Friday, January 30, 2015
Revisioning the Concept of Space. Qualitative Mathematics, Cosmic Space, and Life on Earth
Rudolf Steiner, Stuttgart, January 7, 1921:
You will have seen how we are trying in these lectures to prepare the ground for an adequate world-picture. As I have pointed out again and again, the astronomical phenomena themselves impel us to advance from the merely quantitative to the qualitative aspect. Under the influence of Natural Science there is a tendency, in modern scholarship altogether, to neglect the qualitative side and to translate what is really qualitative into quantitative terms, or at least into rigid forms. For when we study things from a formal aspect we tend to pass quite involuntarily into rigid forms, even if we went to keep them mobile. But the question is, whether an adequate understanding of the phenomena of the Universe is possible at all in terms of rigid, formal concepts. We cannot build an astronomical world-picture until this question has been answered.
This proneness to the quantitative, abstracting from the qualitative aspect, has led to a downright mania for abstraction which is doing no little harm in scientific life, for it leads right away from reality. People will calculate, for instance, under what conditions, if two sound-waves are emitted one after the other, the sound emitted later will be heard before the other. All that is necessary is the trifling detail that we ourselves should be moving with a velocity greater than that of sound. But anyone who thinks in keeping with real life instead of letting his thoughts and concepts run away from the reality will, when he finds them incompatible with the conditions of man's co-existence with his environment, stop forming concepts in this direction. He cannot but do so. There is no sense whatever in formulating concepts for situations in which one can never be.
To be a spiritual scientist one must educate oneself to look at things in this way. The spiritual scientist will always want his concepts to be united with reality. He does not want to form concepts remote from reality, going off at a tangent, — or at least not for long. He brings them back to reality again and again. The harm that is done by the wrong kinds of hypothesis in modern time is due above all to the deficient feeling for the reality in which one lives. A conception of the world free of hypotheses, for which we strive and ought to strive, would be achieved far more quickly if we could only permeate ourselves with this sense of reality. And we should then be prepared really to see what the phenomenal world presents. In point of fact this is not done today. If the phenomena were looked at without prejudice, quite another world-picture would arise than the world-pictures of contemporary science, from which far-fetched conclusions are deduced to no real purpose, piling one unreality upon another in merely hypothetical thought-structures.
Starting from this and from what was given yesterday, I must again introduce certain concepts which may not seem at first to be connected with our subject, though in the further course you will see that they too are necessary for the building of a true world-picture. I shall again refer to what was said yesterday in connection with the Ice Ages and with the evolution of the Earth altogether. To begin with, however, we will take our start from another direction.
Our life of knowledge is made up of the sense-impressions we receive and of what comes into being when we assimilate the sense-impressions in our inner mental life. Rightly and naturally, we distinguish in our cognitional life the sense-perceptions as such and the inner life of ‘ideas’ — mental pictures. To approach the reality of this domain we must being by forming these two concepts: That of the sense-perception pure and simple, and of the sense-perception transformed and assimilated into a mental picture.
It is important to see without prejudice what is the real difference between our cognitional life insofar as this is permeated with actual sense-perceptions and insofar as it consists of mere mental pictures. We need to see these things not merely side by side in an indifferent way; we need to recognize the subtle differences of quality and intensity with which they come into our inner life.
If we compare the realm of our sense-perceptions — the way in which we experience them — with our dream-life, we shall of course observe an essential qualitative difference between the two. But it is not the same as regards our inner life of ideas and mental pictures. I am referring now not to their content but to their inner quality. Concerning this, the content — permeated as it is with reminiscences of sense-perceptions — easily deludes us. Leaving aside the actual content and looking only at its inner quality and character — the whole way we experience it — there is no qualitative difference between our inner life in ideas and mental pictures and our life of dreams. Think of our waking life by day, or all that is present in the field of our consciousness in that we open our senses to the outer world and are thereby active in our inner life, forming mental pictures and ideas. In all this forming of mental pictures we have precisely the same kind of inner activity as in our dream-life; the only thing that is added to it is the content determined by sense-perception.
