# Mathematical Knowledge and the Path to Spiritual Science

## GA 324 — 16 March 1921, Stuttgart

### Lecture I

The spiritual science that underlies this course in anthroposophy, must fight for its validity in the truest sense of the word. This can seem strange to one who has become familiar with the motivating forces of this anthroposophically-oriented spiritual science, for it stands solidly on a common ground with scientific and other cultural demands of our time. It deals with all that is necessary and basic for spiritual life in these times.

One can see, however, that spiritual science must fight, if one takes into consideration the many prejudices that exist at present. Spiritual science is in some ways a natural adversary of certain reactionary forces that remain and can be observed in the souls of human beings of our time.

In these lectures it will be my task to present to you in a direct and scientific manner the significance of what we understand here as spiritual science. I will gradually proceed from relatively elementary things to a real knowledge of man from the point of view of this anthroposophically-oriented spiritual science. I will take pains to introduce some chapters and some special questions by speaking of the methodology, and by the choice of special examples indicate their significance.

Today in this first lecture I would like to point out how present-day scientific thinking has increasingly come to rely on the experiment for its main support. In this regard present-day scientific thinking stands in a certain polarity to older kinds of knowledge acquisition, especially to those which start from simply observing nature and the world as it presents itself.

One can start by observing the established facts of nature and the world, or—as we often do today—by first creating the conditions of an event and then, with the knowledge of these conditions, observing a fact and being led by this to certain scientific results. Along with this methodology, one can see the tendency of this newer scientific thinking to observe the entire field of natural science through mathematics, and with these mathematical thoughts, arrive at mathematical results. You all know the saying by Kant: In every individual science there is only so much real knowledge as there is mathematics.

It is thought that in observation, as well as in experimentation, mathematics must be introduced. Through this, one feels oneself in a secure element, one feels in a position to have an overview of a series of facts with the use of mathematical formulas. This is a totally different relationship to knowledge than when such facts are simply described in their natural state. This feeling of certainty which one has in treating knowledge mathematically, has been characteristic of scientific thinking for a long time. One cannot say we have today a really clear idea of the reasons why one feels so certain and safe with the mathematical handling of the natural world. A clear knowledge of the feeling of certainty accompanying the use of mathematics will lead us to acknowledge the necessity that a spiritual science must come about with an equivalent degree of certainty.

This spiritual science does not have to beg for acceptance from natural science or any other special field. This spiritual science will conform in every discipline to the scientific conscientiousness of modern times; it will, in addition, oppose all that is brought forward by modern science that is suspect, and it will answer questions that often go unanswered. Spiritual science will be on a very sure mathematical foundation.

I only have to ask a very simple question for you to see that this feeling of certainty derived from the mathematical treatment of certain subjects leads quickly to uncertainty. What would we do with a science like history if in every science there were only so much real knowledge as there is mathematics? How shall we understand and get the facts straight in matters of the human soul if we have to struggle to understand what modern psychology, by the use of mathematics, has developed in order also to secure certainty of understanding? One must come to recognize that in this field it is not possible to introduce mathematics into actual knowledge.

One of the first questions that must occupy us is this: What is the significance of this mathematical certainty in the context of human cognition? It is in approaching an answer to this question that we will be led to the justification for spiritual-scientific investigation. I have also said that the newer science prefers the experiment, where one knows the conditions of a process exactly, to outer observation where the determining conditions are more hidden; even in the case of psychology and also the field of education, attempts are made to go over from mere observation to experiment. In saying this, I must emphasize that spiritual science has nothing against the correct use of experimentation in psychology and education. The point I wish to call attention to is this: What draws the scientists in these fields to obtain knowledge by the use of experiment? In these areas we can actually find reasons for the inclination toward the use of experimentation. Let us therefore explore the transition to experimentation in the fields of psychology and education.

