Lecture IV
Stuttgart, January 4, 1921
My Dear Friends,
If I had the task of presenting my subject
purely according to the methods of Spiritual Science, I
should naturally have to start from different premises and we
should be able to reach our goal more quickly. Such a
presentation, however, would not fulfill the special purpose
of these lectures. For the whole point of these lectures is
to throw a bridge across to the customary methods of
scientific thought. Admittedly, I have chosen just the
material which makes the bridge most difficult to construct,
because the customary mode of thought in this realm is very
far from realistic. But in contending against an unreal point
of view, it will become apparent how we can emerge from the
unsatisfying nature of modern theories and came to a true
grasp of the facts in question. Today, then, I should like to
consider the whole way in which ideas have been formed in
modern times about the celestial phenomena.
We must,
however, distinguish two things in the formation of these
ideas. First, the ideas[1] are derived
from observation of the celestial phenomena, and theoretical
explanations are then linked on to the observations.
Sometimes very farreaching, spunout theories have been
linked on to relatively few observations. That is the one
thing, namely, that a start is made from observations out of
which certain ideas have been developed. The other is that,
the ideas having been reached, they are further elaborated
into hypotheses. In this creating of hypotheses, — a
process which ends in the setting up of some definite
cosmology, — much arbitrariness prevails, since in the
settingup of theories, any preconceived ideas existing in
the minds of those who put forward the theory, make
themselves strongly felt.
I will
therefore first call your attention to something which will
perhaps strike you as paradoxical, but which, when carefully
examined, will none the less prove fruitful in the further
course of our studies.
In the whole
mode of thought of modern Science there prevails what might
be called, and indeed has been called, the ‘Regula
philosophandi’. It consists in saying: What has been
traced to definite causes in one realm of reality, is to be
traced to the same causes in other realms. In setting up such
a ‘regula philosophandi’ the startingpoint is as
a rule apparently selfevident. It will be said —
scientists of the Newtonian school will certainly say —
that breathing must have the same causes in man as in the
animal, or again, that the ignition of a piece of wood must
have the same cause whether in Europe or in America. Up to
this point the thing is obvious enough. But then a jump is
made which passes unnoticed, — is taken tacitly for
granted. Those who are wont to think in this way will say,
for example, that if a candle and the Sun are both of them
shedding light the same causes must surely underlie the light
of the candle and the light of the Sun. Or again, if a stone
falls to Earth and the Moon circles round the Earth, the same
causes must underlie the movement of the stone and the
movement of the Moon. to such an explanation they attach the
further thought that if this were not so, we should have no
explanations at all in Astronomy. The explanations are based
on earthly things. If the same causality did not obtain in
the Heavens as on Earth, we should not be able to arrive at
any theory at all.
Yet when you
come to think of it, this ‘regula philosophandi’
is none other than a preconceived idea. Who in the world will
guarantee that the causes of the shining of a candle and of
the shining of the Sun are one and the same? Or that in the
falling of a stone, or the falling of the famous apple from
the tree by which Newton arrived at his theory, there is the
same underlying cause as in the movements of the heavenly
bodies? This would first have to be established. As it is, it
is a mere preconceived idea. Prejudices of this kind enter
in, when, having first derived theoretical explanations and
thought — pictures inductively from the observed
phenomena, people rush headlong into deductive reasoning and
construct worldsystems by deductive methods.
What I am now
describing thus abstractly has, however, become a historical
fact. There is a continuous line of development from what the
great thinkers at the opening of the modern age —
Copernicus, Kepler, Galileo — concluded from
comparatively few observations. Of Kepler — notably of
his third Law, quoted yesterday — it must be said that
his analysis of the facts which were available to him is a
work of genius.
It was a very
great intensity of spiritual force which Kepler brought to
bear when, from the little that lay before him, he discovered
this ‘law’ as we call it, or better, this
‘conceptual synthesis’ of the phenomena of the
universe. Then however, by way of Newton a development set in
which was not derived from observation but from theoretical
constructions, including concepts of force and mass and the
like, which we must simply omit if we only want to hold to
what is given. The development in this direction reaches a
culminating point — conceived, admittedly, with genius
and originality — in Laplace, where it leads to a
genetic explanation of the entire cosmic system (as you will
convince yourselves if you read his famous book
“Exposition du Systeme du Monde”), or again in
Kant, in his “Natural History and Theory of the
Heavens”. In all that has followed in this trend we see
the effort constantly made to come to conclusions based on
the thought pictures that have thus been conceived of the
connections of the celestial movements, and resulting in such
explanations of the origin of the universe as the nebular
theory and so on.
