Franklin Merrell-Wolff was an American Mystic, Philosopher, and Mathematician who combined an extraordinary intellect with profound mystical insight and authenticity. Born in 1887 in Pasadena, California, he was raised in San Fernando as the son of a Methodist minister. Wolff graduated from Stanford University in 1911 with a major in mathematics and minors in philosophy and psychology. He then went on to Harvard graduate school to study philosophy, where he was particularly influenced by the study of Kant's Critique of Pure Reason. As a result of his philosophical studies, Wolff "became convinced of the probable existence of a transcendent mode of consciousness that could not be comprehended within the limits of our ordinary forms of knowledge." Prior to completing his degree at Harvard, he returned to Stanford to teach mathematics. When it became clear to him that he must "reach beyond anything contained within the academic circles of the West" to Realize Transcendental Consciousness, he left his promising career in academia to engage in a spiritual quest. When he married Sarah Merrell, they joined their surnames to symbolize their partnership in a shared spiritual work.
Wolff's twenty years of seeking included deep engagements within the theosophical, Sufi, and Hindu traditions. In the later part of his quest, Wolff was drawn to the philosophical works of the Indian sage Shankara, who founded the Advaita Vedanta school of Hindu philosophy. It was while in deep contemplation of the teachings of Shankara that, in 1936, Wolff's efforts culminated in two Transcendental Realizations which provided the foundation for his philosophy. While the first Realization confirmed the perspective of Shankara's philosophy, the second Realization was unexpected and opened Wolff's philosophical view beyond his understanding of Advaita Vedanta. His books Pathways Through To Space and The Philosophy of Consciousness Without An Object provide a detailed record of Wolff's realizations and a lucid philosophical description of Transcendental Consciousness. Wolff's long life was spent writing, lecturing, teaching, and working the land. He spent his retirement years at the foothills of the eastern Sierra Nevada near Lone Pine, California and died there in 1985 at the age of 98.
Wolff grounds his philosophy in his Realizations, and not in mere rational speculation. In his written report of his mystical unfoldment, Wolff identifies three premonitory recognitions and two fundamental, or transcendental, Recognitions.
Wolff's first premonitory recognition took place in 1922, approximately 14 years prior to his transcendental breakthroughs. Wolff describes this first recognition as a noetic insight into the truth of "I am Atman". The term "Atman" is a Sanskrit term that Wolff uses to refer to the transcendental subject to consciousness (see the discussion above of the second fundamental of the philosophy). Just prior to this insight, Wolff had been engaged in the practice of discrimination of subject (Atman) and object (world). This practice of discrimination is fundamental to the teachings of Shankara, the founder of the Advaita Vedanta school of nondual philosophy. The purpose of this practice is to effect a disidentification and detachment from the objects of consciousness, and a realization of identity with pure subjectivity. Although Wolff previously had been intellectually convinced of the truth of the proposition "I am Atman", this time he suddenly realized its truth at a deeper level than the intellect. Although this was only a veiled Realization, it nevertheless brought a sense of Light and Joy, and had persistent positive effects, such as a certain change in the base of thought, bringing clarity where there had previously been obscurity.
The second premonitory recognition took place in late 1935, approximately 9 months prior to the first fundamental breakthrough. Wolff describes this recognition as the realization that "I am Nirvana". Prior to this noetic insight, his thought upon the subject of Nirvana had been involved in the confusion that Nirvana is a kind of other-world separate from the relative world of subject-object consciousness. While meditating upon Nirvana, however, it suddenly dawned on him that "I am Nirvana", where "I" is understood here to mean the inner core of subjectivity. Like the Atman, Nirvana is never an object before consciousness. It is therefore identical with the subject to consciousness, or the true "I". As with the prior recognition, this insight was accompanied by a sense of Joy and Illumination within the relative consciousness, and had persistent effects. In addition, there was a sense of a Current with profound depth.
The third premonitory recognition took place in late July, 1936, about two weeks prior to the fundamental breakthrough. Prior to this insight, Wolff experienced certain logical difficulties reconciling Transcendent Being with the physical universe. These difficulties arise from the habit of regarding objects of consciousness, i.e., any appearance in consciousness that we can ponder or experience, as in some sense substantial. Although Wolff had a prior intellectual conviction that the Transcendent Being was more substantial, the intellectual idea alone had failed to have a powerful transformative effect on his consciousness. This third premonitory recognition, however, had a profound effect on his consciousness that served to clear the way for the fundamental breakthrough that would follow in a matter of days. Wolff expressed the insight with the following proposition: "Substantiality is inversely proportional to ponderability", or "Reality is inversely proportional to appearance". In other words, the degree of true substance or reality is the inverse or opposite of the degree of ponderability. Thus, concrete objects of experience, which have a high degree of ponderability, are the least substantial. Subtle or abstract objects of experience, on the other hand, which are less ponderable, partake of a higher degree of substantiality and reality. The effect of this insight upon Wolff was an acceptance of substantial reality where the senses reported emptiness, and a greater capacity to realize unreality, or merely dependent or derivative reality, in the material given through the senses. This insight brought about a more profound shift of identification with the transcendent supersensible reality, and a correspondingly profound detachment from the objects of consciousness. This shift was decisive in clearing the way for the fundamental realizations that were to follow.