We can see how until recently investigators in psychology and education have carefully observed the details of the daily life of man, be it fully mature men and women or the transitional developmental life. We might ask: What is fundamentally necessary for an observation of the soul life of the grownup or the developing child? It is to acquire a certain inner relationship to what one observes. Try to put yourselves into the observational methods of olden times, in the fields of psychology and education. You will find that the inner relationship that once existed between human beings has diminished in recent times. We are not so intimately connected in an objective way with the soul life of another human being as was the case in the past. We are no longer aware when our own soul vibrates in sympathetic reverberation with what lives in the soul of another. We are more removed from the objective soul life of the other; formerly it could be directly observed. We are becoming more and more estranged from any really intimate contact with the soul of the other, where in a directly intuitive way one takes part with one's own inner nature in the inner nature of the other soul. Now an effort is made to approach the human soul from the outside through the use of instruments. There is an effort to explore the human soul through the use of apparatus in an external way. This effort is in the character of our time and must be acknowledged as being partially justified. If one has become estranged from a direct perception of the inner activity, then one must accept the outer expression of the inner activity, and at the same time be content with the outer use of experimentation.

It is especially true that when we are estranged from the spirit and soul elements of our fellow man, and yet our experiments are the material expression of this soul-spiritual element, these experiments must be explained in a spiritual sense. They should be wrought throughout with the results of spiritual research. I do not want to speak against experiments as such, but there is a need (I will speak today only in an introductory way) to illuminate the results of these experiments spiritually from within. To explain this properly, I will give you the following example.

Investigations have established that the rate of growth differs between boys and girls. In the development of a boy, it has been shown that in certain phases he grows more slowly, while in the same time period the girl grows faster. One can take notice of these facts even if one only looks at the outer expression of the soul life. But to explain such facts one must know how the soul motivates the growing process, how the soul of the boy is inwardly different, and how the force of the soul expresses itself in different phases of life. Then one will be able to see how the difference of growth rates between boys and girls permits a comprehension of what goes on in the soul of a boy and what goes on in the soul of a girl. It is just here that one can know that a human being who develops very rapidly during the period of 14 to 17 years, develops different forces than those of a human being who grows rapidly in a somewhat earlier period of life.

Especially in our age, in which there is real proficiency in the handling of facts in an outer experimental way, especially now if we are not to be drawn into superficiality, into externalities, what is investigated experimentally must be permeated with the results of spiritual research. This consciousness is opposed to the more mathematical type of consciousness that gives the researcher such a feeling of extraordinary sureness. If one wishes to examine the different ways of research, one might ask oneself the question: How does one actually know things mathematically when one applies mathematics to the facts of the outer sense-accessible world? And what distinguishes this mathematical approach from other modes of dealing with the facts given to us?

Let us start with the fact that the outer objects and events of the world are given to man through his senses. From childhood on, the outer factual world presents itself to us as a kind of chaos. But as time passes we strengthen ourselves inwardly with all kinds of mental images and concepts. (I have set this forth in detail in my booklet Truth and Science.) Through the process of making mental pictures of the outwardly perceived world, we take what may lie far apart in observation and we bring the mental pictures of these observations close together within us. Through this activity we thus create, in our mental life, a certain order in what otherwise is chaotic in the purely sense-perceptible.

We must, however, look very exactly at how we treat the perceptual facts of the world when we do not use our mathematical knowledge. We might ask what happens when we simply observe the outer world and make mental pictures about the connections between the observable facts—for instance, when we use the familiar law of cause and effect. We must acquire some thoughts about what we are doing when we simply observe the facts of the outer world. What do we really do when we bring order into the sense-perceptible chaos? It appears to me that in relation to this question David Hume has spoken quite correctly; however, his fault lies in that he has taken to apply to the universal field of human cognition what is meant only for this particular field, namely, the “observation of outer nature free of mathematics.”

