Mathematics and Occultism 
by Rudolf Steiner
It is well-known that the words above the Platonic Academy were said to have excluded anyone ignorant of mathematics from participating in the teachings of the master. Whatever one may think about the historical truth of this tradition, it correctly expresses Plato’s views on the status of mathematics within the domain of human knowledge. By means of his ‘theory of ideas’ Plato wished to teach his students to move with their knowledge within the world of pure spiritual being. He believed that human beings cannot know anything of the true world as long as their thinking is permeated by the impressions of the senses. He demanded sense-free thinking. A person moves within the world of ideas once they have separated everything out of their thinking that is furnished by the senses. Above all for Plato the question arose: How can human beings free themselves from all sensory perception? He regarded this as an important question for the education of the spiritual life.
Human beings can only liberate themselves from sense perception with difficulty. Self-knowledge teaches us this. Even when a person living in the everyday world withdraws into himself, to let no sense impression work upon him, there are still, nevertheless, traces from the senses present. The undeveloped person stands before a void, a completely empty consciousness, if he disregards the impressions received from the sense world. It is for this reason that certain philosophers maintain there is no such thing as sense-free thinking. They believe that even if a person completely withdraws into the field of pure thought he is still only occupied with the fine shadow pictures of sense perception. But this statement is only valid for the undeveloped person. As soon as a person acquires the ability to develop within himself organs of spiritual perception, just as nature has developed within him organs of sense perception, his thinking ceases to remain empty when all sensory content is separated out. Such sense-free, and yet spiritually-filled thinking, was what Plato demanded from those who wished to understand his theory of ideas. Thus he only demanded something that teachers in all ages have had to demand from their pupils when they have sought to make them into true initiates of higher knowledge. If a person has not experienced in all its ramifications this demand of Plato then he will be unable to attain any insight into the true nature of wisdom.
Plato regarded mathematical conceptions as an educational means for residing within the sense-free world of ideas. For mathematical forms hover on the boundary between the sense world and the purely spiritual world. Think of a ‘circle’. One does not thereby think of this or that sense perceptible circle that one has perhaps sketched on a piece of paper, but any and every given circle that can be drawn or met with in nature. This is the case with all mathematical forms. They relate to the sense world, but are in no way exhausted through anything found in it. They hover over innumerable and manifold sense forms. When I think mathematically, I think about the things of the sense world, but at the same time I do not think within the sense world. It is not the sense perceptible circle that teaches me the laws of the circle, but the ideal circle, that only lives in my mind and of which the sense perceptible one is simply an image. Every other sense perceptible image of the circle can also teach me this. That is what is so significant about the mathematical conception: that a single sense formation leads out beyond itself; and it can only be for me a symbol of a comprehensive spiritual fact. And thus, there also exists the possibility of bringing the spiritual in this sphere into the domain of the sense perceptible. For with mathematical forms I can become acquainted with supersensible facts through sensory means. That is what was so important for Plato. The idea has to be perceived purely spiritually if it is to be known in its true essence. One can arrive at knowledge about these things by exercising the first steps in mathematics, and by becoming aware of what it is that one actually gains from a mathematical form. Learn with the help of mathematics to make yourself free of the senses, for only then can you hope to rise to the sense-free understanding of ideas. – That is what Plato wished to instil into his students.
The gnostics, for example, demanded something similar. “Gnosis is mathesis,” is what they said. They did not mean with this that the essence of the world should be founded on a mathematical conception, but only that the first step in the spiritual education of humanity lay in this view of attaining the supersensible. A person is on the path to spiritual knowledge when he has reached the stage of thinking sense-free about other properties and qualities of the world, just as he has learnt with mathesis to think about geometrical forms and arithmetical number relationships. The gnostics did not strive after mathesis itself, but rather for spiritual knowledge attained in accordance with the model of mathesis. They saw in mathesis a model or standard, and because the geometrical relationships of the world are the most elementary and simple, they are therefore the easiest for a human being to attain. One should learn with elementary mathematical truths to become free of the senses, so that one can also later on become sense-free when dealing with questions of higher knowledge. – Certainly for many the dizzying heights of human knowledge are thereby indicated. However, those whom we can designate as true occultists have always demanded of their students the courage to make these dizzying heights their own. “You may only become a student of the mysteries when you have learned to think about the being of nature and spiritual existence wholly free of any sense-perception, just as a mathematician does regarding a circle and its laws.” That should stand in golden letters before everyone who really seeks the truth. “You will never find a circle in the world that does not confirm within the sense world what you have learned in sense-free mathematical perception; no sense experience can deceive you regarding your spiritual knowledge. You attain, therefore, an imperishable, an eternal knowledge, once you know yourself to be free of the senses.” It is for this reason that Plato, the gnostics and all true occultists have thought of mathematics as an educational tool.
