Mathematics as a Portal to Higher Understanding

     To instruct initiates ¹ in how to achieve a higher state of consciousness, mystical philosophers point to magical portals through which humans can gain an understanding of the metaphysical and conceptual level of being:

• Beauty

• Mathematics

• Art

• Contemplation of concepts, myths, and symbols

• Meditation on concepts, myths, and symbols

• Physical activities such as drama and dance

• Altered states of consciousness

• Mental and sensory exercises
  

  Above these is the search for understanding of the realm of Higher Reason and Higher Intellect which Plato titled dialectic.


Beauty As the Stairway to Higher Knowledge

"The true procedure of spiritual advance is to envision beautiful things of this world as steps along which one mounts upwards to that Higher Beauty." Plato, Symposium

     To understand how Perennialist ² teachings provide an entrée to a supersensible world, we must begin with the most fundamental questions. Why is our physical world not governed by incoherent chaos? Why is this "Cosmos" (as the Greeks termed it) intelligible to humans? What is the essence of these organizing principles: "pattern," "structure," "unity," and "order?"

     Pythagoras and Plato believed that elements in our empirical world are ordered by supersensible Forms residing in a higher dimension. These Forms were necessary to explain the structure we see in the world around us. The only reason the physical universe is intelligible at all is that objects retain the same form, different things take on the same form, and we are able to communicate with one another about the meaning of these patterns and relationships.

     Our bodies, for example, undergo complete change within a seven year cycle--every atom being replaced by new atoms. How it that we retain the same form when everything substantial has been replaced?

     Why is it that completely different objects are somehow "the same" with respect to this or that organizing principle: (height, color, chairness, etc.)? No one would doubt that two chairs are somehow instances of the same thing: the form Chair.

"But now the sight of day and night, and the months and the revolutions of the years have created number and have given us a conception of time, and the power of inquiring about the nature of the universe. And from this source we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man." Plato, Timaeus

     Pythagoras and Plato believed that Forms are neither material objects, aspects of material objects, nor mere concepts in our brains. Forms exist on their own terms, apart from the physical universe, eternal and immutable. Physical objects are what they are by virtue of their participation in specific Forms.

     Aristotle maintained that these Forms are merely "aspects" of physical objects and have no separate being in a supersensible world. Modern scientists such as Rupert Sheldrake have come up with theories similar to Aristotle, seeing forms as merely constituents of objects. Sheldrake called this process of retaining the same form "morphic resonance."

     Most modern scientists ignore metaphysical questions about the essence of Forms. But we cannot avoid this question if we take seriously what Perennialist teachers have maintained: that humans have the capacity to attain a higher state of consciousness through understanding the supersensible world of Forms and mathematical entities.

"Within the human consciousness is the unique ability to perceive the transparency between absolute, permanent relationships, contained in the insubstantial forms of a geometric order, and the transitory, changing forms of our actual world. The content of our experience results from an immaterial, abstract, geometric architecture which is composed of harmonic waves of energy, nodes of relationality, melodic forms springing forth from the eternal realm of geometric proportion."

Robert Lawlor, Sacred Geometry

Spiritual Geometry
     Mathematics and especially geometry were seen by such Perennialist teachers as Pythagoras and Plato as among the most effective means of understanding and entering a spiritual realm composed of eternal, unchanging (invariant) Forms and mathematical entities. In Plato's Commonwealth, Socrates says that only those versed in geometry will be allowed entrance into the ideal state.

"For Pythagoras, mathematics was a bridge between the visible and invisible worlds. He pursued the study of mathematics not only as a way of understanding and manipulating nature, but also as a means of turning the mind away from the physical world, which he held to be transitory and unreal, and leading it to the contemplation of eternal and truly existing things that never vary. He taught his students that by focusing on the elements of mathematics, they could calm and purify the mind, and ultimately, through disciplined effort, experience true happiness."

