The Rudolf Steiner Archive

5. The Pythagorean

Highly Esteemed Attendees!

[ 1 ] I must apologize for my cold; it will be difficult to hear me.

[ 2 ] I have endeavored to show that Pythagoreanism—which flourished in southern Italy in the sixth century B.C.E., this school that was strictly self-contained and yet not entirely self-contained — had a great influence in all the centuries that followed Pythagoreanism, and that this entire tradition was acquired through a difficult course of study that lasted many years. One actually had to do many exercises before reaching the point of grasping things in a purely spiritual way. I [wanted] to show that it is a pity not to have such a school today. I [wanted] to show that the Pythagorean does not stand so isolated, but that in Novalis we have a personality who thought exactly in this way. But there is something else as well.

[ 3 ] It may seem somewhat strange when we hear that the Pythagoreans sought, in the wondrous harmony of numbers, in the numbers themselves, and in the interconnection of numbers, something that represents the primordial foundations of things. It will not surprise us if we assert that our science—ordinary natural science, insofar as it is today physics and chemistry—is on the path to becoming Pythagoreanism. Throughout the nineteenth century, only a certain set of sensory perceptions prevented natural science from merging into Pythagoreanism. We are now facing a reformation of natural science. We are on the verge of chemistry and physics—because only scientific ideas have been adopted—becoming entirely materialistic. Perhaps, however, in a few years we will no longer be able to think about chemical and physical matters in purely materialistic terms. Despite his achievements and tremendous inventions, and although he never moved beyond a certain materialistic framework throughout his life, Helmholtz adopted a kind of idealism toward the end of his life. At the natural scientists’ conference, he announced a lecture on “Apparent Substance and True Movement.” Unfortunately, no records of it have survived in his estate. But we can imagine what Helmholtz wanted to say. A physicist only gives a lecture [with such a title] when he has to. To have to give a lecture on “Apparent Substance and True Motion”—on the fact that everything substantial is only apparently there, and that the spiritual, the motion, is the true reality—is of great significance. For it is not immediately possible for the physicist to ascend to the spiritual. But it is already something when the physicist regards matter not as the real, but only as something apparent. That is a symptom.

[ 4 ] Our entire natural science is, in essence, aimed at confirming the ancient Pythagorean theorem that everything that exists in space can be traced back to numerical ratios. To illustrate what I mean when I say that natural science is Pythagoreanism—that it is on the path to seeing numbers as the decisive factor—let us take chemical substances: oxygen, hydrogen, nitrogen, lead, or any of the various other elements. They form compounds, as we know. When, for example, lead and oxygen combine, what is the determining factor? The number! The number is the determining factor when these two substances combine to form lead oxide. I cannot elaborate on this further, nor can I discuss how the two substances can be separated again, because that would lead too far afield. For us, however, it is sufficient to know that when lead and oxygen combine, they always combine in a specific numerical ratio.

[ 5 ] But it goes even further. Suppose mercury combines with oxygen. In this case, 103 grams of mercury combine with only 8 grams of oxygen. You can see in advance that if you want to combine 8 grams of oxygen, you will need 103 grams of mercury to do so.

[ 6 ] But the fact is this: oxygen always combines with all other elements in such a way that certain weight parts of other elements combine with eight weight parts of oxygen. This is true for all elements. For each of the seventy-three known elements, there is a number that indicates the weight ratio with which it combines with all others. The chemist must acknowledge this from a wealth of facts. That is precisely the point: we have far more facts than the ancients did. They had the same ideas, but fewer facts!

[ 7 ] In lead oxide, 103 parts by weight of lead combine with 8 parts by weight of oxygen. If we combine lead with sulfur, 103 parts by weight of lead combine with 16 parts by weight of sulfur.

[ 8 ] Thus, when you construct the entire world in material terms, you do not create something chaotic and arbitrary, but—as the Pythagoreans imagined—something ordered according to the harmony of numbers.

[ 9 ] And the further we advance in science, the more we see that the organization and structure [of nature] can be expressed through numbers. Modern science confirms the validity of the Pythagoreans’ intellectual construct, namely that we are dealing with numbers in nature.

[ 10 ] Helmholtz, it seems, intended to say the same thing in his lecture. Ultimately, then, we are dealing with the spirit. Whether the spirit is expressed through numbers or viewed in some other way, let us leave that aside for now. If we bring out everything that lies dormant in substance, we ultimately arrive at the conclusion that we are dealing with spirit.

