The Rudolf Steiner Archive

4. The Pythagorean Doctrine

Ladies and gentlemen!

[ 1 ] Last time, I mentioned that I wanted to discuss the Pythagorean teachings. Pythagoras had founded a school in Southern Italy. It was less of a school and more of a community of disciples, with Pythagoras as their spiritual leader. He developed a set of teachings. We can no longer say how much of it belongs to Pythagoras and how much to his students.

[ 2 ] The Pythagorean worldview unfolds before us, and it reveals itself to be one of the most profound worldviews we have. Since it is very important to us to truly introduce you to the subject at hand, I would like—before quoting Pythagoras himself—to present a modern Pythagorean, one who lived in Germany and whose worldview always strikes me as a prelude to Pythagoras.

[ 3 ] In fact, one can understand this worldview much better if one is familiar with the works and views of Baron von Hardenberg—Novalis, a poet of a thoroughly mystical nature. No one familiar with his writings would doubt this.

[ 4 ] Take, for example, his “Apprentices at Sais.” This is something that can only be understood in its esoteric sense. But anyone familiar with Novalis’s personality—he was born in 1772 and died in 1801, having lived to the age of 29—will understand this. This Novalis seems to have remained the most innocent of youths throughout his life. He appears to us more as the revelation of an unearthly individuality than as an earthly personality. It is simply beyond comprehension that such depth, such absorption, could have been attained in such immense youth.

[ 5 ] When we read his Heinrich von Ofterdingen, we find that he drew upon direct sources—the sources of mysticism. He then incorporated these into his novel Heinrich von Ofterdingen, thereby demonstrating his understanding of twelfth- and thirteenth-century mysticism. If we consider his fundamental ideas, we will find a certain similarity to other mystics.

[ 6 ] He was searching for the “Blue Flower.” People have often mocked this “Blue Flower.” We will understand each other better if we recall Goethe’s “The Prophecies of Bakis,” where he speaks of the serpent’s coil and the flower, where he speaks of how a person can walk the path that is long and narrow. When a person walks this path, they see knots ahead of them. They also see the knot in which life gathers. Behind them, they drag a snake. The snake disappears, and the knot transforms before them into a flower.

[ 7 ] This image, which Goethe repeatedly invokes, is egoism—the approach to the highest spirituality or, indeed, the deepest insight. The “Blue Flower” serves as a symbol of this. It also symbolizes the complexities of life that arise for a person as they advance along the path of knowledge. It is this “Blue Flower” that Novalis has his character Heinrich von Ofterdingen envision.

[ 8 ] We also find this flower in the work of Master Klingsohr, who can prophesy. The future lies open before him. Goethe says: “To the one who truly surveys the past in its entirety, the future also lies open.” [...] — Master Klingsohr reveals the future to Heinrich von Ofterdingen. This satisfies him to such an extent that he is able to see the individualized Blue Flower in his daughter, since he has progressed far enough to perceive the Supreme in the female being.

[ 9 ] Mathilde dies, leaving Heinrich von Ofterdingen behind. He resolves to follow his beloved in death. For him, reality transforms into a dream. What he used to regard as a dream—the higher spiritual world—is now reality. He no longer finds this Supreme Being in a single individual, but finds it in other beings as well. He finds a second girl. To him, she is the same. In Cyane, he rediscovers Mathilde. She is like a new incarnation of her. He lives a life in the hereafter.

[ 10 ] We find the idea behind this in his The Apprentices of Sais. A beautiful fairy tale is woven into it about the boy Hyacinth, who loves the girl Roseblossom. Only the trees and birds of the forest know of this love. Then we find Hyacinth transformed. He is overcome by a longing to seek something deeper. He leaves Roseblossom without sufficient reason. Then he comes to the evil old man, who plants within him the longing to seek the Mother of All Things, or the Veiled Virgin. He sets out on a journey to the Temple of Isis, comes upon an image there, and when he unveils it, he finds nothing but Rosen[blüthe]. [Harmony revealed itself to him in one and the same being.] He finds his beloved as the solution to the riddle, as the veiled image at Sais.

[ 11 ] This is reminiscent of the higher interpretation of “Know thyself,” as he expressed it in an epigram. He stands before the veiled image at Sais. He lifts the veil and—miracle of miracles—he finds himself. A magical individualism consists in the ability to find the infinite within the finite, [to make the spirit an immediate reality].

