Space and Counter-Space
With the introduction, in Chapter X, of the peripheral type of
force-field which appertains to levity as the usual central one does to
gravity, we are compelled to revise our conception of space. For in a
space of a kind we are accustomed to conceive, that is, the
three-dimensional, Euclidean space, the existence of such a field with
its characteristic of increasing in strength in the outward direction
is a paradox, contrary to mathematical logic.
This task, which in view of our further observations of the actions of
the levity-gravity polarity in nature we must now tackle, is, however,
by no means insoluble. For in modern mathematics thought-forms are
already present which make it possible to develop a space-concept
adequate to levity. As referred to in Chapter I, it was Rudolf Steiner
who first pointed to the significance in this respect of the branch of
modern mathematics known as Projective Geometry. He showed that
Projective Geometry, if rightly used, carries over the mind from the
customary abstract to a new concrete treatment of mathematical
concepts. The following example will serve to explain, to start with,
what we mean by saying that mathematics has hitherto been used
abstractly.
One of the reasons why the world-picture developed by Einstein in his
Theory of Relativity deserves to be acknowledged as a step forward in
comparison with the picture drawn by classical physics, lies in the
fact that the old conception of three-dimensional space as a kind of
'cosmic container', extending in all directions into infinity and
filled, as it were, with the content of the physical universe, is
replaced by a conception in which the structure of space results from
the laws interrelating this content. Our further discussion will show
that this indeed is the way along which, to-day, mathematical thought
must move in order to cope with universal reality.
However, for reasons discussed earlier, Einstein was forced to conceive
all events in the universe after the model of gravity as observable on
the earth. In this way he arrived at a space-structure which possesses
neither the three-dimensionality nor the rectilinear character of
so-called Euclidean space - a space-picture which, though
mathematically consistent, is incomprehensible by the human mind. For
nothing exists in our mind that could enable us to experience as a
reality a space-time continuum of three dimensions which is curved
within a further dimension.
This outcome of Einstein's endeavours results from the fact that he
tried by means of gravity-bound thought to comprehend universal
happenings of which the true causes are non-gravitational. A thinking
that has learnt to acknowledge the existence of levity must indeed
pursue precisely the opposite direction. Instead of freezing time down
into spatial dimension, in order to make it fit into a world ruled by
nothing but gravity, we must develop a conception of space sufficiently
fluid to let true time have its place therein. We shall see how such a
procedure will lead us to a space-concept thoroughly conceivable by
human common sense, provided we are prepared to overcome the
onlooker-standpoint in mathematics also.
Einstein owed the possibility of establishing his space-picture to a
certain achievement of mathematical thinking in modern times. As we
have seen, one of the peculiarities of the onlooker-consciousness
consists in its being devoid of all connexion with reality. The process
of thinking thereby gained a degree of freedom which did not exist in
former ages. In consequence, mathematicians were enabled in the course
of the nineteenth century to conceive the most varied space-systems
which were all mathematically consistent and yet lacked all relation to
external existence. A considerable number of space-systems have thus
become established among which there is the system that served Einstein
to derive his space-time concept. Some of them have been more or less
fully worked out, while in certain instances all that has been done is
to show that they are mathematically conceivable. Among these there is
one which in all its characteristics is polarically opposite to the
Euclidean system, and which is destined for this reason to become the
space-system of levity. It is symptomatic of the remoteness from
reality of mathematical thinking in the onlooker-age that precisely
this system has so far received no special attention.1
For the purpose of this book it is not necessary to expound in detail
why modern mathematical thinking has been led to look for thought-forms
other than those of classical geometry. It is enough to remark that for
quite a long time there had been an awareness of the fact that the
consistency of Euclid's definitions and proofs fails as soon as one has
no longer to do with finite geometrical entities, but with figures
which extend into infinity, as for instance when the properties of
parallel straight lines come into question. For the concept of infinity
was foreign to classical geometrical thinking. Problems of the kind
which had defeated Euclidean thinking became soluble directly human
thinking was able to handle the concept of infinity.
We shall now indicate some of the lines of geometrical thought which
follow from this.
*
Let us consider a straight line extending without limits in either
direction. Projective geometry is able to state that a point moving
along this line in one direction will eventually return from the other.
To see this, we imagine two straight lines a and b intersecting at P.
One of these lines is fixed (a); the other (b) rotates uniformly about
C. Fig. 7 indicates the rotation of b by showing it in a number of
positions with the respective positions of its point of intersection
with a (P1, P2. . .). We observe this point moving along a, as a result
of the rotation of b, until, when both lines are parallel, it reaches
infinity. As a result of the continued rotation of b, however, P does
not remain in infinity, but returns along a from the other side. We
find here two forms of movement linked together - the rotational
movement of a line (b) on a point (C), and the progressive movement of
a point (P) along a line (a). The first movement is continuous, and
observable throughout within finite space. Therefore the second
movement must be continuous as well, even though it partly escapes our
observation. Hence, when P disappears into infinity on one side of our
own point of observation, it is at the same time in infinity on the
other side. In order words, an unlimited straight line has only one
point at infinity.
It is clear that, in order to become familiar with this aspect of
geometry, one must grow together in inward activity with the happening
which is contained in the above description. What we therefore intend
by giving such a description is to provide an opportunity for a
particular mental exercise, just as when we introduced Goethe's botany
by describing a number of successive leaf-formations. Here, as much as
there, it is the act of 're-creating' that matters.
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 entity 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 space 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 circumference.
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 relationship 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 constitute Space. Which sort
of space this is, depends on the relationship 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 constitute
Euclidean space. By starting in the manner described above, with the
plane as the unit, and conceiving the point from it, we constitute
polar-Euclidean space.
The realization of the reversibility of the relationship between Point
and Plane leads to a conception of Space still free from any specific
character. By G. Adams this space has been appositely called archetypal
space, or ur-space. Both Euclidean and polar-Euclidean space are
particular manifestations of it, their mutual relationship being one of
metamorphosis in the Goethean sense.
Through conceiving Euclidean and polar-Euclidean space in this manner
it becomes clear that they are nothing else than the geometrical
expression of the relationship 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 establishing 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 entities, 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-space
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
passages 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 God, 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
circumference and centre is inverted by the former becoming itself an
'indivisible centre'. In a space of this kind there is no Here and
There, as in Euclidean space, for the consciousness is always and
immediately at one with the whole space. Motion is thus quite different
from what it is in Euclidean space. 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 â - 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 assumes 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 constituted 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 diminishing 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 antithesis 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 entity 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 assimilation
(photosynthesis), so characteristic of the leaf. Through this process
matter passes over from the aeriform condition into that of numerous
separate, characteristically structured solid bodies - the starch
grains. Besides this kind of assimilation we have learnt to recognize a
higher form which we called 'spiritual assimilation'. Here, a
transition of substance from the domain of levity to that of gravity
takes place even more strikingly than in ordinary (physical)
assimilation (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 circumference.
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 relationship 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 space 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 Cassinian
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
space-system corresponding to levity.
2 For particulars of the lemniscate as the building plan of the middle
part of man's skeleton, see K. König, 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.