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Mathematics and Occultism
Schmidt Number: S0870
Online since: 11th June, 2002
A lecture by
Rudolf Steiner
Amsterdam, June 21, 1904
GA 35
The lecture presented here was given in Amsterdam on June 21, 1904. In
the collected edition of Rudolf Steiner's works, the volume containing
the German texts is entitled, Philosophie und Anthroposophie.
Gesammelte Aufsätze 1904 – 1918. Aufsätze und neun
AutoReferate nach Vorträgen in verschiedenen Städten.
(Vol. 35 in the Bibliographic Survey, 1961). Translated from the German
by M. H. Eyre, edited by H. Collison.
An address delivered to the First Annual Congress of the Federation
of European Sections of the Theosophical Society, Amsterdam, June,
1904. Translated by M. H. Eyre, edited by H. Collison. (From the
Transactions of the Congress.)
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Anthroposophical Movement
Weekly News for Englishspeaking Members of the Anthroposophical Society
VOL. V. No. 28 
8th July. 1928 
Published at the Offices of the Anthroposophical Society in Great
Britain, 46, Gloucester Place, London, W.I, England. Subscriptions:
Great Britain and Dominions, 15s. yearly, 3s. 9d. quarterly. Single
Copies, 3d. America, $4.00 yearly, $1.00 quarterly. All rights
reserved.
Mathematics and Occultism
IT is well known that the inscription over the door of Plato's school
was intended to exclude anybody who was unacquainted with the science
of Mathematics, from participating in the teachings of the Master.
Whatever we may think of the historical truth of this tradition, it is
based upon the correct understanding of the place that Plato assigned
to mathematics within the domain of human knowledge. Plato intended to
awaken the perceptions of his disciples by training them to move in the
realm of purely spiritual being according to his “Doctrine of
Ideas.” His point of view was that Man can know nothing of the
“True World” so long as his thought is permeated by what his
senses transmit. He demanded that thought should be emancipated from
sensation. Man moves in the World of Ideas when he thinks, only
after he has purged his thought of all that sensuous perception can
present. The paramount question for Plato was, “How does Man
emancipate himself from all senseperception?” He considered this
to be an allimportant question for the education of the spiritual life.
Of course, it is only with difficulty that Man can emancipate himself
from material perceptions, as a simple experiment on one's own self
will prove. Even when the man who lives in this everyday world does
withdraw into himself and does not allow any material impressions of
the senses to work upon him, the residues of sensuous perception still
linger, in his mind. As to the man who is as yet undeveloped, when he
rejects the impressions which he has received from the physical world
of the senses, he simply faces nothingness — the absolute
annihilation of consciousness. Hence certain philosophers affirm that
there exists no thought free from senseperception. They say,
“Let a man withdraw himself ever so much within the realm of pure
thought, he would only be dealing with the shadowy reflections of his
senseperceptions.” This statement holds good, however, only for
the undeveloped man. When he acquires for himself the faculty of
developing organs which can perceive spiritual truths (just as Nature
has built for him organs of sense), then his thought ceases to remain
empty when it rids itself of the contents of senseperception. It was
precisely such a mind emancipated from senseperception and yet
spiritually full, which Plato demanded from those who would understand
his Doctrine of Ideas. In demanding this, however, he demanded no more
than was always required of their disciples, by those who aspired to
make them true initiates of the Higher Knowledge. Until Man
experiences within himself to its full extent what Plato here implies,
he cannot have any conception of what true Wisdom is.
Now Plato looked upon mathematical science as a means of training for
life in the World of Ideas emancipated from senseperception. The
mathematical images hover over the borderline between the material
and the purely spiritual World. Let us think about the
“circle”; we do not think of any special material circle
which perhaps has been drawn on paper, but we think of any and every
circle which may be represented or met with in Nature. So it is in the
case of all mathematical pictures. They relate to the
senseperceptible, but they are not exhaustively contained in it. They
hover over innumerable, manifold senseperceptible forms. When I think
mathematically, I do indeed think about something my senses can
perceive; but at the same time I do not think in terms of
senseperception. It is not the material circle which teaches me the
laws of the circle; it is the ideal circle existing only in my mind
and of which the concrete form is a mere representation. I could learn
the identical truths from any other sensible image. The essential
property of mathematical perception is this: that a single
senseperceptible form leads me beyond itself; it can only be for me a
representation of a comprehensive spiritual fact. Here again, however,
there is the possibility that in this sphere I may bring through to
senseperception what is spiritual. From the mathematical figure I can
learn to know supersensible facts by way of the senseworld. This was
the allimportant point for Plato. We must visualise the idea in a
purely spiritual manner if we would really know it in its true aspect.
