Cosmic-Earthly-Human Metamorphoses; the necessity of qualitative mathematics for comprehending celestial phenomena

Astronomy in Relation to the Other Scientific Disciplines. Lecture 10.

Rudolf Steiner, Stuttgart, January 10, 1921:

My Dear Friends,

Taking my start yesterday from certain considerations in the realm of form, I showed how the connections should be thought of between the processes of the human metabolic system and the processes of the head, the nervous system, or whatever you wish to call it in the sense of the indications given in my book ‘Riddles of the Soul’ (“Von Seelenratseln”).

It would be regarded as quite out of the question to study the movements of a magnet-needle on the Earth's surface in such a way as to try to explain these movements solely out of what can be observed within the space occupied by the needle. The movements of the magnet-needle are, as you know, brought into connection with the magnetism of the Earth. We connect the momentary direction of the needle with the direction of the Earth's magnetism, that is, with the line of direction which can be drawn between the north and south magnetic poles of the Earth. When it is a question of explaining the phenomena presented by the magnetic needle, we go out of the region of the needle itself and try to enter, with the facts that have been collected towards an explanation, into the totality which alone affords the opportunity to explain phenomena, the manifestations of which belong to this totality. This rule of method is certainly observed in regard to some phenomena, — to those, I should say, the significance of which is fairly obvious. But it is not observed when it is a question of explaining and understanding more complicated phenomena.

Just as it is impossible to explain the phenomena of the magnetic needle from the needle itself, it is equally and fundamentally impossible to explain the phenomena relating to the organism from out of the organism itself, or from connections which do not belong to a totality, to a whole. And just for this reason, because there is so little inclination to reach the realm of totalities in order to find explanations, we arrive at those results put forward by the modern scientific method in which the wider connections are almost entirely left out of the picture. This method encloses the phenomena, whatever they may be, within the field of vision of the microscope; while the celestial phenomena are restricted to what is observable externally, with the help of instruments. In seeking for explanations, no attempt is made to consider the necessity of reaching out to the surrounding totality within which a phenomenon is localised. Only when we become familiar with this quite indispensable principle of method are we in a position to bring our judgment to bear upon such things as I was describing to you yesterday. Only in this way shall we grow able to estimate how such realms of phenomena as are met within the human organism will appear, when truly recognised in the totality to which they properly belong.

Remember what I described at the very beginning of this course of lectures. I drew your attention to the fact that the principle of metamorphosis as it appeared first in the work of Goethe and Oken must be modified if it is truly to be applied to man. The attempt was made — and it was made with genius on the part of Goethe — to derive the formation of the bones of the skull from that of the vertebrae. These investigations were continued by others in a way more akin to 19th-century method, and the progress of the method of investigation (I will not now decide whether it was a step forward or not) can be studied by comparing how this problem of the metamorphosis of one form of bone into another was conceived on the one hand by Goethe and Oken and on the other, for example, by the anatomist Gegenbauer.

These things are only to be set on a real basis if one knows (as I said, I have already mentioned this in the course of these lectures, but we will now link on to it again) how two types of bone in the human organism (not the animal, but the human organism), most widely separated from the point of view of their morphology, are actually related to one another. Bones far removed from one another in the aspect of their form would be a tubular or long bone — femur or humerus, for example, — and a skull-bone. To make a superficial comparison, without really entering into the inner nature of the form and bringing a whole range of phenomena into connection with it, is not enough to reveal the morphological relationship between two polar opposite bones — polar opposite, once more, in regard to their form. We only begin to perceive it if we compare the inner surface of a tubular bone with the outer surface of a skull-bone. Only thus do we get the true correspondence (Fig. 1) which we must have in order to establish the morphological relation. The inner surface of the tubular bone corresponds morphologically to the outer surface of the skull-bone. The skull-bone can be derived from the tubular bone if we picture it as being reversed, to begin with, according to the principle of the turning-inside-out of a glove. In the glove, however, when I turn the outer surface to the inside and the inner to the outside, I get a form similar to the original one. But if in the moment of turning the inside of the tubular bone to the outside, certain forces of tension come into play and mutual relationships of the forces change in such a way that the form which was inside and has now been turned outward alters the shape and distribution of its surface, then we obtain, through inversion on the principle of the turning-inside-out of a glove, the outer surface of the skull bone as derived from the inner surface of the tubular bone. From this you can conclude as follows. The inner space of the tubular bone, this compressed inner space, corresponds in regard to the human skull to the entire outer world. You must consider as related in their influence upon the human being: The outer universe, forming the outside of his head, and what works within, tending from within toward the inner surface of the tubular bone. These you must see to belong together. You must regard the world in the inside of the tubular bone as a kind of inversion of the world surrounding us outside.

