Shortly before he died, Pythagoras had a remarkable dream which was recorded by an anonymous disciple to whom he must have told it. As was his habit, when he felt the need for solitude, the master would retire to the Grotto of Proserpina, where he would meditate and strum upon his lyre. On this occasion, as he struck the chord of the major third, he passed with the dying music into dream, where he found keys as simple as his monochord capable of unlocking all the secret doors of the universe. From the motions of the heavenly bodies to the gyrations of the ultimate particles of matter, some transcendental vibration gave him a vital clue to the few and astonishingly simple laws governing all. He saw clearly that the laws operating at every level of the universe were, like his musical intervals, mathematically and harmoniously the work of one mind, one supreme architectonic mathematician. Soaring through these interdependent spheres on the wings of dreaming omnipresence, Pythagoras abruptly encountered an impenetrable wall of Chaos marking the outer limits of the world.
The dream now became less vivid and the dreamer found himself slipping back into an earlier incarnation, where, as Orpheus, he had charmed even the rocks and trees with the music of his lyre. All danced and swayed around him. He found that he, not the supreme mathematician, was master. The awareness of this disturbed him, and he descended into a deeper dream, where he became again a seeker after ultimate knowledge. His powers of perception superhumanly heightened, Pythagoras resumed his analysis of the universe. He passed along the scale from the tangible things of the senses to the invisible atoms of their substratum and found mathematical laws governing them all. Continuing beyond the atomic realm, he experienced a dissipation of all matter on a plane wherein only thought manifested as number persisted. Pursuing pure thought, he discovered no wall separating reason and chaos, no boundary to the known universe. His analyses of all things of the senses streamed through the fingers of his mind, filling him with fear strangely in the company of a bursting sense of liberation. Impelled by a horror of madness, he searched through the void for the bounding wall. It moved towards him, but not as it had been when he first encountered it. Now it appeared as an infinite mirror without substance, one within which he met himself over and over again until he passed through its unresisting reflections. Beyond the chaos bounding the universe, where was he? He felt he had passed this way before, that he was rapidly approaching a point from which everything had begun, a point which was himself and which was everywhere at once. With a shock he strove to preserve reason, and struggling to find something to which he might cling, he remembered his disciples. With the reality of their need flooding his mind, he awoke in the grotto, the words honouring the sacred Tetraktys issuing forth from his lips.
In his waking thought as in his dream, Pythagoras had arrived at the realization that everything is disposed according to number, an idea with which Plato later concurred, viewing number as the essence of that pristine harmony which is the basis of the cosmos and of man. The Hindu Initiates from whom Pythagoras received insights concerning the mysteries of divine laws taught that numbers constitute the primary substratum of the universe. In their hoary tradition they identified Svabhavat as the mystic essence, the plastic root of physical Nature become number when manifest. If Pythagoras had reached the boundary of the truly unmanifest in his dream, he would have stood at the threshold of this primordial realm from whence proceed the creative, formative and material worlds subject to the rule of Dhyanis, Rishis, pitris and presiding angels. He would have seen number as a fundamental principle, the primary truth manifested as the basis of reason itself. That he perceived this and transmitted his perception to his disciples is revealed in the teachings of his foremost spiritual heir. Plato asserted that because the origin of all things and the underlying harmony of the cosmos is number, it is the basic universal principle that can be traced in the proportionality of the plastic arts and the rhythm of music and poetry, as well as in the ebb and flow of all natural cycles. Expanding the ideas of his spiritual mentor, Plato taught that all forces, forms and modes of growth were subject to laws reducible to one simple numerical ratio. He based his philosophy upon the assumption of an intelligent and repetitive orderliness creating a metaphysical foundation upon which thinking men have striven to build their theories concerning the mathematics of reality for over two thousand years.
For Pythagoras and Plato, numbers expressed not merely quantities but also idea-forces, each having a particular character of its own. As a primary element basic to reason itself, number was seen as both substance (in the Hindu sense) and intelligence, interacting in manifestation as dynamic and ordered expressions of itself. Thus, the world of numbers is the world of reason, for reason itself is built upon this ordered expression. If one begins with an axiom which is believed to reflect the true nature of the universe, subsequent numbers can be expected to unfold in mathematical harmony through the power of number expressed in deductive reasoning. An example of this Platonic view can be illustrated by pinpointing the distinction between thinking on the basis of intervals as opposed to external information. Reason may be unable to deduce the diameter of the earth from data wholly within the human mind. But this defect is wholly negligible when compared to the mind's ability to perceive the existence and properties of the chemical elements, for instance, entirely on the basis of epistemological considerations.
