The Rudolf Steiner Archive

15. The Formation of Concepts and Hegel’s Theory of Categories

[ 1 ] Please consider these lectures on philosophy—of which today’s is one—merely as an episode, an interlude. They are intended, so to speak, to build a bridge between the anthroposophical worldview and the purely philosophical one. I would like to structure today’s lecture in such a way that, through individual remarks appended to the exposition, you will be able to see how the bridge between philosophy and anthroposophy can be built, and how certain philosophical insights and concepts can actually become important to the anthroposophist when he or she enters into practice. Let me begin by stating something that will be useful to us in properly relating the entire philosophical conceptual framework to what is communicated to us through supersensible experience on the path of anthroposophy.

[ 2 ] As a sort of preparation for today’s remarks, you have had the opportunity to attend the lectures on elementary logic that I delivered during the General Assembly session, and many of you have, of course, been well prepared by the course on epistemology and philosophical thinking taught by our dear Mr. Walther. There we recognized thinking as the capacity to engage with the world through a technique of concepts; we characterized it in a certain sense when we attempted to gain an understanding of purely formal logic. We saw there that we can speak of true thinking only when it proceeds in concepts, and at that time we made a strict distinction between perception, representation, and concept. If such a distinction should seem difficult to those who otherwise participate very willingly and with dedication in anthroposophical discussions, one should first consider that, on the foundation of anthroposophy, there must arise what we must recognize as a commitment to rigorous spiritual work—work that is not satisfied with concepts picked up at random, but seeks to ascend to sharp and energetic conceptual contours.

[ 3 ] We have come to understand the concept itself as something that is constructed purely within our own minds, and we have realized that this construction of the concept is a reality, a truth; that all philosophical discussions that view the concept merely as a shade—arising through abstraction—of what we gain as a representation come to a halt halfway, or perhaps a quarter of the way. The concept is not something derived from the representation through abstraction; rather, the concept is something—as was made clear through mathematical figures, such as the concepts of the circle and the triangle—that is arrived at through an inner construction.

[ 4 ] To gain an understanding of the nature of the concept and the conceptual system—the organism of our concepts—let us imagine the relationship this conceptual world has, on the one hand, to the world of sensory perception that surrounds us, and, on the other hand, to the reality that comes to us through supersensible observation in anthroposophy. You can picture the structure, the network of concepts that human beings possess—ranging from mathematical quantities and numerical concepts all the way to the complex concepts that Goethe began to explore in his Metamorphosis, but which are still in their infancy in our Western culture— You can imagine this entire web of concepts as a barrier that forms the boundary between the sensory world on one side and the spiritual world on the other. Thus, we can conceive of ourselves as bounded precisely by this web of concepts: on one side, the sphere of the supersensible, and on the other, the sphere of sensory reality.

[ 5 ] If human beings, as sensory observers of things, were to direct their eyes or other organs of perception solely toward the external environment, they would experience nothing but representations. This had been demonstrated using the following example: If a person were to sail so far out to sea that he saw nothing around him but the surface of the sea and an apparent hemisphere of the sky resting upon that surface, then he would have derived the circle surrounding him—which he perceives as the horizon—through external perception; he would have formed the idea of the circle through external perception. If, on the other hand, they have no such external perception but merely construct in their mind the image that arises when all points on a line are equidistant from a fixed point—the center—then they have—in contrast to the mental image—the concept of a circle. In this way, we could also construct other mathematical concepts internally—for example, the concept of the square, the triangle, the quadrilateral, the ellipse, the hyperbola, and so on. We could go even further, and we could finally rise to a true understanding of Goethe’s morphology, to the concepts of organic form, to the primordial animal, to the primordial plant—concepts that arose in the same way as the concept of the circle and which, as Goethe says, can be applied just as effectively as mathematical formulas.

[ 6 ] When a person approaches sensory reality in this way, he will find that this sensory reality corresponds to the concept he has constructed for himself. For example, they may find that their internally constructed concept of a circle coincides with the circle that presents itself to sensory observation when sailing out to sea. They then begin to understand what is presented to them in perception in comparison to what they have formed for themselves as a concept. Concepts are therefore not derived from perception. This is a prejudice that is very widespread today. Concepts are derived through inner construction. The concept is, so to speak, what a person arrives at precisely when he turns away from all external, sensory reality. And now he can bring into interaction what he has constructed internally with what presents itself to him externally as sensory reality. With this, we have established the relationship of the network of concepts to external, sensory reality. But now we must also ask ourselves: What is the relationship of our network of concepts to supersensible reality? — At first glance, it is no different than with sensory reality. If someone—through the methods of clairvoyance frequently discussed here—opens themselves to supersensible reality and now approaches this reality with their concepts, they will likewise find this conceptual network to coincide with supersensible reality. In exactly the same way, the supersensible facts and beings will act upon his conceptual framework—only from the other side—and he will find it to coincide with them. Thus we can say: In a sense, the supersensible realities cast their rays upon the conceptual framework, just as, on the other side, sensory reality does. Sensory and supersensible reality meet at the conceptual framework.

