The Rudolf Steiner Archive

First Lecture

[ 1 ] Ladies and gentlemen, fellow students! Spiritual Science, as embodied in these anthroposophical university courses, must fight for its rightful place and recognition in the true sense of the word in the present day. At first glance, this might seem surprising, for anyone who has become even slightly acquainted with the driving forces behind this anthroposophically oriented Spiritual Science can see how it stands firmly on the ground of contemporary scientific and other cultural research, and how it fully takes into account all that underlies the spiritual life of the present as a necessity. If it nevertheless has to struggle, one must bear in mind that, as an anthroposophically oriented Spiritual Science, it must first and foremost confront the strongest prejudices present in the present day. It is, in a sense, the natural adversary of everything that still exists as reactionary forces in the souls of people today. And anyone who is able to give this even a little thought knows that there is much that is reactionary in the souls of people today.

[ 2 ] In these lectures, my task will be to set forth, from a scientific perspective, the validity and significance of the Spiritual Science referred to here. I will have to begin with relatively elementary concepts and then, over the course of the lectures, gradually proceed toward a genuine understanding of human nature from the perspective of this anthroposophically oriented Spiritual Science. And although I will primarily be speaking about methodology, I will nevertheless endeavor, through the examples I use in the course of these lectures, to introduce you to individual chapters and specific topics of the Spiritual Science—both in general and in terms of its significance for the various contemporary academic disciplines.

[ 3 ] However, today, in the first of these lectures, I would like to begin by offering a sort of introduction to all of this. I would like to begin by pointing out how contemporary scientific thinking has increasingly come to find its main foundation in the experiment—in the scientific experiment. In this respect, contemporary scientific thinking stands in a certain contrast to older forms of knowledge—those forms of knowledge that were based primarily on observation, as nature and the world present themselves to human beings. It is one thing to start with a fact already presented by nature and the world and to observe it, and quite another to first create the conditions for a process and, with knowledge of those conditions, observe a fact and be led to certain scientific results through this observation. But you also know that the newer scientific way of thinking has increasingly led to the introduction of mathematical ideas and mathematical results into observational data—particularly in the field of the natural sciences themselves. You are all certainly familiar with the statement once made by Kant that in every individual science there is only as much real knowledge, real insight, as there is mathematics present in it. Mathematical concepts and mathematical results are to be introduced into both observation and experimentation. This gives one a sense of security; one feels that one has a completely different relationship to a set of facts—which one is able to grasp through mathematical formulas—than to those facts that are simply described based on their empirical circumstances. This sense of security that one has when dealing with mathematics is something that has long been characteristic of scientific thinking.

[ 4 ] However, one cannot say that there is already, today, a clear, genuine awareness of the reasons why we feel so confident in the mathematical treatment of nature and the world. And it is precisely a clear and distinct recognition of this fact—this sense of confidence within the mathematical approach—that will lead us to acknowledge the necessity of a Spiritual Science in the sense intended here. This Spiritual Science is by no means dependent on, so to speak, having to beg for recognition from the natural sciences or any other scientific discipline. This Spiritual Science intends to measure up in every one of its fields to the scientific rigor of the modern age, and it also intends to gain a secure footing in the face of all the doubts, puzzles, and unresolved questions raised by this modern scientific approach; it specifically aims to place science, in a very specific sense, on a secure foundation—one that is exact in the mathematical sense.

[ 5 ] I need only point out a very simple question to you, and you will see that it is precisely this sense of certainty in mathematical reasoning that—I would say—leads us straight into uncertainty halfway through our scientific pursuit. What, for example, are we to make of a science such as history if every science is supposed to contain only as much real knowledge as there is mathematics in it? And how are we to come to terms with the realities of the human soul if we have all—I might say—endured the drudgery developed by mathematizing psychology, such as that of the Herbartian school, in order to achieve certainty in this field as well? We have come to nothing other than the realization that it is not possible to introduce mathematics into knowledge in this field. This is, in a sense, the first question we must address: What does this mathematical certainty mean in relation to human knowledge? And this question, if we bring it to a certain answer, will lead us directly to the justification of research in Spiritual Science.