The first of Wolff's two fundamental Realizations took place on August 6, 1936. In contrast with the prior insights, which retained objective elements in his own consciousness and thus fell short of genuine identification, the fundamental Realizations unequivocally transcended the subject-object or relative consciousness. Just prior to the first Realization, Wolff had been meditating upon the teachings of Shankara, particularly the discussion of Liberation. Upon meditative reflection, he realized that his efforts to attain Liberation involved a seeking after a subtle object of experience. But any new object of experience, no matter how subtle, was something other than the objectless transcendent consciousness. Thus, Liberation does not necessarily involve any new object of experience or change in the content of consciousness. To seek such a new object or experience, therefore, is a mistake. Genuine Realization, therefore, is a recognition of Nothing -- but a Nothing that is absolutely Substantial and identical with the SELF. The result of this profound realization was the complete and instant cessation of expectation of having any new experience or relative form of knowledge arise. The light of consciousness then turned back upon itself, toward its source, and the pure Atman was realized as absolute fullness and as identical with himself. This Recognition was not an experience of any new content in consciousness, but a Re-Cognition of a Truth that is, was, and always will be. It is a nondual knowledge of identity that transcends space and time. Nevertheless, there were various effects experienced within the relative consciousness, that may be considered expressions of the Recognition. Because the Recognition is not the recognition of any particular effects or phenomena, they should not be confused with the Recognition itself. Some of the effects Wolff experienced were: (1) A shift in the base of reference in consciousness, transplanting the roots of identity from the relative to the transcendent, (2) a transformation of the meaning of self from a point-like principle opposed to objects of experience to a space-like identity with the entire field of consciousness and all its contents, (3) a sense of penetrating knowledge into the depths of reality, (4) a transcendence of space, time, and causality, (4) complete freedom and liberation from all bondage. Also experienced were qualities of joy, felicity, serenity, peace, and benevolence.
Although Wolff's first fundamental Realization was an unequivocal transcendence of the subject-object consciousness, for a period of approximately 33 days there remained certain unresolved tensions preventing it from being a full state of equilibrium. This tension consisted in the contrast in valuation between the superlative Joy, Peace, Rest, Freedom and Knowledge of the Transcendent and the emptiness of the relative world. There was a distinction between being bound to embodied consciousness and not being so bound, with a subtle attachment to being not bound. Counter-acting this subtle attachment, however, was Wolff's prior acceptance of the bodhisattva vow, a commitment to the value of relative manifestation and embodiment, motivated by compassion for all sentient beings. With this motivation, Wolff resisted his strong inclination to retreat into the transcendent bliss of nirvanic consciousness. Instead, he sacrificed his strictly personal enjoyment of those transcendent values in order to maintain a relative embodiment and help liberate all sentient beings. This act of compassion and ultimate renunciation led to an unexpected second fundamental Recognition that resolved the residual tensions between the universe and nirvana. The Realization represented a complete Equilibrium, not only a relative equilibrium between objects, but also an ultimate Equilibrium between relative and absolute levels of consciousness. Because this realization does not give any more valuation to nirvana than to the universe, and recognizes no ultimate difference between the two, Wolff called it the High Indifference. It is the complete resolution of tension between all opposites, the complete transcendence of all distinctions, including the distinction between the transcendent and the relative. At this profoundly deep level of Recognition, all self-identity, both in the highest sense of the transcendental Self and the lower sense of the ego self, was no more. In Wolff's words, "I was no more and God was no more, but only the ETERNAL which sustains all Gods and Selves."
Based on his fundamental Realizations, Wolff developed a transcendental philosophy which he distilled into three fundamental propositions. Wolff emphasizes that these propositions, like his philosophy as a whole, are conceptual symbols of an ineffable Reality. Moreover, Wolff acknowledges that the Realizations upon which his philosophy is based are not necessarily ultimate, and are authoritative only for Wolff and anyone who has had similar Realizations. Nevertheless, the philosophy has value for others who aspire to such Realization. The three fundamentals of his philosophy are as follows.
Wolff's term "Consciousness" here does not mean consciousness as opposed to unconsciousness. Nor does Wolff use the word "Consciousness" here as a consciousness involving any particular structure or mode of experience, such as the structure of intentionality, or the mode of our typical experience based on the distinction between subject and object. Rather, the meaning of the term "Consciousness" here is THAT which is the primordial ground and essential nature of all modes and forms of experience, both subjective and objective. In Wolff's words,The One, nonderivative Reality, is THAT which I have symbolized by 'Consciousness-without-an-object.' This is Root Consciousness, per se, to be distinguished from consciousness as content or as state, on the one hand, and from consciousness as an attribute of a Self or Atman, in any sense whatsoever. It is Consciousness of which nothing can be predicated in the privative sense save abstract Being. Upon It all else depends, while It remains self-existent.Thus, Consciousness is primary, i.e., it is first, prior to everything. Not before or first in the sense of time or temporal sequence, but prior in the sense of not being secondary to or derivative from anything else. Hence, Consciousness is self-existent, i.e., it does not depend upon anything else for its being and is entirely self-sufficient and complete. In particular, Consciousness does not depend upon, and is not derivative from, matter, energy, or any other substance. On the contrary, all experience and all objects are derivative from Consciousness. Thus Consciousness is constitutive of all things, i.e., all things are, in their ultimate nature, nothing but this Primordial Consciousness itself.