It must be
noted that in the historical development of these theories we
have something which is put together from inductions made,
once again, with no little genius in this domain — and
from subsequent deductions in which the special predilections
of their authors were included. Inasmuch as a thinker was
imbued with materialism it was quite natural for him to
mingle materialistic ideas with his deductive concepts. Then
it was no longer the facts which spoke, for one proceeded on
the basis of the theories which had emerged from the
deductions. Thus, for example, inductively men first arrived
at the mental pictures which they summed up in the notion of
a central body, the Sun, with the planets revolving around it
in ellipses according to a certain law, namely: the
radiusvectors describe equal areas in equal periods of time.
By observing the different planets of a solar system, it was
moreover possible to summarize their mutual relations in
Kepler's third law: ‘For different planets the
squares of the periods of revolution are proportional to the
cubes of the radiusvectors’. Here was a certain
picture. The question, however, was not decided, whether this
picture completely fitted the reality. It was in truth an
abstraction from reality; to what extent it related to the
full reality, was not established. From this picture —
not from reality, but from this picture — people
deduced what then became a whole genetic system of Astronomy.
All this must be borne in mind. Modern man is taught from
childhood as if the theories which have been reached in the
past few centuries by deductive reasoning were the real
facts. We will therefore, while taking our start from what is
truly scientific, disregard as far as is possible all that is
merely theoretical and link on to those ideas which only
depart from reality to the extent that we shall still be able
to discover in them a connection with what is real. It will
be my task, in all that I give today, to follow the
direction of modern scientific thought only up to those ideas
and concepts which still permit one to find the way back
again into reality. I shall not depart so far from reality
that the concepts become crude enough to allow of the
deduction of nebular hypotheses.
Proceeding in
this way, — pursuing the modern method of forming
concepts in this particular field, — we must first form
a concept which presented itself inductively to Kepler and
was then developed further I repeat expressly, I will only go
so far in these concepts that even if the picture in the form
in which it was conceived should be mistaken, it has departed
only so far from reality that it will be possible to
eliminate the mistake and return to what is true. We need to
develop a certain flair for reality in the concepts we
entertain. We cannot proceed in any other way if we wish to
throw a bridge across from the reality to the spunout
theories of modern scholarship and science.
Here then, to
begin with, is a concept which we must examine. The planets
have eccentric orbits, — they describe ellipses. This
is something with which we can begin. The planets have
eccentric orbits and describe ellipses, in one focus of which
is the Sun. They describe the ellipses in accordance with the
law that the radius — vectors describe equal areas in
equal periods of time.
A second
essential for us to hold to is the idea that each planet has
its own orbital plane. Although the planets carry out their
evolutions in the neighborhood of each other, so to speak,
yet for each planet there is the distinct plane of its orbit,
more or less inclined to the plane of the Sun's
equator: If this depicts the plane of the Sun's equator
(Fig.1),
an orbital plane of a planet
would be thus; it would not coincide at all with the plane of
the Sun's equator.
[2]
Fig. 1
These are two
very significant mental pictures, to be formed from the facts
of observation. And yet, in the very forming of them we must
take note of something in the real worldpicture, which as it
were, rebels against them. For instance, if we are trying to
understand our solar system in its totality, and only base it
upon the picture of the planets moving in eccentric orbits,
the orbital planes being inclined at varying degrees to the
plane of the solar equator, we shall be in difficulties if we
also take into account the movements of the comets.
The moment we turn our attention to the cometary movements,
the picture no longer suffices. The outcome will be better
understood from the historical facts than from any
theoretical explanations.
Upon these
two thoughtpictures, — that the orbital planes of the
planets lie in the proximity of the plane of the Sun's
equator, and that the orbits are eccentric ellipses, —
Kant, Laplace and their successors built up the nebular
hypothesis. Follow what emerges from this. At a pinch, and
indeed only at a pinch, it is a way of imagining the origin
of the solar system. But the astronomical system thus
constructed contains no satisfactory explanation of the part
played by the cometary bodies. They always fall out of the
theory. This discordance of the comets with the theories
which were formed, as described, in the course of scientific
history, proves that the cometary life somehow rebels against
a concept formed, not from the whole but only from a part of
the whole. We must be clear, too, that the paths of the
comets frequently coincide with those of other bodies which
also play into our system and present a riddle precisely
through their association with the comets. These are the
meteoric swarms, whose paths very frequently — perhaps
even always — coincide with the cometary paths. Here,
my dear friends, taking into account the totality of our
system, we are led to say: A sea of ideas has gradually been
formed from the study of our planetary system as a whole,
— ideas with which we cannot do justice to the
seemingly irregular and almost arbitrary paths of the comets
and meteoric swarms. They simply refuse to be included in the
more abstract pictures that have been reached. I should have
to give you long historical descriptions to show in detail
how many difficulties have arisen in connection with the
concrete facts, when the investigators — or rather,
thinkers — approached the comets and meteoric swarms
with their astronomical theories.