[ 6 ] I have also said that modern scientific methodology prefers experiments—in which the conditions of a process are precisely known—to external observation, in which the conditions, I might say, are more deeply hidden in the depths of existence. Even in fields such as psychology and pedagogy, attempts have been made in recent times to move from mere observation to experimentation. I expressly note that Spiritual Science, as understood here, cannot and should not object to the legitimate claims of experimentation, neither in the field of psychology nor in that of pedagogy. But the point is to understand what exactly underlies this inclination toward experimentation, particularly in such fields of human knowledge. It is precisely in these fields that we can gain a true understanding of what this inclination toward experimentation is based on. Let us therefore, for a moment, take as our starting point the shift toward experimentation specifically in the fields of psychology and pedagogy. There we can see how, until relatively recently, both psychology—the study of the soul—and pedagogy focused on carefully observing the lived experience of human beings, whether adults or developing human beings—that is, children. But what is necessary if one wishes to observe the inner life of both adults and children? It is necessary to be able to relate to what one observes with a certain inner involvement. Let us really try to put ourselves in the shoes of older observational methods in the field of psychology or in the field of education and teaching. One will already perceive there that the inner involvement one person has with another has, in the course of human development, diminished as we approach the present. We are no longer as intimately immersed in objectivity with regard to another person’s soul as educators or psychologists once were. We no longer feel, when our own soul stirrings vibrate, an echo in this vibration of what the other person’s soul is experiencing. We have, I would say, become more distant from the objective soul life of others than people once were—those who ever engaged in observations of soul life at all. And to the same extent that we have become more and more alienated from the intimacies of another’s inner life—to the same extent that we can no longer look into the innermost depths of another’s soul with immediate intuition or with a deep, intimate involvement of our own inner being—to that same extent do we attempt to approach the human soul from the outside with the aid of our admittedly excellent tools. We attempt to record the manifestations of human inner life through instruments; in other words, we attempt—I would say—to approach the human being from the outside. This is certainly justified to a limited extent and must be fully appreciated as legitimate, above all because of the fundamental character of our time. Once one has become directly estranged from the inner world, one must simply appropriate the manifestations of this inner world—the superficial aspects—through external methods and external experimentation.

[ 7 ] But precisely when, in a certain sense, we become estranged from the human spirit and soul and conduct our experiments on the more material manifestations of the spirit and soul, it becomes all the more necessary for us to be able to interpret these experiments ourselves in a spiritual sense and to penetrate them with spiritual inquiry. Therefore, no objection should be raised against experimentation as such; rather, the need will arise—and I wish to speak only by way of introduction today—to shed spiritual light from within on precisely what the experiment reveals. To illustrate this, I would like to present an example to you.

[ 8 ] Experimental pedagogy has rightly observed how growth patterns differ between boys and girls. During school-age, it has been shown that boys and girls grow at different rates during various stages of life—we will have more to say about these matters later—so that there are periods in a boy’s life when he grows, let’s say, more slowly, while girls grow more quickly during the same stage of life. If one merely conducts experiments—if, so to speak, one merely observes the outward manifestations of the soul’s life—one can record this as a fact. But only those who understand how the soul drives the growth process, how the boy’s inner life is fundamentally different, and how the power of this inner life manifests itself in the various stages of life will be able to interpret such a fact with true insight. And then one will be able to see precisely how the difference in growth patterns between boys and girls in turn sheds light on why girls grow at different rates compared to boys during the same stages of life, and it will then shed light on what is taking place in the girl’s soul and what is taking place in the boy’s soul. One will come to understand that a person who, for example, grows particularly rapidly between the ages of fourteen and seventeen develops different forces than a person who, let us say, grows particularly rapidly at a somewhat earlier age.

[ 9 ] Precisely an age that excels in the external, experimental treatment of facts—precisely such an age—must, if it does not wish to sink into superficiality and outward appearances, be capable of penetrating what it explores experimentally with intellectual inquiry. In contrast to all this, the awareness that mathematics offers something that provides a researcher with certainty is extraordinarily decisive. If one wishes to properly appreciate this fact in its essence and significance, then one must indeed ask oneself: How does one recognize precisely through mathematics, how does one apply mathematics to the external, sensually given world of facts, and how does the mathematical treatment differ from other treatments of the factual material given to us?