John Strohmeier and Peter Westbrook,
The Life and Teachings of Pythagoras

     Geometry means "measurement of the earth." When the Nile flooded each year in ancient Egypt, obliterating the property boundaries, priest-mathematicians used geometry to re-establish the markings for specific areas. To the Egyptians and the mystical philosophers, geometry was regarded as a magical science with the power to reveal to humans the properties of given elements (points, lines, angles, surfaces, and solids) that remain invariant under specified transformations. In general, we use geometry to study spatial order through the dimensions and relationships of forms.

     Modern man has returned to a view of the universe held by early Greek philosophers that the most fundamental element is energy, which is perceived by humans through the recognition of specific patterns or forms. Since Einstein's relativity theory, we have come to see matter as a form of energy. According to subatomic physics, there are only patterns of probability. Matter is seen as an illusion of the senses.

"At this stage modern man must face a series of paradoxes of his own making. This 'objective' mathematics, for all its success in mechanics, makes use of abstractions that correspond to nothing in experience. The square root of minus one, the zero, infinity are abstractions corresponding to nothing in that physical realm we call 'reality'. And without these abstractions the formulae do not work. In other words, to describe the phenomenal world 'scientifically', science must have recourse to abstraction. . ."

John Anthony West, Serpent In the Sky:
The High Wisdom of Ancient Egypt

     Seekers begin to move toward a higher understanding when they recognize that the senses perceive only a meager part of reality. Both physics and mathematics demonstrate the existence of elements above the physical realm.

     A number of Perennialist teachers used mathematics, especially geometry, in assisting seekers to move beyond the empirical world to higher dimensions of meaning.

"The implicit goal of [Classical] education was to enable the mind to become a channel through which the 'earth' (the level of manifested form) could receive the abstract, cosmic life of the heavens. The practice of geometry was an approach to the way in which the universe is ordered and sustained. Geometric diagrams can be contemplated as still moments revealing a continuous, timeless, universal action generally hidden from our sensory perception. Thus a seemingly common mathematical activity can become a discipline for intellectual and spiritual insight."

Robert Lawlor, Sacred Geometry

     But the study of geometry must be at the highest possible level to enable students to achieve understanding of a higher realm of being. First they must be fully aware of the various levels of being, the characteristics of each level, and the entities peculiar to each level.

Level of Being	Characteristics	Entities	Faculties	Entry
Spiritual Supersensible	Eternal Unchanging	Forms	Higher Reason Higher Intellect	Dialectic Philosophy
Metaphysical Conceptual	Intellectual Nonmaterial	Concepts Symbols	Reason Intellect	Metaphysics   Mathematics
Material Empirical	Measurable Tangible	Persons Objects	Thought Senses	Reason

     The discerning study of simple geometric concepts enables students to gain an understanding of the second, metaphysical, level of being, since geometry deals with non-material concepts and invariable symbols. For example, the study of a simple circle of any size leads to the discovery of what is called a transcendental number, Pi.

Pi - Π
Any Circle:	Radius (CD) = .5    Diameter (AB) = 1    Circumference = Diameter x pi (3.14159265)
If we take any circle, dividing the circumference by the diameter results in Pi: 3.14159265. Π --Pi is a transcendental number. A transcendental number cannot be the root of any polynomial equation with integer coefficients, meaning that it is not an algebraic number of any degree. Throughout human history there have been many attempts to calculate this number precisely. One of the oldest approximations appears in the Rhind Papyrus (circa 1650 B.C.E.) from ancient Egypt in which a geometrical construction is given where (16/9)2 = Π.

     What is transmitted through geometry is so subtle that it is exceptionally easy to miss it. When we come upon an entity such as Pi, we have arrived at INVARIANCE. No matter how large a circle, dividing its circumference by its diameter invariably gives us Pi. We are able to understand that the cosmos is put together using specific formulae (such as Pi)--we are discovering the very STRUCTURE of the universe!