Most errors and one-sidednesses are based an the application of very correct thinking in one field to the totality of human cognition. This makes it so difficult to take the assertions considered to be universally true. Arguments can be raised for the universal truth being applicable to specific areas, and arguments can also be raised for the opposite point of view. David Hume says: We observe the outer world and we arrange it in a lawful way through our own mental pictures. However, what we then have in our soul as law is not directly representative of something in the objective world. We cannot say that the outer world is always going to follow the course predicted by such a law. We can only say, according to David Hume, that until today we have been able to see the sun rise every morning. That is a statement that fits the facts. We can put these facts into the form of a general law. But in doing so we have no guarantee that we have anything other than a series of events that have happened in the past, of which we made a comprehensive mental picture. What is it really in us that brings about these lawful connections between the sense-perceptible occurrences? What kind of significance do these lawful connections have for the field which we are considering? Is David Hume correct when he says: It lies in the habit of our souls to gather together in a lawful manner the facts as they present themselves to us and, because we respond to this soul habit, we create for ourselves various natural laws? These natural laws are nothing else than what has been gathered together from individual facts through habit of our souls.

Thus one can say: Above all, man develops a practical life by bringing order and harmony into the otherwise chaotic stream of everyday facts; and the more one advances in this knowledge, in this special kind of knowledge, the more one inclines to this characteristic soul habit. This being the situation, one is not inclined to preserve individual phenomena as such; one wants to respond to the soul habit of bringing into uniformity what faces one as sense-perceptible, empirical manifoldness. If one is honest, one has to admit that all the knowledge obtained in this way stands as a closed door to the outer world in that it does not allow the essence of this outer world to enter our cognition. In this kind of cognition we must say: Out there are the material facts; we arrange them habitually into our system of mental pictures, and thus have a comprehensive view of them. We know when a series of facts have happened, that this series will happen a second time in a similar way when the same facts appear again before us. But as long as we remain in this field of knowledge, we cannot see through the outer appearances; we also, of course, do not claim to do so. When we want to present rash metaphysical hypotheses concerning matter, that it consists of this or that, we are attempting to change the state of affairs in which we do not deal with the material itself. We say to ourselves: We cannot see through matter to find out what it really is in its inner being, so what we are inclined to do is to arrange sequences of mental pictures and put these in the form of laws.

By doing so, we remain outside what appears as outer reality; we only create pictures of the external material happenings. Basically, we need this kind of knowledge to maintain our normal human consciousness, and to this end, we concern ourselves with these pictures. Try to think for a moment what it would mean for human consciousness if we were not able to give ourselves up to the kind of knowledge consisting only of pictures of the external world—if every time we wished to know something of the outer world, this world had to flow into us, as it does when we eat or drink, if it had to become part of our soul's apprehension before we could know anything. Just imagine how incompatible such a uniting of the material existence and our inner life would be with what our soul-constitution must be in acquiring knowledge of the outer world! We are in the position where we must tell ourselves: In our activity of knowing, nothing flows into our soul life from the outer world; we form pictures of what we experience in the outer world and these pictures really have nothing to do with the outer world.

Permit me to make an analogy out of the field of art to explain what I have been saying. Suppose I am painting something. The outer world is completely unconcerned about anything I might paint on a canvas. Take, for example, a couple of trees we see out there of which, let's say, I have painted a likeness on a canvas: the trees are completely indifferent as to how I have painted them, or if I do paint them. My picture is added to what is out there as something foreign, something that has nothing directly to do with that outer reality. In the field of theoretical and psychological knowledge it is basically the same as I have just described with the example of painting. If we were not separated from the world as just described, and were to take the content of the world into our soul in a way similar to when we eat or drink, our soul would grow together with, be one with, the world around us, and we would be unable to distinguish ourselves from our surroundings.

We will take up the subject of human freedom at a later time and show that it can only be understood if the way of knowing the material world is as I have characterized it.