I wish only
to point out the directions in which a sound understanding
can be sought. We shall come to such an understanding if we
pay attention to yet another aspect.
Starting in
this way from concepts which still have a remnant of reality
in them, we will now try to go back a little towards what is
real. It is indeed always necessary to do this in relation to
the outer world, in order that our concepts may not stray too
far from reality, — for this is a strong propensity of
man. We must go back again and again to the reality.
There is
already no little danger in forming such a concept as that
the planets move in ellipses, and then beginning at once to
build a theory upon this concept. It is far better, after
forming such a concept, to turn back to reality in order to
see if the concept does not need correcting, or at least
modifying. This is important. It is very clearly seen in
astronomical thinking. Also in biological and especially in
medical thought, the same failing has led people very far
astray. They do not take into account, how necessary it is
directly they have formed a concept, to go back to reality in
order to make sure that there is no reason to modify it.
The planets,
then, move in ellipses. But these ellipses vary; they are
sometimes more circular, sometimes more elliptical. We find
this if we return to reality with the ellipse idea. In the
course of time the ellipse becomes more bulging, more like a
circle, and then again more like an ellipse. So I by no means
include the whole reality if I merely say, ‘the planets
move in ellipses’. I must modify the concept and say:
The planets move in paths which continually struggle against
becoming a circle or remaining one and the same ellipse. If I
were now to draw the elliptic line, to be true to the reality
I should have to make it of indiarubber, or form it flexibly
in some way, continually altering it within itself. For if I
had formed the ellipse which is there in one
revolution of the planet, it would not do for the next
revolution, and still less for the following one. It is not
true that when I pass from reality to the rigid concept I
still remain within the real. That is the one thing.
The other is:
We have said that the planes of the planetary orbits are
inclined to the plane of the Sun's equator. Where the
planets cross the point of intersection of their orbits (with
the Ecliptic) in an upward or downward direction, they are
said to form Nodes. The lines, joining the two Nodes
(KK 1 in Fig. 1), are variable. So too are the
inclinations of the planes to oneanother, so that even these
inclinations, if we try to express them in a single concept,
bring us to a rigid concept which we must immediately modify
in face of the reality. For if an orbit is inclined at one
time in one way, and at another time in another way, the
concept we deduce in the first instance must afterwards be
modified. To be sure, once such a point has been reached, we
can take an easy line and say that there are
‘disturbances’ and that the reality is only
grasped ‘approximately’ with our concepts. We
then go on swimming comfortably in further theories. But in
the end we swim so far that the fanciful and theoretic
pictures we are constructing no longer correspond to the
reality, though they are meant to do so.
It is easy to
agree that this mutability of the eccentric orbits, and of
the mutual inclination of the planes of the orbits, must
somehow or other be connected with the life of the
whole planetary system, or shall we say, with its continuing
activity. It must be connected in some way with the living
activity of the whole planetary system. That is quite
evident. Starting from this, one might again try to form the
concept, saying: Well now, I will bring such mobility into my
thoughts that I picture the ellipses continually bulging out
and contracting, the planes of the orbits ascending,
descending and rotating, and then from this startingpoint I
will build up a worldsystem according to reality. Good. But
if you think the idea through to the end, then precisely as
the outcome of such logical thought, the result is a
planetary system which cannot possibly go on existing.
Through the summation of the disturbances which arise
especially through the variability of the Nodes, the
planetary system would move towards its own ultimate death
and rigidity. Here there comes in what philosophers have
pointed out again and again. While such a system can
be thought out, in reality it would have had ample time to
reach the ultimate finale. There is no reason why it should
not. The infinite possibility would have been fulfilled;
rigidity would long ago have set in.
We enter here
into a realm where thought apparently comes to a standstill.