[ 7 ] This does not yet answer the question of where the network of concepts itself comes from within our soul. Today we will, so to speak, treat this question as a given, for the answer to it can really only emerge through patiently following the logical-metaphysical path that we may yet be able to walk together, if we are able to continue these lectures. We will then delve deeper and deeper into the supersensory reality. Today we want to use an image to clarify where this web of concepts—which human beings know they weave, so to speak, inwardly in the mind—where this web of concepts actually comes from. We can best understand this by imagining the image of a shadow cast onto a wall. When you see that a hand casts a shadow on the wall, you will say: If the hand were not there, the shadow would not arise either. The shadow resembles its original form, but it has a peculiar characteristic—it is actually nothing! For it is precisely because the hand blocks the light—by allowing non-light to take the place of light—that the shadow arises. Thus, the shadow image arises through the hand’s obstruction of the light. This is exactly how our concepts actually arise. We merely think that we spin them out of ourselves. They arise because the supersensible reality stands behind our thinking soul and casts its shadow images upon that soul. And the concept is actually nothing other than the projection of the supersensible reality onto the wall of our soul. And because our concepts resemble the archetypes of the supersensible world—just as the shadow of the hand resembles its archetype—concepts are something that can evoke in a person a sense of the supersensible realities. The fact that people believe they spin the web of concepts out of themselves stems from the fact that they initially have no direct perception of this supersensible world. But it is there and at work; it casts its shadows. Wherever it encounters the perception of the sensible world, these shadows arise, and the concepts are nothing other than these shadows. Thus, we do not have a supersensible reality in the concepts, any more than we have the hand itself in the shadow of the hand, but we have, so to speak, shadows of it. We have thus defined the network of concepts, so to speak, as the boundary between sensory and supersensory reality, while at the same time recognizing that concepts flow into the soul not from the sensory world but from the supersensory world. Such is the fact.

[ 8 ] Now we must ask ourselves: How can human beings actually arrive at true concepts, even if they have no insight into supersensible reality? If they had only external, sensory reality and were to contemplate it, they would arrive only at images, never at concepts. Concepts must be constructed within the soul and added to the images provided by external, sensory reality. It is entirely possible to live in concepts and yet not ascend into the supersensible reality. The seer, however, who can ascend into the supersensible reality, can in fact more easily arrive at a complete world of concepts, because he comes to know the forces that flow in and give rise to the concepts. You will find the spiritual-scientific basis for what I am now saying in philosophical terms in my book Theosophy, where the Devachan is discussed and the devachanic life of the shadow-images is dealt with in detail. A person arrives at the web of concepts by literally allowing the concepts to flow down upon them. For the seer, it is the case that he can look up to the archetypes, where reality lies. How, then, is it possible for a person who [cannot himself look up to supersensible reality] to create a network of concepts for himself? After all, the majority of people have only arrived at pure concepts through mathematics. Most people believe that one can arrive at concepts only by having perceptions, deriving mental images from them, subtracting perception from the mental image, and then finding the general, the concept—that is, through abstraction. This, of course, is not how a concept arises. Even independent thinkers are unclear about how concepts are formed.

[ 9 ] When I attempted in my Philosophy of Freedom to clarify the necessity of constructing concepts, I had a very peculiar experience. In my Philosophy of Freedom, drawing on Spencer’s ideas in a contrary way, I explain how it is a wholly inadequate philosophical approach to attempt to form the concept of a concept solely on the basis of the external, sensory world of facts.

[ 10 ] “The concept cannot be derived from observation. This is evident from the fact that a growing person only slowly and gradually forms concepts of the objects that surround him. The concepts are added to the observation.