This, however, is not so when I know something mathematically. Let's start by imagining how you know something of a mathematical nature, whether it is in the field of arithmetic, algebra, higher mathematics, or in the field of analytical or synthetic geometry. There we are not confronted by an outer world, we live directly and immediately in the objects of our mathematical knowledge. We form mathematical objects inwardly with all their interconnections and relationships, and when at times we sketch these forms, it is only for our own ease and comfort. What we refer to as mathematical is never some part of the outer world which we perceive with the senses, it is always something inwardly constructed. It is something that only lives in the part of our soul life that is not concerned with the senses as such. We build up, we inwardly construct, the mathematical content of our soul. There is a radical difference between the field of knowledge concerned with the empirical outer world presenting itself to the senses and that of the mathematical. In the external given world the objects of our knowledge remain strictly outside of us. In mathematical knowledge we stand with our whole soul within the objects of our knowledge, and what is observed as substance is the result of an experience in our soul of what we ourselves constructed.

Here we have a significant problem which forms, as it were, the first stage to what will be the next higher stage of considerations: How does one arrive at the anthroposophical spiritual science when starting from the familiar science of the present day? I don't believe anyone will be able to answer this question in a truly scientific way who cannot first answer the question: How is our knowledge of a purely observational kind raised to the kind of knowledge of nature that is permeated with mathematics?—how is this knowledge related to mathematical knowledge as such?

Now a further question arises which the scientist can answer himself, out of his own experience with scientific work. I have already mentioned what Kant called our attention to, that in every science there is only so much knowledge as there is mathematics contained in it. And, I repeat, this is a one-sidedness, because it is only applicable to a certain field. Kant's error lies in the fact that he takes a specialized truth and tries to make it into a universal law. We have a tendency not to want to leave the facts alone as they are presented to us, but rather to color them with what we have created as mathematical formula, so that we may measure and compare them.

What really lives in us when we strive in this direction, when we don't want to remain standing still, habitually combining the outer facts with general rules, when we permeate the given facts with what we have formulated in full consciousness mathematically as objects in our soul life? It is clear that anyone who has experience in the field of objective observation will admit that the whole of nature surrounding his own being is felt, in regard to its materiality, as something foreign. Please notice that, in a sense, we can submerge ourselves into what we feel as a foreign material element, with the help of what we have ourselves inwardly constructed as mathematical formulas. What we describe in a mathematical way actually seems as if what happens in nature has occurred according to the mathematical formula that we have constructed. What is at the basis of this perception? It is the fact that we desire above all else to become one with what we perceive at first as foreign surroundings. We group what is presented to us externally in order to be able to reconstruct it in the same way that we construct something in the purely mathematical realm. We strive to experience what presents itself to us externally in an inwardly exact manner.

This internalization of the outer world with the wish to experience exactness is what motivates a mathematical explanation of nature. This is especially characteristic of our present-day scientific efforts in the direction of technology. Today's science has an intense longing to penetrate outer occurrences with mathematical concepts. This means that we bring something we have created in our own soul out into what presents itself to us in raw perception. We do this so that we may understand what is perceived, but in doing so we can have the impression that the outer occurrence actually proceeds in the way we portray it mathematically. When we have gone so far that we have achieved this ideal, as we have in the field of optics and light theory, where every phenomenon is represented in terms of a formula, what really have we done? What really is the content of our soul when instead of plain external appearances a sum of mathematical formulas seem to present themselves? What does our soul receive from this? We look at this edifice, the world portrayed as mathematical relationships, and then we turn our gaze to the actual outer world and we find something strange. We find that all that we look at, all that we consider outer material world, appears inwardly dark until it is brightened by the introduction of mathematical concepts. But at the same time we cannot deny the fact that the picture we have created of the outer world no longer contains reality, no longer the reality which presented itself to us originally.

Take, for example, optical appearances, the whole field as it presents itself to our eyes; contrast this with what we have, to a certain extent, correctly constructed as mathematical geometric optics, full of rules. If one uses just a little objectivity, it is clear that in what is constructed as a mathematical picture there is nothing left of the abundance of color. Everything that our senses first offered us, namely, actual outer reality, has been pressed out of the picture. The picture of the outer world is in sharp contrast to what is really out there; it lacks reality, it lacks the tremendous abundance that actually exists in the world.