     Along with the discovery of the ordering principles through which our universe is constructed, in geometry we enter a fascinating world of CERTAINTY. In the ordinary empirical world of shoes and ships and sealing wax, we experience constant uncertainty, having to settle for approximations at best. Students of geometry can feel the difference in worlds as they enter the domain of mathematics; they are suddenly in a world of certainty; they will arrive at conclusions that are not just probable but undeniable.

      The world of mathematics is one where all entities are defined and if we follow its principles then our conclusions are unequivocal. We say of it that it is a "closed" world--because most of its elements are defined, but we must remember that the world of mathematics also contains symbols which point beyond it: irrational numbers, transcendental numbers, infinity, point, zero, etc. Students of geometry learn to breathe in a world of certainty and understand that in the higher world of Forms, this sense of certainty is unlimited: all in that world is eternal, unchanging, unified.

     As Plato points out, through the higher study of mathematics we can achieve the ability to think abstractly--attaining independence from sensible objects in our contemplation. The freeing of thought from dependence upon the sensible image is an accomplishment of the very greatest magnitude. Until thought has achieved this power, it cannot penetrate into the realm of imageless consciousness.

     Through following the specific exercises prescribed by the Perennialist teacher, initiates gain an awareness of the second level of being, the metaphysical-conceptual domain. Having understood the second, metaphysical, level of being, the seeker is able to move upward to an understanding of the highest world of being: the dimension of Forms.

"The true use of [mathematics] is simply to draw the soul towards being.  . . .

"The philosopher, because he has to rise out of the sea of change and lay hold of true being . . . must be an arithmetician.

"Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument.  . . .

"The knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient.  . . .

"Geometry will draw the soul toward truth, and create the spirit of philosophy."

Plato, The Commonwealth, Book VII

"The immediate content of the Higher Consciousness can not be cross-translated, but certain formal properties can be through the use of systematic symbols. . .  In fact, if the consciousness-equivalents of the entities and operations of pure mathematics were realized, we would find that, in that great science and art, cross-translation in a lofty sense already exists." Franklin Merrell-Wolff, Pathways Through To Space

Mental and Sensory Exercises: Stopping the World

     In the fascinating account of the teachings of don Juan, Carlos Casteneda depicted how he had to learn a new description of the world in a total sense and pit it against the old description, breaking the dogmatic certainty we all share that our interpretation of reality is the only interpretation.

"'Stopping the world' was indeed an appropriate rendition of certain states of awareness in which the reality of everyday life is altered because the flow in interpretation, which ordinarily runs uninterruptedly, has been stopped by a set of circumstances alien to that flow. In my case the set of circumstances alien to my normal flow of interpretations was the sorcery description of the world."

     It takes some effort to understand that Socrates, an advanced Perennialist teacher, employed fantastic practices which seemed to many of his interlocutors sheer sorcery.

"Menon: Well now, my dear Socrates, you are just like what I always heard before I met you: always perplexed yourself and shaking the beliefs of everyone else. And now you seem to me to be a veritable wizard, casting your spells over me, and I am truly getting bewitched and enchanted, and YOU HAVE STOPPED MY WORLD. And if I may venture to make a jest about you, you seem to me both in your appearance and in your power over others to be very like the flat torpedo fish, who torpifies those who come near him and touch him, as you have now STOPPED MY WORLD.  . . . And I think that you are very wise in not venturing away from home, for if you performed your necromancy in other places as you do in Athens, you would be cast into prison as a sorcerer."

Plato, Meno (80a) ³

Beware: Fatal Danger Ahead

     Throughout our study of portals to a higher state of consciousness, we must constantly retain an essential mood: the sense that what Socrates, Plato, Jesus, Rumi, and all Perennialist teachers spoke about and what we're now examining are LIFE-AND-DEATH-ISSUES.