[ 11 ] Herbert Spencer, a widely read contemporary philosopher, describes the mental process we undergo in response to observation as follows:

[ 12 ] “If, while walking through the fields one day in September, we hear a sound a few steps ahead of us and see the grass moving along the side of the ditch from which it seemed to come, we will probably head toward that spot to find out what caused the sound and the movement. As we approach, a partridge flutters into the ditch, and with that our curiosity is satisfied: we have what we call an explanation of the phenomena. This explanation, mind you, boils down to the following: because we have experienced countless times in life that a disturbance in the静止 state of small bodies is accompanied by the movement of other bodies situated among them, and because we have therefore generalized the relationships between such disturbances and such movements, we consider this particular disturbance to be explained as soon as we find that it provides an example of precisely this relationship.” On closer inspection, the matter presents itself quite differently from how it is described here. When I hear a sound, I first seek the concept corresponding to this observation. It is this concept that points me beyond the sound itself. Anyone who does not think further simply hears the sound and is satisfied with that. Through my reflection, however, it becomes clear to me that I must conceive of a sound as an effect. Thus, only when I connect the concept of effect with the perception of the sound am I prompted to go beyond the individual observation and search for the cause. The concept of effect gives rise to that of cause, and I then search for the causative object, which I find in the form of the partridge. However, I can never arrive at these concepts—cause and effect—through mere observation, no matter how many instances I examine. Observation challenges thought, and it is thought alone that shows me the way to connect one individual experience to another.

[ 13 ] If one demands that a “strictly objective science” derive its content solely from observation, one must at the same time demand that it renounce all thinking. For thinking, by its very nature, goes beyond what is observed.” [“Philosophy of Freedom,” pp. 58–59 of the 14th ed., 1978]

[ 14 ] If one were to follow Spencer’s line of thought, one would conclude that concepts arise only through the crystallization of the general from the particular details of observations. As long as I respond to a sound only in the way Spencer describes, I cannot arrive at any understanding at all. Something else must be added.

[ 15 ] At the time, I also presented an author’s copy of this book to a very prominent contemporary philosopher, who then wrote to me saying that he had so much to say about the book’s content that he did not want to write it all in a letter; instead, he had written all his comments in the margins. In the margin of the relevant sentence—which deals with the concept of sound—he wrote: “The hare certainly doesn’t do that!”—and sent the book back to me. However, such an investigation is not about the philosophy of the hare, but rather that of humans. [Gap in the postscript.]

[ 16 ] We must realize that our soul must be capable of forming a network of concepts even when it is not in a position to have it before it through the immediate perception of the world. Thus, it cannot form this network by relying on external perceptions and the ideas derived from them. The soul would never arrive at concepts if it were to merely direct its gaze outward toward external perception and form ideas from it. The methods—even if they are the most scientific methods used to form ideas about the world through external experience—cannot serve to enable the human soul to construct the network of concepts internally on its own.

[ 17 ] There must therefore be a method that is, on the one hand, independent of external observation and, on the other hand, also independent of clairvoyant observation. For, as we assume, the human soul should already be able to form concepts before it ascends to clairvoyance. [Gap in the postscript.]

[ 18 ] He thus moves from one concept to another; he remains within the realm of concepts and can now move from one concept to another within the network of concepts. For this to be possible—for the soul to move from one concept to another—we must assume a method that has nothing to do with external sensory observation and nothing to do with clairvoyant observation, which is intended only for verification.

[ 19 ] This process of thinking in pure concepts is called, in the sense of the great philosopher Hegel, the “dialectical method,” whereby human beings live solely in concepts and enable themselves to allow one concept to emerge from another, as it were, to grow out of it. Thus, human beings live in a sphere in which they turn away from the external, sensory world, and from what lies beyond it—the supersensory world. The soul moves from concept to concept, and the force that drives it forward from concept to concept causes one concept to emerge from another. This method is called the dialectical method, the method of the self-moving concept.

[ 20 ] We have thus pointed out what the soul does as it moves forward within its network of concepts. It weaves concept upon concept—we will soon be able to imagine this in more detail in terms of the dialectical method. This dialectical method leads the soul from concept to concept. We will see that we must start somewhere with these self-propelling concepts, but then we will be led onward from concept to concept. What, then, should result from this? If the soul begins somewhere by positing a concept and then allows concepts to grow out of one another, it would construct the sum of all concepts; in this way, it would form the sum of all concepts adapted to the universe—both downward to the sensory world and upward to the supersensory world.