In the coming lectures I will be speaking of a comparison, that to begin with I would like you to consider as an analogy. When we permeate empirical facts with mathematics, our activity consists of two stages: First we must look at the empirical facts, let's say the facts of the eye. The second is the arrangement of these percepts into mathematical formulas. In a certain way, as a result of this we have essentially an experience of mathematical formulating. We no longer view the empirical world of phenomena. This process can be compared to our inhaling life-sustaining oxygen; we saturate our whole organism with it. The oxygen then combines with carbon and we exhale carbon dioxide, which is no longer the life-sustaining air. But the combined process was necessary for our inner life. We had to inhale the life-strengthening oxygen and combine it with something in us. What is produced in this way is something killing; we can contrast it with what was inhaled, which was life-sustaining.

For the time being, this should only be considered as a picture of the way in which we pursue the knowledge of nature. We take something into ourselves that is presented to the senses and try to unite it intimately with something we produce only in ourselves, with mathematical construction. We feel that something is created by this union. Nature is not contained in what we have created; the living quality is not there, just as the life force is no longer in the air we exhale. We can say that our perception of the outer world is like an inhaling by the soul of what then is changed into the opposite. If one looks closely at this process of striving for mathematical knowledge of nature, it is proof of the fact that mathematical knowledge is something completely different from the merely perceptual knowledge of nature. This mere perceptual knowledge of nature contrasts with the habitual state of our soul, which consists of a feeling of competence derived from the use of inwardly formed mathematical knowledge. This state of soul wishes to have something that will explain the outer world in accordance with our own being, to unite something inner with something outer.

When one realizes how the longing for mathematical explanations of nature are based on this soul habit of longing to take inner possession of the outer world, then it will also be clear that what one attains by this is completely different from the content of sense experience. One goes more deeply into human inner life with mathematical knowledge. One believes that one gets correspondingly closer to the outer world through an inner representation of the nature of the outer world. One has an inner experience of what has been changed into mathematical formulas; at the same time, one has basically lost the fullness of the outer world. One must, however, be conscious of the fact that what the outer world has given has been connected with something constructed purely inwardly.

One must really experience what goes on in one's soul when one makes mathematical formulas; one must experience this correctly. One must see that a mathematical formula actually is constructed within us. One must realize that this inner human construction has been achieved apart from the outer world, and yet in a sense it has brought one closer to the outer world. Even so, this inner mathematical construction cannot be regarded as inner reality as compared to what we find in the outer world. If this were not true, we would have the feeling that this mathematical construction contained true reality instead of a bland version of the outer world which it does actually present to us. Think what the situation would be if in our spiritual contemplation of a mathematical construction we had the whole content of the eyes' original experience in all its color intensity. If this were the case, we would experience in the formula itself the lighting up, the intensity of colors, when considering the wave theory, or “interference phenomena,” in mathematical form. This we certainly do not see. The fact that we do not see this proves that with our mathematical formulas we penetrate only to some degree into the outer world. We do come closer to it, but at the same time we no longer have the full reality of it.

We have shown a progression from an ordinary sense-based knowledge to a knowledge of inner mathematical construction. The question then arises: Can this progression be continued further in human soul life? First, we have an outer world before us; then we confront it in such a way that the laws which we create, based on observation, are entirely different from it in form. We go through this and we can do so because we become inwardly separated from the outer world. We are inwardly completely separated from the outer world while experiencing these mathematical formulas. We do gain a certain penetration through these mathematical formulas, but it is obvious that they are not filled with reality or we would see the whole outer reality recreated in the formulas.