Fig. 1

There, for the bones in the first place, you have the true principle of metamorphosis! The other bones are intermediary forms; morphologically, they mediate between the two opposite extremes, which represent a complete inversion, accompanied by a change in the forces determining the surface. The idea must however be extended to the entire human organism. In one way, it comes to expression most clearly in the bones; but in all the human organs we must distinguish between two opposing factors, — that which works outward from an unknown interior, as we will call it for the moment, and that which works inward from without. The latter corresponds to all that surrounds us human beings on the planet Earth.

Fig. 2

Fig. 3

The tubular bone and the skull-bone represent indeed a remarkable polarity. Take the tubular bone and think of this centre-line (Fig. 2). This line is in a way the place of origin of what works outward, in a direction perpendicular to the inner surface of the bone (Fig. 3). If you now think of what envelops the human skull, you have what corresponds to the central line of the tubular bone. But how must you draw the counterpart of this line? You must draw it somewhere as a circle, or more exactly, as a spherical surface, far way at some indeterminate distance (Fig. 4). All the lines which can be drawn from the centre-line of the tubular bone towards its inner surface (Fig. 3) correspond, in regard to the skull-bone, to all the lines which can be drawn from a spherical surface as though to meet in the centre of the Earth (Fig. 4). In this way you find a connection — approximate, needless to say — between a straight line, or a system of straight lines, passing through a tubular bone and bearing a certain relation to the vertical axis of the body, the direction of which coincides, in fact, with that of the Earth's radius and a sphere surrounding the Earth at an indeterminate distance. In other words, the connection is as follows. The radius of the Earth has the same cosmic value in regard to the vertical posture of the human organism, perpendicular to the surface of the Earth, as a spherical surface, a cosmic spherical surface, has in regard to the skull organisation. This, however, is the same contrast which you experience within yourself if you make yourself aware of the feeling of being inside your own organism and of experiencing the outer world at the same time. This is the polarity you reach if you compare your feeling of self — that feeling of self which is really based on the fact that in normal life you can depend upon your bodily organisation, that you do not become giddy, but keep a right relation to the force of gravity — with all that is present in your consciousness in connection with what you see around you through the senses, even as far away as the stars.

Fig. 4

Putting all this together, you will be able to say: There is the same relation between this feeling of being in yourself and the feeling of consciousness you have in perceiving the outer world as there is between the structure of your body and of your skull. We are thus led to the relationship between what we might call: Earthly influence upon man, of such a character that it works in the direction of the Earth's radius, and what we might call: The influence which makes itself felt in the entire circumference of our life of consciousness, and which we must look for in the sphere, in what really is for us the inner wall, the inner surface, of a hollow sphere. This polarity prevails in our normal day-waking conscious life. It is this polarity which, roughly speaking — if we leave out of account what is in our consciousness as a result of observing our earthly environment — we may look upon as the contrast between the starry sphere and earthly consciousness, earthly feeling of ourselves, — Earth-impulse living in us. If we compare this impulse of Earth, this radial Earth-impulse, to our consciousness of the vast sphere, — if we observe how this polarity prevails in normal waking consciousness — we shall perceive that it is always there, living in us, playing its part in our conscious life. We live far more in this polarity than we are wont to think. It is always present and we live within it. The connection between the forming of mental images and the life of will can be really studied in no other way than by considering the contrast between ‘sphere’ and ‘radius’. In psychology, too, we should come to truer results with regard to the connection of our world of ideas and mental pictures, manifold and extensive as it is, with the more unified world of our will, if a similar relationship were sought between them as is symbolised in the relation of the surface-area of a sphere with the corresponding radius.

Now, my dear friends, let us look at all this which is at work in our day-waking consciousness, forming the content of our soul-life, let us now consider how it takes its course when we are in quite a different situation. In effect, how does it work upon us during the time of the embryonic life? We can well imagine — indeed we must imagine — that the same polarity will be at work here too, only in another way. During the embryonic period we do not direct towards the outer world the same activity which afterwards dims down this polarity to a pictorial one; at this time, the polarity affects all that is formative in our organisation in a much more real way than when, in picture form, it becomes active in our life of mind and soul. If therefore we project the activity of consciousness back in time to the embryonic period, then one might say that in the embryonic life we have what we otherwise have in the activity of consciousness, but we have it at a more intensive, more realistic stage. Just as we clearly see the relation of sphere and radius in our consciousness, so to reach any real result, we must look for this same polarity of heavenly sphere and earthly activity in what happens in the embryonic life. In other words, we must look for the genesis of human embryonic life by finding a resultant between what takes place out in the starry world — an activity in the ‘sphere’ — and what takes place in man as a result of the radial Earth-activity.