To understand this philosophical proposition, we need to first clarify Wolff's use of the terms subject and object. Our experience is normally conditioned or structured by the distinction between a subject to consciousness and objects of consciousness. The subject to consciousness is that which is aware of objects or appearances in consciousness. Objects of consciousness are distinct states or appearances in consciousness, ranging from the most concrete to the most subtle. A concrete object in consciousness might be a visual perception of a chair or a sensation of pain in our foot. More subtle objects are appearances in consciousness such as a thought or memory, an intuition about something, or a state of consciousness such as an experience of the world that is permeated by a subtle sense of bliss. It is important to note that the term "object" as used here by Wolff includes our thoughts, feelings, and other inner experiences. Such inner phenomena are still objects in consciousness just as much as outer phenomena are.In contrast to objects in consciousness, the subject to consciousness is the principle or aspect of consciousness by which there is awareness of objects. Because an object cannot be reasonably said to be in consciousness if it is not an object of awareness, the existence of any object in consciousness necessarily implies a subject to consciousness. At the basis of our relative experience, therefore, is a distinction between subject and object. The second fundamental of the philosophy states that the subject transcends the object, i.e., that the subjective principle or aspect of consciousness is more fundamental to consciousness than the objective appearances in consciousness. This philosophical proposition derives from the insight that, on the one hand, the objective appearances of consciousness vanish in the transcendent nirvanic state of consciousness, while, on the other hand, the subjective principle of consciousness, i.e., the capacity of awareness, is common to both relative and transcendent levels of consciousness. The subjective principle is therefore transcendental, while the objective principle is not.
The third fundamental of Wolff's philosophy is an affirmation of a third way of knowing, or a third organ of knowledge. Secular philosophy in the west admits only two modes of knowledge: perception and conception. Perception includes all sensory knowledge we derive from seeing, hearing, feeling, smelling, and tasting. Conception includes all knowledge we derive from thought, memory, imagination, and the like. If we admit only these two forms of knowledge, then our knowledge of reality is forever limited to our hypothetical, conceptual speculations about what reality might be "behind" our perceptual appearances. If we are limited to conception and perception alone, any certain, categorical knowledge of reality and truth is not possible, and there is no rational way to understand the possibility of mystical realization or transcendental consciousness. The third fundamental, however, affirms the existence of a third way of knowing, which Wolff calls "introception". The introceptive capacity is normally latent or partially latent, but can be activated partially or fully, through intentional effort, spontaneously or both. When activated, introception provides immediate, categorical knowledge that transcends the subject-object distinction, i.e., it is not a relational knowledge of something by something else, but a knowledge through identity in which there is only knowledge itself that includes and transcends both knower and known. The third fundamental, in short, affirms that, in addition to the capacity of perception and conception, there is also a capacity for transcendental knowledge.
In addition to using more traditional forms of philosophic expression, Wolff also expressed his Realization in the form of poetry and aphorisms. Regarding his Aphorisms, Wolff writes:
There are two lines of approach to, and employment of, the aphorisms. They may be regarded as seeds to be taken into the meditative state, in which case they will tend to arouse the essentially inexpressible Meaning and Realization which they symbolize. This we may call their mystical value. On the other hand, they may be regarded as primary indefinables upon which a systematic philosophy of the universe and its negation, Nirvana, may be developed. In this case, they may be viewed as a base of reference from which all thought and experience may be evaluated.
In the following aphorisms, Wolff uses the terms "Consciousness-without-an-object" to refer to "the Sole Reality upon which all objects and all selves depend and derive their existence" (aphorism 54).
Although Wolff gave expression to his Realization primarily in philosophical form, he also wrote some beautiful poetry. Two of his shorter poems, Compassion and The Nameless, give a taste of this side of Wolff.