When we take a closer look we see that not only are they not real in themselves but in fact they have the effect of destroying reality. The question now arises: would it be possible to strengthen our capacity to make these inner mathematical constructions by which we then penetrate the sense-perceptible world? Is it possible that what is first experienced mathematically as pale abstractions can be made stronger? In other words, could the force which we have to use to attain a mathematical knowledge of nature be used more effectively?—with the result not just a mathematical abstraction, but something inwardly, spiritually concrete? In that case, we would not just see a re-created version of the outer world or an abstract mathematical picture, but we would have something formed in an entirely different manner. We would have gained something with the full character of reality, but obtained similarly to the way we obtain mathematical pictures. We would then have before us spiritually a reality that shines out toward us in the same way that the outer sense-perceptible world streams toward us. But we would have this from pictures filled with reality, not from mathematically abstract pictures. We would have lifted ourselves, through strengthening our mathematical capacity, to a higher level, and in doing so we would reveal more of our own inner reality. This we can see as a third step in our attainment of knowledge. The first step would be the familiar grasping of the real outer world. The second step would be the mathematical penetration of the outer world, after we have first learned inwardly to construct the purely mathematical aspect. The third would be the entirely inner experience, like the mathematical experience but with the character of spiritual reality.

So we have before us: The ordinary outer empirical knowledge of nature, then mathematical knowledge, and finally, spiritual knowledge. We have, as the last step, through an inwardly creative activity, spiritual worlds before us .

As preparation for viewing these worlds as real, we start by creating mathematical, pictorially-abstract elements. We use this mathematics in relation to the outer world, but if we are honest we must say: What we construct mathematically is still not a reality in itself; it does not bring reality up out of the depths of our souls, rather it is a picture of reality. In spiritual science we gain the ability to bring out of the depths of our souls what is not just a picture of the outer existence, but reality itself, true reality.

The three levels of human knowledge are: Knowledge of physical nature, mathematical knowledge, and spiritual knowledge. This is not just taking spiritual science out of thin air with the purpose of constructing a spiritual science method; rather, it arises naturally. Starting from merely empirical research we come to a mathematical approach, and the continuation of this leads us to study an anthroposophically-oriented spiritual science.

This, my dear friends, is what I wanted to say today as an introduction to this course of lectures. I wanted to show you that this anthroposophical spiritual science knows where its place is in the whole system of sciences. It is not born out of some kind of subjective caprice, some kind of dilettantism; it is born out of an exact theory of knowledge. It is born out of the knowledge that must be used even to understand the correct use of mathematics. It was not for nothing that Plato demanded of his pupils that they must first of all have a good grounding in the knowledge of geometry and mathematics. Plato did not require an arithmetical or geometric knowledge of some particular kind, but rather a sound understanding of what really happens in a man when he does mathematics or geometry. This is based an a seemingly paradoxical but deeply meaningful saying of Plato: “God geometrizes.” He did not mean by this that God just created with mathematics, or with five- or six-sided figures; rather, He creates with the force of which we can only make pictures to ourselves, in our mathematical abstract thinking. Therefore I believe that he who understands the place of mathematics in the whole field of the sciences, will also understand the correct place of spiritual science. Spiritual science will battle for its right to exist, no matter what adversaries it may have, for it builds on an exact foundation thoroughly in accord with historical necessity. Therefore I can say: We welcome any and all opponents who will seriously enter into what spiritual science has to say; we welcome any serious dialogue. Spiritual science has no fear of opposition because it is well supplied with all the scientific weapons of ordinary science and it knows how to use them. It would only not like to be continuously interrupted by those who don't understand it, due to their dilettantism and uninformed opinions. Spiritual science as we mean it here is actually a necessity for the other special sciences. The borders of these other special sciences must be crossed over with the help of spiritual science. We must inwardly resolve at least to confront those who, without reason, oppose this spiritual science, and sometimes even be a bit rude with them. There is a fundamental need for humanity to adopt this spiritual science as quickly as possible, and in all seriousness. This can really happen if only we bring good will to the understanding of it.