Precisely by following my thinking through to the very last,
I arrive at a worldsystem which is still and rigid. But that
is not reality.
Now, however,
we come to something else, to which we must pay special
attention. In pursuing these things further — you can
find the theory of it in the work of Laplace; I will only
relate the phenomena — one finds that the reason why
the system has not actually reached rigidity under the
influence of the disturbances — the variability of the
Nodes, etc., — is that the ratios of the periods of
revolution of the planets are not commensurable. They are
incommensurable quantities, numbers with decimals to an
infinite number of places. Thus we must say: If we compare
the periods of revolution of the planets in the sense of
Kepler's Third Law, the ratios of these periods cannot
be given in integers, nor in finite fractions, but only in
incommensurable numbers. Modern Astronomy is clear on this.
It is to the incommensurability of the ratios between the
periods of revolution of the several planets (in
Kepler's third Law) that the planetary system owes its
continued mobility. Otherwise, it must long ago have come to
a standstill.
Observe now,
what has happened. In the last resort, we are obliged to base
our thoughts about the planetary system upon numbers which in
the end elude our grasp. This is of no little importance.
We are
therefore led, by the very requirements of scientific
development, to think of the planetary system mathematically
in such a way that the mathematical results are no longer
commensurable. We are at the place, where in the mathematical
process itself we arrive at incommensurable numbers.
We have to let the number stand, — we come to a stop.
We can write it in decimals no doubt, but only up to a
certain place. Somewhere or other we must leave off when we
come to the incommensurable. The mathematicians among you
will be clear about this. You will see that in dealing with
incommensurable number I reach the point where I must say: I
calculate up to here and then I can go no further. I can only
say (forgive my using a somewhat amusing comparison for a
serious subject) that this coming to an inevitable halt in
mathematics reminds me of a scene in which I was once a
participator in Berlin. A fashion in Varietyentertainment
came about through certain persons, one of whom was Peter
Hill. He had founded a kind of Cabaret and wanted to read his
own poems there. He was a very lovable person, in heart and
soul a Theosophist, he had rather gone to seed in Bohemian
circles. I went to a performance in which he read his own
poems. The poem had got so far that single lines were
finished, and so he read it aloud:
The Sun came up.
... etc. (The first line.)
The Moon rose.
... etc. (That was the second line.)
At each line he said ‘etc.’
That was a reading I once attended. As a matter of fact it
was most stimulating. Everyone could finish the line as he
chose! Admittedly with incommensurable numbers [you] cannot do
this, yet here too you can only indicate the further process.
You can say that the process continues in a certain
direction, but nothing is given by which you might form an
idea as to what numbers may yet be coming. It is important
that precisely in the astronomical field we are led into
incommensurabilities. We are forced by Astronomy to the very
limits of mathematising; here the reality escapes us. Reality
escapes us, we can say nothing else; reality eludes our
grasp.
What does
this mean? It means that we apply the most secure of our
sciences, Mathematics, to the celestial phenomena, and in the
last resort the celestial phenomena do not submit; the moment
comes where they elude us. Precisely where we are about to
reach their very life, they slip away into the
incommensurable realm. Here then, our grasp of reality comes
to an end at a certain point and passes over into chaos.
We cannot say
without more ado, what this reality, which we are trying to
follow mathematically, actually does when it slides away into
the incommensurable. Undoubtedly this is related to its power
of continued life. To enter the full astronomical reality we
must take leave of what we are able to master mathematically.
The calculation plainly shows this; the very history of
science shows it.
Such are the
points which we must work towards, if we would proceed in a
realistic spirit. Now I would like to set before you the
other pole of the matter. If you follow it physiologically
you can begin from any point you like in embryonic
development, whether it be from the development of the
human embryo in the third or second month, — or the
embryo of some other creature. You can follow the development
back as far as ever you can with the means of modern science.
(it is in fact only possible to a limited extent, as those of
you who have studied it will know.) You can trace it back to
a certain point, from which you cannot get much further,
namely to the detachment of the ovum — the fertilized
ovum. Picture to yourselves how far you can go back. If you
wished to go still further back you would be entering the
indeterminate realm of the whole maternal organism. This
means that in going back you come into a kind of chaos. You
cannot avoid this, and the fact that it cannot be avoided is
shown by the course of scientific development. Think of such
scientific hypotheses as the theory of
“Panspermia” for instance, where they speculated
as to whether the single germcell was prepared out of the
forces of the whole organism, which was more the point of
view of Darwin, or whether it developed in a more segregated
way in the purely sexual organs. You will see when you study
the course of scientific development in this field that no
little fantasy was brought to bear on the attempt to explain
the underlying genesis, when tracing backward the arising of
the germ cell from the maternal organism. You come into a
completely indeterminate realm. There is little but
speculation in the external science of today as to the
connection between the germcell and the maternal
organism.