We will now consider what some outstanding personalities have had to say on the relationship between mathematics and natural science. There is only so much true science in our knowledge of nature, as there is mathematics in it – has said, for example, Kant and many like him. This indicates nothing else except to point out that a knowledge is attained through the mathematical formulation of the processes of nature which transcends sense perception, a knowledge which indeed comes to expression via sense perception, but is, however, comprehended in the mind. I have first understood the way a machine works when I have expressed it in mathematical formulae. It is the ideal of mechanics, physics and is becoming ever more the ideal of chemistry, to express in such formulations those processes lying spread out before the senses. But only what lives itself out in space and time, only what has extension in this sense, can be expressed mathematically in this way. For as soon as one ascends into the higher worlds where it is not a question of extension in this sense then mathematics too has to renounce its immediate form. However, one should not dispense with the mode of perception that lies at the basis of mathematics. We have to acquire the capacity to speak about living things, the soul and so on, so freely and independently of their single observable forms, just as we speak of a circle independently of one drawn on a piece of paper.
Just as it is true that all our knowledge of nature contains only so much real knowledge as there is mathematics in it, so it is also true that in all higher spheres knowledge can only be attained when it is carried out in accordance with the model of mathematical knowledge. Mathematical knowledge has made very significant progress in more recent times. It has undertaken within its own domain an important step into the supersensible. This has taken place in the analysis of the infinite, which we owe to Newton and Leibniz. Thus we have obtained another mathematics in addition to the one we call Euclidean. Euclidean mathematics can only express in mathematical formulae those things that can be presented and constructed within the field of the finite. What I state regarding a circle, a triangle or in numerical relationships in the sense of Euclidean mathematics, can be constructed within the finite field and in a sense perceptible way. This is no longer possible with the differential calculus with which Newton and Leibniz have taught us to calculate. The differential has all the qualities that make it possible to carry out calculations with it, but it is as such not visible to sense perception. In the differential, sense perception is first brought to a vanishing point, and we then obtain a new sense-free foundation for our calculation. The sense perceptible is calculated out of what is no longer visible to the senses. Thus the differential is what is infinitely small in relation to the finite-sensible. The finite is mathematically led back to something wholly dissimilar from it – to the truly infinitely small. With infinitesimal calculus we stand at a very important boundary. We are mathematically led beyond the sense perceptible but remain so much within reality that we can compute the imperceptible. Once we have calculated, the perceptible then proves to be the result of our calculation out of the imperceptible. With the application of infinitesimal calculus in mechanics and physics to the processes of nature we are really carrying out nothing more than the calculation of the sensible from out of the supersensible. We comprehend the sensible out of its supersensible beginning or origin. For the outer senses the differential is a point or a zero. For a spiritual understanding, however, the point becomes alive, the zero becomes a cause. Thus, for a spiritual understanding space itself becomes enlivened. If we comprehend space with the outer senses, its points, its infinitely small parts are dead; but if we understand the points as differential quantities, there streams inner life into the dead co-existing parts. Extension itself becomes the product of the extension-less. Through infinitesimal calculus life came into our knowledge of nature. The sensible is led back to the point of the supersensible.
