Man or Matter Part 20

The following exercise will help us towards further clarity concerning the nature of geometrical infinity.

We imagine ourselves in the centre of a sphere which we allow to expand uniformly on all sides. Whilst the inner wall of this sphere withdraws from us into ever greater distances, it grows flatter and flatter until, on reaching infinite distance, it turns into a plane. We thus find ourselves surrounded everywhere by a surface which, in the strict mathematical sense, is a plane, and is yet one and the same surface on all sides. This leads us to the conception of the plane at infinity as a self-contained ent.i.ty although it expands infinitely in all directions.

This property of a plane at infinity, however, is really a property of any plane. To realize this, we must widen our conception of infinity by freeing it from a certain one-sidedness still connected with it. This we do by transferring ourselves into the infinite plane and envisaging, not the plane from the point, but the point from the plane. This operation, however, implies something which is not obvious to a mind accustomed to the ordinary ways of mathematical reasoning. It therefore requires special explanation.

In the sense of Euclidean geometry, a plane is the sum-total of innumerable single points. To take up a position in a plane, therefore, means to imagine oneself at one point of the plane, with the latter extending around in all directions to infinity. Hence the journey from any point in s.p.a.ce to a plane is along a straight line from one point to another. In the case of the plane being at infinity, it would be a journey along a radius of the infinitely large sphere from its centre to a point at its circ.u.mference.

In projective geometry the operation is of a different character. Just as we arrived at the infinitely large sphere by letting a finite sphere grow, so must we consider any finite sphere as having grown from a sphere with infinitely small extension; that is, from a point. To travel from the point to the infinitely distant plane in the sense of projective geometry, therefore, means that we have first to identify ourselves with the point and 'become' the plane by a process of uniform expansion in all directions.

As a result of this we do not arrive at one point in the plane, with the latter extending round us on all sides, but we are present in the plane as a whole everywhere. No point in it can be characterized as having any distance, whether finite or infinite, from us. Nor is there any sense in speaking of the plane itself as being at infinity. For any plane will allow us to identify ourselves with it in this way. And any such plane can be given the character of a plane at infinity by relating it to a point infinitely far away from it (i.e. from us).

Having thus dropped the one-sided conception of infinity, we must look for another characterization of the relations.h.i.+p between a point and a plane which are infinitely distant from one another. This requires, first of all, a proper characterization of Point and Plane in themselves.

Conceived dynamically, as projective geometry requires, Point and Plane represent a pair of opposites, the Point standing for utmost contraction, the Plane for utmost expansion. As such, they form a polarity of the first order. Both together const.i.tute s.p.a.ce. Which sort of s.p.a.ce this is, depends on the relations.h.i.+p in which they are envisaged. By positing the point as the unit from which to start, and deriving our conception of the plane from the point, we const.i.tute Euclidean s.p.a.ce. By starting in the manner described above, with the plane as the unit, and conceiving the point from it, we const.i.tute polar-Euclidean s.p.a.ce.

The realization of the reversibility of the relations.h.i.+p between Point and Plane leads to a conception of s.p.a.ce still free from any specific character. By G. Adams this s.p.a.ce has been appositely called archetypal s.p.a.ce, or ur-s.p.a.ce. Both Euclidean and polar-Euclidean s.p.a.ce are particular manifestations of it, their mutual relations.h.i.+p being one of metamorphosis in the Goethean sense.

Through conceiving Euclidean and polar-Euclidean s.p.a.ce in this manner it becomes clear that they are nothing else than the geometrical expression of the relations.h.i.+p between gravity and levity. For gravity, through its field spreading outward from an inner centre, establishes a point-to-point relation between all things under its sway; whereas levity draws all things within its domain into common plane-relations by establis.h.i.+ng field-conditions wherein action takes place from the periphery towards the centre. What distinguishes in both cases the plane at infinity from all other planes may be best described by calling it the all-embracing plane; correspondingly the point at infinity may be best described as the all-relating point.

