“Do you want to take down scientific materialism, the scientific anti-philosophical paradigm?
Do you want philosophy to use mathematics and physics better than science can, and to make philosophy once more reign supreme in intellectualism and to assert its supremacy over science?
Perhaps something more current, more part of the social zeitgeist, will help.
Unless you have an event more interesting.”
“So I came across this reference to Pythagoras:”
“There is an inherent short circuit in that kind of reasoning though.
Unless there is an observation there is no point to have a formula about it, as there is nothing to formulate, right?
So, empiricism is the first step, correct interpretation of the observation the second and most crucial step and only then can maths come into the equation.
Also consider this quote from Galileo:
Philosophy is written in this grand book – I mean the universe — which stands continually open to our gaze. But it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.
— Galileo Galilei (1564-1642), Il Saggiatore (1623)
“Sure, it can be written in mathematics by humans, but the observation still comes first and the correct interpretation second!
Always remember Phlogiston: observation + incorrect interpretation = a mess. No amount of maths could make phlogiston a real substance.”
“Yes what you are getting at is a well-historied story of the idealists vs. the empiricists. Your remarks have been made by the greatest philosophers who have ever lived and your concerns are discussed at various places in Hockney’s God Series books.
The empiricists say: Nothing in the mind that wasn’t first in the senses.
The idealists say: Nothing in the mind that wasn’t first in the senses, except the intellect itself.
The reason the idealists say that is because the senses cannot be detected without a mind to detect them in the first place. The empiricist says that nothing can be in the mind without first being put there by the senses; the idealist says that that statement therefore admits that mind has come first, because you first need a mind to put senses into.
We must be careful for these two positions not to talk past each other. The concern of the idealist is the fundamental nature of reality itself, to understand and justify the very substance, the very most fundamental essence or starting point of existence, that which existence itself is made of. That is their concern. And they are right, that mind must come first before the senses, since the senses need a mind to detect them. And so, it is correct: mind is the fundamental essence of existence. Existence itself is made of mind. The Pythaogreans further concluded that mind must be structured by numbers, by mathematics, since only mathematics and numbers implicitly have structure; mind would be insane, and could do nothing, detect nothing, make sense of nothing, if mind was made of 2 + 2 = 5, 2*2 = 7, 1 – 1 = anything. Mind has to be structured, and 1 + 1 = 1 + 1 which we then denote as 2 is fundamentally structured, and, we see, empirically, reality being described by number everywhere. So, empiricism here supports idealism in confirming that reality is structured by numbers, by mathematics. If it wasn’t, it would just be insane noise. The trick, which humanity has not pursued since Pythagoras, although it was the next obvious step in the evolution of Pythagorean thought, is to determine the fundamental singular structure of mathematics itself which reality corresponds to. That is, is there a form of mathematics which is identical to nature? Is there special field of mathematics that all of reality reduces to?
So, the idealists are concerned with establishing the fundamental basis of reality. The basis of reality is not the senses, not empiricism, because empiricism requires mind to detect things. The empiricists are concerned with something slightly different, and that is simply, how does one sensorily experience reality? That is, how do we know what is all out there? How do we know how mind/mathematics will manifest itself to the senses? You see, that is a different concern, and so the two positions talk past each other much of the time because they don’t appreciate either positions concern.
The idealists are right: mind comes first, and as the Pythaogreans said, this means that math comes first since mind has to be structured by numbers. In theory, it would be possible to work out the set of mathematics which create reality just by thinking about this, just by thinking about what particular math reality needs to be based on. That is the entire project of Hockney’s books, and, he succeeds.
But he succeeds now with the knowledge of empiricist science and the type of mathematics that it has developed. That is, if you’re not smart enough to work out all of the universal maths on your own, then let nature be your guide as Galileo said. In theory it could be done, but in practice it probably isn’t done all that often. A species who did do it would be a very special species indeed.