This also helps us realize that man's life of ideation — his forming of mental pictures — is a more inward process than sense-perception. Even the structure of our sense-organs — the way they are built into the body — shows it. The processes in which we live by virtue of these organs are not a little detached from the rest of the bodily organic life. As a pure matter of fact, it is far truer to describe the life of our senses as a gulf-like penetration of the outer world into our body (Fig. 1) than as something primarily contained within the latter. Once more, it is truer to the facts to say that through the eye, for instance, we experience a gulf-like entry of the outer world. The relative detachment of the sense-organs enables us consciously to share in the domain of the outer world. Our most characteristic organs of sense are precisely the part of us which is least closely bound to the inner life and organization of the body. Our inner life of ideation, on the other hand — our forming of mental pictures — is very closely bound to it. Ideation therefore is quite another element in our cognitional life than sense-perception as such. (Remember always that I am thinking of these processes such as they are at the present stage in human evolution.)
Now think again of what I spoke of yesterday — the evolution of the life of knowledge from one Ice Age to another. Looking back in time, you will observe that the whole interplay of sense-perceptions with the inner life of ideation — the forming of mental pictures — has undergone a change since the last Ice Age. If you perceive the very essence of that metamorphosis in the life of knowledge which I was describing yesterday, then you will realize that in the times immediately after the decline of the Ice Age the human life of cognition took its start from quite another quality of experience than we have today. To describe it more definitely: while our cognitional life has become more permeated and determined by the senses and all that we receive from them, what we do not receive from the senses — what we received long, long ago through quite another way of living with the outer world — has faded out and vanished, ever more as time went on. This other quality — this other way of living with the world — belongs however to this day to our ideas and mental pictures. In quality they are like dreams. For in our dreams we have a feeling of being given up to, surrendered to, the world around us. We have the same kind of experience in our mental pictures. While forming mental pictures we do not really differentiate between ourselves and the world that then surrounds us; we are quite given up to the latter. Only in the act of sense-perception do we separate ourselves from the surrounding world. Now this is just what happened to the whole character of man's cognitional life since the last Ice Age. Self-consciousness was kindled. Again and again the feeling of the “I” lit up, and this became ever more so.
What do we come to therefore, as we go back in evolution beyond the last Ice Age? (We are not making hypotheses; we are observing what really happened.) We come to a human life of soul not only more dream-like than that of today, but akin to our present life of ideation rather than to our life in actual sense-perception. Now ideation — once again, the forming of mental pictures — is more closely bound to the bodily nature than is the life of the senses. Therefore what lives and works in this realm will find expression rather within the bodily nature than independently of the latter. Remembering what was said in the last few lectures, this will then lead you from the daily to the yearly influences of the surrounding world. The daily influences, as I showed, are those which tend to form our conscious picture of the world, whereas the yearly influences affect our bodily nature as such. Hence if we trace what has been going on in man's inner life, as we go back in time we are led from the conscious life of soul deeper and deeper into the bodily organic life.
In other works; before the last Ice Age the course of the year and the seasons had a far greater influence on man than after. Man, once again, is the reagent whereby we can discern the cosmic influences which surround the Earth. Only when this is seen can we form true ideas of the relations — including even those of movement — between the Earth and the surrounding heavenly bodies. To penetrate the phenomena of movement in the Heavens, we have to take our start from man — man, the most sensitive of instruments, if I may call him so. And to this end we need to know man; we must be able to discern what belongs to the one realm, namely the influences of the day, and to the other, the influences of the year.
Those who have made a more intensive study of Anthroposophical Science may be reminded here of what I have often described from spiritual perception: the conditions of life in old Atlantis, that is before the last Ice Age. For I was there describing from another aspect — namely from direct spiritual sight — the very same things which we are here approaching more by the light of reason, taking our start from the facts of the external world.
We are led back then to a kind of interplay between the Earth and its celestial environment which gave men an inner life of ideation — mental pictures — and which was afterwards transmuted in such a way as to give rise to the life of sense-perception in its present form. (The life of the senses as such is of course a much wider concept; we are here referring to the form it takes in present time.)
But we must make a yet more subtle distinction. It is true that self-consciousness or ego-consciousness, such as we have it in our ordinary life today, is only kindled in us in the moment of awakening. Self-consciousness strikes in upon us the moment we awaken. It is our relation to the outer world — that relation to it, into which we enter by the use of our senses — to which we owe our self-consciousness. But if we really analyze what it is that thus strikes in upon us, we shall perceive the following. If our inner life in mental pictures retained its dream-like quality and only the life of the senses were added to it, something would still be lacking. Our concepts would remain like the concepts of fantasy or fancy (I do not say identical with these, but like them). We should not get the sharply outlined concepts which we need for outer life. Simultaneously therefore with the life of the senses, something flows into us from the outer world which gives sharp outlines and contours to the mental pictures of our everyday cognitional life. This too is given to us by the outer world. Were it not for this, the mere interplay of sensory effects with the forming of ideas and mental pictures would bring about in us a life of fantasy or fancy and nothing more; we should never achieve the sharp precision of everyday waking life.