[ 12 ] So in Novalis we undoubtedly find a mystical personality. If, then, we assume that we are dealing with a profoundly mystical nature in Novalis, and if we then get to know him, he does not appear to us as a mystic, as he has just been described, but as a reborn disciple of the Pythagoreans.

[ 13 ] When we let Novalis pass us by, when he then seems more like a memory, and when we then see how this breath of the earthly, how this personality, is nevertheless firmly rooted in life—possessing inclinations that we would least expect to find in such romantically inclined natures—then we are reminded of the Pythagoreans, as if they were fleeting ghosts.

[ 14 ] We must by no means equate this view and philosophical perspective—as we find it in his Romantic work—with the views of other Romantics, his contemporaries, who lack any depth. Friedrich Wilhelm Schlegel or Tieck, [E.T.A.] Hoffmann, and so on must not be confused [with him]. But anyone who allows Novalis’s work to take hold of them will not be led astray into such a confusion. What is astonishing about Novalis—despite his [poetic] nature—is that he is one of the most enthusiastic admirers of all things mathematical. He possesses a thoroughly cultivated, mathematical psyche—a direct revelation of what he calls the “magical” in nature. In this he finds the law of the spirit. That which those who wish to venture into the higher realms might wish to set aside is precisely what we find in Novalis as the central concern—the very thing that led him to emphasize the magical in his [idealism]. In the interconnection of fundamental mathematical concepts, he sees the most captivating revelation of the world’s mystery. He sees free matter at the very core of things. Mathematics is the foundation upon which existence rests; it is therefore nothing other than the highest form, the purest form of spirituality.

[ 15 ] If we find this to be the basis of his view, then he appears to us as a representative of Pythagoreanism. We can understand Pythagoreanism much better if we imagine it as Novalis does. One must imagine the Pythagorean soul in this way; then we arrive at the same position as Novalis; [just as] Pythagoras was able to arrive at the view that, in the relationship between numerical quantities and spatial dimensions, this harmony indeed embodies the fundamental structure, the fundamental essence, and the fundamental spirit of the universe.

[ 16 ] If we wish to gain insight, starting from the most elementary foundations, into a Pythagorean-minded soul, we must imagine it in the following way. The student was guided, step by step, toward the insights he was meant to attain. He was guided with great care. First came mathematical insights, then astronomical ones. Astronomy was, above all, a matter of mathematics. Regularity emerged in the numerical ratios within the cosmos. He was first introduced to these numerical ratios. Then he was gradually led on to an understanding of humanity itself. The fulfillment of the longing “Know thyself” [came] last. First, he was introduced to mathematics.

[ 17 ] How can one imagine that human beings could actually arrive at the idea that mathematics is the spiritual foundation of the entire universe? How can this be conceived as taking the form of harmony in space and time? When we delve into those realms of space and time that outwardly already exhibit a regular pattern—such as the movement of the celestial bodies—and when we immerse ourselves in them, then what we have in this structure of the celestial vault, which we construct in our minds, is essentially nothing other than mathematics made tangible, calculation made tangible.

[ 18 ] No one can actually find anything in the world or in reality that corresponds to a mathematical structure—a spatial arrangement of geometric figures—unless they have first formed these mathematical figures in their mind. If someone were to describe a circle or an ellipse, we would not know what it is—what object they are describing. We would be able to trace the line at various points in space and connect those points. But we would not be able to associate a concept with the entire line that describes the object unless we had already formed that concept. We can draw a star and then consider what kind of line the star describes. But we can only identify the figure once we already have it in our minds. The same is true for other things as well, even when we consider numerical relationships. We will recognize objects out in space in their specific mutual numerical relationships—in their numerical diversity—only if we have formed these relationships in our minds. If we know that 2 × 2 = 4, then we can also recognize this in the external world. We would be unable to connect any concepts with reality; we would be unable to grasp them at all; they would flit past us like nothingness, not even existing for us, if we had not formed these images in a purely mental way within our psyche.

[ 19 ] Thus, the Pythagoreans were able to say: What I see outside must also, in a certain way, be contained within my mind. What emerges from the source of my soul is the very same thing that I perceive outside as the very foundation of the world itself. The Pythagoreans reflected more deeply on this and said to themselves: It is impossible for two things that are completely separate from one another—the mind on the outside and the world on the inside—to [merely] exist side by side [without corresponding]. This correspondence would only have meaning if what is in the mind were exactly the same as what is out there in space. If the circle, the ellipse that I perceive within myself, and the numerical ratios are the same as those outside—which I see in the external world—then it makes no sense at all unless [the Pythagorean] has something within himself that he cultivates. If he sees the spirit of things and possesses it within himself, then that alone has meaning.