| Compassion |
| O Compassion! More than the other loves of men, less than
the High Indifference; Calmly standing by and waiting; years, centuries, millennia; Taking to Thyself the suffering of all; transforming toward Joy; With Light restraining Darkness; with good, evil; Refusing release while others are bound; melting differences; Accepting impurity, giving purity; Bound by no law, yet acquiescing in bondage; Available for all as the light of the sun, yet forced on no man against his will; Needing nought for Thyself, though giving to all in need; The Base of all hope for this humanity so low; Pure Radiance Divine. Sweet art Thou, unutterably sweet; melting within me all hardness; Stirring inclusion of the low as the high; the evil as the good; the weak as the strong; the unclean as the pure; the violent as the considerate; none left out; Awaking new understanding and patience beyond Time; Arousing forgetfulness of the petty in the grand sweep of the noble; Equalizing regard, yet exalting true worth; Reaching beyond all contradiction. To Thee I sing, glorious Spirit; grandest God mankind can know. |
| The Nameless |
|
Above, below, to right, to left, all-encompassing, Before and after and all between, Within and without, at once everywhere, Transforming and stable, ceaselessly; Uncaused, while fathering all causes, The Reason behind all reasoning, Needing nought, yet ever supplying, The One and Only, sustaining all variety, The Source of all qualities, possessing no attributes, Ever continuous, appearing discrete, Inexpressible, the base of all expression, Without number, making possible all number, Containing the lover and the beloved as one, Doing nought, remaining the Field of all action— The actor and the action not different— Indifferent in utter completion; Diffused through all space, yet in the Point concentrated, Beyond time, containing all time, Without bounds, making bounds possible, Knowing no change; Inconceivable, yet through It all conceiving becoming; Nameless ever and unmastered; THAT am I, and so art Thou. |
Here is an account of the enlightenment experience and its consequences written by a trained philosopher and mathematician who is also a master of English prose. Merrell-Wolff experienced enlightenment, became established in the state, and wrote clearly about the value and nature of the knowledge he attained. This is a record of transformation in consciousness written during the actual process itself, supplying an unusually intimate view.
The author faces the epistemological problem directly—the problem of demonstrating the reality and value of knowledge springing from mystical roots. He gives serious attention to the philosophical and psychological criticism, writing with an eye to the pitfalls indicated by such criticism. He did not write only for those who believe easily.
Publisher's Summary:This book presents a philosophy that includes the enlightenment experience—that embraces the wider ranges opened by the door of realization—while not excluding the contents of the more common experience. A realization in consciousness that finds no place or adequate recognition in philosophical systems proves the inadequacy of those systems. The author first briefly surveys the principal schools of modern Western philosophy in order to show how they fall short. He then presents his philosophy grounded on the authority of direct realization resulting from a transformation in consciousness.
Please direct inquiries Sept. to May to
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Publisher's Summary:This book provides a critical exposition of the philosophy of Franklin Merrell-Wolff, a twentieth-century mystic and philosopher—an exceedingly rare and fruitful combination. Wolff's training in philosophy and science convinced him that it was important to ground his thought in immediate awareness to avoid the pitfalls of mere intellectual speculation. As a mystic, he included firsthand accounts of his experiences and transformations, the sort of invaluable primary data that is most often lacking in a mystic's writings.
Ron Leonard discusses Wolff's influences and realizations and uses phenomenological and analytic methods to explore the implications of his work within the contemporary philosophical context. In particular, Leonard focuses on Wolff's two primary claims: (1) that Consciousness, transcending the subject-object structure, is primary, and (2) that there is in mystical experience a means of knowing other than sensation and conception. This book explores the accounts of Wolff's grounding in the immediacy of his Realizations, and the nature and philosophical significance of mysticism for our understanding of knowledge, reality, and ourselves.
"Ron Leonard does an outstanding job of explicating the philosophy of Wolff, as well as bringing this philosophy to bear on central issues in the study of mysticism. He also does an excellent job of weaving contemporary philosophical programs—such as phenomenology, theories of the self, egological/nonegological theories of consciousness, and the religious use of language into his analysis." —Robert A. Holland, Hofstra University
"I like the author's ability to express Wolff's mystical experiences and the philosophy following these experiences clearly and his ability to integrate these experiences into a coherent philosophical system. Leonard's ability to analyze Wolff's use of pure mathematics in his transcendental experiences is admirable." —Joan Price, Mesa Community College
Author's Summary:I am not afraid of trying to do the impossible, and in writing Authentic Knowing I was not afraid to try to penetrate beneath the usual level of discourse concerning spirituality and science. Talked into writing it by one of my students, by the time I had finished, I realized that I had succeeded beyond my expectations. This book is not for those who are satisfied with the usual scientistic or new age ideas about reality but for those who are looking for a more comprehensive discussion. Do I have all the answers? Of course not. As Jeffrey Mishlove, author of The Roots of Consciousness has said "This extraordinary book invites its readers to wrestle with questions of spiritual awareness, in the light of modern science and humanistic thought." Instead of answers, I invite the reader to join me in an exploration of the deep questions concerning life in order to seek to achieve greater clarification. In a review published in the Journal of Consciousness Studies, Thomas Draper has said "Thoughout, the author proves to be a worthy and mature guide for a journey that contemplates both the spoken and the unspoken, the seen and the unseen, and keeps its sojourners within light's bounds." Allan Combs, author of Synchronicity: Science, Myth, and the Trickster has summed it up as follows: "This book is about the search for authentic personal and spiritual truth in an age of confusion and distraction. In it Baruss moves . . . toward that which is essential and valid, in other words toward truth. Most satisfying. In the tradition of Franklin Merrell-Wolff, Imants Baruss is a mystic, mathematician, philosopher, and above all a seeker."