Then at a
certain point in its development this germ appears in a very
definite way, in a form which can be grasped at least
approximately by mathematical or at any rate geometrical
means. Diagrams can be made from a certain point onward. Many
such diagrams exist in Embryology. The development of the
germcell and other cells can be delineated more or less
exactly. So one begins to picture the development in a
geometrical way, representing it in forms similar to purely
geometrical figures. Here we are following up a reality which
in a way is the reverse of what we had in Astronomy. There we
pursued a reality with our cognitional process and came to
incommensurable numbers; the whole thing slips into
chaos through the process of knowledge itself. In Embryology
we slip out of chaos. From a certain moment onward
we can grasp what emerges from chaos through forms that are
like purely geometrical forms. Thus in effect, in employing
Mathematics in Astronomy we come at one point into chaos. And
by pure observation in Embryology we have at a certain point
nothing before us but chaos; it all seems chaotic at first,
observation is impossible. Then we come out of chaos into the
realm of Geometry. It is therefore an ideal of certain
biologists — a very justifiable ideal — to grasp
in a geometrical form what presents itself in Embryology; not
merely to make illustrations of the growing embryo
naturalistically, but to construct the forms according to
some inherent law, similar to the laws underlying geometrical
figures. It is a justifiable ideal.
Now therefore
we can say: When in Embryology we try to follow up the real
process by observation, we emerge out of a sphere which lies
about as near to our understanding as that which is beyond
the incommensurable numbers. In Astronomy on the one hand, we
proceed with our understanding up to the point where we can
no longer follow mathematically. In Embryology on the other
hand our understanding begins at a certain point,
where we are first able to set to work with something
resembling Geometry.
Think the
thought through to its conclusion. You can do so,
since it is a purely ‘methodological’ thought,
that is to say the reality of it is in our own inner
life.
If in
arithmetic we reach the incommensurable numbers, — that
is, we reach a point where the reality is no longer
represented by a number that can be shown in its complete
form — then we should also begin to ask whether the
same thing may not happen with geometrical form as with
arithmetical analysis. (We shall speak more of this in the
next lecture.) The analytical process leads to
incommensurable number. Now let us ask: How do geometrical
forms image the celestial movements? Do not these images
perhaps lead us to a certain point. Similar to that to which
arithmetical analysis is leading when we reach
incommensurable number? Do we not in our study of the
heavenly bodies — namely the planets — come to a
boundary, at which we must admit we can no longer use
geometrical forms as a means of illustration; the facts can
no longer be grasped with geometrical forms? Just as we must
leave the region of commensurable numbers, it may well be
that we must leave the region where reality can still be
clothed in geometrical (or again arithmetical, algebraic,
analytical) forms, such as in drawings of spirals and other
figures derived from Geometry. So, in Geometry too, we should
be coming into the incommensurable realm. In this sense it is
indeed remarkable that in Embryology, though arithmetical
analysis is not yet of much use, Geometry makes its presence
felt pretty strongly the moment we begin to take hold of the
embryological phenomena as they emerge from chaos. Here we
are dealing, not indeed with incommensurable number
but with something that tends to pass from incommensurable
into commensurable form.
We have thus
sought to grasp reality at two poles: On the one hand where
the process of cognition leads through analysis into the
incommensurable, and on the other where observation leads out
of chaos to a grasping of reality in ever more commensurable
forms. It is essential that we bring these things before our
minds with full clarity, if we would add reality to what is
presented by the external science of today. In no other way
can we reach this end.
I should now
like to add a methodical reflection, from which we can
tomorrow make our way into more realistic problems.