Now let us look at the different phenomena quite simply in Goethe's way, or — as has since been said, rather more abstractly — in Kizchhoff's way. Before doing so I must however make another incidental remark. Scientists nowadays speak of a “physiology of the senses”, and even try to build on this foundation a “psychology of the senses”, of which there are different schools. But if you see things as they are, you will find little reality under these headings. In effect, our senses are so radically different from one another that a “physiology of the senses”, claiming to treat them all together, can at most be highly abstract. All that emerges, in the last resort, is a rather scanty and even then very questionable physiology and psychology of the sense of touch, which is transferred by analogy to the other senses. If you look for what is real, you will require a distinct physiology and a distinct psychology for every one of the senses.
Provided we remember this, we may proceed. With all the necessary qualifications, we can then say the following. Look at the human eye. (I cannot now repeat the elementary details which you can find in any scientific textbook.) Look at the human eye, one of the organs giving us impressions of the outer world, — sense-impressions and also what gives them form and contour. These impressions, received through the eye, are — once again — connected with all the mental pictures which we then make of them in our inner life.
Let us now make the clear distinction, so as to perceive what underlies the sharp outline and configuration which makes our mental images more than mere pictures of fancy, giving them clear and precise outline. We will distinguish this from the whole realm of imagery where this clarity and sharpness is not to be found, — where in effect we should be living in fantasies. Even through what we experience with the help of our sense-organs — and what our inner faculty of ideation makes of it — we should still be floating in a realm of fancies. It is through the outer world that all this imagery receives clear outline, finished contours. It is through something from the outer world, which in a certain way comes into a definite relation to our eye.
And now look around. Transfer what we have thus recognized as regards the human eye, to the human being as a whole. Look for it, simply and empirically, in the human being as a whole. Where do we find — though in a metamorphosed form — what makes a similar impression? We find it in the process of fertilization. The relation of the human being as a whole — the female human body — to the environment is, in a metamorphosed form, the same as the relation of the eye to the environment. To one who is ready to enter into these things it will be fully clear. Only translated, one might say, into the material domain, the female life is the life of fantasy or fancy of the Universe, whereas the male is that which forms the contours and sharp outlines. It is the male which transforms the undetermined life of fancy into a life of determined form and outline. Seen in the way we have described in today's lecture, the process of sight is none other than a direct metamorphosis of that of fertilization; and vice-versa.
We cannot reach workable ideas about the Universe without entering into such things as these. I am only sorry that I can do no more than indicate them, but after all, these lectures are meant as a stimulus to further work. This I conceive to be the purpose of such lectures; as an outcome, every one of you should be able to go on working in one or other of the directions indicated. I only want to show the directions; they can be followed up in diverse ways. There are indeed countless possibilities in our time, to carry scientific methods of research into new directions. Only we need to lay more stress on the qualitative aspects, even in those domains where one has grown accustomed to a mere quantitative treatment.
What do we do, in quantitative treatment? Mathematics is the obvious example; ‘Phoronomy’ (Kinematics) is another. We ourselves first develop such a science, and we then look to find its truths in the external, empirical reality. But in approaching the empirical reality in its completeness we need more than this. We need a richer content to approach it with than merely mathematical and phoronomical ideas. Approach the world with the premises of Phoronomy and Mathematics, and we shall naturally find starry worlds, or developmental mechanisms as the case may be, phoronomically and mathematically ordered. We shall find other contents in the world if once we take our start from other realms than the mathematical and phoronomical. Even in experimental research we shall do so.
The clear differentiation between the life of the senses and the organic life of the human being as a whole had not yet taken place in the time preceding the last Ice Age. The human being still enjoyed a more synthetic, more ‘single’ organic life. Since the last Ice Age man's organic life has undergone, one might say, a very real ‘analysis’. This too is an indication that the relation of the Earth to the Sun was different before the last Ice Age from what it afterwards became. This is the kind of premise from which we have to take our start, so as to reach genuine pictures and ideas about the Universe in its relation to the Earth and man.