Just as reason finds its source in number, so also the Platonic view assigns greater power to numbers closest to the source of unity. The farther a number is from unity the greater its involvement in matter. Thus, the first ten numbers in the Greek system were treated as entities, archetypes and symbols of moving energy. Once the first number emanates from unity, an engendering power is released, causing all subsequent numbers to tend to exceed or surpass their own limits through a confrontation between even and odd (yin and yang) forces. Numbers lying beyond ten are basically products of combinations of these forces, expressing varying mixtures of the conflict associated with even numbers and the solution believed to be inherent in the odd. The assignment of such fundamental characteristics to number is not incompatible with the reasoning behind modern symbolic logic and the theory of groupings which goes back to the idea that the quantitative is the basis for the qualitative. First numbers, then the characteristics. The difference lies, of course, in the fact that the classical system was based upon the assumption of a divinely intelligent source of those characteristics, whereas modern theories approach the question from a more inductive perspective, beginning with differences and similarities perceived in the sensory world. The question of a quantitative basis for qualities is an interesting one which need not be limited to any one perspective and which sharpens the deeper analysis necessary to afford a glimpse into the purely abstract realm visited by the very few thinkers in human history who have asserted with confidence that "everything is number".
In questioning the source of number, one comes to contemplate, perhaps with hindsight, the concept of the zero. Along with the basic nine integers familiar to much of the world, the ancient Hindus developed the idea and form of the zero, which completed the decad system and liberated digits from the counting board so that they could stand alone. They perceived that the zero enabled them to signify that in a particular position or class, nothing is there. It was a positive statement about nothing and they termed it shunya ('empty') or shunya bind ('empty dot'). Taken over by the Arabs, the zero became known as as-sifr ('the empty'), which in Latin emerged as cifra, the immediate source of the English 'cipher' (or 'digit'). This is why it is said that Pythagoras, returning from India, was the first teacher of ciphering in the West, the Greeks treating the one and the nought as the first and final ciphers. Having no positive value in itself, the zero can give a higher value to other numbers, following a system based upon classificatory distinctions of powers identified with position. It is easy to see that the reasoning behind such a concept would, of its own nature, necessarily have participated in relatively high levels of abstraction residing at close hand to the realization of unity and the source of number itself. In arcane tradition this level of thought is lofty indeed, for there the zero is associated with Parabrahm, Ain-Soph or the Eternal No-Thing which the devotee strives to approach by making of himself a zero. Far from being a dark negation of something, the zero is a circle of eternal light. It is androgynous, and the number one that proceeds from it as a point upon its unsullied disc is its shadow or reflection, the male from the female in the androgyne. In the realm of further manifest numbers, the echo of this archetypal cipher becomes a positive power when one of the nine figures precede it, but there is no number to stand before the primordial zero.
If the zero is non-being, suggesting the latency of the void, the notion of unity is a fullness which is complete within itself. Basic to the idea of number, the conditions of unity and multiplicity express themselves in a ceaseless action of centripetal and centrifugal motion which ramifies through every thought, process of growth, planet or atom in the universe. Unity is the idea of oneness that awakens in the zero of non-being and heralds the emanation of the point which becomes the one out of which multiplicity proceeds. From the one comes the duad, the not-one or the 'other' which expresses conflict, reflection and primordial duplication. Counting begins with the first awareness of the 'not-I', marking the beginning of arithmetic and the first glimmering awareness of objectivity, of contrasts, of a classificatory potential. To the ancients the duad also marked the loss of goodness and the beginning of strife, a stalemate between opposing forces. It is a liberation to move on to the ternary formula for the creation of worlds and the solution of conflict posed by dualism. The three is the product of the action of the one upon duality, a creative process that will repeat itself in generation after generation of numbers. The quaternary is the fourth, the symbol of the terrestrial order which, when added to the spiritual triad, yields the sacred septenary key of the whole of occult Nature and embodied mankind.