What I have just described must be taken into account with the same inner necessity of method as the Earth's magnetism is in connection with the magnetic needle. There may be much that is hypothetical even in this, but I will not go into it now. I only wish to point out: We have no right to restrict our considerations to the embryo alone, — to explain the processes taking place within it simply out of the embryo itself. In just the same way as we have no right to explain the phenomenon of the magnet out of itself alone, so too, we have no right to explain the form and development of the embryo purely on the basis of the embryo itself. In attempting to explain the embryo we must take these two opposites into account. As we take the Earth's magnetism into account in connection with the magnet, so must we observe the polarity of sphere and radial activity, in order to understand what is developing in the embryo, — which, when the embryo is born, fades into the pictorial quality of the experience of consciousness. The point is, we must learn to see the relationship which exists in man between tubular or long bone and skull-bone in the other systems too — in muscle and nerve, and so on; — and when we do study this polarity, we are led out into the life of the Cosmos. Consider how closely related (as described in my book “Riddles of the Soul”) is the whole essence and content of the human metabolic system with what I have now characterised as being under the influence of the ‘radial’ element, and how closely related is the head system to what I have just described as being under the influence of the ‘sphere’. Then you will say: We must distinguish in the human being what conditions his sensory nature and what conditions his metabolic life; moreover, these two elements are related to one another as heavenly sphere to earthly activity.

We must therefore look for the product of the celestial activity in what we bear in our head organisation and for what unites to a resultant with this, the activity belonging to the Earth — tending, as it were, towards the centre of the Earth — in our metabolism. These two realms of activity and influence fall apart in man; it is as thought they represent two Ice Ages, and the middle realm, the rhythmic realm, mediates between them. In the rhythmic system we actually have something, — if I may so express myself, — which is a realm of mutual interplay between Earth and Heaven.

And now if we wish to go further, we must consider various other relationships which reveal themselves to us in the realm of reality. I will now draw your attention to something very intimately connected with what I have just been describing.

There is the familiar membering of the outer world which surrounds us and to which we as physical man belong; we divide it into mineral kingdom, plant kingdom, animal kingdom, and regard man as the culmination of this external world of Nature. Now, if we would obtain a clearer view of what we have described in connection with the working of the celestial phenomena, we must turn our attention to yet another thing.

It is not to be denied — it is indeed quite obvious to any prejudiced observer — that with our human organisation as it is now, in the present phase of the cosmic evolution of humanity, we are, in regard to our capacities of knowledge, entirely adapted to the mineral kingdom. Take the kind of laws we seek in Nature; and you will agree that we are certainly not adapted to all aspects of our environment. To put it curtly, all that we really understand is the mineral kingdom. Hence all the efforts to refer the other kingdoms of Nature back to the laws of the mineral domain. After all, it is because of this that such confusion has arisen with regard to mechanism and vitalism. To the ordinary view which is ours toady, life remains either a vague hypothesis, as it was in earlier times, or else its manifestations are explained in terms of the mechanical, the mineral. The ideal, to reach an understanding of life, is unaccompanied by any recognition of the fact that life must be understood as life; on the contrary, the fundamental aim is to refer life back to the laws of the mineral realm. Precisely this betrays a vague awareness of the fact that man's faculties of knowledge are only adapted to understand the mineral kingdom and not the plant nor animal.

Now when we study on the one hand the mineral kingdom itself and on the other hand its counterpart, namely, our own knowledge of the mineral kingdom, in that these two correspond to one another, we shall be compelled, — since as explained just now we must relate all our life of knowledge to the heavenly sphere, also to bring into connection with the heavenly sphere, in some way, that to which our knowledge is related, namely the mineral kingdom. We must admit: In regard to our head organisation, we are organised from the celestial sphere; therefore what underlies the forces of the mineral kingdom must also be organised from the celestial sphere in some way. Compare then what you have to your sphere of understanding — the whole compass of your knowledge of the mineral kingdom — with what is actually there in the mineral kingdom in the outer world, and you will be led to say: What is thus within you relates to what is in the mineral kingdom outside you, as picture to reality.