     We're not in an ivy-leafed academic world where we merely joust with words or play with concepts: PEOPLE HAVE ALREADY DIED IN THIS ARENA. What we're engaged in is deadly serious business; we're dealing with critical issues for which men such as Socrates, Jesus, Servetus, and Bruno gave their very lives!

     In the same vein, the practice of stopping the world in an initiate's psyche is a radical, intense psychophysical procedure which can be dangerous if carried out incorrectly. The Perennialist teacher possesses, among her other capabilities, the discernment of the initiate's state of mind. ⁴

      In some of the advanced (and esoteric) practices involved in initiation (rebirth), there comes a point where a person must be willing to say: "I am so intent on entering a New Life that if I have to die to this present life I am willing to do so." The rebirth experience is one in which the initiate is not sure whether he will continue to live in the physical world or not. He must be ready to actually give up his life if that is required.

Conclusion
     Through the portals of art, mathematics, mental and sensory exercises, contemplation, meditation, physical activities, or altered states of consciousness, the initiate is able to enter into the world of Higher Being.

"When a person starts on the discovery of the absolute by the light of reason only, and without any assistance of sense, and perseveres until by pure intelligence he arrives at the perception of the absolute good, he at last finds himself at the end of the intelligible world, as in the case of sight at the end of the visible."

Plato, The Commonwealth, Book VII

Notes:

¹ Initiation

² The Perennial Tradition: the secret legacy, the single stream of initiatory teaching flowing through all the great schools of mysticism

³ The above is my own translation from the Greek text

Thomas Taylor's translation: Meno: "Socrates, I heard, before I had conversed with you, that the only part you take in conversation is this:--You pretend to be at a loss and doubtful yourself upon all subjects, and make others too no less to be at a loss what to think and say. You seem to be now playing the same conjurers tricks upon me; you manifestly use incantations to bewitch me, and to fill me with such perplexity that I know not what to say. If you will allow me to joke a little, I think you resemble exactly, not only in form but in other respects also, that broad sea-fish called the cramp-fish; for that too never fails to give a numbness to every person who either touches or approaches it. You seem to have done some such thing at present to me, and to have benumbed me. For I actually suffer a kind of numbness and stupidity, both in mind and body, and find myself disabled from giving you any answer; and yet have I a thousand times discoursed much about virtue, and to many persons, and extremely well too, as I thought; but I am now not in the least able to tell so much as what virtue is. I think you have acted very prudently in never going out of your own country either by sea or land. For if you was to behave in this manner in any other city where you are a stranger, you would run a risque of being driven thence as a magician or enchanter."

⁴ See the chapter in The Perennial Tradition entitled Regeneration Into A Higher Consciousness

Understanding: "You have always confused understanding with knowing or having information. But to know and to understand are two quite different things and you must learn to distinguish between them. In order to understand a thing, you must see its connection with some bigger subject, or bigger whole, and the possible consequences of this connection. Understanding is always the understanding of a smaller problem in relation to a bigger problem. . . You cannot understand and disagree. In ordinary conversation we very often say: I understand him but I do not agree with him. From the point of view of the system we are studying, this is impossible. If you understand a man, you agree with him; if you disagree with him, you do not understand him. "To understand" means to agree. People who understand one another must not only have an equal knowledge, they must also have an equal being. Only then is mutual understanding possible.

"Another wrong idea which people have or which belongs particularly to our times, is that understanding can be different, that people can, that is, have the right, to understand the same thing differently. This is quite wrong from the point of view of the system. Understanding cannot be different. There can only be one understanding, the rest is non-understanding, or incomplete understanding. But at the same time people often think that they understand things differently. We can see examples of this every day. How can we find an explanation of this seeming contradiction? In reality, there is no contradiction. Understanding means understanding of a part in relation to the whole. But the idea of the whole can be very different in people according to their knowledge and being. This is why the system is again necessary. People learn to understand by understanding the system and everything else in relation to the system."

Ouspensky, Fourth Way