In outer nature the all-embracing plane is as much the 'centre' of the earth's field of levity as the all-relating point is the centre of her field of gravity. All actions of dynamic ent.i.ties, such as that of the ur-plant and its subordinate types, start from this plane. Seeds, eye-formations, etc., are nothing but individual all-relating points in respect of this plane. All that springs from such points does so because of the point's relation to the all-embracing plane. This may suffice to show how realistic are the mathematical concepts which we have here tried to build up.

When we set out earlier in this book (Chapter VIII) to discover the source of Galileo's intuition, by which he had been enabled to find the theorem of the parallelogram of forces, we were led to certain experiences through which all men go in early childhood by erecting their body and learning to walk. We were thereby led to realize that man's general capacity for thinking mathematically is the outcome of early experiences of this kind. It is evident that geometrical concepts arising in man's mind in this way must be those of Euclidean geometry.

For they are acquired by the will's struggle with gravity. The dynamic law discovered in this way by Galileo was therefore bound to apply to the behaviour of mechanical forces - that is, of forces acting from points outward.

In a similar way we can now seek to find the source of our capacity to form polar-Euclidean concepts. As we were formerly led to experiences of man's early life on earth, so we are now led to his embryonic and even pre-embryonic existence.

Before man's supersensible part enters into a physical body there is no means of conveying to it experiences other than those of levity, and this condition prevails right through embryonic development. For while the body floats in the mother's foetal fluid it is virtually exempt from the influence of the earth's field of gravity.

History has given us a source of information from these early periods of man's existence in Traherne's recollections of the time when his soul was still in the state of cosmic consciousness. Among his descriptions we may therefore expect to find a picture of levity-s.p.a.ce which will confirm through immediate experience what we have arrived at along the lines of realistic mathematical reasoning. Among poems quoted earlier, his The Praeparative and My Spirit do indeed convey this picture in the clearest possible way. The following are relevant pa.s.sages from these two poems.

In the first we read:

'Then was my Soul my only All to me, A living endless Ey, Scarce bounded with the Sky Whose Power, and Act, and Essence was to see: I was an inward Sphere of Light, Or an interminable Orb of Sight, Exceeding that which makes the Days . . .'

In the second poem the same experience is expressed in richer detail.

There he says of his own soul that it -

... being Simple, like the Deity, In its own Centre is a Sphere, Not limited but everywhere.

It acts not from a Centre to Its Object, as remote; But present is, where it doth go To view the Being it doth note ...

A strange extended Orb of Joy Proceeding from within, Which did on ev'ry side display Its force; and being nigh of Kin To G.o.d, did ev'ry way Dilate its Self ev'n instantaneously, Yet an Indivisible Centre stay, In it surrounding all Eternity.

'Twas not a Sphere; Yet did appear One infinite: 'Twas somewhat everywhere.'

Observe the distinct description of how the relation between circ.u.mference and centre is inverted by the former becoming itself an 'indivisible centre'. In a s.p.a.ce of this kind there is no Here and There, as in Euclidean s.p.a.ce, for the consciousness is always and immediately at one with the whole s.p.a.ce. Motion is thus quite different from what it is in Euclidean s.p.a.ce. Traherne himself italicized the word 'instantaneous', so important did he find this fact. (The quality of instantaneousness - equal from the physical point of view to a velocity of the value ?? - will occupy us more closely as a characteristic of the realm of levity when we come to discuss the apparent velocity of light in connexion with our optical studies.)

By thus realizing the source in man of the polar-Euclidean thought-forms, we see the discovery of projective geometry in a new light. For it now a.s.sumes the significance of yet another historical symptom of the modern re-awakening of man's capacity to remember his prenatal existence.

We know from our previous studies that the concept of polarity is not exhausted by conceiving the world as being const.i.tuted by polarities of one order only. Besides primary polarities, there are secondary ones, the outcome of interaction between the primary poles. Having conceived of Point and Plane as a geometrical polarity of the first order, we have therefore to ask what formative elements there are in geometry which represent the corresponding polarity of the second order. The following considerations will show that these are the radius, which arises from the point becoming related to the plane, and the spherically bent surface (for which we have no other name than that again of the sphere), arising from the plane becoming related to the point.