[ 20 ] Therefore, the Pythagorean did not initially think in the same way as the philosophers of the nineteenth century, who were influenced by Kant. He did not ask: How is it that my inner conception corresponds to the things outside? My experience is quite different. For me, there is an undeniable unity between what is outside and what is in my mind. That is how the Pythagorean thinks.

[ 21 ] It makes no difference whether I use the concepts of Pythagorean astronomy or apply the new ones. It doesn’t matter at all. So when the Pythagorean sees a celestial body tracing an elliptical path, it is an immediate experience for the Pythagorean that the ellipse he perceives within himself and the ellipse that exists outside as the path of a star are not two ellipses, but only one. And that is experience.

[ 22 ] Schelling also articulated this, and it clarifies the matter in the simplest way. He drew on the “force of attraction” that physicists have always [known] about. It was imagined that objects exert a force of attraction on one another. The Earth attracts the Moon, and the Sun attracts the Earth. If the Sun attracts the Earth, then it acts upon the Earth. It is difficult, then, to attribute an effect to a body where it is not at all. But the fact is this: if a body acts upon the Earth, then it is on the Earth. A body is where it acts. The boundary of light is not the boundary of the actual Sun. The Sun is present throughout the entire space where it exerts its gravitational pull. The space occupied by the Earth is part of the Sun’s space.

[ 23 ] You should imagine this Schellingian concept as [already] underlying Pythagorean doctrine. The human spirit fills the entire universe. It is not confined to a single organism. The spirit is wherever it perceives.

[ 24 ] For the nineteenth-century philosophers who follow Kant, the question is this: How is it that the mind perceives what is outside itself? — The Pythagorean does not ask at all: How is it that the mind perceives what is outside itself? The Pythagorean says: When the mind perceives an ellipse in the sky, it is a fact that the mind is not confined to the organism, that it is not where it perceives through the senses, but rather that it is where it perceives [mentally]. The boundary of the mind is not the sense; rather, the mind is where it perceives. — There is a separation between the numerical ratios existing in space and those that exist in our minds as numerical ratios—a separation that does not exist for the Pythagoreans. The Pythagoreans are unfamiliar with the notion that human beings are, first and foremost, sensory, finite beings, enclosed along with the psyche in a fabric that connects the senses to the external world. This gives modern people the illusion that the spirit, too, is enclosed within [a] vessel.

[ 25 ] While other philosophers take this to be reality and ask, “How is it that we perceive external things?”, the Pythagoreans see the matter the other way around. They do not ask: How is it that the mind is enclosed within such an organism? — It is perhaps better that I not say “individual” but rather “singular being.” This then leads to an understanding of a worldview such as the Pythagorean one. It leads to a conception that can only be grasped if one sees in mathematics what constitutes the fundamental structure of the universe—and what, when one conceives of the entire world as imbued with the mind, constitutes the fundamental structure of the mind itself.

[ 26 ] Thus, at the very foundation—deep down, on a lower level—of what is perceptible to the senses within the spacetime of the universe, we find, in the commonalities expressible through spatial dimensions and numerical ratios, that which appears to the spirit on a higher level. The spirit has a numerical, geometric foundation. The spirit has its origin where order prevails. The spirit grows out of the mathematically constructed world. Therefore, [the Pythagorean] seeks the primordial foundations of existence in the mathematically constructed world.

[ 27 ] I pointed out that there is a difference between the Greek worldview, as represented by Heraclitus, and the Pythagorean one. At the time, I structured my remarks in such a way that they came back to Goethe’s fundamental view. I said then that Goethe maintains that the seed and the plant are one and the same being. The material seed contains everything that is still within it, completely hidden. It is the same as the fully developed plant. Although the plant is not physically contained within it, the point is that, in a spiritual sense, the plant is the same in every form as it is in another form; thus, the plant—with its leaves and petals, with its entire fruit, and with everything within it—is to be regarded as the material, physical manifestation of what exists within the seed in an ideal form. Goethe therefore says that the seed is the entire plant, except that the spirit is still hidden within it. What is ideal in the seed becomes material reality in the entire plant.

