlds of phenomena and noumena. Passage B describes the nature of the Real and the Intellectual-Principle, emphasizing the infinite and unchangeable nature of true Being.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 645
“Section 6 6. But how can that higher soul have sense-perception? It is the perception of what falls under perception There, sensation in the mode of that realm: it is the source of the soul's perception of the sense-realm in its corresponde…”
Both passages explore the concept of infinity and its relationship to mathematics and reality. They discuss the limitations of finite mathematics and the existence of a higher, more real mathematics that deals with infinite and variable magnitudes. The passages also touch on the idea of transfinite numbers and their properties, showing a genuine conceptual parallel across different mystical traditions.