In Euclidean geometry the sphere is defined as 'the locus of all points which are equidistant from a given point'. To define the sphere in this way is in accord with our post-natal, gravity-bound consciousness. For in this state our mind can do no more than envisage the surface of the sphere point by point from its centre and recognize the equal distance of all these points from the centre. Seen thus, the sphere arises as the sum-total of the end-points of all the straight lines of equal length which emerge from the centre-point in all directions. Fig. 8 indicates this schematically. Here the radius, a straight line, is clearly the determining factor.

We now move to the other pole of the primary polarity, that is to the plane, and let the sphere arise by imagining the plane approaching an infinitely distant point evenly from all sides. We view the process realistically only by imagining ourselves in the plane, so that we surround the point from all sides, with the distance between us and the point diminis.h.i.+ng gradually. Since we remain all the time on the surface, we have no reason to conceive any change in its original position; that is, we continue to think of it as an all-embracing plane with regard to the chosen point.

The only way of representing the sphere diagrammatically, as a unit bearing in itself the character of the plane whence it sprang, is as shown in Fig. 9, where a number of planes, functioning as tangential planes, are so related that together they form a surface which possesses everywhere the same distance from the all-relating point.

Since Point and Plane represent in the realm of geometrical concepts what in outer nature we find in the form of the gravity-levity polarity, we may expect to meet Radius and Sphere as actual formative elements in nature, wherever gravity and levity interact in one way or another. A few observations may suffice to give the necessary evidence.

Further confirmation will be furnished by the ensuing chapters.

The Radius-Sphere ant.i.thesis appears most obviously in the human body, the radial element being represented by the limbs, the spherical by the skull. The limbs thus become the hieroglyph of a dynamic directed from the Point to the Plane, and the skull of the opposite. This indeed is in accord with the distribution in the organism of the sulphur-salt polarity, as we learnt from our physiological and psychological studies. Inner processes and outer form thus reveal the same distribution of poles.

In the plant the same polarity appears in stalk and leaf. Obviously the stalk represents the radial pole. The connexion between leaf and sphere is not so clear: in order to recognize it we must appreciate that the single plant is not a self-contained ent.i.ty to the same degree as is the human being. The equivalent of the single man is the entire vegetable covering of the earth. In man there is an individual centre round which the bones of his skull are curved; in the plant world the equivalent is the centre of the earth. It is in relation to this that we must conceive of the single leaves as parts of a greater sphere.

In the plant, just as in man, the morphological polarity coincides with the biological. There is, on the one hand, the process of a.s.similation (photosynthesis), so characteristic of the leaf. Through this process matter pa.s.ses over from the aeriform condition into that of numerous separate, characteristically structured solid bodies - the starch grains. Besides this kind of a.s.similation we have learnt to recognize a higher form which we called 'spiritual a.s.similation'. Here, a transition of substance from the domain of levity to that of gravity takes place even more strikingly than in ordinary (physical) a.s.similation (Chapter X).

The corresponding process in the linear stalk is one which we may call 'sublimation' - again with its extension into 'spiritual sublimation'.

Through this process matter is carried in the upward direction towards ever less ponderable conditions, and finally into the formless state of pure 'chaos'. By this means the seed is prepared (as we have seen) with the help of the fire-bearing pollen, so that after it has fallen to the ground, it may serve as an all-relating point to which the plant's Type can direct its activity from the universal circ.u.mference.

In order to find the corresponding morphological polarity in the animal kingdom, we must realize that the animal, by having the main axis of its body in the horizontal direction, has a relations.h.i.+p to the gravity-levity fields of the earth different from those of both man and plant. As a result, the single animal body shows the sphere-radius polarity much less sharply. If we compare the different groups of the animal kingdom, however, we find that the animals, too, bear this polarity as a formative element. The birds represent the spherical (dry, saline) pole; the ruminants the linear (moist, sulphurous) pole.