[ 11 ] Natural science also provides us with proof of this. Day by day, natural science increasingly refutes materialism. The notion that natural science requires materialism may still exist in some minds, but in truth, nothing refutes materialism more than natural science. I would like to give you an example from contemporary science to show you how this science is overcoming materialism day by day, bit by bit.

[ 12 ] You know that the human eye consists of a slightly flattened spherical capsule or shell filled with a glass-like, watery fluid, the vitreous humor, and that seeing an object is made possible by a small crystalline lens embedded in this mass, which projects the image onto the retina. Light enters through the pupil. This light is refracted by the lens, creating a small image, and this image is the basis of our vision. Eyes of similar construction are also found in various animals, vertebrates, and a number of invertebrates.

[ 13 ]The octopus also has an eye that appears to be constructed similarly to that of humans. It has an eye with a vitreous body; embedded in the vitreous body is a lens that enables vision. What is interesting here is that in humans, the crystalline lens develops in a completely different way than in the octopus, even though both are crystalline lenses. In the octopus, the crystalline lens forms as it grows within the aqueous fluid of the eye [through secretion and thickening]. Now you might say to yourself: The eye, with its aqueous fluid and lens, resembles a camera obscura. In humans, however, the lens does not form through secretion and thickening, but rather by the surface drawing the lens out from within itself, which is then inverted into the eye. This crystalline lens has, in one instance, formed from within the eye, and in another instance, formed in a different way from the outside and was then inverted into the vitreous body!

[ 14 ] Can we still say, when we see how a material structure can arise in such different ways, that it is the material that forms nature? Must we not say that nature is not built upon material but upon purely spiritual forces? The nature of the material is completely irrelevant here. Sometimes it is shaped from the inside, other times from the outside. The material as such is therefore, in essence, completely irrelevant to the structure of organisms. It is the spirit that constructs things.

[ 15 ] The further we advance in the natural sciences, the more we find that we can never say what would come into being if we took [only] matter into account. It is the spirit that uses matter to create forms. That is why the Pythagoreans divide the world into these two components: On the one hand, they conceive of the world as that which makes it perceptible, and on the other hand as that which is not perceptible in the world—or, to put it better, that which is perceptible and recognizable only to the mind; they have clearly distinguished these two things in the human and animal eye; they see how the same eye is constructed in one case and in the other.

[ 16 ] This law—indeed, the entire spiritual realm that makes use of the material—cannot be perceived by the senses; it can only be perceived in a spiritual way. But space could not be filled if the spirit could not make use of the perceptible.

[ 17 ] The Pythagorean distinguishes between these two things: on the one hand, there is the eternally creative, the spiritual, which makes use of matter to create countless forms; then, on the other hand, there is matter, which is not active in itself, but only serves to make visible something that is not perceptible to the senses.

[ 18 ] The Pythagorean constructs the entire world from the perceptible and the imperceptible. For them, the One is the number one. It forms the boundary for them. For us, something exists by becoming one, by becoming individual. But this is only apparent; it is connected to all other things in the world. And the “how” is conveyed by numerical symbolism and numerical relationships. And so we can understand it because the Pythagoreans had this connection; they saw how much the number governs when it comes to material things and forms. But they also saw how the Pythagorean strives to seek out the wondrous connection within the numbers, and to find what he has discovered inwardly confirmed outwardly.

[ 19 ] Anyone who does not take this into account, who is not clear that it was the harmony of the inner and the outer, who does not consider this, that it was the delight for the Pythagorean, can very easily find mere games in the Pythagorean teachings on numbers. But this lives in the minds only when they cannot perceive the inner delight, when they do not understand, when they cannot grasp how, for the Pythagoreans, the whole process of counting becomes something else once we have reached ten.

[ 19 ] If one does not understand this purely as the Pythagoreans understood it, one cannot grasp what the Pythagoreans meant. As long as we count up to ten, we are dealing with units, where we add one number to another. But when we reach ten, we continue counting. We do not, however, count only units, but the tens: 10 = 1 × 10, 20 = 2 × 10, 30 = 3 × 10. While we have 10, we add 10 more to it. So we count what we used to count materially in a spiritual way. We count the numbers themselves starting from 10, so that in the Pythagorean sense, when we count to 10, we are actually counting materially. But when we continue counting the tens, we can disregard the material aspect. We do this by saying: 1 × 10, 2 × 10, and so on. And then we come to the hundreds. The further we count upward, the more we forget the material basis and what previously served us for counting. It is precisely in counting, the Pythagorean believes, that we have a means of lifting ourselves higher and higher into the spiritual realm.