We can train ourselves to this if we only avail ourselves of the first
steps in mathematical knowledge for this purpose, and understand
clearly what it is that we really gain from a mathematical figure.
“Learn to emancipate thyself from the senses by mathematics,
then mayest thou hope to rise to the comprehension of ideas
independently of the senses”: this was what Plato strove to
impress upon his disciples.
The Gnostics desired something similar. They said, “Gnosis is
Mathesis.” They did not mean by this that the essence of the
world can be based on mathematical ideas, but only that the first
stages in the spiritual education of Man are constituted by what is
supersensible in mathematical thought. When a man reaches the stage of
being able to think of other properties of the world independently of
senseperception in the same way as he is able to think mathematically
of geometrical forms and arithmetical relations of numbers, then he is
fairly on the path to spiritual knowledge. They did not strive for
Mathesis as such, but rather for supersensible knowledge after the
pattern of Mathesis. They regarded Mathesis as a model or a
prototype, because the geometrical proportions of the World are the
most elementary and simple, and such as Man can most easily
understand. He must learn through the elementary mathematical
truths to become emancipated from sense in order that he may reach,
later, the point where the higher problems are appropriately to be
considered. This will certainly mean, for many, a giddy height of
human perceptive faculties. Those, however, who may be considered as
true Occultists have in every age demanded from their disciples the
courage to make this giddy height their goal: — “Learn to
think of the essence of Nature and of Spiritual Being as independently
of senseperception as the mathematician thinks of the circle and its
laws, then mayest thou become a student of Occult Science”
— this is what everyone who really seeks after Truth should keep
before his mind as if written in letters of gold. “Thou wilt
never find a Circle in the World, which will not confirm for thee in
the realm of sense what thou hast learned about the Circle by
supersensible mathematical perception; no experience will ever
contradict thy supersensible perception. Thus dost thou gain for
thyself an imperishable and eternal knowledge when thou learnest to
perceive free of the senses.” In this way did Plato, the
Gnostics and all Occultists conceive mathematical science as an
educational means.
We should consider what eminent persons have said about the relation
of mathematics to natural science. Kant and many others like him, for
example, have said that there is as much of true science as there is
mathematics in our knowledge of Nature. This implies nothing else
than that by reducing to mathematical formulae all natural
phenomena, a science is obtained transcending senseperception —
a science which, although expressed through senseperception, is
visualised in the spirit. I have visualised the working of a machine
only after I have reduced it to mathematical formulae. To express by
such formulae the processes presented to the senses is the ideal of
mechanics and physics and is increasingly becoming the ideal of
chemistry.
But it is only that which exists in space and time and has extension
in this sense, which may be thus mathematically expressed. As
soon as we rise to the higher worlds where it is not only in this
sense that Extension must be understood, the science of Mathematics
itself fails to afford any immediate expression. But the method of
perception which underlies mathematical science must not be lost.
We must attain the faculty to speak of the realms of Life and Soul,
etc., quite as independently of the particular objective entity, as we
are able to speak of the “circle” independently of the
particular circle drawn upon paper.
As it is true that only so much of real knowledge exists in Natural
Science as there is Mathematics in It, so it is true that on all the
higher planes knowledge can be acquired only when it is fashioned
after the pattern of mathematical science.
Now, within the last few years, mathematical science has made
considerable progress. An important step has been taken within the
realm of mathematics itself, towards the supersensible. This has come
about as the result of the Analysis of Infinity which we owe to Newton
and Leibnitz. Thus another branch of mathematical science has been
added to that which we call “Euclidian.” Euclid expresses by
mathematical formulae only what can be described and constructed
within the field of the “finite.” What I can state in terms
of Euclid about a circle, a triangle or about the relations of
numbers, is within the field of the finite, it is capable of
construction in a senseperceptible manner. This is no longer possible
with the Differential Calculus with which Newton and Leibnitz taught
us to reckon. The Differential still possesses all the properties that
render it possible for us to calculate with it; but in itself as such,
it eludes senseperception. In the Differential, senseperception is
brought to a vanishing point and then we get a new basis — free
from senseperception — for our reckoning. We calculate what is
perceptible by the senses through that which eludes senseperception.