[ 10 ] You see, to begin with, the external realities of the world are presented to human beings through their senses. Just as we enter this world of external facts from childhood onward, it actually presents us, at first, with a kind of chaos in relation to our subjectivity; and only by mustering the inner strength to form all sorts of mental images and concepts—I have described this in more detail in my little book Truth and Science — do we link one fact to another, group facts together, and sometimes conceptually bring together facts that appear very distant from one another in external observation; in this way, we create a certain ideal, conceptual order within the chaos of immediate sensory experience.

[ 11 ] Now we must look closely at how our treatment of the external sensory world of facts proceeds at first, when we do not apply mathematics to our understanding—that is, when we simply observe the external world and form mental images about the relationships among external natural facts, for example, according to the common laws of cause and effect and so on, as well as according to other laws. We must form mental images about what our approach to the external world actually entails. What are we actually doing when we bring order to the sensory chaos? It seems to me that, with regard to what we are doing there, David Hume did, after all, make a very accurate observation. His great mistake lies solely in the fact that he thought what applies specifically to this one area—that is, to the non-mathematical observation of external nature—was valid for the entire scope of human knowledge.

[ 12 ] Most errors and one-sidedness in scientific thinking stem from the fact that what is entirely valid in one field is then applied to the universality of human knowledge. That is why it is sometimes so difficult to refute something that is, after all, erroneous in relation to the universal, because—let me state the paradox — there is scarcely any proposition that could be called an error in the universal sense that would not have its justification in a specific context here or there, so that, in that specific context, one can certainly cite good reasons for something that must be challenged when a claim is made that it should have universal validity. This is the case when David Hume says: We observe the external world; we organize it in a systematic way through our mental images. But what we then have present in our minds as a law is by no means such that we can readily say it corresponds to something objective in the external world, or that the external world must always unfold in its facts exactly as such a law prescribes. According to David Hume, one can really only say that, so far, one has seen the sun rise every morning. That is a proposition based on experience. One can express all these facts of the same kind that have presented themselves to us in a general law, but there is no guarantee that one has anything other than a series of experienced facts summarized in a mental image. — What, then, is it within us that forms lawful connections from the facts observed through the senses, and what significance do these lawful connections have for the realm just described? Is David Hume correct when he says: It is in the nature of our minds to summarize the facts presented to us in a systematic way, and because we act in accordance with this mental habit, we form certain laws of nature. These laws of nature, however, are nothing other than what has been synthesized from the individual facts by our mental habits.

[ 13 ] This leads one to say to oneself: Human beings initially develop in their empirical lives in such a way that they become inwardly accustomed to bringing order and harmony into the chaos of empirical facts; and it must be said that the further one advances in knowledge—especially in the field limited to this discussion—the more one tends toward this characteristic habit of the soul. One cannot then strive to preserve disjointed facts; one wants to conform to the aforementioned mental habit; one wants, if possible, to bring unity into that which confronts us as sensory, empirical diversity. However, if one examines this entire process of cognition with unbiased self-reflection, one must admit that, when one perceives in this way, one faces the external world in such a way that this external world does not actually enter into our cognition. In this field of cognition, one must always tell oneself: Out there are the material facts. We habitually incorporate them into our system of concepts; we then also take an overview of them; we know that if a series of facts has played out a certain number of times, it will, when the first facts reappear before us, also play out in a similar manner in its second phase. And yet, if we remain within this realm, we do not penetrate the external world; in fact, we do not even initially claim to be able to penetrate it. However, if we wish to formulate rash metaphysical hypotheses, we will speak of matter as being this or that. But if we do not wish to formulate such rash metaphysical hypotheses, we will, so to speak, leave matter aside. We will tell ourselves that we do not fathom what matter actually is in its innermost essence, but what it presents to us, so to speak, on the side it turns toward us—that we arrange into certain lines of thought, into certain laws of thought. We thus remain outside of external reality in such a way that we form images of the course of events on the outer surface of material occurrences. And fundamentally, for our true human consciousness, we also need this awareness that we are dealing with images. Just think for a moment what it would actually mean for human consciousness if we could not accept the fact that, at first, we are dealing only with images of the external world. If, every time we find ourselves in the realm we are describing here, we had to tell ourselves that something from the external world is flowing into us, just as something flows into us when we drink or eat. Just think for a moment how little such a merging of inner and material existence would correspond to what our human soul’s disposition must be in perceiving the external world. We may find ourselves having to say: Nothing flows from the external world into our inner life in the process of cognition. What we experience in the external world, we shape into images that, strictly speaking, have nothing at all to do with that external world.

