The Rudolf Steiner Archive

5. The Pythagorean

[Ladies and gentlemen!]

[ 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, at the same time, not entirely self-contained—had a profound influence in all the centuries that followed, and that this entire school of thought was mastered through a rigorous course of study lasting many years. One actually had to do many exercises before reaching the point where one could grasp things in a purely spiritual way. I [wanted] to show that it is a pity we do not have such a school today. I [wanted] to show that the Pythagorean does not stand so isolated, but that in Novalis we have a figure who thought exactly in this way. But there is something else as well.

[ 3 ] It may seem somewhat strange to hear that the Pythagoreans sought, in the remarkable harmony of numbers, in the numbers themselves, and in the interrelationship of numbers, something that represents the fundamental principles of all things. It will come as no surprise to us if we assert that our science—conventional 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 converging with Pythagoreanism. Today we stand on the threshold of a reformation of natural science. We are on the verge of chemistry and physics—because only scientific ideas have been adopted—becoming entirely materialistic. But perhaps, 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 even though he never moved beyond a certain materialistic framework throughout his life—Helmholtz adopted a form of idealism toward the end of his life. At the natural scientists’ conference, he announced a lecture on “Apparent Substance and True Motion.” Unfortunately, no notes on this have survived in his estate. We can, however, imagine what Helmholtz intended to say. A physicist gives a lecture [with such a title] only when he has to. Having to give a lecture on “Apparent Substance and True Motion”—on the fact that everything substantial is only seemingly there, and that the spiritual, the motion, is what is true—is of great significance. For it is not immediately possible for a physicist to ascend to the spiritual realm. But it is already significant 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 consider chemical substances: oxygen, hydrogen, nitrogen, lead, or any of the various other elements. As we know, they form compounds. 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 take us too far afield. For our purposes, however, it suffices to know that when lead and oxygen combine, they always do so 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'll need 103 grams of mercury.

[ 6 ] But here’s the thing: Oxygen always combines with all other elements in such a way that certain parts by weight of other elements combine with eight parts by weight 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 at which it combines with all the others. The chemist must acknowledge this based on 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 and form the entire world as a material entity, you do not create something chaotic and arbitrary, but—as the Pythagoreans believed—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 accuracy of the Pythagoreans’ intellectual construct—that we are dealing with numbers in nature.

[ 10 ] Helmholtz seems to have meant to say the same thing in his lecture. So, ultimately, we are dealing with the mind. Whether the mind is expressed through numbers or viewed in some other way, let us set 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 the mind.

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

[ 12 ] You know that the human eye consists of a slightly flattened spherical shell filled with a glassy, watery fluid—the vitreous humor—and that we are able to see an object because embedded within this mass is a small crystalline lens that projects the image onto the retina. Light enters through the pupil. It is refracted by the lens, producing a small image, and this image is what enables us to see. Various animals, including vertebrates and a number of invertebrates, have eyes of similar structure.

[ 13 ] The octopus also has an eye that appears to be structured 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 humor of the eye [through secretion and thickening]. Now you might say to yourself: The eye, with its aqueous humor and lens, resembles a camera obscura. In humans, however, the lens does not form through secretion and thickening, but rather because the surface pulls the lens out from within itself, and it is then folded inward into the eye. This crystalline lens is formed in one case from within the eye and in the other case in a different way from the outside, and is then folded inward into the vitreous body!

[ 14 ] Can we still say—when we see how a material form can arise in so many different ways—that it is the material that constitutes nature? Must we not say, then, that nature is structured not according to material but according to purely spiritual forces? The nature of the material itself is entirely irrelevant. Sometimes it is shaped from within, other times from without. The material as such is therefore, in essence, completely irrelevant to the structure of organisms. The spirit 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 mind 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 imperceptible in the world, or, to put it better, that which is perceptible and recognizable only to the mind; they have clearly distinguished two things in the human and animal eye; they see how the same eye is structured in one case and in the other.

[ 16 ] This law—indeed, the entire spiritual realm, which 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 were not able to 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; on the other hand, there is matter, which is not active in and of itself, but only serves to make visible that which 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. For them, it forms the boundary. 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 recognized this connection; they saw how deeply the number governs material things and forms. But they also saw how the Pythagorean strives to uncover the wondrous connections within the numbers and to find external confirmation of what he has discovered inwardly.

[ 19 ] Anyone who fails to take this into account, who does not realize that it was the harmony between the inner and the outer, who does not consider that it was a source of delight for the Pythagoreans, may very easily come to view the Pythagorean teachings on numbers as mere games. But this lives on in people’s minds only when they are unable to 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 entirely once we reach the number ten.