Now we must think of this relationship more concretely than in the form of picture and reality, and we are helped to do so by what I said before. Our attention is drawn to what underlies the human metabolic system and to the forces active there, forces which are connected with the pole of earthly activity, typified by the radius. In seeking for the polar opposite, within ourselves, to that part of our organisation which forms the basis for our life of knowledge, we are directed from the encompassing Sphere to the Earth. The radii converge to the middle point of the Earth. In the radial element we have something by which we feel ourselves, which gives us the feeling of being real. This is not what fills us with pictures in which we are merely conscious; this is what gives us the experience of ourselves as a reality. When we really experience this contrast, we come into the sphere of the mineral kingdom. We are led from what is organised only for the picture to what is organised for the reality. In other words: In connection with the cause and origin of our life of knowledge, we are led to the wide, encompassing sphere, — we conceive it in the first place as a sphere, — whereas, in following the radii of the sphere towards the middle of the Earth, we are led to the middle point of the Earth as the other pole.

Thinking this out in more detail, we might say: Well, according to the Ptolemaic conception for example, out there is the blue sphere, on it a point (Fig. 5) — we should have to think of a polar point in the centre of the Earth. Every point of the sphere would have its reflected point in the Earth's centre. But, of course, it is not to be understood like that. (I shall speak more in detail later on; to what extent these things correspond exactly is not the question for the moment.) The stars, in effect, would be here (Fig. 6). So that in thinking of the sphere concentrated in the centre of the Earth, we should have to think of it in the following way: The pole of this star is here, of this one here, and so on (Fig. 6). We come, then, to a complete mirroring of what is outside in the interior of the Earth.

Fig. 5

Fig. 6

Picturing this in regard to each individual planet, we have, say, Jupiter — and then a polar Jupiter within the Earth. We come to something which works outward from within the Earth in the way that Jupiter works in the Earth's environment. We arrive at a mirroring (in reality it is the opposite way round, but I will now describe it like this), a mirroring of what is outside the Earth into the interior of the Earth. And if we see the effect of this reflection in the forms of the minerals then we must also see the effect of what works in the cosmic sphere itself in forming our faculty of understanding the minerals. In other words: We can think of the whole celestial sphere as being mirrored in the Earth: We conceive the mineral kingdom of the Earth as an outcome of this reflection, and we conceive that what lives within us, enabling us to understand the mineral kingdom, comes from what surrounds us out in the celestial space. Meanwhile the realities we grasp by means of this faculty of understanding come from within the Earth.

You need only follow up this idea and then cast a glance at man, at the human countenance, and, if you really look at this human countenance, you will hardly be able to doubt that in it something is expressed of the celestial sphere, and that there also appears in it what is present as pictorial experience in the soul, namely the forces which rise up into the realm of soul activity from the realm of bodily activity, after having been at work more intensively in this bodily realm during embryonic life. Thus we find a connection between what is outside us in outer reality, and our own organisation for the understanding of this outer reality. We can say: The cosmos produces the outer reality, and our power to understand this outer reality is organised physically by virtue of the fact that the cosmic sphere is only active in us now for our faculty of knowledge. Therefore we must distinguish, in the genesis of the Earth as well, between two phases: One in which active forces work in such a way that the real Earth itself is created, and then a later phase of evolution, in which the forces work so as to create the human faculty for understanding the realities of the Earth.

Only in this way, my dear friends, do we really come near to an understanding of the Universe.

You may say: Well and good, but this method of understanding is less secure than the method used today with the aid of microscope and telescope. It may be that to some people it appears less secure. But if things are so constituted that we cannot reach the realities with the methods in favour today, then we are faced with the absolute necessity of comprehending the reality with other modes of understanding and we shall have to get used to developing those other methods. It is of no avail to say that you will have nothing to do with such lines of thought, since they appear too uncertain. What if this degree of certainty alone were possible! However, if you really follow up this line of thought, you will see that the degree of certainty is just as great as in your conception of a real triangle in the outer world when you take hold of it in thought with the inner idea of construction of a triangle. It is the same principle, the same manner of comprehending outer reality, in the one case as in the other. This should be borne in mind.

Certainly, the question arises: Taking these thoughts, as I have here developed them, it is possible to become clear in a general way about such connections, but how can one reach a more definite comprehension of these things? For only in a much more definite form can they be of use in helping us to grasp the realm of reality. In order to go into this, I must draw your attention to something else.

Let us return to what I said yesterday, for example, in regard to the Cassini curve. We know that this curve has three, or, if you like, four forms. You remember, the Cassini curve is determined as follows. Given two points A and B, I will call the distance between them 2a; then any point of the curve will be such that AM — MB = b2, that is, a constant. And I obtain the various forms of the Cassini curve according to whether a, that is, half the distance between the foci, is greater than, equal to, or less than b. I obtain the lemniscate when a = b, and the discontinuous curve when a is greater than b.