Author's Summary:This essay presents Merrell-Wolff's writings on the spiritual function of mathematics, selected from his three major works, Pathways Through To Space, The Philosophy of Consciousness Without An Object, and Introceptualism. Mathematics, according to Wolff, functions as a bridge between the relative and transcendent states of consciousness. It serves, on the one hand, as a vehicle for crossing from the transcendent to the relative by providing a highly subtle and precise language for expressing the immediate contents of transcendent states with minimal distortion. On the other hand, it also serves as a vehicle for crossing from the relative to the transcendent by providing highly abstract and universal symbols for generating insights through contemplation. Wolff emphasizes, however, that although the structure of this mathematical bridge is provided by the highly subtle forms of thought, an actual crossing of the bridge requires the motivating power of love and devotion.
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If one searches the historical record for evidence of rational and logical thought, one finds among the most highly developed intellects the spiritual philosophers such as Shankara, Nagarjuna, and Plato, for whom the primary function of the intellect is to serve the ends of spiritual realization. Moreover, mathematics, perhaps the most subtle and rigorous form of thought, traces its origins back to Pythagoras and Plato, for whom mathematics is first and foremost a spiritual activity. Although today the spiritual function of mathematics, and rational thought in general, has been largely forgotten, yet there are a small number among us who remember; perhaps the most notable to live in our century is Franklin Merrell-Wolff.
This essay presents Merrell-Wolff's writings on the spiritual function of mathematics, selected from his three major works, Pathways Through To Space, The Philosophy of Consciousness Without An Object, and Introceptualism. Mathematics, according to Wolff, functions as a bridge between the relative and transcendent states of consciousness. It serves, on the one hand, as a vehicle for crossing from the transcendent to the relative by providing a highly subtle and precise language for expressing the immediate contents of transcendent states with minimal distortion. On the other hand, it also serves as a vehicle for crossing from the relative to the transcendent by providing highly abstract and universal symbols for generating insights through contemplation. Wolff emphasizes, however, that although the structure of this mathematical bridge is provided by the highly subtle forms of thought, an actual crossing of the bridge requires the motivating power of love and devotion.
Wolff calls the expression of the content of transcendent states in the forms of relative thought the process of cross-translation. "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."[51] Yet Wolff acknowledges that cross- translation is not a trivial task, for the Higher Consciousness "is not an easy Sea in which to think, when one tries to retain correlation with the outer consciousness. Whatever success I do attain in navigating in this compound of the Sea and relative consciousness, I owe very largely to the years of training in higher mathematics."[52]
But why should mathematics be so well suited for this task of cross- translation? Put simply, it is because "our most abstract language is the best vehicle of ultimate truth."[53] This is an immediate consequence of Wolff's fundamental principle that substantiality is inversely proportional to ponderability.[54] "The universal. . .is in closer affinity to the Illuminative cognition than is the particular. . .Since it is from the general or universal concepts that the largest deductive development is possible, it follows that a philosophy grounded on the Illuminative Cognition would elaborate itself mainly as a deductive system. . .Here we can see the possibility of a mathematic which is a revelation of truth."[55] In other worlds, since the abstract or universal is closer to Truth than the concrete or particular, it introduces the least distortion when used to symbolize or embody that truth; and since mathematics constitutes our most abstract language, it has the potential to carry the purest revelation of truth.
Although abstract concepts are excellent vehicles for carrying transcendent values, they are still, in themselves, objects in the relative world and can therefore never completely convey the non- relative content of transcendent states. "There is an irreducible incommensurability which forever makes correct cross-translation impossible. . .Still, there is such a thing as approximation to cross-translation in much the sense that a mathematician can give an approximate rational evaluation of an irrational number, such as the square root of 2."[56] To completely cross-translate the square root of 2 would mean to completely specify all the digits of its decimal expansion; but the decimal expansion of the square root of 2 has an infinite number of digits with no repeating pattern; thus to completely specify all the digits in its decimal expansion would require an infinite amount of time; it is therefore impossible to completely cross-translate the square root of 2 within the temporal bounds of the relative world. It is the same, Wolff says, with attempts to express the transcendent within the bounds of the relative world.
