The Rudolf Steiner Archive

4. The Pythagorean Doctrine

Highly Esteemed Attendees!

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

[ 2 ] The Pythagoreans’ worldview emerges before us, and it reveals itself to us as one of the deepest worldviews we possess. Since it is very important to us to truly introduce you to the matters at hand, I would like, before I introduce Pythagoras himself, to present a modern Pythagorean, a Pythagorean who lived in Germany himself and whose worldview always strikes me as a prelude to Pythagoras.

[ 3 ] One understands this worldview much better, in fact, if one is familiar with the works and views of Baron von Hardenberg—Novalis, a poet of a thoroughly mystical nature. No one who knows his writings will doubt this.

[ 4 ] Take his “The Disciples of Sais.” This is something that can only be understood in its esoteric meaning. But anyone familiar with the personality of Novalis—he was born in 1772 and died in 1801, having lived to the age of 29—will grasp 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 incomprehensible that such depth, such contemplation, could have been attained in such immense youth.

[ 5 ] When we read his “Heinrich von Ofterdingen,” we find that he drew from immediate sources, from the sources of mysticism. He then incorporated these into his novel “Heinrich von Ofterdingen,” thereby demonstrating that he understood the mysticism of the twelfth and thirteenth centuries. If we consider his fundamental ideas, we will find a certain similarity with other mystics.

[ 6 ] He sought the “Blue Flower.” People have often mocked this “Blue Flower.” We will understand each other better if we recall Goethe’s “Prophecies of Baki,” 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 before them. They also see the knot in which life gathers. Behind him, he drags a snake. The snake disappears, and the knot transforms before him into a flower.

[ 7 ] This image, which Goethe repeatedly invokes, represents egoism, the approach to the highest spirituality or deepest insight. The “Blue Flower” serves as a symbol for this. It also symbolizes what presents itself to humanity as the entanglement of life when one advances along the path of knowledge. It is this “Blue Flower” that Novalis envisions for his Heinrich von Ofterdingen.

[ 8 ] We also find this flower in the figure of Master Klingsohr, who possesses the gift of prophecy. The future lies open before him. Goethe says: For those who truly and completely survey the past, 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 advanced so far that he can perceive the Supreme in the female being.

[ 9 ] Mathilde dies, leaving Heinrich von Ofterdingen behind. He resolves to follow his beloved in death. Reality transforms into a dream for him. What he was previously inclined 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 the same in other beings as well. He finds a second girl. She is the same to him. He finds Mathilde again in Cyane. She is like a new incarnation of her. He lives a life of the beyond.

[ 10 ] We find the idea of this in his “The Disciples 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 changed. 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]. [He saw the correspondence in one and the same being.] He finds the beloved as the solution to the riddle, as the veiled image at Sais.

[ 11 ] This recalls the higher conception 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 fact that one can find the infinite in the finite, [that one can make the spirit an immediate reality].

[ 12 ] Thus, in Novalis we undoubtedly find a mystical personality. If we therefore assume that we are dealing with a deeply rooted, 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 revived disciple of the ancient Pythagoreans.

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

[ 14 ] We must by no means equate this view and philosophical perspective, as we find it in his Romanticism, 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 to work upon them will not be led 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 educated, 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 ascend to higher realms would prefer to set aside, we find precisely in Novalis as the central matter, as that which 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 root 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 foundation of his view, then he appears to us as a representative of Pythagoreanism. We can understand Pythagoreanism much better if we have a mental image of it as Novalis does. One must have in one's mental image 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 actually constitutes the fundamental structure, the fundamental essence, the fundamental spirit of the universe.

[ 16 ] If we wish to gain insight into a Pythagorean-minded soul starting from the first elementary principles, we must create a mental image of it in the following way. In a step-by-step sequence, the student was led up to the insights he was to attain. He was guided in a very careful manner. The first were the mathematical insights, the second the astronomical ones. Astronomy became, above all, 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 the knowledge of the human being 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 mental image that mathematics is the spiritual foundation of the entire universe? How can this be conceived as harmony, formed in space and time? When we delve into those realms of space and time that already exhibit a regular pattern outwardly—such as the movement of the celestial bodies—when we delve into that, then in this structure of the celestial vault that we construct in our minds, we have essentially nothing other than embodied mathematics, embodied calculation.