[ 21 ] One must take into account what he means by the so-called “gnomons.” By this he means nothing other than the inner lawfulness that governs our world of numbers when we study this world of numbers in the right way.

[ 22 ] Consider the following: If you take the consecutive numbers and multiply each number by itself, you obtain the so-called square numbers: 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25.

[ 23 ] So when we multiply the individual numbers by themselves, we obtain the square numbers. Now, however, there is a remarkable connection between the numbers and the square numbers. The Pythagoreans were the first to investigate this relationship.

[ 24 ] Now take what is not yet contained in 2x2; take what is new for 4—the 5—and add it, and you get the second perfect square. Strangely enough, this law holds true continuously throughout all the square numbers. 4 is the first square number, 9 is the second, 16 is the third.

[ 25 ] 2 + 2 = 4; the next new number is 5, so plus 5 = 9.

[ 26 ] Let’s take 3 + 3 = 6; the next new number is 7, so 3 × 3 = 9, plus 7 = 16.

[ 27 ] You can continue this process. And the Pythagoreans called this the “gnomon.”

[ 28 ] Take the 4: 4 × 4 = 16. 16 + (4 + 4 + 1 = 9) = 25, the square of 5.

[ 29 ] Take 5: 5 × 5 = 25. 25 + (5 + 5 + 1 = 11) = 36, the square of 6.

[ 30 ] Take the 6: 6 × 6 = 36. 36 + ( 6 + 6 + 1 = 13 ) = 49[, which is the square of 7].

[ 31 ] You can find this inner regularity throughout the entire number line. Here you gain an inner insight into the connection itself. This was what compelled the Pythagorean to believe that the numerical realm possesses an inner law of its own. He found this in things like gnomons, so that the Pythagorean could say to himself: The same thing that I form within my mind, I find out there in the cosmos, and the same thing that forms the inner connection stands in inner harmony with the cosmos. If we have 3 x 3 things out there [in the cosmos] and have arranged them in this way, then they correspond to what we have brought forth in our minds. We can develop the entire mathematics in our minds; we could construct the entire theory of numbers; we would need to know nothing of the external world. If we then close our eyes, the external world will obey the laws we have devised!

[ 32 ] This is what led Pythagoras to recognize a numerical regularity—and all that lies within it.

[ 33 ] I would just like to draw attention to the great chemical discoveries of Lothar Meyer and the Russian Mendeleev, which are a complete confirmation of what the Pythagorean intended with his views. I have said that the elements all combine in specific numerical ratios: Hydrogen always combines with oxygen in a ratio of 8 or a multiple of 8. If we now examine the intervals between the individual given numbers, we find a complete regularity. We proceed from oxygen with 16 or also from the element that has 7. The elements cannot combine with others according to any numerical ratio other than their own; rather, there is a regularity in these combinations.

[ 34 ] Lothar Meyer gave an interesting lecture on these matters at a meeting of natural scientists regarding lithium, potassium, and sodium, which combine with the other elements in the following weight ratios:

— lithium in a weight ratio of 7,
— sodium in a weight ratio of 23,
— potassium in a weight ratio of 39.

[ 35 ] When these numbers are added together, remarkable correlations emerge. If we take the weight ratio of 7 to 23, we get a difference of 16. If we take the ratio of 23 to 39, we again get a difference of 16. Such consecutive triads of three consecutive substances occur frequently in chemistry. We could also take three other elements and find that the same numerical intervals occur between them.

[ 36 ] If we took all the elements, we would always find intervals that can be expressed in numbers. We have certain substances that group together quite nicely in sequence. But there is a number missing in between. Suppose we had lithium and sodium; we would also have other substances and know that there is a certain gap between them; but then something is missing in between! Now the chemist has already become accustomed to something; he no longer says: Here is an irregularity, but rather he says: It is because we do not yet know the substance that has this numerical ratio. Many substances have been discovered recently. At first, one only suspected that they must exist. Later, however, they were found. The confidence that the mind would find it led to the mind actually discovering it.

[ 37 ] [Uranus was discovered through observation. Neptune was predicted on the basis of calculations.] People observed the laws governing the motion of the planets. These laws did not apply to Uranus, and people concluded: There must be another body at a specific point in space [that deflects Uranus from its calculated orbit]. People have more faith in the spiritual law than in sensory reality, and this has fully confirmed the mind’s insights, proving the mind right.

[ 38 ] This harmony between spirit and matter, as established by the Pythagoreans, is being confirmed bit by bit by natural science. What we perceive as material in it will increasingly be overcome. It will increasingly lead to the apparent substance dissolving into spiritual relationships.