Nevertheless, even though we can never completely capture the irrational with the rational, we can obtain increasingly precise approximations by taking more digits in the expansion. Using abstract mathematical symbols to cross-translate transcendent states thus corresponds to a very precise approximation of an irrational number by a rational. Thus, using mathematical concepts, it is possible to express subtleties of transcendent states with much greater precision than with ordinary language. In one of his efforts at cross-translating, for example, Wolff says that "we are here dealing with a very profound conception where, again, it seems that only mathematics can help us."[57]
Wolff repeatedly uses mathematical symbols and metaphors where conventional language seems to fail completely. His most frequently used mathematical symbols are Infinity, Space, Point, and Zero. Infinity was most notably used in reference to the High Indifference. "Just as in mathematics there are infinitudes of higher orders infinitely transcending lower infinities, so it is in the Transcendent World. . .I have found an Infinite World, and then another Infinite consuming the first. . .It is an Infinity of some higher order, that is an INFINITY which comprehends lesser Infinities."[58] Here Wolff refers to Cantor's mathematical theory of transfinite numbers. In his famous diagonalization argument, Cantor proves that one transfinite number can be larger than another, in particular, that the number of points on the real number line between zero and one is greater than the number of positive integers. The proof begins by supposing that these two infinite numbers are equal, i.e. that the points on the number line can be put in one-to-one correspondence with the positive integers. If, as supposed, we can number each point with a positive integer, then we can make a sequential list of all the points, writing the decimal expansion of each real number next to its corresponding positive integer.
| 1 | 0.029384710... |
| 2 | 0.470128374... |
| 3 | 0.234928702... |
| 4 | 0.872938472... |
| 5 | 0.295629056... |
| 6 | etc. |
By supposition, this list completely enumerates every real number. Yet it is not complete since there are numbers not on the list. For example, construct a number that has its first digit different from the first digit of the first number, its second digit different from the second digit of the second number, its third digit different from the third digit of the third number, and so on. This will result in a number that differs from every number on the list and yet it specifies a point on the real number line. Therefore, the number of points on the real number line must be greater than the number of positive integers. Since both numbers are infinite, there are thus two infinities of different magnitudes. In fact, there are infinities beyond infinities in transfinite set theory. These infinities are just two of the smallest infinities.
Another aspect of the High Indifference is the Realization of two-fold egolessness, i.e. the relativity of both the personal and transcendental egos, both the self and the Self. To communicate this notion, Wolff uses the mathematical conception of a parameter. "With respect to a specific entity, the invariable identity is the Self, but with respect to all creatures and all modes of consciousness, the Self becomes a parameter that varies. Behind and supporting this parameter is the ultimate invariant, Pure Consciousness Itself."[59] The mathematical meaning of parameter to which Wolff refers is as follows. Take, for example, the equation x^2 + y^2 = r^2, which describes the set of points (x, y) in the plane that are at a constant distance r from the point (0, 0), i.e. a circle with radius r centered about the origin.
The description just given takes the variable r to be constant. For this fixed value of r, the equation yields a single circle through the variation of the values of x and y. Yet we may change the value of r to yield another circle having a different radius. Thus, through the variation of r the equation yields a whole family of circles. A variable like r which has a fixed value with respect to other variables is called a parameter. The Self, which is a relative invariant with respect to the rounds of cyclic existence, corresponds to the value of r, which is a relative invariant with respect to the values of x and y around the circle. Yet the value of r, like the Self, is only a relative invariant since it can change to yield different rounds of cyclic existence. The absolute invariant, which all the circles have in common, is their common centerpoint and the entire Space in which they reside.
Corresponding to twofold egolessness is a twofold emptiness or zero. On the one hand, Wolff uses zero as a symbol for the Self stripped of all objects. "Here we have a notion which stands for nothing, and yet becomes the most vital unifying conception of mathematics. . .Upon zero we build our systems of reference, which is merely a way of saying that with nothing as a center we have the fulcrum for control of all elaborations in form. . .The pure 'I' is the zero-point of organized consciousness."[60] This first egolessness, this bare point-I, is then transformed into the Space-I. "The Transcendent 'I' is as much a space as a point. Its mathematical symbol would be both 0 (zero) and oo (infinity)."[61] Yet in the High Indifference, this Self blows out leaving only Consciousness- Without-An-Object (and Without-A-Subject). "Consciousness-Without-An- Object is the universal solvent within which the centers of tension, or objects, have their field of play. All tendency in that play is counter-balanced by its counter-tendency, the culminating effect being an expression equated to zero. It is zero that symbolizes the durable Reality, or Consciousness-Without-An-Object."[62] This is the higher sense of zero or emptiness, symbolizing the cancellation of the positive and negative polarizations corresponding to the Universe and Nirvana.[63]
Wolff coined the term 'inverse cognitions' to describe cognitions polarized in the opposite sense as objective cognitions, i.e. toward the Nirvanic. "Awareness of the absence of objects, in its purity, is not a cognition of an object, but another form of consciousness that is not concerned with objects. However, a reflection of this state of consciousness may be produced so that a special cognition arises, of such a nature that its content is definable as the inverse of all objects. . .illustrated very well in mathematics in connection with the development of the notions of negative, imaginary, infinitesimal, and transfinite numbers. All these may be regarded as of the nature of inverse cognitions."[64] Since inverse cognitions are of a non-objective nature, they involve mystical recognition. Wolff also mentions the continuum as another inverse cognition, polarized opposite to the discrete nature of the Universe of objects. "It is true that man has arrived at the notion of continuity, although. . .he never really thinks it. . .Continuity belongs to the hinterland of Consciousness. This simply illustrates the eternal fact, i.e. that the actual consciousness of man continually operates in a Nirvanic as well as in a Sangsaric sense."[65] Thus these inverse cognitions found in mathematics are profound vehicles for recognizing the Nirvanic pole of consciousness.