[ 14 ] Perhaps I can use an image from the world of art to illustrate what we’re discussing here. Imagine I’m painting something. What I then put on the canvas is something that this external world—which is, so to speak, being transferred onto the canvas—truly doesn’t care about. A few trees out there that I paint on the canvas—they are completely indifferent to whether I paint them or how I paint them on the canvas. My painting is added to what is out there as something foreign, something that has not the slightest internal connection to this external reality. Psychologically and epistemologically, this is essentially true of all cognition that operates within the field I am speaking of right now. We would immediately find ourselves in a situation where we could no longer distinguish ourselves from the external world—merging with it, just as we do when we eat and drink—if what takes place in our soul during this process of cognition did not prove to be something entirely foreign to the external material world. We shall see later how that which is to be grasped as human freedom can only be understood on the condition that knowledge of the material external world behaves as I have just described.

[ 15 ] But that is not the case when I engage in mathematical understanding. Just imagine for a moment how mathematical understanding takes place, whether in the field of arithmetic, algebra, or any of the higher branches of analysis, or whether in the field of analytical or synthetic geometry. We are not confronted with some external world that we cannot grasp; rather, in mathematical cognition, we live directly within everything that is our object. We form the mathematical objects and their relationships internally, and if we draw these mathematical structures in any way, it is, I might say, merely for our convenience. What we mean is, after all, never something that in any way belongs to that external world which we perceive with our senses; rather, what we mean in mathematical cognition is entirely an internally constructed entity—something that lives solely in that activity of our soul which provides a glimpse of all that is inaccessible to the senses as such. We construct internally as we build up the field of mathematical science. This is a very radical difference from the first field of cognition—the cognition of the empirical external world of the senses. There, what is the object remains strictly outside of us. In mathematical cognition, we are immersed with our entire soul in the objective realm, and whatever emerges as the content of mathematical science is the result of a construction experienced and carried out purely within our soul.

[ 16 ] This is a significant problem, and I would like to say that this problem is the preliminary stage to the other problem, which will then become the advanced stage: How does one ascend from the ordinary sciences to anthroposophical Spiritual Science? I do not believe that anyone will be able to answer the latter question properly and in a truly scientific manner unless they have first answered the question: How does external, pure observation and the understanding of nature that governs observation rise to the level of an understanding of nature that is interwoven with mathematics, and how does it relate to mathematical understanding as such?

[ 17 ] But this raises another question—one that scientists must answer for themselves based on their experience with scientific work. As I have already said, Kant pointed out that in every science there is only as much genuine knowledge as there is mathematics involved. This, too, is a one-sided view, for it applies only to a certain field. Kant’s error lies in the fact that he took what applies to a particular field and assumed it to be universal. But with regard to certain aspects of external natural existence, as conveyed to us through our senses, we must indeed say: We carry within us the need—we will speak of this need in greater detail later—we carry within us what may quite rightly be called a scientific need to penetrate the facts that present themselves to us mathematically, not merely to measure them and compare the measurements, but to penetrate them with that which we ourselves have first constructed in mathematical formulas.