From the One to the many, numbers reveal critical stages of manifesting life. They measure and separate and fragment and never quite add up to the wholeness of unity. But they bear within them the oneness which is its reflection and so express an ordering power which must necessarily find its source in a single universal impulse. It is because of this essential quality of oneness in every number that caution must be exercised in considering a quantitative basis for manifest qualities. After all, in the realm of human action, abstract mathematics and morality are bound to meet.
The Pythagoreans noticed that no even number could be decomposed into a sum of three numbers, of which the middle one was the number one and the first and last were the same as each other. This truism became for them the numerological essence of the metaphysics of the limited and the unlimited, of the finite and infinite, and of time and eternity. The double duad of the terrestrial four was not capable of engendering growth and evolution on its own. They did not believe that a double or triple or any other multiple degree of duality possessed in its essential nature an adequate impulse of the One Life to ensure generation. Only the odd number which, when divided, preserved the one at its centre could act to bring this about, could spring forth out of the duad precipitating a new generation. The essentially latent and unchanging condition of the duad was identified by the Pythagoreans with the feminine notion of infinity, whilst the masculine and dynamic odd numbers were seen as finite, bounded and terminated. Thus, the odd numbers carried forth the nature of that which, arising out of an infinite field, is limited by time and space, has a beginning and an end, and carries in it the recapitulating cyclic impulse capable of emanating from and acting upon the field. The duad, like a reflection of unity, represents the field itself, carried forth in all numbers capable of an even division by two.
Meditating upon the fundamental character of these numbers, the Pythagoreans and followers of Plato perceived in them abstract qualities such as truth, beauty, goodness and justice. Thus the quaternary of justice added to the ternary of truth yielded beauty and goodness in the world, a manifestation of the fully formed human potential which combines eternity and time in a self-conscious transcendence of both. It is painful to trace the degradation of such pristine ideas which took place under the label of numerology in the Christian era. Elaborate systems were evolved in order to prove or disprove controversial theological propositions, the nature of which, in themselves, defied logic and common sense and provoked thinkers like Giordano Bruno to open rebellion. Following a Platonic mode of thought, both he and Galileo supported the Copernican view of the universe in opposition to that supported by orthodox Christian numerology, which placed the earth at the centre of our solar system. Ecclesiastical abuses of number theories gave the whole of numerology a bad name, which unfortunately obscured the fact that individual philosophers like Bruno pursued a Platonic appreciation of numbers and the qualities associated with them. This, together with the highly fragmentary nature of what has been preserved from the teachings of Pythagoras, has encouraged modern mathematicians to dismiss the whole subject. This is a great pity, for along with this dismissal go the pregnant ideas concerning the qualities assigned, for example, to the finite and the infinite. We are left with views of 'rational' or 'irrational' magnitudes which have little to do with the philosophical questions concerning the hidden and causal realms that lie within or beyond the physical world.
Whether one accepts the Platonic view that the infinite is irrational, a negative formless principle capable of limit only when subordinated to the One (number) which gives rise to precise, quantitative ideas (also numbers), or that it is capable of being described in terms of the transfinite aggregates of Cantorian theory, Voltaire's witticism remains apt. For despite the contrivances of modern mathematical theory, the problem of consciously grasping infinity remains, and the common man is reduced to a rather pathetic fascination with enormous numbers measuring size or distance or time. But the mere utterance of terms like quadrillion (1015) or vigintillion (1063) does not enable one to grasp the enormity of such numbers themselves, let alone the idea of infinity. The Pythagoreans have been scoffed at by modern thinkers for not seeming to come to terms with the problem of incommensurability. Their consternation with the later proof that the diagonal of a square is incommensurable with its side has been taken as a sign of philosophical naïveté and superstitious inflexibility. Pythagoras taught that those who "brought out the irrationals from concealment" would be shipwrecked, to a man, and that, in the words of Proclus, "those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves".