Moreover our attention is here drawn to another question, my dear friends. To what extent is ‘Euclidean space’ real? The name, of course, does not matter — I mean the space which is characterized by three rigid directions at right angles to each other. This, surely, is a rough and ready definition of Euclidean space. I might also call it ‘Kantian space’, for Kant's arguments are based on this assumption. Now as regards this Euclidean — or, if you will, Kantian — space we have to put the question: Does it correspond to a reality, or is it only a thought-picture, an abstraction? After all, it might well be that there is really no such thing as this rigid space. Now you will have to admit: when we do analytical geometry we start with the assumption that the X-, Y- and Z-axes may be taken in this immobile way. We assume that this inner rigidity of the X, Y and Z has something to do with the real world. What if there were nothing, after all, in the realms of reality, to justify our setting up the three coordinate axes of analytical geometry in this rigid way? Then too the whole of our Euclidean mathematics would be at most a kind of approximation to the reality — an approximation which we ourselves develop in our inner life, — a convenient framework with which to approach the reality in the first place. It would not hold out any promise, when applied to the real world, to give us real information.
The question now is, are there any indications pointing in this direction, — suggesting, in effect, that this rigidity of space cannot, after all, be maintained? I know that what I am here approaching will cause great difficulty to many people of today, for the simple reason that they do not keep step with reality in their thinking. They think you can rely upon an endless chain of concepts, deducing one thing logically from another, drawing logical and mathematical conclusions without limit. In contrast to this tendency in science nowadays, we have to learn to think with the reality, — not to permit ourselves merely to entertain a thought-picture without at least looking to see whether or not it is in accord with reality. So in this instance, we should investigate. Perhaps after all, by looking into the world of concrete things, there is some way of reaching a more qualitative determination of space.
I am aware, my dear friends, that the ideas I shall now set forth will meet with great resistance. Yet it is necessary to draw attention to such things. The theory of evolution has entered ever more into the different fields of science. They even began applying it to Astronomy. (This phase, perhaps, is over now, but it was so a little while ago.) They began to speak of a kind of natural selection. Then as the radical Darwinians would do for living organisms, so they began to attribute the genesis of heavenly bodies to a kind of natural selection, as though the eventual form of our solar system had arisen by selection from among all the bodies that had first been ejected. Even this theory was once put forward.
So too it came about that man was simply placed at the latter end of the evolutionary series of the animal kingdom. Human morphology, physiology etc. were thus interpreted. But the question is whether this kind of investigation can do justice to man's organization in its totality. For, to begin with, it omits what is most striking and essential even from a purely empirical point of view. One saw the evolutionists of Haekel's school simply counting how many bones, muscles and so on man and the higher animals respectively possess. Counting in that way, one can hardly do otherwise than put man at the end of the animal kingdom. Yet it is quite another matter when you envisage what is evident for all eyes to see, namely that the spine of man is vertical while that of the animal is mainly horizontal. Approximate though this may be, it is definite and evident. The deviations in certain animals — looked into empirically — will prove to be of definite significance in each single case. Where the direction of the spine is turned towards the vertical, corresponding changes are called forth in the animal as a whole. But the essential thing is to observe this very characteristic difference between man and animal. The human spine follows the vertical direction of the radius of the Earth, whereas the animal spine is parallel to the Earth's surface. Here you have purely spatial phenomena with a quite evident inner differentiation, inasmuch as they apply to the whole figure and formation of the animal and man. Taking our start from the realities of the world, we cannot treat the horizontal in the same way as the vertical. Enter into the reality of space — see what is happening in space, such as it really is, — you cannot possibly regard the horizontal as though it were equivalent or interchangeable with the vertical dimension.
Now there is a further consequence of this. Look at the animal form and at the form of man. We will take our start from the animal, and please fill in for yourselves on some convenient occasion what I shall now be indicating. I mean, observe and contemplate for yourselves the skeleton of an mammal. The usual reflections in this realm are not nearly concrete enough; they do not enter thoroughly enough into the details.
Consider then the skeleton of an animal. I will go no farther than the skeleton, but what I say of this is true in an even higher degree of the other parts and systems in the human and animal body. Look at the obvious differentiation, comparing the skull with the opposite end of the animal. If you do this with morphological insight, you will perceive characteristic harmonies or agreements, and also characteristic diversities. Here is a line of research which should be followed in far greater detail. Here is something to be seen and recognized, which will lead far more deeply into realty than scientists today are wont to go.
It lies in the very nature of these lectures that I can only hint at such things, leaving out many an intervening link. I must appeal to your own intuition, trusting you to think it out and fill in what is missing between one lecture and the next. You will then see how all these things are connected. If I did otherwise in these few lectures, we should not reach the desired end.