The problem now arises in that science rejects the idea of a fundamental rational order underlying existence, and so empiricism has completely rejected the otherwise totally valid and correct conclusions of the idealists. Empiricism thinks that it can exist on its own without the guidance of idealism. This is wrong, and it has gotten empiricist materialist science into all manner of irrationalities, climate change being one of them, and Einstein’s relativity and Copenhagen quantum mechanics being others. None of these things are logically compatible with each other. Even Darwinian Evolution falls into fatal paradox in relation to these other fields of science.
The solution is for empiricism to be at the service of idealism, because only idealism can save pure-empiricism from the paradoxes it runs itself into.
You are right: interpretation is crucial! There is nothing more important, in fact, than interpretation. Empiricism thinks that it can get by without performing the appropriate philosophical interpretation, and this is why it creates so many internal inconsistencies and contradictions within its own various fields of pursuit. Empiricism must turn to idealism in order for its sensory findings to be correctly logically fitted in to the rest of known physics. Phlogiston was an empiricist supposition, created by the uber-empiricist practice of alchemy. Any idealist philosopher at the time would have ridiculed the idea that fire is created by a substance which has a virtue of being on fire. It is like saying that opium creates sleepiness because of its dormitive virtue. These seem to be explanations, but only to an empiricist because the explanations are in fact completely empty of deeper meaning or truth.
You could say that mind and observations are also implicitly connected in any case. Mind implicitly, immediately, recognizes its own existence through the observation of its own existence. Hence Descartes: “I think, therefore I am.” Mind, which IS thought, is its own observation of its own existence. It then creates a universe to figure out why it exists, and what its optimal expression is, which it then seeks to express. And the ultimate, optimal expression of mind is God.
Hence we, and every other point of existence itself, as individual nodes of mind, are all becoming God.
Written by Joseph E Postma, Astrophysicist
The Pythagorean SCIENCE OF NUMBERS
IT was an auspicious day for the student at Crotona when Pythagoras received him into his own dwelling and welcomed him as a disciple. The candidate could now look back upon his eight years of probationary discipline with gratitude, for he knew that they had prepared him for the study of Nature’s hidden secrets and placed him on the Path leading to Adeptship.
Pythagoras began his instructions by establishing certain universal principles, proceeding from them into particulars. The key to the whole Pythagorean system, irrespective of the particular science to which it is applied, is the general formula of unity in multiplicity, the idea of the One evolving and pervading the many. This is commonly known as the Doctrine of Emanations. Pythagoras called it the Science of Numbers.
Pythagoras taught that this science — the chief of all in occultism — was revealed to men by “celestial deities,” those godlike men who were the Divine Instructors of the Third Race. It was first taught to the Greeks by Orpheus, and for centuries made known only to the “chosen few” in the Mysteries. Just before the Mysteries began to degenerate, Pythagoras instituted this teaching in his School, thus preserving under the name of “philosophy” the ancient science which, as Plato truly says, is “the greatest good that was ever imparted to men.” In his Life of Pythagoras, Iamblichus repeats the statement of Plato that the study of the science of Numbers tends to awaken that organ in the brain which the ancients described as the “eye of wisdom” — the organ now known to physiology as the pineal gland. Speaking of the mathematical disciplines, Plato says in the Republic (Book VII), “the soul through these disciplines has an organ purified and enlightened, an organ better worth saving than ten thousand corporeal eyes, since truth becomes visible through this alone.”
The present mode of teaching mathematics does little to arouse the higher mind. Even geometry, although based on the Elements of Euclid, is studied only for the purpose of acquiring a knowledge of the other parts of mathematics dependent upon it,
…without having even a dreaming perception of its first and most essential use, that of enabling its votary, like a bridge, to pass over the obscurity of a material nature, as over some dark sea, to the luminous regions of perfect reality. (Thomas Taylor: Theoretic Arithmetic of the Pythagoreans.)