[ 28 ] The same image can be applied to the entire world. One can understand the world by observing it in its highest state, by immersing oneself in its blossoms and fruits—in the human soul—and by studying the principle of “Know thyself” and applying it to human beings. Where the purely spiritual-soul-level then appears directly—that is, in deep contemplation, in direct immersion in the self—one can first seek a worldview, a philosophy of life. But one can also examine a seed. One can find ways and means to examine the seed. One might surmise that what lies within the seed is already hinted at there, and that the worldview derived from the human being is the highest. The Pythagoreans do not seek out the human being where he is soul, nor where he appears as spirit, but rather where he seemingly is not spirit at all, where he seemingly does not exist at all. Through neutral numbers, the Pythagorean seeks a certain reality. And that is why he seeks the spirit where he already knows the spirit to be. That is also why he finds in mathematics the primordial source, the fundamental structure of existence.

[ 29 ] I simply meant to say that this Pythagorean worldview can only be understood if one understands Novalis’s contemplation—which must be understood mathematically—of Novalis, who was, after all, thoroughly poetic in nature and, as such, was what literary history calls a “Romantic,” yet was so deeply rooted in such laws that he could regard rigorous mathematics as the primal source of existence. That is why the Pythagoreans, too—because their spirit was powerful enough—were able to find spirit in numerical ratios. They started from the lowest level of the spiritual. Just as a seed is not yet a plant but can become one, so they ascended from the seemingly non-spiritual to the spiritual.

[ 30 ] This is what can help us understand the entire Pythagorean worldview. The Pythagorean worldview is usually presented as if it were the numerical nature of the world that led the Pythagoreans to regard number as the origin of all things. And it is difficult to imagine exactly what they meant by that. I must admit that if we simply follow and read what is written in the textbooks—that the Pythagoreans regarded number as the origin of all things—it would seem meaningless to me. Only when I imagine how it actually is, when I assume that they grew up within an entirely different epistemology, can I understand what they meant. Their view is simply described by the following words: “The Pythagoreans did not seek the spirit where it appears to be a sensory entity, but rather where they perceive it—as something that fills all of space.”

[ 31 ] That is one aspect of the Pythagorean worldview; that is why they delved into numbers and geometric figures. On the other hand, the reason is also that they found something in these numbers and geometric figures that spoke to them as spirit.

[ 32 ] What do we mean by geometric or mathematical relationships? We cannot say that someone who can only imagine a circle or an ellipse when they are drawn on the blackboard has a concept of real geometric or mathematical relationships. If that person has to place five peas or beans on the table in order to imagine the number “5,” we cannot say that he has a concept of real numbers.

[ 33 ] We are, in fact, well aware that what we call a circle, what we call an ellipse, can only be represented approximately in material reality. We know that the material circle we draw is only an approximate representation of what we can create in our minds. We also know that the paths described by the celestial bodies in outer space are only approximately circular. However, the law that governs the unfolding of the universe is the same as the law that governs us when we imagine a circle in our minds, when we no longer need to model the spiritual after the physical. That is why mathematics would also be the best means of introducing us to the spiritual realm. That is also why the Pythagoreans placed the highest value on mathematics. Whoever truly wishes to understand the spirit must therefore be able to look beyond all that is physical. One must be able to realize that what one draws with chalk on the blackboard is not a real circle, but rather what remains in the mind without the chalk drawing on the blackboard. Using a cube of salt, one could demonstrate that the cube is something entirely different from the [salt] cube. In this way, students could be shown that the spiritual aspect—including that of other things—can only be grasped when the sensory aspect is set aside. This is easy to demonstrate with the cube of salt. The spiritual content is not the same as the external cube.

[ 34 ] But when we understand this in relation to the entire sum of worldly phenomena—when we understand that the spiritual can be separated from the material—this leads us upward to higher levels. Everyone admits that mathematics has nothing to do with the things of the world, but rather with the spiritual. But when this goes further, people confuse the spirit with reality

[ 35 ] Just recently, a curious document has been published that exemplifies the confusion of the mind with reality. A book has been published under the title Critique of Language by Fritz Mauthner, in which he seeks to show how all our knowledge hangs in the air, how nothing is given to us except the sensory world, and that when we look beyond the sensory world, we have nothing left in our world of ideas but empty words.

