Tertium Organum (P.D. Ouspensky) · 159 of 225 · tr. Nicholas Bessaraboff and Claude Bragdon
Chapter XX (part 3)
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.
Themes
◆Gnosis Direct Knowledgesoteriology
Salvation through direct experiential knowledge of the divine nature, not through faith, ritual, or moral works alone.
The text implies a need for knowledge beyond logic and words to understand the true nature of reality.
All passages →◆Logoscosmology
The divine reason, word, or ordering principle through which the universe was created; mediates between the transcendent God and the material world.
The term 'Logos' is explicitly defined as 'the word' and the object of logic, linking it to ordering principles though not explicitly divine creation here.
All passages →◇Apophatic Theologytheology
Describing the divine only by negation—God transcends all positive categories and concepts; the via negativa.
[auto] The passage emphasizes that not everything can be expressed in concepts or words, aligning with the via negativa regarding true reality beyond logic.
All passages →◇Cosmic Dualismtheology
Ontological opposition between two primal principles (light/darkness, truth/lie, spirit/matter) as the fundamental structure of reality.
[auto] The text distinguishes between the phenomenal world (subjective/unreal) and a higher reality where standard axioms fail, establishing an ontological opposition.
All passages →◇Opposites Transcendedanthropology
The dissolution of binary oppositions (male/female, inner/outer, above/below) as a mark of spiritual completion.
[auto] It explicitly states that binary logical axioms like 'A is not Not-A' are untrue in relation to infinite and variable magnitudes.
All passages →◇Paradox As Teachingethics
Using apparent contradiction, reversal, or riddle to destabilise ordinary understanding and induce insight.
[auto] The author uses paradoxical statements such as 'mathematics... is entirely illogical' to challenge ordinary understanding of axioms.
All passages →Related passages · 8 parallels
◆neoplatonism|Select Works of Plotinus|Select Works, Section 131
“Section 9 9. But how can we conceive a thing having existence without having magnitude? We have only to think of things whose identity does not depend on their quantity- for certainly magnitude can be distinguished from existence as can man…”
Both passages explore the limitations of logical and mathematical principles in describing reality. Passage A discusses how logic and mathematics are derived from observing the phenomenal world and are limited in their applicability to infinite and variable magnitudes, as well as to emotions and ideas that cannot be expressed in words. Passage B similarly argues that magnitude is an abstract principle that does not inherently possess quantity, highlighting the distinction between the abstract and the concrete.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 133
“Section 11 11. "But, given Magnitude and the properties we know, what else can be necessary to the existence of body?" Some base to be the container of all the rest. "A certain mass then; and if mass, then Magnitude? Obviously if your Base…”
Both passages explore the nature and limitations of mathematical and logical axioms, discussing how they apply to different realms of existence. They both highlight the distinction between finite/constant and infinite/variable magnitudes, though from different philosophical perspectives.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 512 (part 1)
“Section 4 4.
We are told that number is Quantity in the primary sense, number together with all continuous magnitude, space and time: these are the standards to which all else that is considered as Quantity is referred, including motion wh…”
Both passages explore the nature and limitations of logic and mathematics, discussing how they are derived from observations of the phenomenal world and their applicability to different aspects of reality. They both highlight the conditional and subjective nature of these systems.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 512 (part 2)
“True, greatness and smallness are not claimed to be quantities, but relations: but it is by their apparent possession of quantity that they are thought of as relations.
All this, however, needs more careful examination.
In sum, we hold th…”
Both passages explore the nature of quantity and its relationship to other concepts. Passage A discusses the distinction between primary and secondary quantities, while Passage B examines the limitations of mathematics and logic in describing certain aspects of reality, particularly those that are infinite or variable. Both texts delve into the philosophical underpinnings of their respective subjects, highlighting the complexities and nuances involved.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 517 (part 2)
“Doubtless the truth is that every quality performs its own function independently of a standard; for in no case could it produce an effect outside of its power.
Even beauty would seem to have a power of its own.
Does this apply to triangu…”
Both passages discuss the nature and limitations of mathematics and logic, exploring how they are derived from observations of the phenomenal world and their applicability to different aspects of reality. They both highlight the conditional and subjective nature of these systems.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 518
“Section 11 11. But if these considerations are sound, why has Quality more than one species? What is the ground for distinguishing between habit and disposition, seeing that no differentia of Quality is involved in permanence and non-perman…”
Both passages explore the limitations and nature of logic and mathematics, particularly in how they relate to concepts and quantities. They both argue that these systems are derived from observations of the phenomenal world and have inherent limitations when applied to other domains, such as emotions or infinite variables.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 572
“Section 11 11. Passing to Quantity and the quantum, we have to consider the view which identifies them with number and magnitude on the ground that everything quantitative is numbered among Sensible things or rated by the extension of its s…”
Both passages discuss the nature of quantity and its relationship to other concepts. Plotinus explores the distinction between absolute and relative quantities, while the second passage discusses the foundations of mathematics and logic, highlighting their similarities and limitations. Both delve into the philosophical underpinnings of their respective subjects, showing a conceptual parallel in their exploration of fundamental principles.
◆neoplatonism|Select Works of Plotinus|Select Works, Section 581
“Section 20 20. We have to ascertain whether there is not to every quality a contrary. In the case of virtue and vice, even the mean appears to be contrary to the extremes. But when we turn to colours, we do not find the intermediates so rel…”
Both passages discuss the limitations and scope of logic and mathematics, exploring how these systems are derived from observations of the phenomenal world and their applicability to different aspects of reality. They both highlight the conditional and subjective nature of these systems.