[ 21 ] All such concepts that are formed through self-movement—through the emergence of one concept from another—and thereby represent to us what corresponds to both the sensible world and the supersensible world; all such concepts are called “categories” in the broadest sense of the word. Categories are thus those concepts that are obtained through the dialectical method—that is, through the emergence of one concept from another. Thus, fundamentally, the entire network of concepts is composed of categories. One might just as well say that all concepts are categories as one might say that all categories are concepts. Of course, we have become accustomed to applying the term “categories” to the main concepts—the nodal points, the most important ones, the root concepts—namely because formal logic has always drawn on Aristotle, who [was the first to speak of categories and to specify ten such “nodal points”]. In the strict sense, however, the terms “concept” and “category” can be used interchangeably, so that we can call the sum of all our concepts—“the doctrine of categories”—provided we are dealing with proper conceptual constructions, that is, when the concepts are internally constructed and developed through self-movement, when the concepts have grown out of themselves. And what Hegel calls the “Science of Logic” in the first part of his philosophy—logic, derived from logos, which also means “concept”—is actually a theory of categories. If we form only individual concepts, we do not have all the categories; but if we weave the network of concepts internally, placing each concept in its proper place within the overall organism of concepts, then we have all the categories. Now, Hegel himself has already said: If one thus determines the entire scope of the web of concepts, one has within it the content of the world as it exists in the thought of the divine essence before the creation of the world. - Since we find the concepts within the world, they must have been originally placed there. When we trace the concepts, we find within them the thoughts of the Deity. If we think correctly through inner [construction] according to the dialectical method, we find the world in the content of the concepts, in the content of the categories.

[ 22 ] I cannot go into the historical development of the theory of categories today—as Aristotle formulated it and as Kant further developed it—but I would like to say a few words about how Hegel, the great master of the theory of categories, developed the theory of concepts. Hegel is perhaps the least understood philosopher today. This is evident in the academic literature; what is written about Hegel there is downright terrible. For instance, people still say today what was already said during his lifetime: that he wanted to develop the entire world out of concepts. One particularly clever gentleman was the Leipzig philosopher Wilhelm Traugott Krug, who wrote an entire library of philosophical works. He understood Hegel as though he wanted to spin the world out of concepts—for example, to deduce the concept of the rose from a concept, from an idea. Krug once pointedly remarked to Hegel that he should try deducing his quill pen from a concept. Hegel replied: “In this regard—and in other respects as well—Mr. Krug, in a quite naive sense, once challenged natural philosophy to perform the feat of deducing nothing but his quill pen. One might have given him hope of achieving this feat—and of the corresponding glorification of his quill pen—if science were someday to have advanced so far and to have come to terms with all that is important in heaven and on earth, in the present and the past, that there would be nothing more important left to comprehend.”

[ 23 ] It is of extraordinary importance for the anthroposophist to immerse themselves in these pure concepts, through which one gradually [gains the entire network of concepts]. It is extraordinarily useful and constitutes an extraordinarily fruitful form of meditation to live within Hegel’s crystal-clear concepts; it is an important means of education for the soul. At the same time, it is a means of education against all carelessness and sloppiness in concepts, which are thoroughly eradicated by Hegelian dialectics. Once one has trained the mind in Hegel’s dialectic, one often gets the impression of sloppy concepts when reading books by modern writers.

[ 24 ] Of course, one must have a starting point; one must begin with something. Naturally, this must be the simplest concept, the one with the least content and the greatest extension. Formal logic explains why the content of a concept is in contrast to its extension. The concept that has the least content and the greatest extension is the concept of being. It is, in fact, the concept that is applicable throughout the entire scope of our world; it has the greatest extension and the least content. When we speak of being in the abstract, nothing is said about the nature of being. Hegel takes the concept of being as his starting point. Now the question arises: How do we move beyond this concept of Being? We cannot remain at this concept; otherwise, we will not obtain a system of concepts. We must have the possibility of developing a system of concepts by allowing concepts to grow out of other concepts. How do we find a starting point for this? We find this starting point precisely in the dialectical method, namely when we realize how every concept contains within itself something other than what it initially appears to be. A concept is like a root. The root actually contains the entire plant, which has not yet grown out of it but is still inside it. When we look at the root, we do not yet see everything that is there. We do not see the plant itself, which is inside the root. If we look at the root only with our physical eyes, we do not see what the plant is pushing out of the root. In the same way, there is something within every concept that can grow out of it, just as there is something within the root that can grow out of it; specifically, the concept of being contains its opposite—nothingness—within it. When we grasp the concept of being, it encompasses everything possible that can appear in the sensible and supersensible worlds. Because it encompasses everything, it simultaneously encompasses “nothingness.” “Nothingness” is contained within “Being”; it sprouts forth from “Being.” When we contemplate “Being” inwardly, we can already see the concept of “Nothingness” growing out of the concept of “Being.” If we wish to form a conception of the concept of Nothingness, this is as difficult as it is important. Many people, including philosophers, will say that it is utterly impossible to form a concept of “Nothing.” But this is something that is immensely important within the conceptual world for the anthroposophist, and a time will come when anthroposophy will delve more deeply into concepts; much will depend on the concept of “Nothing” being grasped in the correct way. Theosophy suffers from the fact that the concept of “nothingness” is defined unclearly. That is why Theosophy has become a kind of “doctrine of emanation,” [gap in the postscript] as if what came later had emerged from what came before.