[ 36 ] Well, ladies and gentlemen, this is something that someone who is unable to grasp the essence of things at a higher level of reality—as one can do with mathematical constructs—can very easily fall into. Anyone who lacks intuition, who does not truly possess—from the very source of his spirit—what he needs to set against things, who is sterile and barren, who cannot fill his soul with spiritual realities, believes that when he goes beyond [the sensory world], he has nothing more than words. Instead of a “Critique of Knowledge,” he writes a “Critique of Language.”

[ 37 ] The book consists of two volumes. It strikes me as if someone wanted to write a critique but did not have a firm grasp of what he intended to critique. He confuses what the mind adds to the forms. What Mauthner offers—compared to what spiritual content is capable of and ought to provide—would amount to a critique of pencil drawing. It demonstrates just how well a pencil can draw circles. Thus, sterile views cling to those who are unable to grasp the true content. He does not know that the spirit gradually acquires the ability to ascend into the higher realms of existence, and that at every stage of spiritual life it is conscious of its distinctiveness from material things, just as the mathematician is able to detach the spiritual and the soulful from things, thus advancing from that which is not yet spirit at all to the immediate presence of God in the world.

[ 38 ] This was something the Pythagoreans sought to achieve step by step by attempting to guide the student from the lower to the higher. They were convinced that, by ascending from the lower to the higher, a person did not merely have an experience within themselves, but fulfilled a task in the universe itself. They were convinced that human beings contribute something to the world; they were so convinced of this that they even compared this ascent to the numerical ratios themselves. They said to themselves: The individual human being who perceives is seemingly a duality—the perceiver and the perceived. For the Pythagoreans, these two great opposites stood at the foundation of their ladder of knowledge.

[ 39 ] But they said to themselves: All this is only apparent because human beings do not stand on the highest level of perfection, but on the lower levels. The perceiver and the perceived must be transcended if they are to become one. Thus, the Pythagorean imagines that, just as in human knowledge today, unity triumphs over duality—over the separated elements in the world—and that the Pythagorean must conceive of everything according to numerical ratios, and specifically in such a way that what is a duality when separated appears to him as a unity.

[ 40 ] Now, the Pythagorean is convinced that the entire diversity of the world—the fact that there are many things in the world—stems solely from the fact that human beings initially see appearance, not the thing itself; that they do not see things as they are, but rather see them as they are not, due to the limitations of their own existence. He sees that when he overcomes this appearance, this multiplicity then reveals itself in reality, in truth, as unity. What human beings ultimately attain is the primordial unity, the primordial One of the world, and the Pythagorean regards this at the same time as the foundation from which everything springs.

[ 41 ] This is what enables human beings to perceive something in space. This is the universal unity of the world, to which, however, human beings can only ascend gradually. What is revealed last is what was there first, precisely because it is a part of this diversity. After having been set aside for a time, it integrates itself into the structure of the world and becomes one with the harmony of the world. The numerical harmony, the geometric regularity of the world picture, encompasses human beings as well. And so they find it by integrating themselves into the structure of numbers. That is why the Pythagoreans can say that all that is good, all virtue, consists in human beings overcoming appearances and finding the numerical, geometric regularity, through which they integrate themselves into the great existence of the world.

[ 42 ] In this way, human beings appear to themselves as a note in a harmony, and because they appear to themselves as a note in a harmony, they must establish the right note and the right relationship. They do not fulfill a task for their own sake, but rather fulfill a moral task. If they do not fulfill it, then they are not in the correct numerical proportion. They have something to contribute not to themselves, but to the entire structure of the world. Through every transgression, human beings take on unlimited responsibility, and, recognizing this, they should strive more and more to attain the mood they are meant to fulfill in the great music of the world.

[ 43 ] Thus, to the Pythagorean, what is spread out in space and time appears as a moral task in and of itself. For the Pythagoreans, the moral task is not to be understood as a mathematical one at a higher level. The mathematical task is for him to discover the universe, but in such a way that he encompasses himself within it, that he is integrated into it like a note in the music of the universe, like a number in the laws governing numbers. He then discovers that when he does something—because he is not merely his own savior—this is not merely significant for himself, but is something that concerns the entire universe. The spirit is not only within me, but also wherever it acts. He then realizes: The spirit must not only work toward its own moral perfection, but must also work toward the harmonization of the entire universe. When the Pythagorean conceives of the harmony of the universe in such a way that he imagines the world permeated by musical tones, by the music of the spheres analogous to music itself, this is because music is based on tonal relationships.