Imagine now that I wanted not only to solve this geometrical problem, assuming two constant magnitudes a and b and then setting up equations to determine the distances of M from A and B. Suppose I wanted to do more than this, namely, to move in the plane from one form of line or curve to another by treating as variable magnitudes those magnitudes which remain constant for a particular curve. In the picture, after all, we only envisaged certain limiting positions with a greater or smaller than b. Between these there are an infinite number of possibilities. I can pass over quite continuously to the construction of one form of the Cassini curve after another. And I shall obtain these different forms if, let us say, to the variability of the first order, say between y and x, I add a variability of the second order; that is, if I allow my construction of the curves as they pass over from one to the other continuously to take its course in such a way that a remains a function of b.

What am I doing when I do this? I am constructing curves in such a way that I create a continuous, moving system of Cassini curves passing over via the lemniscate into the discontinuous forms, not at random, but by basing it on a variability of the second order, in that I bring the constants of the curves themselves into relationship with one another so that a is a function of b, a = φ(b). Mathematically, it is of course perfectly feasible. But what do we obtain by it? Just think: by means of it I obtain the condition for the character of a surface such that there is a qualitative difference even mathematically speaking, in all its points. At every point another quality is present. I cannot comprehend the surface obtained like this in the same way as I comprehend some abstract Euclidean plane. I must look upon it as a surface which is differentiated within itself. And if by rotation I create three-dimensional forms then I should obtain bodies differentiated within themselves.

If you think of what I said yesterday, namely, that the Cassini curve is also the curve in which a point must move in space if, illuminated from a point B, it reflects the light to a point A with constant intensity; and if you also bear in mind that the constancy underlying the curve here brings about a relation between the effects of light at different points; then, just as in this instance certain light-effects result from the relation of the constants, so one can also imagine that a system of light-effects would follow if a variability of the second order were added to the variability of the first. In this way you can create, even in mathematics itself, a process of transition from the quantitative to the qualitative aspect.

These attempts must indeed be made in order to find a way of transition from quantity to quality, — and this endeavour we must not abandon. For a start can be made from what it is that we are really doing when we form an inner connection between the function within the variability of the second order and the function within variability of the first order. (It has nothing to do with the expression “order”, as it is familiarly used; but you will understand me, as I have explained the whole thing from the beginning.) By turning our attention to this relationship between what I have called first and second order, we shall gradually come to see that our equations must be formed differently according to whether we are taking into account, for example, what in an ordinary bodily surface lies between the surface and our eye, or what lies behind the surface of the body. For a relationship not unlike this between the variabilities of the first order and of the second order exists between what I must consider as being between myself and the surface of a quite ordinary body and what lies behind the surface of the body. For example, suppose we are trying to understand the so-called reflection of the rays of light, — what we observe when there is a reflecting surface. It is a process taking place, to begin with, between the observer and the surface of the body. Suppose that I conceive this as a confluence of equations taking their course between me and the surface of the body in a variability of the first order, and then, in this connection consider what is at work behind the surface so as to bring about the reflection as an equation in the variability of the second order. I shall arrive at quite other formulae than are now applied according to purely mechanical laws, — omitting phases of vibration and so on — when dealing with reflection and refraction.

In this way the possibility would be reached of creating a form of mathematics capable of dealing with realities; and it is essential for this to happen, if we would find explanations particularly in the realm of astronomical phenomena. In regard to the external world, we have before us what takes place between the surface of the Earth-body and ourselves. When, however, we contemplate the celestial phenomena — say, a loop of Venus — trivially speaking we also have before us something which takes place between us and some other thing; yet the reality confronting us in this case is in fact like the realm beyond the sphere in its relation to what is within the central point. However we look to the phenomena of the heavens, we must recognise that we cannot study them simply according to the laws of centric forces, but that we must regard them in the light of laws which are related to the laws of centric forces as is the sphere to the radius.

If, then, we would reach an interpretation at all of the celestial phenomena, we must not arrange the calculations in such a way that they are a picture of the kind of calculations used in mechanics in the development of the laws of centric forces; but we must formulate the calculations, and also the geometrical forms involved, so that they relate to mechanics as sphere relates to radius. It will then become apparent (and we will speak about this next time) that we need: In the first place, the manner of thinking of mechanics and phoronomy which has essentially to do with centric forces, and secondly, in addition to this system, another, which has to do with rotating movements, with shearing movements, and with deforming movements. Only then, when we apply the meta-mechanical, meta-phoronomical system for the rotating, shearing, and deforming movements — just as we now apply the familiar system of mechanics and phoronomy to the centric forces and centric phenomena of movement — only then shall we arrive at an explanation of the celestial phenomena, taking our start from what lies empirically before us.

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