[ 25 ] Imagine yourself confronted with an external reality—for example, two people—and observe them from a perspective that depends solely on you. And consider, for example, two people—one tall and one short—and form an idea about them; create a concept that would never have been formed if you had not encountered them. It doesn’t matter what you think about these two people, but the concept would not have been formed if you had not encountered them. Suppose the two had lived in America; then, as a European, you would never have encountered them. But because you did encounter them, the concepts of “tall” and “short” arose within you. It is therefore not up to you that the concept of the “great” and the “small” person has formed; you will find nothing within yourself that would have led to your concept of “great” and “small.” On the other hand, you will not find the underlying causes that should have led to this concept in those two people either. You first had to encounter them. Thus, it is not up to you what has formed as a concept here, nor is it up to the “great” or “small” person; it is something that has been brought about purely by the relationship of things to one another, by their constellation. Now, however, this concept, which arose out of nothing, becomes a factor that continues to exert an influence within you. You cannot conceive of it any other way than that this concept can emerge out of nothing through the relationship of things to one another, through their constellation. From this relationship, from this constellation, a continuous force thus shapes something that then continues to exert an influence. That is to say, something arises out of nothing. Nothingness is thus a very real factor in world events, and you can never comprehend these world events if you have not grasped nothingness in this real sense. You would also understand the concept of “nirvana” better if you had a clear concept of nothingness, if you had ever meditated on the concept of nothingness, which is something quite effective.

[ 26 ] We have thus derived the concept of nothingness from the concept of being. The next concept is found by combining these two concepts. When one combines “being” and “non-being,” “becoming” emerges. “Becoming” is a richer concept that already contains the other two within itself. “Becoming” is a continuous transition from non-being to being; what precedes it passes away, and what follows it comes into being. Thus, in the concept of “becoming,” you have the interplay of the two concepts “being” and “nothing.” Starting from the concept of becoming, you then arrive at the concept of “existence.” It is what follows immediately after becoming: the solidification of becoming is “existence,” a completed becoming. “Existence” must be preceded by becoming. What do we gain, then, when we have internally developed these four concepts and arrived at them in this way? We gain a great deal. When we think of the concept of “becoming,” we now think of nothing other than what we have come to know here as the content of the concept. We must exclude everything that does not belong to the concept. Anyone who is properly trained in dialectics, when speaking of “becoming,” sees in this concept nothing other than the interplay of “being” and “nothing.” When the dialectically trained thinker speaks of “becoming,” it is just as specific a concept as when he speaks of the concept of “triangle.” Thus, dialectics is precisely the most wonderful discipline of thought.

[ 27 ] Being -> Nothingness -> Becoming -> Existence

[ 28 ] We have already developed four consecutive categories here: the categories “Being,” “Nothing,” “Becoming,” and “Existence.” Now we could go further and let all manner of concepts sprout from the concept of “Existence,” and we would obtain a richly structured conceptual system from the concept of “Existence” along this one line. But we can also proceed in a different way. “Being” allows concepts to grow out of it in two directions. It is something very fruitful. There is something that is already there before Being bursts into reality. The pure thought of Being is already given before Being has burst out of thought into reality. At the moment when Being becomes itself, when it becomes content in itself, at that moment we must designate what we then grasp as “essence,” so that in this way we have formed the concept of “essence” from the concept of “Being.” Thus, on the one hand, we have derived the concepts of “nothing,” “becoming,” and “existence” from the concept of “Being,” and on the other hand, we have derived the concept of “essence” from the concept of “Being.”