Thus the Differential is an Infinitesimal as against the finitely
sensible. The “finite” is mathematically referred back to
something quite different from it, namely to the real
“infinitesimally small.” In the Infinitesimal Calculus
we stand on an important boundary line. We are mathematically led out
beyond what is perceptible to the senses, and yet we remain so much
within the real that we calculate the “Imperceptible.” And
when we have calculated, the perceptible proves to be the result of
our calculation from the imperceptible. Applying the Infinitesimal
Calculus to natural processes in Mechanics and Physics, we accomplish
nothing else, in fact, than the calculation of the sensible from the
supersensible. We comprehend the sensible by means of its
supersensible beginning of origin. For senseperception, the
Differential is but a point, a zero. For spiritual comprehension,
however, the point becomes alive, the zero becomes an active Cause.
Thus, for our spiritual perception, Space itself is called to life.
Materially perceived, all its points, its infinitesimally small parts,
are dead; if, however, we perceive these points as differential
magnitudes, an inner life awakens in the dead “sidebyside.”
Extension itself becomes the creation of the extensionless. Thus did
life flow into Natural Science through Infinitesimal Calculus. The
realm of the senses is led back to the point of the supersensible.
It is not by the usual philosophical speculations upon the nature of
differential magnitudes that we grasp the full range of what is
mentioned here, but rather by realising in true
“selfknowledge” the inner nature of our own spiritual
activity when from the infinitely small we attain an understanding of
the finite through Infinitesimal Calculus. Here we find ourselves
continually at the moment of the genesis of something
senseperceptible from something no longer senseperceptible. This
spiritual activity in the midst of supersensible proportions and
magnitudes has become in recent years a powerful educational means for
the mathematician. And for what has been accomplished in the realms
lying beyond the limits of ordinary physical perception by intellects
such as Gauss, Riemann and our contemporary German thinkers Oskar
Simony, Kurt Geissler, as well as many others, we are indebted
precisely to this. Whatever may be objected in particular against
these attempts: the fact that such thinkers extend the conception of
space beyond the threedimensional compass; that they reckon in terms
that are more universal and more comprehensive than the space of the
senses; these are simply the results of mathematical thought
emancipated by Infinitesimal Calculus from the shackles of
senseperception.
In this way important indications have been set for Occultism. Even
when mathematical thought ventures beyond the limits of
senseperception, it yet retains the strictness and sureness of true
thoughtcontrol. Even if errors do creep in this field, they will
never act so misleadingly as do the undisciplined thoughts of the
nonmathematical student when he penetrates into the realms of the
supersensible.
Plato and the Gnostics only recognised in mathematical science a good
means of education, and no more than this is here implied about
the mathematics of the infinitely small; nevertheless to the Occultist
it does present itself as a good educational means. It teaches him to
effect a strict mental selfeducation where senseperceptions are no
longer there to control his wrong associations of ideas. Mathematical
science teaches the way to become independent of senseperception, and
at the same time it teaches the surest path; for though indeed its
truths are acquired by supersensible means, they can always be
confirmed in the realm of the senses. Even when we make a mathematical
statement about fourdimensional space, our statement must be such
that when we leave the fourth dimension out and restrict the result to
three dimensions, our truth will still hold good as the special case
of a more general proposition.
No one can become an Occultist who is not able to accomplish within
himself the transition from thought permeated with sense to thought
emancipated from senseperception. For this is the transition where we
experience the birth of the “Higher Manas” from the
“Kama Manas.” It was this experience which Plato demanded
from those who wished to become his disciples. But the Occultist who
has passed through this experience must go through one still higher.
He must also find the transition from thought emancipated from
senseperception in form, to formless thought. The idea of a triangle,
of a circle, etc., is still qualified by form, even though this form
is not an immediately sensible one. Only when we pass over from what
is limited by finite form to that which does not yet possess any form,
but which contains within itself the possibility of formcreation,
only then are we able to understand what is the realm of Arupa
in contrast to the realm of Rupa. On the lowest and most
elementary plane we have an Arupa reality before us in the
Differential. When we reckon in Differentials we are always on the
borderline where Arupa gives birth to Rupa. In
Infinitesimal Calculus, therefore, we can train ourselves to grasp the
idea of Arupa and the relation of this to the Rupa. We
need but once integrate a differential equation with full
consciousness; then we shall feel something of the abounding power
that exists on the borderline between Arupa and Rupa.