In the seventh book of the Republic Plato indicates the possibilities lying behind the knowledge of numbers. He would make it compulsory for those who manage the affairs of state to study mathematics, “not in a common way, but till by intelligence itself they arrive at the survey of the nature of numbers.” This science, he assures us, should not be used merely for buying and selling, but “for facility in the energies of the soul itself.”
The Pythagorean student approached the science of mathematics from the universal point of view. By applying mathematics to both the Macrocosm and the Microcosm he was able to grasp the secrets of evolution in their minutest details. Quoting from the Neo-Pythagorean Moderatus, Porphyry says that the numerals of Pythagoras were “hieroglyphic symbols, by means whereof he explained ideas concerning the nature of things,” or the origin of the universe.
Plato, summarizing the Pythagorean formula, says that “Deity geometrizes.” The universe evolves from within outward. From the “point” a radiation equal in all directions begins, establishing a circumference, or sphere, within which all activities of the “point” are confined. The point, extending horizontally, becomes a diameter dividing the sphere into positive and negative hemispheres — the basis for action and reaction. The vertical extension of the point into a line crossing the horizontal makes the cross within the circle, and so on ad infinitum. The eleventh Chapter of The Bhagavad-Gita is a dissertation on the Pythagorean Science of Numbers, couched in Eastern terminology. There Krishna shows Arjuna the “vital geometry” of his Divine Form, with all the living lines of force therein and the countless lesser forms produced by them, representing the powers and elements that go to make up the universe.
Pythagoras described the indivisible Unity lying behind all manifestation as “No Number,” in this way repeating the statement in the Stanzas of Dzyan that “there is neither first nor last, for all is one: number issued from no number.” The plane above, therefore, can be indicated only by the nought or Circle, which Pythagoras said is the most appropriate symbol of Divinity.
On the plane below, the Monad or first number appears, and from this number the geometry of the universe emerges. Pythagoras called the Monad, or One, the first odd and therefore divine number. It is through the misinterpretation of the Pythagorean Monad that the various “personal Gods” of the different religions arose, most of whom are represented as a Trinity. In the phenomenal world the Monad becomes the apex of the manifested equilateral triangle, or the “Father.” The left line of the triangle becomes the Duad or “Mother.” This represents the origin of all the contrasts in nature, the point at which the roads of good and evil bifurcate. This being the case, the Pythagoreans are said to have “hated” the Binary. Considering the number Two as a representation of the law of polarity, they stressed its positive aspect by entering a temple on the right side and by putting on the right shoe first. The right line of the triangle represents the “Son,” described in every ancient cosmogony as one with the apex or “Father.” The line at the base of the triangle stands for the universal plane of productive nature, in which “Father-Mother-Son” are unified on the phenomenal plane as they were united in the supersensuous world by the apex.
The triangle is the most profound of all geometrical symbols. As a cosmic symbol representing the Higher Trinity of the universe it became the root of the word Deity. The ancient Greeks called the letter D (the triangular delta) “the vehicle of the Unknown Deity.” The Boeotians wrote the word Zeus with a delta, from which came the Latin Deus. The triangle is also a basic form in Nature. When the molecules of salt deposit themselves as a solid, the first shape they assume is that of a triangle. A flame is triangular in shape; hence, the word pyramid from the Greek pyr, or fire. The triangle is also the form assumed by the pine, the most primitive tree after the fern period.
The Pythagoreans called the number Four the “Key-bearer of Nature.” As a cosmic symbol it represents the universe as chaotic matter before being informed by Spirit. The cross made by the intersection of the vertical line of Spirit and the horizontal line of matter represents spiritual man crucified in the flesh, while the four-pointed star is a symbol of the animal kingdom.
The five-pointed star, the pentacle, is the symbol of man, not only of the physical man with his four limbs and head, but also of conscious, thinking man, whose fifth principle is Manas. The Pythagoreans associated the number Five with the fifth element, Ether. They called Five the “beam of the balance,” which suggests the power of choice and perhaps the final “moment of choice” for our humanity in the middle of the Fifth Round.
