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 Auto-Referate 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.)
Thanks to an anonymous donation, this lecture has been made available.
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Mathematics and Occultism By Rudolf Steiner.
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 sense-perception?” He considered this to be an all-important 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 every-day 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 sense-perception. 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 sense-perceptions.” 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 sense-perception. It was precisely such a mind emancipated from sense-perception 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 sense-perception. The mathematical images hover over the border-line 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 sense-perceptible, but they are not exhaustively contained in it. They hover over innumerable, manifold sense-perceptible 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 sense-perception. 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 sense-perceptible 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 sense-perception what is spiritual. From the mathematical figure I can learn to know super-sensible facts by way of the sense-world. This was the all-important 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 super-sensible in mathematical thought. When a man reaches the stage of being able to think of other properties of the world independently of sense-perception 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 super-sensible 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 sense-perception 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 super-sensible mathematical perception; no experience will ever contradict thy super-sensible 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 sense-perception — a science which, although expressed through sense-perception, 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 super-sensible. 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 sense-perceptible 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 sense-perception. In the Differential, sense-perception is brought to a vanishing point and then we get a new basis — free from sense-perception — for our reckoning. We calculate what is perceptible by the senses through that which eludes sense-perception. 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 super-sensible. We comprehend the sensible by means of its super-sensible beginning of origin. For sense-perception, 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 “side-by-side.” 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 super-sensible.
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 “self-knowledge” 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 sense-perceptible from something no longer sense-perceptible. This spiritual activity in the midst of super-sensible 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 three-dimensional 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 sense-perception.
In this way important indications have been set for Occultism. Even when mathematical thought ventures beyond the limits of sense-perception, it yet retains the strictness and sureness of true thought-control. Even if errors do creep in this field, they will never act so misleadingly as do the undisciplined thoughts of the non-mathematical student when he penetrates into the realms of the super-sensible.
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 self-education where sense-perceptions are no longer there to control his wrong associations of ideas. Mathematical science teaches the way to become independent of sense-perception, and at the same time it teaches the surest path; for though indeed its truths are acquired by super-sensible means, they can always be confirmed in the realm of the senses. Even when we make a mathematical statement about four-dimensional 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 sense-perception. 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 sense-perception 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 form-creation, 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 border-line 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 sense-perception cannot even divine. For the man who knows nothing beyond sense-perception, the words of the Occultist must at first seem devoid of all meaning.
A science which is gained in realms where the support of sense-perception 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 semi-conscious longing. The Occultist and the Mystic live in the super-sensible 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 all-embracing 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 plant-forms 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 self-training above-mentioned, 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 sense-free 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.
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