Here, of course, it is at first only in an elementary manner that one
has grasped what the advanced Occultist is able to perceive in higher
realms of being. But one here has the means to see at least an
idea of what the man who is limited to senseperception cannot
even divine. For the man who knows nothing beyond senseperception,
the words of the Occultist must at first seem devoid of all meaning.
A science which is gained in realms where the support of
senseperception is necessarily removed, can be understood in the most
simple manner at the stage where man emancipates himself most easily
from such perception. And such is the case in mathematics. The latter,
therefore, constitutes the most easily mastered preliminary training
for the Occultist who will raise himself to the higher worlds with
definite enlightened consciousness and not in dim sensuous ecstasy or
in a semiconscious longing. The Occultist and the Mystic live in the
supersensible with the same enlightened clearness as the elementary
geometrician enjoys in the realm of his laws of triangles and circles.
True Mysticism lives in the light, not in the darkness.
When the Occultist, who starts from a point of view like that of
Plato, calls for research in the mathematical spirit, he can easily be
misunderstood. It might be objected that he overrates the mathematical
spirit. This is not the case. Such an overrating rather exists on the
part of those who admit exact knowledge only to the extent to which
mathematical science reaches. There are students of natural science at
the present time who reject as not being scientific in the full sense
of the word any statement which cannot be expressed in numbers or
figures. For them vague faith begins where mathematics end; and
according to them, all right to claim objective knowledge ceases at
this point. It is precisely those who oppose this overrating of
mathematics itself who can most thoroughly value the true enlightened
research which advances in the spirit of mathematics even where
mathematical science itself ceases. For in its direct meaning
mathematical science after all has to do only with what is
quantitative; where the qualitative begins, there its domain ends.
The point is, however, that we should also be able to research (in the
exact sense of the word) in the domain of the qualitative itself. In
this sense Goethe set himself with particular emphasis against an
overrating of mathematics. He did not want to have the qualitative
bound and fettered by a purely mathematical method of treatment.
Nevertheless, in all things he wanted to think in the spirit of
the mathematician, according to the model and pattern of the
mathematician. This is what he says: — “Even where we do not
require any calculation, we should go to work in such a manner as if
we had to present our accounts to the strictest geometrician. For it
is the mathematical method which on account of its thoroughness and
clearness reveals each and every defect in our assertions, and its
proofs are really only circumstantial explanations to the effect that
what is brought into connection has already been there in its simple,
single parts and in its entire sequence; that it has been perceived in
its entirety and established as incontestably correct under all
conditions.” Goethe wishes to understand the qualitative in the
forms of plants with the accuracy and clearness of mathematical
thought. Just as one draws up mathematical equations in which one only
has. to insert special values in order to include under one general
formula a multiplicity of single cases, so does Goethe seek for the
primordial plant which is qualitatively allembracing in spiritual
reality. Of this he writes to Herder in 1787: “I must further
assure you that I am now very near to the secret of the generation and
organization of the plant, and that it is the very simplest thing that
can be imagined ... The prototype of the plant (Urpflanze)
will be the most wonderful creation of the world, for which Nature
herself shall envy me. With this model and the key thereto one can
then discover plants without end, which will necessarily be
consistent, that is to say, which — even if they do not exist
— could yet exist.” That is to say, Goethe seeks the as yet
formless protoplant, and he endeavours to derive therefrom the actual
plantforms just as the mathematician gets from an equation the
special forms of lines and surfaces. In these realms Goethe's trend of
thought was really tending towards true Occultism. This is known to
those who learn to know him intimately.
The point is that by the selftraining abovementioned, Man should
raise himself to a perception emancipated from the senses. It is only
through this, that the gates of Mysticism and Occultism are thrown
open to him. Through the schooling in the spirit of mathematics lies
one of the paths to the purification from life in the senses. And just
as the mathematician is consistent in life, just as he is able to
construct bridges and bore tunnels by virtue of his training —
that is to say, he is able to command the quantitative reality, in the
same way, only he will be able to understand and rule the
qualitative, who can make himself master in the ethereal
heights of sensefree perception. This is the Occultist. Just as the
mathematician builds the shapes of iron into machines according to
mathematical laws, so does the Occultist shape life and soul in the
world according to the laws of these realms which he has understood in
the spirit of mathematical science. The mathematician is led back to
real life through his mathematical laws; the Occultist no less so
through his laws. And just as little as he who is ignorant of
mathematics is able to understand how the mathematician builds up the
machine, even so little can he who is not an Occultist understand the
plans by which the Occultist works upon the qualitative forms of life
and soul.