The number six illustrates the six directions of extension of all solid bodies. The interlaced triangles picture the union of spirit and matter, male and female. The Pythagoreans considered this number as sacred to Venus, since “the union of the two sexes, and the spagyrisation of matter by triads are necessary to develop the generative force … which is inherent in all bodies.” (Rayon: Potency of the Pythagorean Triangles.)
Pythagoras called seven a perfect number, making it the basis for “Music of the Spheres.” Regarding seven as a compound of three and four, he gave a twofold account of its meaning: On the noumenal plane the triangle is Father-Mother-Son, or Spirit, while the quaternary represents the ideal root of all material things; applied to man, the triangle represents his three higher principles, immortal and changeless, while the quaternary refers to the four lower principles which are in unstable flux. Seven not only governs the periodicity of the phenomena of life on the physical plane, but also dominates the series of chemical elements, as well as the world of sound and color, as shown by the spectroscope.
The Pythagoreans called the number eight “Justice.” In that symbol we find an expression of the eternal spiral motion of cycles, the regular inbreathing and outbreathing of the Great Breath. They called the number nine the “Ocean” and the “Horizon,” as all numbers are comprehended by and revolve within it. If we consult the Table of the Yugas on page 125 of The Ocean of Theosophy, we shall observe that all the figures may be resolved into the number nine.
Ten, or the Decade, brings all these digits back to unity, ending the Pythagorean table. In both the Microcosm and the Macrocosm the three higher numbers of the Decade stand for the invisible and metaphysical world, while the lower seven refer to the realm of physical phenomena.
The Tetraktys of Pythagoras — composed of ten dots arranged in four rows to form a triangle — was the sacred symbol upon which the Pythagoreans took their most binding oath:
I swear by him who the Tetraktys found,
Whence all our wisdom springs and which contains
Perennial Nature’s fountain, cause and root.
Theon of Smyrna says that this symbol was honored by the Pythagoreans “because it appears to contain the nature of all things.” H.P.B. indicates the extraordinary philosophical value of the Tetraktys in The Secret Doctrine (I, 612). According to Iamblichus, the PythagoreanTetraktys had eleven forms, each one applying to some one particular phase of cosmic or terrestrial life.
Pythagoras applied the Science of Numbers to music, giving the Western world the mathematical basis of its present musical system. The abstract Circle of music is Sound. The mathematical point within that circle, from which the music of our earth emerges, is the “Tone of Nature,” called Kung by the ancient Chinese. The “line” of music, derived from the ratio 2:3, is what is now called the “perfect fifth.” The rotation of this line forms the “Circle of Fifths,” which gives the basis of all key relationships.
The music of this planet, according to Pythagoras, is but a small copy of the “Music of the Spheres.” The seven tones of the musical scale correspond to the seven sacred planets, each of which is characterized by a certain tone. As Shakespeare makes Lorenzo say in The Merchant of Venice, “There’s not the slightest orb which thou beholdest but in its motion like an angel sings.” The study of music was obligatory in the Pythagorean School, not only as a science but also as a healing agent. Iamblichus informs us that “Pythagoras believed that music greatly contributed to health, if it was used in the proper manner.” Pythagoras taught that the purest type of sound comes from stringed instruments and that wind instruments tend to excite the lower nature rather than to quiet it, an observation later corroborated by Plato.
The study of astronomy was a duty of the School. Pythagoras taught the heliocentric system and the sphericity of the earth; he declared that the moon is a dead planet which receives its light from the sun and described the composition of the Milky Way. More than a thousand years later both Bruno and Galileo derived their theories of astronomy from Pythagorean fragments.