[ 18 ] What is it, actually, that lives within us, causing us to strive for such things—not merely to stop at habitually linking external empirical facts to general rules, but to penetrate these empirical facts with what we first construct internally, what we ourselves have formulated with our entire inner life while fully engaged with the mathematical object? Well, it is quite evident—as anyone who has undergone scientific experience in this field can tell themselves through unbiased self-observation—that there is, without a doubt, the fact that we do indeed feel that, at first, the whole of nature around us actually stands opposed to our human being as something materially alien to us. Now, when we realize that we can, so to speak, immerse ourselves in this materially foreign world using what we have first constructed internally—that we can express in a mathematical formula that which otherwise presents itself to us only from the outside—and that, by expressing it mathematically, the natural process itself unfolds according to this mathematical formula—what lies at the root of this? Well, apparently nothing else underlies this than that, through this process, we fully internalize the external natural process—which is initially foreign to us—by striving, as it were, to become one with it. Are we doing anything other than grasping that which initially presents itself to us entirely externally in such a way that we can reconstruct its unfolding just as we construct it within the realm of pure mathematics? Are we doing anything other than striving to experience inwardly—and indeed, precisely inwardly—that which at first merely stares at us from the outside? This internalization of the external is what drives the mathematical explanation of nature; it is the very lifeblood of the mathematical explanation of nature. It is particularly characteristic of modern scientific endeavor and its relationship to technology—a topic we will also have to discuss—that the longing is so intense to incorporate as much of the mathematical into external events as possible; that is to say, to incorporate what has first been constructed internally; that is, to fully comprehend what presents itself to our gaze by observing it, as if we ourselves had given it its form—the forms of its occurrence. And once we have brought it as far as possible—in a sense, to the fulfillment of a certain ideal—of applying mathematics to external natural phenomena, once we succeed in taking this as far as it has gone today—where this is no longer pursued with the same intensity as it once was by the atomists, who, for example, in the field of light phenomena had attempted to penetrate everything that presented itself externally with mathematical formulas—if we have then come to the point of having fulfilled this ideal as far as possible in a certain field, of having brought mathematics into the external world, what do we have then? Then we can look at what we have. We can ask ourselves: What have we achieved with this? — We can clearly bring to mind: What, in fact, is the content of our soul when, instead of observing the external world—let’s say, instead of observing the phenomena of polarization—it makes a sum of mathematical formulations present within itself? What does our soul gain from this? We look at this construct—this external world, so to speak, entirely rendered in mathematical formulations—we look at it, and when we then observe it impartially and are equally able to direct our gaze toward the external world with the same impartiality, we discover something quite peculiar. We find that, although we have taken into account all the material conditions the external world has provided us with in order to link our mathematical formulations to it, we find, as it were, that we initially had something that seemed obscure to us internally, and which now appears clear—namely, illuminated by mathematical concepts. We find this to be the case. But we can no longer deny the fact that, at the same time, we are now holding up to nature—to the external world—an image that no longer contains anything of the reality that first presented itself to us.

[ 19 ] Take optical phenomena in their fullness, in their intensity; take them just as they first present themselves to us when the eye observes them, and contrast them with what, from a certain point of view, quite rightly brings forth an inner mathematical construction—let us call it mathematical optics—as the image, as the image of these visual phenomena shaped according to mathematically formulated rules. You will have to admit—it only takes a little open-mindedness—that: This mathematical image contains nothing of the richness of color phenomena. Everything that the senses—that is, external reality—initially presented has been squeezed out of this image. We hold up to the external world an image that no longer possesses its inner, intense richness, one that lacks its intense reality.

[ 20 ] I would like to use a comparison here, which—as you will see in the following lectures—is not merely a comparison. But as a pure analogy, I ask that you accept it for now. For when we mathematically analyze empirical facts, our understanding breaks down into two stages. The first is this: We must observe the empirical facts—that is, let’s say, the facts perceived by the eye. The second stage is this: We translate these observations into mathematical formulations, and then, in a sense, we are left with these mathematical formulations as the result. We then no longer look to the world of empirical facts. It is exactly—or, to put it more precisely—as if we were inhaling life-giving oxygen and allowing it to permeate our organism. It combines within us with carbon to form carbon dioxide; we exhale the carbon dioxide—it is no longer life-giving air—but the entire process was necessary for our inner life. We had to inhale the life-giving oxygen; we had to combine it within ourselves with something that is already within us. When we now contrast what has arisen in this way with the external world, it is, in the same sense, a life-destroying force for that external world as the inhaled air is a life-giving one. For now, let this serve merely as an illustration. This is how we proceed in the mathematical understanding of nature. We take into ourselves what presents itself to our senses; we seek to unite it intimately within ourselves with something that is within us—with something found only within us—with what we have constructed mathematically within ourselves. Through this—the union of what is empirically known with what we have constructed within ourselves—something arises: precisely the result of the mathematical understanding of nature. Let us hold this up against nature: nature is no longer contained within it in its liveliness, just as the life force of the inhaled air is no longer present in the exhaled air. It is, in a sense, a spiritual inhalation of the external world, but an inhalation counterbalanced by an exhalation that, transformed in a certain way, has turned into its opposite that which was inhaled and thus connected to the organism of the soul. We must look closely at this process, which takes place within us when we strive for mathematical knowledge of nature; for it proves to us that this mathematical knowledge of nature is indeed different from merely empirical knowledge of nature. Merely empirical knowledge of nature extends only as far as our inner soul habits; mathematical knowledge of nature, on the other hand, does not merely set a habit alien to the external world in opposition to that external world, but rather sets in opposition something internally effective, something experienced, something internally formed—and it seeks within this internally formed reality something that explains the external world according to its own essence; it thus seeks, so to speak, to connect the inner with the outer.