[ 18 ] No one can actually find anything of a mathematical structure, of a spatial structure of geometric figures, in the world and in reality, 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 they are describing as an object. We would be able to trace the line through the 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 look at a star and then think about what kind of line the star describes. But we can only find the figure if 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 the objects out there 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 it out there in space. We would not be able to connect any concepts with reality at all; we would not be able to grasp them; they would flit past us like nothingness, not even exist for us, if we had not formed the images in a purely mental way within our psyche.

[ 19 ] It is therefore the case that the Pythagoreans could say: What I see outside must also, in a certain sense, 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 outside and the world within—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 behold in the external world, then it makes no sense at all if [the Pythagorean] does not have 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 under the influence of Kant. He did not ask: How is it that my mental image corresponds to the things outside? My experience is quite different. For me, there is the unquestionable unity of what is outside and what is in my mind. That is how the Pythagorean thinks.

[ 21 ] It makes no difference whether I take the mental images of Pythagorean astronomy or apply the new ones. It doesn’t matter at all. So when the Pythagorean sees a celestial body tracing a path in the shape of an ellipse, 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 expressed this, and it clarifies the matter in the simplest way. He drew on the “force of attraction” that physicists have always [known]. A mental image was created of objects exerting a force of attraction on one another. The Earth attracts the Moon, the Sun the Earth. When the Sun attracts the Earth, it acts upon the Earth. It is difficult to attribute an effect to a body where it is not at all. But the fact is: 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 force. The space that the Earth occupies is part of the Sun’s space.

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

[ 24 ] For nineteenth-century philosophers who follow Kant, the question is this: How is it that the spirit perceives what is outside of itself? — The Pythagorean does not ask: How is it that the spirit perceives what is outside of 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 with the senses, but that it is where it perceives [mentally]. The boundary of the spirit is not the sense; rather, the spirit is where it perceives. — There is a separation between the numerical relationships existing in space and those existing in our minds as numerical relationships, a separation that does not exist for the Pythagoreans. The Pythagoreans do not have a mental image of human beings as primarily sensory, finite beings, enclosed with the psyche in a fabric that connects the senses to the external world. This gives rise to the illusion for modern people that the mind, too, is enclosed within [a] shell.

[ 25 ] While other philosophers take this as reality and ask, “How is it that we perceive external things?”, the Pythagoreans view the matter the other way around. They do not ask: How is it that the spirit is enclosed within such an organism? — It is perhaps better that I do not say “individual” but “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 the entire world as filled with spirit—constitutes the fundamental structure of the spirit itself.

[ 26 ] Thus, at the very foundation of what lies deep below—on a lower level, perceptible to the senses—in the spacetime of the universe, we find, in 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 things proceed in a regular manner. 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 have 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 so that they returned to Goethe’s fundamental view. I said there that Goethe states that the seed and the plant are one and the same being. The material seed contains everything that is still within it, in complete concealment. It is the same as the fully developed plant. The plant is not actually contained within it, but the point is that, in a spiritual sense, the plant is the same in every form as it is in another form, so that 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 is present in the seed in an ideal form. Goethe therefore says that the seed is the whole plant, only that the spirit is still hidden behind it. What is ideal in the seed becomes material reality in the whole plant.

[ 28 ] The same image can be applied to the whole world. One can understand the world by observing it in its highest state, by immersing oneself in its bloom and fruit, in the human soul, by studying the “ Know Thyself” and applies it to humanity. Where the purely spiritual-soul-related then appears directly—that is, in deep contemplation, in the direct immersion into 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 the human being where he is soul, nor where he appears as spirit, but where he seemingly is not spirit at all, where he seemingly is not 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 why he also finds in mathematics the primal source, the fundamental structure of existence.

[ 29 ] I merely wanted to say that this worldview of the Pythagoreans can only be understood if one understands the contemplation of Novalis, which must be understood mathematically—of Novalis, who was indeed of a thoroughly poetic nature and, as such, was what literary history calls a “Romantic,” yet was rooted in such laws that he could regard rigorous mathematics as the primal source of existence. That is why the Pythagoreans, 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 the 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. Usually, the Pythagorean worldview is presented as if it were the numerical aspect of the world that led the Pythagoreans to regard number as the origin of things. And one cannot quite form a mental image of what they meant by this. I must admit that if we follow and read what is written in the textbooks—that the Pythagoreans regard number as the origin of all things—it would seem meaningless to me. Only when I have a mental image of how it really is, when I assume that they grew up in a completely different epistemology, can I understand what they meant. Their view is simply described by the words: The Pythagorean did not seek the spirit where it is apparently a sensory entity, but where he perceives it, as something that fills the entire space.