The conceptions of negative, imaginary, infinitesimal, and transfinite numbers, as well as the conception of the continuum, are all symbols for inverse cognitions by virtue of their common transcendence of the conventional universe of positive integers. We have already seen how the infinity of the continuum transcends the infinity of positive integers with respect to the size of the infinities present in each type of number. The transcendence, however, is also a transcendence of the kind of number. It is a liberation not only from quantitative bounds, but also from qualitative bounds into new forms of number having more universal laws. For example, within the bounds of positive integers the laws of subtraction are limited by the requirement that the result must be positive. With the birth of the conception of negative number, however, no such limits need be imposed upon the laws of subtraction. Similarly, within the bounds of real numbers the laws for taking square roots must be restricted by the requirement that square roots may only be taken of positive numbers. With the birth of imaginary numbers, however, no such limits need be imposed. Similarly, in the case of transfinite and infinitesimal numbers the laws become less restrictive and more universal. What was impossible before is now possible. For example, a transfinite number can be divided in two without decreasing in size. These new forms of number thus symbolize a way of consciousness where the restrictive forms of our conventional universe are transcended, where the impossible becomes the possible, and where new forms of reality are realized where there was nothing before.
The latent Nirvanic powers of mathematical conceptions provide valuable symbols for the transformation of the mind. "It is possible to take the symbol itself as an object of thought and use it for the purposes of philosophical and general mystical integration. . .There is some reason to believe that such a method of procedure is possible within the setting of Western culture. . .This possibility I see as growing out of our peculiar mathematical development."[66] Moreover, "the greatest achievement of western genius has been in the development of the abstract thought which has its crown in higher mathematics. The freeing of thought from dependence upon the sensible image is an accomplishment of the very greatest difficulty. Until thought has won this power, it cannot penetrate into the Realm of Imageless Consciousness. Now, once it is realized how much has been accomplished in this direction in the field of higher mathematics, it is easy to see what a powerful instrument in the practice of Dhyana we have forged."[67] Because of its very abstract nature, mathematics can carry one to the utmost limit of subtle form where the leap to the Transcendent is relatively small. "The labor whereby a man attains the point of working with objects of highest tenuity actually implies much of the austerity requisite for the achievement of true objectlessness."[68] The activity of mathematics, thus is a powerful method for training the mind to be receptive to subtle forms and, consequently, formlessness. "In my own experience, thought on the level of Imageless Consciousness was possible by employing the intellectual capacities unfolded during the years of mathematical discipline."[69]
There are other equally important contributing factors to the transformation of the mind by mathematics. "The validity of mathematics is established upon a basis that is quite impersonal and universal. . .In its purity it deals only with transcendental or ideal objects of the very highest order of thinkable abstraction or universality. In high degree, the consciousness of the mathematician qua mathematician is not concerned with either a self or objects. . .in higher degree than anywhere else, except perhaps in states of samadhi of a high order. Herein is revealed the power of pure mathematics as an instrument of consciousness- transformation on a very lofty level."[70]
Here we begin to touch upon important aspects of the spiritual power of mathematics: its universality, impersonality, and selflessness. When we enter the realm of mathematics, we transcend the realm of the personal human and enter a realm of universals. "Man will have long since ceased to be human, in the restricting meaning of the term, by the time he has awakened in terms of Consciousness at the most advanced levels represented by mathematical concepts and symbolic formulae. Mathematics thus constitutes a thread to the Beyond that has never been lost."[71] Moreover, "for him who penetrates deeply into the roots of logic itself, the Recognition can be aroused. . .Let a man unveil this Recognition and make It immediately and consciously his own, and then he will find in logic a power which, if followed with a single eye, will take him through to the Higher Consciousness."[72] Furthermore, "pure mathematics is the only real invariant that we have in the ever- changing phantasmagoria of experience. . .the law that governs the flow of consequences [in a mathematical system] is tougher than tempered steel and harder than the hardest rock. Save in the Self, here, as nowhere else, is there something to which human consciousness may tie and give its trust. . .and this invulnerable core carries straight through to Consciousness-Without-An-Object."[73]