[ 39 ] Matter is not merely frozen spirit, nor is it merely spirit in appearance. It dissolves bit by bit, so that the spirit perceives not something else, but itself. Thus, science provides us with the proof for what is emphasized today from the most diverse perspectives.

[ 40 ] Annie Besant drew attention to similar connections between her views and natural science. Starting from these fundamental views, the Pythagoreans delved into the world and arrived at a perspective that must be particularly valuable to us because Goethe, in more recent times, has arrived at a similar perspective based on his scientific principles!

[ 41 ] The Pythagoreans conceived of the constitution of the world in such a way that this spirit, which they sought to capture in numbers, is limited by the infinite and, by being limited, becomes perceptible.

[ 42 ] Now the Pythagoreans imagined that this entire process of becoming is the continuous interpenetration of the finite and the infinite. They conceived of the appearance of the infinite as a boundary as a kind of matter. Matter is the indifferent, which causes nothing other than the spirit to become visible. To form beings out of matter, the spirit inhales matter and exhales it again into the cosmic space. Therefore, the spirit [acts] in a continuous inhaling and exhaling, as a breathing process.

[ 43 ] Goethe also uses this image, creating a mental image of this process as the inhalation and exhalation of the airspace. Goethe conceives that the Earth, together with the atmosphere, influences itself from the outside, literally contracting into itself, inhaling what it needs from outer space, and then exhaling again what it has processed within itself. There is a difference in air pressure when one inhales and exhales—a becoming stronger and also a becoming weaker. People have abandoned this line of thought because they did not want to believe in such regular mental images. [Goethe] sought to show that the fluctuations of the barometer are not arbitrary, but essentially trace back to fundamental fluctuations, to something that is entirely regular. In the seemingly irregular processes, one can perceive a regularity that stems from the fact that the Earth inhales air and then exhales it again, a regularity that then produces regular fluctuations, indicating that we are dealing with an inhalation and exhalation on Earth.

[ 44 ] In Goethe, this is presented to us in an interesting way. I do not regard this as trivial; rather, it seems to me that it is immensely important that our mental images be spiritualized by such ideas. We can constantly observe the processes in the outer world, such as barometric fluctuations. If we have not grasped them, at least in terms of their direction, we will not even become aware of them. Such regular fluctuations can also be found when we examine the irregular and then subtract the regular. What remains is then irregular.

[ 45 ] My intention was to point out that what the Pythagoreans taught is not something outdated, but rather that it finds application precisely in today’s natural science. [In astronomy, the Pythagoreans proceeded by applying the spatial conception of the finite and the infinite to the cosmos as well.]

[ 46 ] Every era can only comprehend the various perceived phenomena in accordance with the state of the relevant science of perception—astronomy, for example, and so on—at that time. Research is conducted within the realm of experience. The Pythagoreans, too, had to reckon with [the mental images of their time]. They formed a mental image of the central fire, which represents unity, as the primal source of the world; that which continually breathes matter in and out and thereby brings the world into being. The stars, with their regular motion, represented to them a regular multiplicity expressed in numerical ratios. And what took place in the sphere below the moon was a process of becoming irregular, a constant struggle between the finite and the infinite. But in what lies above, the struggle is balanced into a great harmony. There the celestial bodies move in regular orbits. Consequently, the regular orbits have taken the place of the point, the sharply defined One. We are no longer dealing with the One, but with Unity. This struggle, which constantly unfolds before our eyes, takes place between the Earth and the Moon, unfolding as an eternal struggle. On Earth, harmony and disharmony are constantly alternating. Human beings stand in this struggle because they are a unity. They seek to reconnect, having been torn from the harmony of the world, by striving to return to harmony with the world through what the Pythagoreans call virtue. You see that the Pythagorean penetrates everything from the lowest phenomena up to those of human life with numerical mental images.

[ 47 ] Now, from the Pythagorean worldview, the most important things remain for us, namely those elements of spiritual life to which the ethical sphere rises by the spirit attempting the deepest immersion into its own inner self. The Pythagoreans also held certain views on this, which were derived from cosmological mental images.

[ 48 ] In seeking to establish a harmony that leads extraordinarily deeply into the world—one that bridges what separates human beings from the entire cosmos—the Pythagoreans arrived at the mental image of reincarnation, of the connection between the various incarnations of all beings.