The irony of this is profound, for with the advent of irrationals came the decline of Pythagorean mysticism, to be replaced with Aristotelian empiricism and religious superstition, two opposing lines of thought which would toss thinking man on the waves of a shallow dichotomy for the next two thousand years. That the unwillingness to trot out the dimensions of the irrational and sacred has been crudely misinterpreted by moderns should be apparent to the unbiased seeker of truth, leaving open in his or her mind the question of whether all aspects of the physical universe can be measured by the same standard and at what point number has ceased to describe the merely physical and entered into the realm of the metaphysical. That the modern mind is capable of probing levels of abstract thought rendered communicable only through numbers or integers of some sort helps to revive the old Platonic assertion that reason is number and both exist prior to the visible world. What Pythagoras and his followers sought to protect from desecration was the mystery of how the invisible noumenon became the phenomenon of the universe, something known only to those who travelled in meditation or visionary dreams to the threshold dividing the two worlds. They knew that there was an enormous gap between the cleverest juggling of abstract labels and true knowledge born of experience.
he Platonic view that number, unlike qualities experienced by the senses, was abstract and had to be experienced by the mind through a contemplation of that which independently exists on an archetypal plane of ideation persisted until the latter part of the nineteenth century. At this point the idea that mathematical reality lies outside man, whose function it was to discover it, was replaced with the notion that mathematical truths have no existence independent and apart from the human mind.
Considering these opposing perspectives, one is prompted to ask questions such as whether deductive reasoning itself was discovered or invented, or whether we are stuck with an absolute order or with an endlessly relative flux. To some mathematicians their discoveries and inventions seem to have been waiting for them in the knowable future. A so-called rationalist might say that this mathematician is merely projecting himself into an illusory time of his own devising. "The future he imagines he is penetrating", they might say, "is his own present of abstraction and proof, the substance and spirit of mathematics." Others merely find in this irrefutable proof of the existence of a supreme and changeless intelligent order pervading the universe. Of course, the concepts of the 'knowable future' and one's own 'present of abstraction' are readily resolvable into one notion of a timeless reality. The real difference of perspective lies in the question of whether the immortal soul of man knows all and is one with all and man must consciously discover this in the process of becoming immortal, or whether man is a physical being with a mental facility great enough to enable him to take his ego games to such high levels of abstraction as to ensure an infinitesimally small cheering section for his achievements. If the latter presents a ludicrous picture, it is no more ridiculous than the notion that one can explain reasoning in terms of physical responses. Until philosophers, physicists and mathematicians confront this more fundamental question squarely and with complete lack of bias, their efforts will participate in sustaining a huge intellectually and spiritually shipwrecking muddle.
With the acceptance of irrational and negative numbers, the path to the development of modern 'man-made' mathematics was cleared. The immeasurable 'sacred', as it was understood, was considered measurable, and by the seventeenth century the deductive approach was no longer considered capable of standing alone. It had to be checked and revised against observation and experiment. With justification it has been pointed out that the successes of this 'scientific' method have so outweighed its failures that it spawned a technological application that wrought greater change in western civilization within two centuries than had four thousand years of wars and social upheavals. While there is no denying the magnitude and rapidity of these developments, it is wise to remember that Pythagoras would have viewed it dispassionately and simply enquired as to whether it had brought man closer to an understanding of his true nature. Perhaps because this question has lurked unanswered in the background of human thought, some heretical thinkers have dared to ponder it and even advance dramatically innovative propositions into various scientific fields. But the vast majority of mathematicians and logicians continue to dismiss the notion that discoverable numbers exist independently and address themselves to the question of what might be the more fundamental concept from which the invented idea of number is derived. This question is approached either through the reductive analysis of numerical propositions or through the study of the emergence of numerical ideas during the psychological development of childhood. The more conceptual approach, whilst not Platonic, assumes that number must be independent from gross sensible objects. The psychological approach, however, treats number as a quality of sensible objects themselves and places a great deal of faith in a sort of innate or primitive notion of order.
Basically, all of this breaks down into the so-called classificatory and relational theories of number. The fundamental question, according to those backing the class theory, is whether numbers serve to bring things together under a common heading or keep things separate and distinct. Philosophers like Bertrand Russell felt that the idea of number springs from the idea of manyness rather than from the idea of order, that it could be associated with the perceived manyness of collections of things rather than the positions of their progressions. The relational theory argues that basic to the idea of number is an innate recognition of the simple progressions of ordinal terms, order instead of class being the intuitionist priority. The Swiss psychologist Jean Piaget tried to soften the difference between the two views when he wrote that "finite numbers are therefore necessarily at the same time cardinal and ordinal, since it is of the nature of number to be both a system of classes and of asymmetrical relations blended into one whole". But his psychological tests with children persisted in bearing out the claims made by the relational or ordinal theorists. One could argue that the recognition of ordinal numbers is innate and that of cardinal numbers is learnt, but this would be an oversimplification of the deeper questions still unanswered about the nature of the human mind.