Diagrammatically now (Fig. 2), let this be the animal form. If after going into an untold number of intervening links in the investigation, you put the question: ‘What is the characteristic difference of the front and the back, the head and the tail end due to?’, you will reach a very interesting conclusion. Namely you will connect the differentiation of the front end with the influences of the Sun. Here is the Earth (Fig. 3). You have an animal on the side of the Earth exposed to the Sun. Now take the side of the Earth that is turned away from the Sun. In one way or another it will come about that the animal is on this other side. Here too the Sun's rays will be influencing the animal, but the earth is now between. In the one case the rays of the Sun are working on the animal directly; in the other case indirectly, inasmuch as the Earth is between and the Sun's rays first have to pass through the Earth (Fig. 3).
Expose the animal form to the direct influence of the Sun and you get the head. Expose the animal to those rays of the Sun which have first gone through the Earth and you get the opposite pole to the head. Study the skull, so as to recognize in it the direct outcome of the influences of the Sun. Study the forms, the whole morphology of the opposite pole, so as to recognize the working of the Sun's rays before which the Earth is interposed — the indirect rays of the Sun. Thus the morphology of the animal itself draws our attention to a certain interrelation between Earth and Sun. For a true knowledge of the mutual relations of Earth and Sun we must create the requisite conditions, not by the mere visual appearance (even though the eye be armed with telescopes), but by perceiving also how the animal is formed — how the whole animal form comes into being.
Now think again of how the human spine is displaced through right angle in relation to the animal. All the effects which we have been describing will undergo further modification where man is concerned. The influences of the Sun will therefore be different in man than in the animal. The way it works in man will be like a resultant (Fig. 4). That is to say, if we symbolize the horizontal line — whether it represent the direct or the indirect influence of the Sun — by this length, we shall have to say; here is a vertical line; this also will be acting. And we shall only get what really works in man by forming the resultant of the two.
Suppose in other words that we are led to relate animal formation quite fundamentally to some form of cosmic movement — say, a rotation of the Sun about the Earth, or a rotation of the Earth about its own axis. If then this movement underlies animal formation, we shall be led inevitably to attribute to the Earth or to the Sun yet another movement, related to the forming of man himself, — a movement which, for its ultimate effect, unites to a resultant with the first. From what emerges in man and in the animal we must derive the basis for a true recognition of the mutual movements among the heavenly bodies.
The study of Astronomy will thus be lifted right out of its present limited domain, where one merely takes the outward visual appearance, even if calling in the aid of telescopes, mathematical calculations and mechanics. It will be lifted into what finds expression in this most sensitive of instruments, the living body. The forming forces working in the animal, and then again in man, are a clear indication of the real movements in celestial space.
This is indeed a kind of qualitative mathematics. How, then, shall we metamorphose the idea when we pass on from the animal to the plant? We can no longer make use of either of the two directions we have hitherto been using. Admittedly, it might appear as though the vertical direction of the plant coincided with that of the human spine. From the aspect of Euclidean space it does, no doubt (Euclidean space, that is to say, not with respect to detailed configuration but simply with respect to its rigidity.) But it will not be the same in an inherently mobile space. I mean a space the dimensions of which are so inherently mobile that in the relevant equations, for example, we cannot merely equate the x- and the y-dimensions: y = ƒ(x). (The equation might be written very differently from this. You will see what I intend more from the words I use than from the symbols; it is by no means easy to express in mathematical form.) In a coordinate system answering to what I now intend, it would no longer be permissible to measure the ordinates with the same inherent measures as the abscissae. We could not keep the measures rigid when passing from the one to the other. We should be led in this way from the rigid coordinate system of Euclidean space to a coordinate system that is inherently mobile.
And if we now once more ask the question: How are the vertical directions of plant growth and of human growth respectively related? — we shall be led to differentiate one vertical from another. The question is, then, how to find the way to a different idea of space from the rigid one of Euclid. For it may well be that the celestial phenomena can only be understood in terms of quite another kind of space — neither Euclidean, nor any abstractly conceived space of modern mathematics, but a form of space derived from the reality itself. If this is so, then there is no alternative: it is in such a space, and not in the rigid space of Euclid, that we shall have to understand them.
Thus we are led into quite other realms, namely to the Ice Age on the one hand and on the other to a much needed reform of the Euclidean idea of space. But this reform will be in a different spirit than in the work of Minkowski and others. Simply in contemplating the given facts and trying to build up a science free of hypotheses, we are confronted with the need for a thoroughgoing revision of the concept of space itself. Of these things we shall speak again tomorrow.