[ 20 ] If one does not interpret this exactly as the Pythagoreans did, one cannot understand 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 once we reach ten, we continue counting. But we do not merely count units; we count 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 in a material way, now in a mental way. We count the numbers themselves starting from 10, so that, in the Pythagorean sense, when we count to 10, we are actually counting in a material way. But when we continue counting in tens, we can look beyond the material. 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 foundation and what previously served as our basis for counting. According to the Pythagoreans, it is precisely in counting that we have a means of elevating ourselves more and more into the spiritual realm.

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

[ 22 ] Consider the following: If you take the consecutive numbers and multiply each number by itself, you get what are called square numbers: 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25.

[ 23 ] So, if we multiply each number by itself, we get the squares. Now, there is a curious relationship between the numbers and their squares. The Pythagoreans were the first to investigate this relationship.

[ 24 ] Now take what is not yet included in 2x2; take what is new for the 4—the 5—and add it to that, and you will get the second perfect square. Curiously, this law holds true for all square numbers in sequence. 4 is the first square number, 9 is the second, and 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 4: 4 × 4 = 16. 16 + (4 + 4 + 1 = 9) = 25, which is the square of 5.

[ 29 ] Take 5: 5 × 5 = 25. 25 + (5 + 5 + 1 = 11) = 36, which is 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 system. Here you gain an inner insight into the connection itself. This was what compelled the Pythagorean to believe that the numerical order possesses an inner law of its own. He found this in things like gnomons, so that the Pythagorean could say to himself: What I form within my mind, I find outside in the cosmos, and what constitutes the inner connection stands in inner harmony with the cosmos. If we have 3 × 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 field of mathematics in our minds; we could construct the entire theory of numbers—we would need to know nothing of the outside world. If we then close our eyes, the outside world will obey the laws we have devised!

[ 32 ] This is what led Pythagoras to recognize a numerical law—and all that it entails.

[ 33 ] I would just like to draw attention to the great chemical discoveries of Lothar Meyer and the Russian Mendeleev, which fully confirm what the Pythagoreans sought to demonstrate with their theories. I have said that all elements 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, which has a value of 16, or from the element that has a value of 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 topics at a meeting of naturalists, discussing 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 you add these numbers together, you discover some curious relationships. 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 elements occur frequently in chemistry. We could also take three other elements and find that the same numerical intervals exist between them.

[ 36 ] If we were to consider all the elements, we would always find gaps that can be expressed as numbers. We have certain substances that line up quite neatly in a row. But there is a number missing in between. Let’s suppose we have lithium and sodium, along with other elements, and we know there is a certain gap between them; but then something is missing in between! Now the chemist has already grown accustomed to this; he no longer says, “Here is an irregularity,” but rather, “It is because we do not yet know the element that has this numerical ratio.” Many substances have been discovered recently. At first, people only suspected that they must exist. Later, however, they were found. The confidence that the mind would find them led to the mind actually discovering them.

[ 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 hold true for Uranus, and it was concluded that there must be another body at a specific point in space [that deflects Uranus from its calculated orbit]. People have more confidence in spiritual law than in sensory reality, and this has fully confirmed the spirit and proven it right.

[ 38 ] This harmony between spirit and matter, as defined by the Pythagoreans, is being confirmed bit by bit by the natural sciences. The material aspects we perceive in it will increasingly give way to it. 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. Natural science thus provides us with proof of what is emphasized today from a wide variety of perspectives.

[ 40 ] Annie Besant drew attention to similar connections between her views and the natural sciences. Starting from these fundamental views, the Pythagorean delved deeply into the world and arrived at a perspective that must be particularly valuable to us because Goethe, in more recent times, has in turn arrived at a similar perspective, based on his own 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. In order to form beings out of matter, the spirit inhales matter and exhales it back into the cosmic space. Therefore, the spirit [acts] in a continuous inhalation and exhalation, forming a breathing process.

[ 43 ] Goethe also uses this image, imagining this process as the atmosphere inhaling and exhaling. Goethe imagines that the Earth, together with the atmosphere, influences itself from the outside, literally contracting within itself, inhaling what it needs from outer space, and then exhaling what it has processed within itself. There is a difference in air pressure between inhaling and exhaling—a process of becoming stronger and also of becoming weaker. People have abandoned this line of thought because they did not want to believe in such regular patterns. [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 these seemingly irregular processes, one can perceive a regularity that stems from the Earth inhaling air and then exhaling it again—a regularity that gives rise to periodic fluctuations, indicating that we are dealing with a process of inhalation and exhalation on Earth.

[ 44 ] In Goethe’s work, this becomes evident to us in an interesting way. I do not regard this as insignificant; rather, it seems to me that it is immensely important for our conceptions to be spiritualized by such conceptions. We can constantly track the processes in the external world, such as fluctuations in the barometer. 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 point was to emphasize that what the Pythagoreans taught is not something outdated, but rather that it finds application precisely in today’s natural sciences. [In astronomy, the Pythagoreans applied their spatial conception of the finite and the infinite to the cosmos as well.]