[ 44 ] The Pythagorean conveys this by saying: Just as the ratios of tones are perceptible to our senses as a harmony of sounds, so too is there a harmony of sounds—a music of the spheres—in the world, which functions like the numerical ratios in the world. But if he does not find within himself the correct numerical ratio, the correct tonal ratio to the world, then he disturbs the harmony of the world.

[ 45 ] Therefore, the Pythagoreans’ insights must have led to the most rigorous educational system. The Pythagorean is aware that when he teaches an individual this or that, he takes on a responsibility not only toward that person, but toward the entire universe.

FAQ:

[ 46 ] Everyone is capable, through their unique disposition, of attaining spiritual knowledge. The Pythagoreans strove to make this possibility available to everyone.

[ 47 ] [Mathematical concepts are easy to prove only because they are simple, with almost no substance.]

[ 48 ] But for those who are not naturally suited to delving into the content of the world, the best and surest path will be to study mathematics. Plato therefore required his students to have a thorough knowledge of mathematics. Otherwise, it might not have been possible for everyone. I would like to illustrate the case of someone who has gone through the Pythagorean school as follows: Let us imagine a person who can only feel. Such a person could perceive geometric figures and also arrive at a conception of numbers. In fact, blind and deaf people have been taught these concepts and have become accomplished mathematicians. Such a person can also arrive at music through mathematical means. The numerical ratios appear to him only in a shadowy way. Now let us imagine that such a person suddenly gains the sense of hearing. He will then perceive the same thing he had previously grasped. He now perceives it with his ears. The same is true for the blind person. Through an explanation of the vibrations of the world, he can gain a conception of colors through the numerical ratios. The Pythagorean, however, is also supposed to awaken the higher senses. It is the same as when a mathematician approaches a composer who is constructing his own work and calculates the matter for him. Then the composer can say: “Let that be far from me.” If one possesses the necessary receptivity, one can have perceptions even without the mathematical representation.

[ 49 ] I have contrasted two schools of thought. One school within Hellenism, originating with Heraclitus, and the other, originating with Pythagoras. Heraclitus and Pythagoras stand before us as two who share the same subject matter. Heraclitus, as it were, is the composer; Pythagoras is the one who mathematically verifies his work. It is the same with us as it is in Pythagoreanism. One must first teach the blind and the deaf, and then one can lead them to higher levels.

[ 50 ] Mathematical concepts devised by humans are often confirmed in the external world. In the case of electricity, one calculates that this or that must be a certain way. When one then carries out an experiment to test this in reality, the results must agree with the calculation.

[ 51 ] I would like to quote a famous conversation between Schiller and Goethe here. Goethe and Schiller were leaving a lecture on the natural sciences together and struck up a conversation about what they had heard. During the conversation, Goethe took a piece of paper and drew a symbolic plant—an ideal plant—saying: “This plant is actually present in every plant.” Every plant is, in fact, an individual manifestation of this universal plant.” Schiller replied, “Yes, but that’s just an idea!” To which Goethe responded, “Then I see my ideas with my own eyes.”

[ 52 ] [Or let’s take a] triangle [it is presumably drawn]: The angles add up to 180 degrees. Having seen a triangle, we can form a quadrilateral by connecting the blue and green sides. This can be extended in the mind. We can move from the triangle to the quadrilateral. But we cannot move from one shade of color to another. What belongs to the sensory world, we can perceive only through the senses. In mathematics, the spiritual is easiest to grasp. Mathematics is the most spiritual.

[ 53 ] Don't you know how to perceive the notes from the numerical ratios? Tones are not perceived [with the ears], but only conceived. Composers who become deaf therefore have only a surrogate. It is as if we were inferring one mathematical structure from another. It is not [sensory] perception, but a mental experience.

[ 54 ] The sensual is transformed [into the spiritual]; it is elevated.

[ 55 ] It is not the study of mathematics that matters here, but rather the recognition of the essence of mathematics. Even the most superficial person merely dabbles and splashes about in its primordial essence. Yet even such a person may have studied mathematics. Goethe studied very little mathematics. But no one understood the essence of mathematics better than he did. Goethe arrived at his magnificent world of metamorphoses precisely because he had such a magnificent conception of the essence of mathematics, even though he was only able to apply it to the [gap in the notes] theorem.

[ 56 ] Someone who can make razors may not be able to shave, and someone who can shave usually cannot make razors. Similarly, a mathematician who knows mathematics [only] in terms of its form does not need to know its meaning or its application to the primordial being.