The esoteric students of Pythagoras were given the Mystery teachings in regard to the nature of the soul, its relation to the body and its ultimate destiny. Pythagoras taught that the soul of man is derived from the World-Soul; hence is immortal and cannot be destroyed by death. The soul of man, he said, accomplishes its evolution by means of numberless incarnations on earth. He frequently spoke to his pupils about their own former lives, and when asked about himself said that he had come into the region of mortality to benefit mankind. He also taught the doctrine of Karma, saying that all the seeming injustices on earth are explained by the fact that every life on earth is but a reward or punishment for deeds performed in previous lives. No outside circumstances are to blame for our unhappy lives, he said, since “men draw upon themselves their own misfortunes, voluntarily and of their own free choice.”
Applying the Science of Numbers to the problem of good government, Pythagoras first made himself a “point” in which great spiritual forces were focused, and from that “point” the radii of their influence extended. The Pythagorean School eventually became a small model city, its form of government being adopted by Crotona. From Crotona the sphere of Pythagorean influence expanded to include the neighboring towns, where legislative systems based upon Pythagorean principles lasted for generations.
When Pythagoras was almost a hundred years old he went to Delos to attend the funeral ceremonies of an old friend. One evening, when the Teacher and forty of his pupils were talking together, some of his former pupils who had been expelled from his School set fire to the building where they were assembled, and Pythagoras, with thirty-eight of his pupils, were consumed in the flames.
After the death of the Teacher the School at Crotona was closed and the students departed from Italy. Fearing that the very word philosophy — a word which Pythagoras had coined — would disappear from the Greek language, some of these loyal disciples collected the writings of the older Pythagoreans and wrote down many things which Pythagoras himself had said. These writings were passed down from teacher to pupil, or from father to son, for many generations.
The direct successor to Pythagoras — if such a man could be said to have a successor — was his pupil Aristæus. After him came Pythagoras’ son Mnesarchus, who was named after his grandfather. The Pythagorean fragments were preserved by two hundred and thirty-five of his loyal disciples, two hundred and eighteen of whom were men, the other seventeen women. At the present day all that remains of his ethical precepts is found in the Golden Verses.
COMPILER’S NOTE: The following is a separate item which followed the above article but was on the same page. I felt it was useful to include it here:
THE FATAL SEPARATION
The fruit of the Tree of Knowledge gives death without the fruit of the Tree of Life. Man must know himself before he can hope to know the ultimate genesis of beings and powers less developed in their inner nature than himself. So with religion and science; united two in one they were infallible, for the spiritual intuition was there to supply the limitations of physical senses. Separated, exact science rejects the help of the inner voice, while religion becomes merely dogmatic theology — each is but a corpse without a soul.
(Pages 301-306; Size: 18K)
(Number 51 of a 59-part series)
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This book presents the mysterious formula that rules art, nature, and science. This book deals with the Divine Proportion, a secret code that rules art, nature, and science. It is known by many names: Golden Mean, Sacred Cut and Phi are only a few; and it is not by chance that the Divine Proportion was given its name. It has been called divine because over thousands of years it has been deemed to be so.The Divine Proportion reveals a number of simple patterns: It is seen in the seed patterns of fruits, the family tree of bees, the pyramids of Egypt, Gothic cathedrals, Renaissance paintings, the human body, shells…the list is endless.Mathematicians use the Greek symbol F to represent the Divine Proportion and equate it to a number that is defined by the ratio (1 + sqrt5) / 2 or 1.6180339… Numbers do little, however, in describing this unique ratio that is found everywhere in nature and for 2500 years has been an aesthetic guide in art and architecture.Beginning with calculations found on clay tablets in ancient Babylon, the story of Divine Proportion can be traced alongside the history of numbers to the fractals of the digital age. As its many forms unfold we uncover the Golden Rectangle in the Parthenon, Golden Spirals in the human inner ear, a Golden Angle in the petal patterns of a rose, and the Fibonacci numbers in lilies, daisies, pineapples, and in our own DNA.With its natural balance and elegant beauty, the Divine Proportion is a perpetual reminder that our hope for regeneration and continuity lies in realizing the meaningful and harmonious relationship of all the parts to the whole.