[ 31 ] This is one aspect of the Pythagorean worldview; this is the reason why they descended to numbers and geometric figures. On the other hand, the reason is also that they found in these numbers and geometric figures something they could address as spirit.

[ 32 ] What do geometric or mathematical relationships mean? We cannot say that someone who can only form a mental image of a circle or an ellipse when they are drawn on the blackboard has a conception of real geometric or mathematical relationships. If they have to place five peas or beans on the table in order to imagine the number ‘5’, we cannot say that they have a mental image of real numbers.

[ 33 ] Rather, we are 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 what the celestial bodies describe in outer space is only approximately a circle. However, the law that governs the unfolding of the universe is the same as the law that governs us when we create a mental image of a circle in our minds, when we no longer need to copy the spiritual from the physical. That is why mathematics would also be the best means of introducing us to the spiritual. That is why the Pythagoreans also placed the highest value on mathematics. So whoever truly wishes to know the spirit must be able to look beyond all the physical. One must be able to realize that it is not what one draws with chalk on the blackboard that is a real circle, but rather what remains in the mind without the chalk drawing on the blackboard. Using the salt cube, one could demonstrate that the cube is something entirely different from the [salt] cube. In this way, students could be shown that the spiritual—including that of other things—can only be grasped when the sensory is set aside. This is easy to demonstrate with the salt cube. The spiritual content is not the same as the outer cube.

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

[ 35 ] Precisely in our day, a remarkable document of the confusion of the spirit with reality has appeared. A book has been published under the title “Critique of Language” by Fritz Mauthner, in which it is intended to be shown how all our knowledge floats in the air, how nothing is given to us except the sensory world, and if we disregard the sensory world, we have nothing left in our world of ideas but empty words.

[ 36 ] Well, ladies and gentlemen, this is something that can very easily happen to someone who is unable to detach the spirit of things from a higher level of reality, as one can do with mathematical constructs, can very easily fall into. Whoever lacks intuition, whoever does not truly possess, from the very source of his mind, what he must hold up against things, whoever is sterile and barren, whoever cannot fill his soul with spiritual realities—such a person 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 comprises two volumes. It strikes me as if someone wanted to write a critique but does not master what he intended to critique. He confuses what the spirit adds to the concepts. What Mauthner offers—compared to what spiritual content is capable of and ought to provide—would be a critique of pencil drawing. It demonstrates how much the pencil is capable of depicting circles. Thus sterile views cling to the one who is unable to sense the true content. He does not know that the spirit gradually acquires the ability to ascend into the higher realms of existence, and is aware at every stage of spiritual life of the difference 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 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, in ascending from the lower to the higher, the human being did not merely have an experience within themselves, but fulfilled a task within the universe itself. They were convinced that he contributes something to the world; they were so convinced of this that they even compared the ascent itself to numerical ratios. They said to themselves: The individual human being who perceives is seemingly a duality—the perceiver and the perceived. These two great opposites stood at the foundation of the Pythagoreans’ table of knowledge.

[ 39 ] But they said to themselves: All of this is only apparent because humanity does not stand on the highest level of perfection, but on the lower levels. The perceiver and the perceived must be overcome, if they are to become one. Thus, the Pythagorean imagines that, just as in human cognition today, unity triumphs over duality, over the separated in the world; the Pythagorean must create a mental image of everything according to numerical ratios and, specifically, in such a way that what is a duality when separated appears to him as 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 this multiplicity, once he overcomes the appearance, then presents itself in reality, in truth, as unity. What man ultimately attains 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 humans to perceive something in space. This is the universal unity of the world, to which, however, humans can only ascend gradually. What is revealed last is there first, precisely because it is a link in this diversity. After having been set aside for a time, it integrates itself into the structure of the world, becoming one with the harmony of the world. The numerical harmony, the geometric regularity of the world picture, encompasses human beings as well. And so he finds it by integrating himself into the structure of numbers. Hence the Pythagorean can say that all good, all virtue, consists in man overcoming illusion and finding the numerical, geometric regularity, through which he integrates himself into the great existence of the world.