The carnivorous quadrupeds form the intermediary (mercurial) group. As ur-phenomenal types we may name among the birds the eagle, clothed in its dry, silicic plumage, hovering with far-spread wings in the heights of the atmosphere, united with the expanses of s.p.a.ce through its far-reaching sight; among the ruminants, the cow, lying heavily on the ground of the earth, given over entirely to the immensely elaborated sulphurous process of its own digestion. Between them comes the lion - the most characteristic animal for the preponderance of heart-and-lung activities in the body, with all the attributes resulting from that.

Within the scope of this book it can only be intimated briefly, but should not be left unmentioned for the sake of those interested in a further pursuit of these lines of thought, that the morphological mean between radius and sphere (corresponding to Mercurius in the alchemical triad) is represented by a geometrical figure known as the 'lemniscate', a particular modification of the so-called Ca.s.sinian curves.2

1 For further details, see the writings of G. Adams and L. Locher-Ernst who, each in his own way, have made a beginning with applying projective geometry on the lines indicated by Rudolf Steiner. Professor Locher-Ernst was the first to apply the term 'polar-Euclidean' to the s.p.a.ce-system corresponding to levity.

2 For particulars of the lemniscate as the building plan of the middle part of man's skeleton, see K. Konig, M.D.: Beitrage zu einer reinen Anatomic des menschlichen Knochenskeletts in the periodical Natura (Dornach, 1930-1). Some projective-geometrical considerations concerning the lemniscate are to be found in the previously mentioned writings of G. Adams and L. Locher-Ernst.

CHAPTER XIII

'Radiant Matter'

When man in the state of world-onlooker undertook to form a dynamic picture of the nature of matter, it was inevitable that of all the qualities which belong to its existence he should be able to envisage only those pertaining to gravity and electricity. Because his consciousness, at this stage of its evolution, was closely bound up with the force of gravity inherent in the human body, he was unable to form any conception of levity as a force opposite to gravity. Yet, nature is built bipolarically, and polarity-concepts are therefore indispensable for developing a true understanding of her actions. This accounts for the fact that the unipolar concept of gravity had eventually to be supplemented by some kind of bipolar concept.

Now, the only sphere of nature-phenomena with a bipolar character accessible to the onlooker-consciousness 'was that of electricity. It was thus that man in this state of consciousness was compelled to picture the foundation of the physical universe as being made up of gravity and electricity, as we meet them in the modern picture of the atom, with its heavy electro-positive nucleus and the virtually weightless electro-negative electrons moving round it.

Once scientific observation and thought are freed from the limitations of the onlooker-consciousness, both gravity and electricity appear in a new perspective, though the change is different for each of them.

Gravity, while it becomes one pole of a polarity, with levity as the opposite pole, still retains its character as a fundamental force of the physical universe, the gravity-levity polarity being one of the first order. Not so electricity. For, as the following discussion will show, the electrical polarity is one of the second order; moreover, instead of const.i.tuting matter as is usually believed, electricity turns out to be in reality a product of matter.

We follow Goethe's line when, in order to answer the question, 'What is electricity?' we first ask, 'How does electricity arise?' Instead of starting with phenomena produced by electricity when it is already in action, and deriving from them a hypothetical picture, we begin by observing the processes to which electricity owes its appearance. Since there is significance in the historical order in which facts of nature have come to man's knowledge in the past, we choose as our starting-point, among the various modes of generating electricity, the one through which the existence of an electric force first became known. This is the rousing of the electric state in a body by rubbing it with another body of different material composition. Originally, amber was rubbed with wool or fur.

By picturing this process in our mind we become aware of a certain kins.h.i.+p of electricity with fire, since for ages the only known way of kindling fire was through friction. We notice that in both cases man had to resort to the will-power invested in his limbs for setting in motion two pieces of matter, so that, by overcoming their resistance to this motion, he released from them a certain force which he could utilize as a supplement to his own will. The similarity of the two processes may be taken as a sign that heat and electricity are related to each other in a certain way, the one being in some sense a metamorphosis of the other. Our first task, therefore, will be to try to understand how it is that friction causes heat to appear in manifest form.