In all that
we have spoken of hitherto, we have been taking it for
granted that the cosmic phenomena have been approached from
the standpoint of Mathematics. It appeared that at one point
the mathematician comes up to a limit — a limit he
encounters too in purely formal Mathematics. Now there is
something underlying our whole way of thinking in this realm,
which perhaps passes unnoticed because it always wears the
mask of the ‘obvious’ and we therefore never
really face the problem. I mean the whole question of the
application of mathematics to reality. How do we proceed? We
develop Mathematics as a formal science and it appears to us
absolutely cogent in its conclusions; then we apply it to
reality, without giving a thought to the fact that we are
really doing so on the basis of certain hypotheses. Today
however, sufficient ground has already been created for us to
see that Mathematics is only applicable to outer reality on
the basis of certain premises. This becomes clear when we try
to continue Mathematics beyond certain limits. First, certain
laws are developed, — laws which are not obtained from
external facts, as for example are Kepler's Laws, but
from the mathematical process itself. They are in fact
inductive laws, developed within Mathematics. They
are then employed deductively; highly elaborate
mathematical theories are built upon them.
Such laws are
those encountered by anyone who studies Mathematics. In
lectures given recently in Dornach by our friend Dr. Blumel,
significant indications were given of this line of
mathematical research. One of the laws in question is termed
the Commutative Law. It can be expressed in saying:
It is obvious that a + b equals b + a , or a x b equals b x a
. This is a selfevident fact so long as one remains within
the realm of real numbers: But it is merely an inductive law
derived from the use of the implicit postulates in the
arithmetic of real numbers.
The second
law is the Associative Law. It is expressed as
(a + b) + c = a + (b + c). Again this is a law, simply derived
by working with the implicit postulates in the arithmetic of
real numbers.
The third is
the socalled Distributive Law, expressible in the
form: a (b + c) = ab + ac. Once more, it is a law obtained
inductively by working with the implicit postulates in the
arithmetic of real numbers.
The fourth
law may be expressed as follows: ‘A product
can only equal zero if at least one of the factors
equals zero.’ This law again is only an
inductive one, derived by working with the implicit
postulates in the arithmetic of real numbers.
We have,
then, these four laws; the commutative law, the associative
law, the distributive law, and this law about the product
being equal to zero. These laws underlie the formal
Mathematics of today, and are used as a basis for further
work. The results are most interesting, there is no question
of that. But the point is this: These laws hold good so long
as we remain in the sphere of real numbers and their
postulates. But no thought is ever given to the question, to
what extent the real facts are in accord with them. Within
our ordinary formal modes of experience it is true, no doubt
that a + b = b + a, but does it also hold good in outer
reality? There is no ascertainable reason why it should. We
might be very astonished one day to find that it did not work
if we applied to some real process the idea that a + b equals
b + a. But there is another side to it. We have within us a
very strong inclination to cling to these laws; with them
therefore. We approach reality and everything that does not
fit in escapes our observation. That is the other side.
In other
words: We first set up postulates which we then apply to
reality and take them as axioms of the reality itself. We
ought only to say: I will consider a certain sphere of
reality and see how far I get with the statement a + b = b +
a. More than that, I have no right to say. For by approaching
reality with this statement we meet what answers to it, and
elbow aside anything that does not. We have this habit too in
other fields. We say for example, in elementary physics:
Bodies are subject to the law of inertia. We define
‘inertia’ as consisting in the fact that bodies
do not leave their position or alter their state of motion
without a definite impelling force. But that is not an axiom;
it is a postulate. I ought only so say: I will call a body
which does not alter its own state of motion
‘inert’, and now I will seek in the real world
for whatever answers to this postulate.
In that I
form certain concepts, I am therefore only forming guiding
lines with which to penetrate reality, and I must keep the
way open in my mind for penetrating other facts with other
concepts. Therefore I only regard the four basic laws of
number in the right way if I see them as something which
gives me a certain direction, something which helps me
regulate my approach to reality. I shall [be] wrong
if I take Mathematics as constituting reality, for
then in certain fields, reality will simply contradict me.
Such a contradiction is the one I spoke of, where incommensurability
enters in, in the study of celestial phenomena.
Notes:
1. Note by translators: In
the first few pages of this lecture, the word
Vorstellungen has been translated, either as
“mental pictures” or
“thoughtpictures”, or by the word
“ideas” as in Prof. Hoernle's original
English edition of Dr. Steiner's Philosophie
der Freiheit. In other translations, including the
later editions of this book, the word is rendered
“representations”, or again, “Mental
presentations”. Dr. Steiner's use of
Vorstellung corresponds, we believe, to the colloquial,
workaday meaning of the word “idea” in
current English. (Where Idea is meant in its deeper, more
spiritual meaning — German Idee — it
can be distinguished by the use of a capital.)
2. Note by Editor: This
plane is inclined at an angle of about seven degrees to
the plane of the ecliptic.