[ 21 ] If one truly understands how the longing for a mathematical explanation of nature is based on this inner appropriation of the external world, then one can no longer fail to see that one possesses a completely different kind of knowledge in mathematics than in purely external, sensory, and empirical cognition—which synthesizes sensory experience through reason. Mathematical cognition takes us deeper into the human inner world, and it is precisely through this that we believe we are drawing closer to the external world; it is precisely through this that we believe we are inwardly experiencing what the essence of the external world represents. Only then do we realize that, by transforming something into a mathematical formulation, we have, in essence, lost the full richness of the external world. We must be aware that: one is connecting what the external world gives one with something constructed purely within. And one must be able to truly experience what is actually taking place in the soul when one creates mathematical formulations. One must realize that mathematics is an inner product of humanity, and that one nevertheless—even though one is dealing in this mathematical realm with an inner product of humanity— one nevertheless has a feeling—we will see later that it is an insight—that with this internally constructed mathematical structure, which is constructed entirely apart from the external world, there is something that brings us closer to the external world than we would otherwise be. But nevertheless, again, this internally constructed mathematical structure cannot be an inner reality—at least not an immediate inner reality in relation to the real external world. For otherwise, when one has the mathematical image before one as the result of one’s research, one would not have the feeling that the full richness of the external world has now faded away and only the mathematical formulation remains; rather, one would have the feeling that in this mathematical formulation one has something that is, in fact, all the more real internally. Just imagine how different the situation would be if we had the entire range of visual experiences before us, with all the intense sensations of color, and if, after formulating a mathematical expression, we could then mentally see in that mathematical expression the very thing we had just perceived. In our mathematical formulas of wave theory, we would see colors flash and glow, experience interference phenomena, and so on. We do not do this. The fact that we do not do this shows us that, while we penetrate the external world with our mathematical formulations in such a way that we come closer to it, we do so at the same time at the expense of no longer actually having the full reality of the external world.

[ 22 ] Since we have now advanced from ordinary, habitual knowledge to knowledge formulated internally in mathematical terms—a process that must be preceded by the formulation of purely internally experienced mathematical structures—the question must inevitably arise: Can this development not be continued further within the life of the human soul? — At first, we have an external world before us; we face it in such a way that the rules and laws we form based on our observations of it are entirely foreign constructs here. We move forward, and we can do so only by experiencing the mathematical formulations inwardly, entirely apart from the external world. We then penetrate this external world solely with these mathematical formulations. Internally, they are clearly not imbued with reality; otherwise, we would also possess reality. When we then set them before us, when we examine them closely, when we reflect on them: they cannot be real, for on the contrary, they erase the reality to which we have applied them. The question arises at this point in our considerations: Is it possible that what we formulate mathematically within ourselves—what we do first in order to penetrate the sensory external world more intimately—is what gives us strength? Could it be possible that what is initially a mathematical inner experience—I would say, such a pale abstraction that it causes reality itself to fade for us—could it be that what is experienced there in mathematical structures might be made more powerfully present within us? In other words: Could the effort we must exert to arrive at a mathematical understanding of nature—as a preparatory step toward that understanding—be applied with far greater intensity, so that we might shape within ourselves not merely mathematical abstractions but concrete spiritual realities? Then, although we would not initially see the external sensory world re-emerge in what we have formed internally, we would have before us something that is certainly not a mathematical construct—for mathematical constructs are, by definition, abstract—but must be formed differently. We would have before us something that has been attained in the same way as mathematical constructs, but attained with the character of reality. And before us, viewed spiritually, would stand something of which we could say that it shines before us just as much in reality as external sensory reality does, but we have attained it by drawing forth from within ourselves not merely mathematically abstract constructs, but real constructs. We have strengthened our capacity for mathematical thinking and have thereby risen to the point of bringing reality forth from within ourselves. That would then be a third stage of our cognition.