[ 46 ] Every era can understand the various perceived things only in accordance with the state of the relevant science of perception—astronomy, for example, and so on—at that time. Research is conducted in the realm of experience. The Pythagoreans, too, had to reckon with [the ideas of their time]. They envisioned the Central Fire, which represents unity, 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 that found expression in numerical ratios. And what took place in the sphere below the Moon was a process of becoming irregular, a ceaseless struggle between the finite and the infinite. In the sphere above, however, this struggle is balanced into a great harmony. There, the celestial bodies move in regular orbits. As a result, 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; it unfolds as an eternal struggle. On Earth, harmony and disharmony are constantly alternating. Human beings stand within this struggle, since they are a unity. He seeks to regain his connection, having been torn away from the harmony of the world, by striving to return to harmony with the world through what the Pythagoreans call virtue. You can see that the Pythagoreans penetrate everything—from the lowest phenomena up to those of human life—with numerical concepts.

[ 47 ] Now, all that remains for us from the Pythagorean worldview are the most important things—namely, those elements of spiritual life to which the ethical sphere rises as the spirit endeavors to sink as deeply as possible into its own innermost being. The Pythagoreans also held certain views on this subject, which were derived from cosmological concepts.

[ 48 ] In seeking to establish a harmony that penetrated extraordinarily deeply into the world—one that addressed what separates human beings from the entire universe—the Pythagorean arrived at the concept of reincarnation and the interconnectedness of the various incarnations of all beings.

[ 49 ] This is a topic we will address in particular next time. I wanted to discuss the fundamental question that was only revealed after a long period of training through Pythagorean physics and ethics, which were shaped by the world of ideas just described. One must not imagine that the teaching of reincarnation or re-embodiment was conveyed 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 and 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.

Q&A:

Question: [?]

[ 51 ] Dr. Steiner’s answer: The way a person breathes is closely connected to the fact that a person can speak. If a person did not have a vertically positioned lung and a vertically positioned larynx, they could not speak. A dog could never speak words. In general, dogs are, after all, 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 positioned upright. The production of articulated sounds can only occur with a larynx and lungs positioned upright. Monkeys cannot produce articulated sounds, even if they can be made to walk somewhat upright. Their bodies are not built that way. The upright larynx inhales and exhales air in a distinctive manner. This inhalation and exhalation produces 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 beings that walk upright. Only when the body was able to walk upright could the mind take hold. Creatures that still use their front limbs to propel themselves forward cannot harbor a mind. During the Tertiary Period—specifically the Diluvial Era—gibbons lived among us; they were highly intelligent creatures. After the climate grew colder, they migrated, but then returned and were forced to live in colder climates where the plant life was not as abundant. They had to use their front limbs as tools, then gradually learned to walk upright, and the mind was then able to take possession of the brain.

Question: Does a squid look like a human?

[ 52 ] Answer: Kurd Laßwitz wrote a fairy tale that addresses this question. It is highly likely that what occurs in a human’s eye also occurs in an octopus’s eye. One makes the crystalline lens out of glass, the other out of rock crystal or a similar material, and the same thing occurs through the crystalline lens. In humans, the same image is formed in a different material than in an octopus. But even if the octopus has the same image as a human, the image must first be spiritualized and processed. Just because you see a circle outlined in red in your eye does not mean you have an object. It must be connected to other ideas. We do not know whether the octopus can do this.

[ 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 governed by laws. The indifference of matter is what physicists see as the conservation of energy or matter. What is used as an 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 turn one into the other, one would have to demand that one could see with one’s ears and hear with one’s 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 lose their sense of harmony so completely?

[ 55 ] [Answer:] The Pythagorean holds the view that there is no perfect harmony on Earth, but rather a process by which harmony emerges from 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 find the detached being that yearns to return to harmony. The celestial bodies as a whole follow 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 them. Christ says, “Where two or three are gathered together, I am in their midst.” This is a Pythagorean concept. And it is this: The Pythagorean sees the beginning in the One; in the Two, he sees the indeterminate being added; and the perfected being is present 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 number three. The number three has a center; the number two has only a juxtaposition. The number three must bring about balance. It is the spirit that connects them.

Question: [?]

[ 56 ] Answer: The Bible is composed of elements from all over the world. Nothing is easier than interpreting the Bible. With esoteric interpretation, one can get roughly the right idea, but one cannot know whether one has arrived at the historically [correct] interpretation.

Question: [2]

[ 57 ] Answer: Neptune would not be the last planet. [Dawes] already pointed that out in 1839. Kunowsky is a welcome addition.