Mathematics is not only a thread by which we may know the Beyond, but also a means by which we may commune with it. Just as mathematics gives us deeper vision into the outer structures of the objective world, so "the 'eye' of the mathematician actually sees into the deep structure of the subjective psyche. . .May it not be that the mathematical thought is the speech of the Divinity in the inner consciousness of man? Then the mathematical thought is inner communion."[74] Here we begin to see a glimmer of the heart and soul of mathematics. "Without the Recognition of Soul, in some sense, such as the soul of mathematics or of logic or in some other form, formal demonstration proves merely possibility or the hypothetical imperative but never arrives at the categorical imperative. The only knowledge that can possibly Liberate man is categorical."[75]
Thus the transformative power of mathematics lies not only in its form and logic, but also in its soul where it is felt with the heart. Indeed, Wolff points out, in regard to the use of mathematical symbols for spiritual transformation, that "to step from the symbol to that which is symbolized, though this does afford a peculiarly exacting demand upon acuity of thought, yet requires much more. Here, feeling, in the best sense, must fuse with thought. The thinker must learn also to feel his thought, so that, in the highest degree, he thinks devotedly. It is not enough to think clearly, if the thinker stands aloof, not giving himself with his thought. The thinker arrives by surrendering himself to Truth, claiming for himself no rights save those that Truth herself bestows upon him. In the final state of perfection, he possesses no longer opinions of his own nor any private preference. Then Truth possesses him, not he, Truth. He who would become one with the Eternal must first learn to be humble. He must offer, upon the sacrificial alter, the pride of the knower. He must become one who lays no possessive claim to knowledge or wisdom. This is the state of the mystic ignorance -- of the emptied heart."[76]
Thus the transformation of the heart is an essential part of the method which employs mathematics for spiritual realization. Mathematics can itself aid in this heart transformation by virtue of its embodiment of Truth in a particularly pure form. Mathematics thus provides an excellent object of devotion. This played an important role on Wolff's path. "A desire for the transcendent Self and a love of universals also tend toward the required melting of the egoistic feeling. In this part of the discipline I found that my already established love of mathematics and philosophy was an aid of radical importance."[77] Indeed, "in the cognitive activity of pure mathematics. . .the desire is almost wholly directed toward Truth and Beauty, with little or no attachment to any preconception of what Truth may ultimately prove to be. Cognitive activity of this type. . .may well prove to be one of the most powerful subsidiary aids for those who can make use of it."[78]
The essential key to employing mathematics as a transformative discipline, then, lies in the heart we put into it. "The two great factors which implement the motivation underlying the drive toward Mystical Realization are (1) Love of Truth, and (2) Compassion. . .Compassion and the Love of Truth are the only valid and effective motivations, and the Compassion must be utterly self-disregarding, and the seeking of Truth must be so pure that every preconception is offered up on the alter of sacrifice."[79] Thus, "one can raise a study to the status of an effective transforming agent only by giving himself to it with the same completeness that is characteristic of the more intense religious natures. . .It is just the subtle change implied in the difference between secular and sacred which makes all the difference in the world. In principle, anything whatsoever can acquire the sacred value; it is simply important that the attitude of sacredness shall exist and shall absorb the predominant portion of the interest. Sacredness implies self-giving, while secularity implies self- withholding. In the transformative process, everything else is incidental to the attaining of the self-giving attitude."[80]
Thus mathematics, when practiced with great devotion and humility, necessarily involves the transformation of both mind and heart and leads inevitably to both Compassion and Wisdom. It is one path, albeit not for many, which combines logic with love and conception with compassion.
End Notes
[51] Franklin Merrell-Wolff, Pathways Through To Space, Julian, 1973, Ch. LXXIX
[52] Merrell-Wolff, Pathways, Ch. LXXXVIII
[53] Merrell-Wolff, Introceptualism, Phoenix Philosophical Press, 1970, p. 180
[54] Merrell-Wolff, Pathways, Ch. LXI
[55] Merrell-Wolff, Introceptualism, p. 67
[56] Merrell-Wolff, Pathways, Ch. XXXVI
[57] Merrell-Wolff, Introceptualism, p. 149
[58] Merrell-Wolff, Pathways, Ch. LII
[59] Franklin Merrell-Wolff, The Philosophy of Consciousness Without an Object, Julian, 1973, p. 263
[60] Merrell-Wolff, Introceptualism, p. 149
[61] Merrell-Wolff, Pathways, Ch. XLI
[62] Merrell-Wolff, Philosophy, p. 225
[63] Merrell-Wolff, Philosophy, p. 200
[64] Merrell-Wolff, Philosophy, p. 195
[65] Merrell-Wolff, Philosophy, p. 161
[66] Merrell-Wolff, Philosophy, p. 172
[67] Merrell-Wolff, Pathways, Appendix A
[68] Merrell-Wolff, Introceptualism, p.159
[69] Merrell-Wolff, Pathways, Appendix A
[70] Merrell-Wolff, Philosophy, p.172
[71] Merrell-Wolff, Pathways, Ch. LXXXII
[72] Merrell-Wolff, Pathways, Ch. LIII
[73] Merrell-Wolff, Philosophy, p. 173
[74] Merrell-Wolff, Introceptualism, p. 257
[75] Merrell-Wolff, Pathways, Ch. LIII
[76] Merrell-Wolff, Philosophy, p. 178
[77] Merrell-Wolff, Philosophy, p. 27
[78] Merrell-Wolff, Pathways, Ch. LXXVII
[79] Merrell-Wolff, Introceptualism, p. 225
[80] Merrell-Wolff, Introceptualism, p. 143
END
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