[ 23 ] The first stage would be the habitual acceptance of the real external world. The second stage would be the mathematical penetration of this external world, after we have first developed pure mathematics. The third would be an experience of the spirit—an inner experience, necessarily as intimate as the mathematical experience, but with the character of spiritual reality. And so we would have before us the external, empirical knowledge of nature; mathematical knowledge; and spiritual knowledge—that knowledge in which, by becoming creatively active within ourselves, we have spiritual worlds before us.

[ 24 ] We are already prepared to regard these worlds as real in that we apply what we find in our mathematically formed—though still figuratively abstract—knowledge to an external reality, and thus tell ourselves: The way we construct mathematically does not in itself possess any reality; it does not draw a reality up from the depths of our souls, but it does draw up something that is an image of a reality. In spiritual science, we draw up from the depths of our souls that which is not merely an image of an external reality, but which is reality itself, which is actuality. These three stages of human knowledge exist: first, knowledge of physical nature; second, mathematical knowledge; and third, Spiritual Science. And it is not merely on the basis of mere assumptions that a method of Spiritual Science is constructed as a necessity; rather, as you can see, it presents itself—especially to those who understand how mathematization emerges from purely empirical research—as a further progression toward spiritual knowledge, even though this does not yield a mathematical world but something entirely different: a true spiritual world. And it must be said that whoever understands how mathematics arises will also be able to rise to an understanding of how Spiritual Science—referred to as anthroposophically oriented—arises.

[ 25 ] That is what I wanted to say to you today as an introduction to these lectures. I wanted to show you that this anthroposophical Spiritual Science knows its place within the entire system of the sciences. It does not arise from some subjective whim or from dilettantism; rather, it is born of a thorough theory of knowledge—it is born of the very knowledge that must first be applied in order to understand the validity of mathematics itself. It was not without reason—albeit based on older, instinctive scientific insights—that Plato required his students to first have a solid foundation in geometry or mathematics. His aim was not to enable his students to acquire specific arithmetic or geometric knowledge, but rather to foster a thorough understanding of what actually takes place within a person when they engage in mathematical or geometric reasoning. And this is essentially the basis for Plato’s seemingly paradoxical but profoundly significant statement: “God geometers.” - He did not mean that God merely creates pentagons or hexagons in mathematics, but that He creates with an inner power such as we can already envision—albeit only figuratively and abstractly—in mathematical thinking. That is why I believe that precisely those who thoroughly understand where mathematics lies within the realm of the sciences will also understand where Spiritual Science lies.

[ 26 ] This Spiritual Science will fight for its rightful place, no matter how its opponents may act, for it has no intention whatsoever of building on any frivolous or amateurish foundations, but rather seeks to be built precisely on true precision and historical thoroughness. That is why I can already say: Any opponents who genuinely and seriously engage with what the Spiritual Science has to say in its own defense are always welcome, for it is happy to discuss matters with them. It has no fear of them, for it is equipped with all the scientific tools available in the other sciences and is capable of using them in debate. It simply does not wish to be constantly held back by those who understand nothing of it and who oppose it precisely out of their sheer dilettantism and petty-mindedness; for Spiritual Science, as it is understood here, believes that it is a necessity for the other specialized sciences, and it has, in essence, no time to lose, for it is necessary that the boundaries within which the specialized sciences find themselves confined everywhere be transcended by the Spiritual Science. Therefore, we must first resolve, at least to some extent, to deal rather harshly at times with those who oppose the Spiritual Science without reason, out of motives that may be alluded to here and there in the course of these lectures. For, fundamentally speaking, it is precisely today that humanity must be informed about Spiritual Science as quickly as possible—seriously, very seriously, and with exact scientific rigor—and this can indeed happen if one is only willing to engage with it.