[ 29 ] His -> nature

[ 30 ] Essence is being held within itself, being that permeates itself. The easiest way to grasp the “essence” of a thing is to consider what is essential and what is inessential about it. Essence is the Being at work within, the Being that is strengthened through work itself. We refer to this as “essence.” We speak of the “essence” of a human being when we consider both their higher and lower faculties together, and we regard the concept of “essence” as the concept directly linked to “Being.”

[ 31 ] From the concept of “essence,” you derive [organically as the next step] the concept of “appearance”—the manifestation outward—the opposite of “essence,” the opposite of what the essence contains within itself. “Essence” and “appearance” are two contradictory concepts that relate to one another in a manner similar to the concepts of “being” and “nothingness.” If we now combine the two concepts of “essence” and “appearance,” we obtain the appearance that, in turn, contains the essence within itself. [Gap in the postscript.] In a certain sense, there is a contradiction between inner essence and outer appearance. But when inner essence overflows into appearance, so that the appearance itself contains the essence, we speak of “reality.”

[ 32 ] Essence -> Appearance -> Reality

[ 33 ] No one trained in dialectics will speak of the concept of “reality” in any other way than by saying: In the concept of reality, appearance lives, permeated by essence. — The convergence of “essence” and “appearance” yields the concept of “reality.” Thus, all discourse on reality must be permeated by these concepts.^{1Incomplete text in the manuscripts; see note.}

[ 34 ] We can now go even further and ascend to even richer concepts. This leads us to say: “Essence” is “Being” that is in itself, that has come to itself within itself, that has manifested itself. - If this Being not only manifests itself, but also extends its lines to the surrounding world—that is, if it expresses not only itself internally but also attempts to express something else—then we arrive, through dialectical reasoning, at the concept of the “concept” itself. Thus, we ascend from “Being” through “Essence” to the “Concept.”

[ 35 ] Its -> Essence -> Concept

[ 36 ] Now recall what I said about the concept from a purely formal-logical perspective. When we consider our own “essence,” it applies to us; it prevails within us. But when we allow the “concept” to prevail within us, we have something in it that points outward and encompasses the other—the external world. Thus, through an inner construction, we have been able to proceed from “being” through “essence” to the “concept.”

[ 37 ] Being -> Nothingness -> Becoming -> Existence

[ 38 ] and from

[ 39 ] Essence -> Appearance -> Reality

[ 40 ] If we now derive further concepts from the “concept” in the same way that we have derived the previous ones, we arrive at the following: We have now seen how the “concept” operates formally and logically in the figure of reasoning. There, the concept remains within itself. But now it can step outside itself, and we then speak of a concept that reflects the nature of things to us. We arrive at “objectivity.” In contrast to subjective concepts, which are subject to the techniques of thought, we now have objective concepts. Just as “appearance” relates to “essence,” so does “objectivity” relate to “concept.” One has truly grasped the concept of “objectivity” only when one conceives of it in this way, emerging from the “concept.” And now, when we connect “concept” with “objectivity,” we arrive at what is an inner concept for us, yet one that simultaneously contains its own reality within itself—one that is at once a subjective concept and, as such, objective. This is the “idea.” Just as “reality” relates to “phenomenon,” so does the “idea” relate to “objectivity.”

[ 41 ] Concept -> Objectivity -> Idea

[ 42 ] Here is a small example of how, in dialectics, we can derive concepts from the root concept of “Being.” We could have derived many other concepts from “Being” in this way.

[ 43 ] Thus we see how, through the movement of concepts, this transparent, diamond-clear, crystalline, spiritual world of concepts emerges, and that only when equipped with this faculty of cognition—trained in precise concepts—should human beings once again approach the sensory world. It then becomes evident how the concepts derived through dialectics correspond, on the one hand, to sensuous reality and, on the other, to supersensuous reality, and how human beings arrive at the harmony between concept and reality in which true knowledge first exists.

[ 44 ] Some people say: We’re entering a world that’s cold and frosty. So I’d like to conclude by sharing a little story. Many years ago, I had to study graphical statics, a very abstract science… It involves tracing possible movements and states of equilibrium using only lines. But I knew someone who presented it with boundless enthusiasm whenever he drew a line on the blackboard, while the other students were asleep—and most of the time, the teacher was asleep too. I’ve even heard people recite lyric poetry who lacked that enthusiasm. These are simply people who cannot feel the way Novalis did, who called mathematics a great poem. Mathematics is a way of giving us a glimpse of how one can experience the world of concepts with beautiful clarity and bright light.