So many mysteries surround the human involvement with number. Why is it, for example, that a tribal woman can know instantly when one dog is missing out of a large pack and yet not have numbers to count? Animals have a 'number sense' enabling a mother to know unerringly if all her brood is present. Is this what is operating in the case of the tribal woman? Or is her awareness a product of a knowledge of just how much space certain animals occupy when standing or walking or lying down? Does she, or mankind in general, have the ability to learn instantly to read shape or is the ability innate? This is like asking if a child, blind from birth, can classify or count. What if they have no sense of hearing as well, or of touch? Will they be able to classify or count? All these empirically based questions activate hordes of psychologists and anthropologists eager to provide the ultimate answers, but the deeper question still remains: Is everything in the universe disposed according to numbers, as Pythagoras taught, and is number the base of reasoning itself? If this is so, then the very concepts of ordinals or cardinals, innate knowledge or learned knowledge, are the product of number and not the other way around.
Presaging a heretical and slowly growing return to a sort of neo-Pythagoreanism, such statements as the above rush like a fresh wind through musty corridors clogged with empirical data. They open doors leading at once into the past and future,unleashing the quantum leaps of thought required to cut through the biases of the past few centuries. Though few in number, such thinkers as Einstein, Eddington or Capra represent reincarnations, as it were, of an ancient marriage of mathematics and mysticism, unsullied by the tug of war between religion and science which forced so many otherwise open-minded thinkers into a materialistic corner. Freed from this, there is nothing to prevent the unbiased investigator from plunging his or her mind once again into waters capable of matching the breadth and depth of their greatest potential. The metaphysical statement made by Proclus so long ago that "before the mathematical numbers, there are the Self-moving numbers; before the figures apparent - the vital figures, and before producing the material worlds which move in a Circle, the Creative Power produced the invisible Circles" suggests far more abstract notions about the nature of mind and cosmos than conventional mathematics can deal with. One of the most important contributions of the so-called new physics lies in the realization that consciousness cannot be separated from the perceived world. The world is shaped by the former just as consciousness itself is affected by the nature of the substances and forms enveloping it. But the mind is potentially far more powerful than form and substance. The reasoning mind can understand the order and structure of the universe because it is that universe. The Divine Intelligence which informs it rests in the immaculate field of No-Thing (zero) long before and after number or the reasoning potential arises.
Thus, in speaking of number we are speaking of mind and cosmos merged together in man or writ large in the universe. The sameness and difference of classes, the arithmetical and geometrical progressions, the endless combinations of oneness and manyness, these are all expressions of mind and matter in manifested existence. In the ancient Hindu science of number Brahma is the first Oneness in Unity, the One which separates out of the primordial Egg of Brahma (zero) to become Brahma Vach, the six-in-one, or the septenary root from which all proceeds. From this comes the five, the Dhyan Chohans forming the Ring Pass Not (Dhyanipasa) separating the phenomenal from the noumenal world. Their number is the three, the one, the four, the one and the five, or pi, the ratio of a circle's diameter to its circumference, the ideational ratio capable of converting the One from the zero into the number of the phenomenal world. Hidden from all worldly comprehension, classes of these great Dhyanis speak through numbers and say:
They speak and reveal that the Three, the One, the Four, the One and the Five, as well as the One, the Zero, the Six and the Five, are the Essences, the Elements, the formless and formed Numbers summing up the divine cosmic Man from whom emanates the Army of the Voice.
Thus number and being are one and the same, the nature of each reflecting in the world of form as the points, lines, triangles, cubes and spheres of conditioned existence. These compose the body of man, the environment that surrounds him, but they are merely an expression of number issuing from No-Number, conscious intelligence issuing from the Logoic progeny of the Divine Darkness which is No-Thing. If the divine ancestry of man is sacred, so too numbers are sacred. "Regard them well," say the Dhyanis, "discover their harmony in Nature and in your mind. For their magnitude, their intervals, their combinations of cluster and pause will reveal to you rhythmic keys, strings upon a cosmic lyre whose melody will fill your heart with intelligence, your mind with grace. Regard them well, O Disciple, for in their very progression, their dependency upon one another, their power of division and multiplication, lies the mystical point of the One from which they all sprang."