[ 49 ] This is a subject we will address in particular next time. I wanted to discuss the fundamental question that, through Pythagorean physics and ethics—which were developed from a conceptual framework such as the one just described—was only communicated after a long period of training. One must not form a mental image of the teaching of reincarnation or re-embodiment being imparted immediately! The student was first trained through concepts expressed in the regularity of the world of numbers, which then led him deeply into the subject. Then he was also shown what he must do to attain world harmony, to eliminate the so-called original sin from the world. The problem: How can the separation of the individual soul from the universe be justified? — became the great Pythagorean question.

[ 50 ] And this question led the Pythagoreans to resolve it within Pythagoreanism, and that is the fundamental feature of Pythagorean training, which we will discuss next time.

Questions and Answers:

Question: [?]

[ 51 ] Dr. Steiner’s answer: The way a person breathes is intimately connected with the fact that a person can speak. If a person did not have vertically positioned lungs and a vertically positioned larynx, they could not speak. A dog could never speak words. Generally speaking, dogs are more intelligent than parrots and starlings, yet parrots and starlings can be taught to speak more easily than dogs. This is related to the entire constitution of the human lungs and larynx, and specifically to the fact that the lungs and larynx are upright. The production of articulated sounds can only occur with an upright larynx and upright lungs. Monkeys cannot produce articulated sounds, even if they can be made to walk somewhat upright. Their organism is not built that way. The upright larynx breathes air in and out in a peculiar way. This inhalation and exhalation enables articulated speech. And since we know that articulated speech is connected to the spirit, we have scientifically established the possibility that the spirit takes hold within the organism. This possibility exists only in upright-walking beings. Only when the body was able to walk upright could the spirit take hold. Beings that still use their front limbs to move forward cannot harbor a spirit. During the Tertiary period, specifically the Diluvial epoch, gibbons lived among us—highly intelligent beings. After it grew colder, they migrated, but returned and then had to live in colder climates where the plant life was not as abundant. They had to use their front legs as tools, then gradually learned to walk upright, and the spirit was then able to take possession of the brain.

Question: Does an octopus see the same way a human does?

[ 52 ] Answer: Kurd Laßwitz wrote a fairy tale that addresses this question. It is highly probable that what comes into being in the eye of a human also comes into being in the eye of an octopus. One makes the crystalline lens out of glass, the other out of rock crystal or another similar substance, and the same thing comes into being through the crystalline lens. In humans, it is imprinted in a different substance than in the octopus. But even if the octopus has the same image as the human, the image must first be spiritualized, processed. Just because you have a red-bordered circle in your eye does not mean you have an object. It must be connected with other mental images. Whether the octopus can do this, we do not know.

[ 53 ] One might assume that the same animal comes into being in very different ways. In the case of the dog, we are dealing with matter that is organically structured and permeated by laws. The indifference of matter is what physicists see in the conservation of energy or matter. What is used for the eye today may serve another purpose tomorrow. Matter is indeterminate. It would be different if matter contained the laws within itself. The same thing can consist of different material prerequisites. Regularity does not depend on matter; rather, it can be conceived by us. This is what delights the Pythagoreans. The matter of the eye is the same as the matter of the ear. On the other hand, if one could make everything out of everything, one would have to demand that one could see with the ears and hear with the eyes! Disharmonies arise, which then balance each other out.

Question: [2] Answer: We no longer know the sublunar circle today.

[ 54 ] [Question:] How is it that people fall so completely out of harmony?

[ 55 ] [Answer:] The Pythagorean holds the view that there is no perfect harmony on Earth, but rather a becoming of the harmonious out of disharmony. There is only a continuous falling out of and being drawn back into harmony. This is what constitutes the individual being. In the individual being, we have the detached being that longs to return to harmony. The celestial bodies as a whole describe their orbits, which do not interfere with one another. This is perfect harmony. Even in hunger and satiety there is harmony and disharmony: balance is life. There are beings at rest within themselves, in contrast to those in motion within themselves. The [individualists] have restlessness within themselves. Christ says: “Where two or three are gathered together, I am in their midst.” This is a Pythagorean mental image. And it is this: The Pythagorean sees the beginning in the One, in the Two he sees the indeterminate added, and the perfected being is where the Three is added. Imagine two points; connect them and you have a line. With three points, you have a plane, the triangle. Something two-dimensional is defined only by the Three. The Three has a center; the Two has only a juxtaposition. The Three must bring about balance. It is the spirit that connects them.

Question: [?]

[ 56 ] Answer: The Bible is composed of the whole world. Nothing is easier than interpreting the Bible. With esoteric interpretation, one can get roughly the right idea, but one cannot know whether one is hitting the historical [truth].

Question: [2]

[ 57 ] Answer: Neptune would not be the last planet. [Dawes] already reported this in 1839. Kunowsky is a welcome development.