[ 42 ] Through this, human beings appear to themselves as a note in the harmony, and because they appear to themselves as a note in the harmony, they must give themselves the right note and the right proportion. They do not fulfill a task for their own sake, but fulfill a moral task. If he does not fulfill it, then he is not in the correct numerical ratio. He has something to contribute not to himself, but to the entire structure of the world. Through every transgression, the human being takes on unlimited responsibility, and, recognizing this, he should strive more and more to attain the mood he is meant to fulfill in the great music of the world.

[ 43 ] Thus, to the Pythagorean, what is spread out in space and time appears itself as a moral task. For the Pythagoreans, the moral task is not to be understood as a mathematical one at a higher level. The mathematical task is that he discovers the cosmic space, but in such a way that he encompasses himself within it, that he is to be integrated into it like a tone in the music of the universe, like a number in the law of 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 where it acts. He then sees: The spirit must not only work on its moral perfection, but it must work on the harmonization of the entire universe. When the Pythagorean has a mental image 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 happens because music is based on tonal relationships.

[ 44 ] The Pythagorean conveys this by saying: Just as tonal ratios become perceptible to our senses as a harmony of tones, so too is there a harmony of tones, a music of the spheres, in the world that acts 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 insights of the Pythagoreans must have led to the strictest 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.

Answering Questions:

[ 46 ] Everyone is capable, through their unique disposition, of attaining spiritual knowledge. The Pythagoreans strove to create this opportunity for everyone.

[ 47 ] [Mathematical mental images are easy to prove only because they are simple, almost devoid of content.]

[ 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 this for those who have gone through the Pythagorean school as follows: Let us imagine a person who can only feel. Such an organism could perceive geometric figures and also arrive at the mental image of numbers. In fact, blind and deaf people have been taught these relationships and made into accomplished mathematicians. Such a person can also arrive at music through mathematical means. The numerical relationships present themselves to him only in a shadowy manner. Now let us imagine that such a person were to suddenly hear. He would then perceive the same thing he had previously grasped. He now perceives it with his ears. It is the same with the blind person. Through an explanation of the vibrations of the world, he can form a mental image of colors through the numerical relationships. But the Pythagorean is also supposed to awaken the higher senses. It is the same as when a mathematician comes to a composer who is constructing his own work and calculates the matter for him. Then the composer can say: “Leave me alone with that.” If one has the necessary receptivity, one can have perceptions even without the mathematical representation.

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

[ 50 ] Mathematical constructs devised by humans often find confirmation in the external world. In the case of electricity, one calculates that this or that must be so or so. When one then carries it out in reality as an experiment, it must correspond [to the calculation].

[ 51 ] I would like to cite here a famous conversation between Schiller and Goethe. Goethe and Schiller left a lecture on natural science together and struck up a conversation regarding 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 actually an individual manifestation of this general plant. To this, Schiller replied: Yes, but that is only an idea! To which Goethe answered: Then I see my ideas with my eyes.

[ 52 ] [Or let us take a] triangle [it is presumably drawn]: The angles together measure 180 degrees. By having seen a triangle, we can form a quadrilateral by connecting the blue side with the green one. This can be extended in the mind. From the triangle we can move on to the quadrilateral. But we cannot move from one shade of color to another. What belongs to the sensory world, we can only perceive sensually. In mathematics, the spiritual is the easiest to grasp. Mathematics is the most spiritual.

[ 53 ] You don’t know how to perceive tones from numerical ratios? Tones are not perceived [with the ears], but only thought. Composers who go deaf therefore have only a surrogate. It is like when we infer one mathematical structure from another. It is not [sensory] perception, but a spiritual experience.

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

[ 55 ] Studying mathematics is not what matters here, but rather the recognition of the essence of mathematics. The most superficial person merely muddles and splashes about in the primordial essence. Yet even such a person may have studied mathematics. Goethe studied little mathematics. But no one has understood the essence of mathematics more than he did. Goethe arrived at his magnificent world of metamorphoses precisely because he had such a magnificent mental image of the essence of mathematics, even though he was only able to bring it down to the [gap in the transcript] theorem.

[ 56 ] Someone who can make razors may not be able to shave, and someone who can shave usually cannot make razors. Thus, the mathematician who knows mathematics [only] in form need not know its meaning and its application to the primordial essence.