The consequences of everything we have mentioned here cannot be grasped through customary philosophical speculations on the nature of differential quantities, but rather by realising through self-knowledge how to proceed in one’s own spiritual activity, when with infinitesimal calculus one attains the finite from out of the infinitely small. One continually stands before the moment of the coming into being of something sensible from out of what is no longer sensible. It is therefore entirely comprehensible that this spiritual life in supersensible, mathematical relationships of magnitude, has become a powerful educational tool for the mathematician of more recent times. We are indebted to what spirits such as Gauss, Riemann and more recently the German thinkers Oskar Simony and Kurt Geissler, among many others, have accomplished in this field that lies beyond normal sense perception. One can always object in detail to these attempts. But the fact that these thinkers have extended our concept of space beyond the three-dimensional, and they calculated in conditions that are more general and comprehensive than physical space – are all a result of mathematical thinking that is emancipated from the sense world by infinitesimal calculus.
Very important indications for occultism are thereby hinted at. For mathematical thinking continues to possess, even in those regions where it ventures beyond the sense-perceptible, the exactness and certainty that belong to genuine disciplined thinking. Aberrations may of course also occur in this domain, but their effects are never as disastrous as when the undisciplined thoughts of the non-mathematically trained penetrate into the supersensible. Just as little as Plato or the gnostics viewed mathematics as anything other than an educational tool, so we too maintain nothing more with respect to the mathematics of the infinitely small. However, it is indeed an educational tool for the occultist. It teaches him to bring strict mental discipline into those regions where false chains of reasoning can no longer be corrected with the help of normal sense perception. Mathematics teaches us to become free of the senses. However, it does this via a very sure path, because although its truths are gained in a supersensible manner, they can always be confirmed through sensory means. Even if we state something mathematically regarding four-dimensional space, the statement must be of such a nature that if we were to leave four dimensions and specialise our result to three, our truth would still remain a special case of a general statement.
No one can become an occultist who cannot carry out within himself the transition from sense-filled, to sense-free thinking. For this is the transition where we experience the birth of ‘higher manas’ out of ‘kama-manas’. This is what Plato required those to experience who wished to become his students. The occultist, however, who has already passed through this experience, must still undergo something higher. He must also find the transition from sense-free thinking in forms, to form-less thinking. The thought of a triangle, a circle and so on, always has a form even though this form is not a directly material one. Only when we are able to pass from what exists in a finite form, to something that does not have any form, but contains the possibility of form-production within it, are we able to grasp what the arupa [formless] realm is in contrast to that of the rupa [formed]. At the lowest, most elementary level we have an arupa reality before us in the differential. If we calculate using the differential, we always stand at the point where the arupean gives birth to the rupean. With infinitesimal calculus, therefore, we can learn to comprehend the arupean and understand what sort of relationship it has to the rupean. One has to only carry out in full consciousness the integration of a differential equation to feel something of the source of power that exists at the boundary between the arupean and the rupean. However, one has still only grasped in the most elementary way what the advanced occultist is able to perceive in higher realms of being. Nevertheless, one has a means for beholding, and at least an indication of those things of which the person who remains bound to the sense world cannot have any idea. For a person only confined to the outer senses the words of the occultist must at first seem to be devoid of all content and meaning.
Knowledge attained in those regions where the crutches of sense perception are lacking can be most readily understood where the freeing from such a perception is most easily gained. This is the case with mathematics. Mathematics is therefore the most easily acquired preparatory training for the occultist who seeks to rise to bright and radiant clarity in the higher worlds, and not to a dim sentient form of ecstasy or to dreamy premonitions. The occultist and mystic live in the same light-filled clarity within the supersensible world, as the elementary geometer does in working with the laws of triangles and circles. True mysticism lives in light not in darkness.
It can also be easily misunderstood when the occultist who is speaking out of a conviction that can be called Platonic demands that research be carried out in the manner of mathematics. One could assume that mathematics is being overrated. This is not the case. However, they are guilty of such an overestimation who only allow something to count as exact knowledge to the extent that it falls within the realm of mathematics itself. There are natural scientific researchers of the present time who reject every statement in the fullest sense as unscientific which cannot be expressed in numbers or figures. For them, where mathematics ends, vague belief begins; and therefore every claim to objective knowledge must cease. It is precisely those who object to the overestimation of mathematics who are able to become the true judges of that genuine crystal-clear type of research that also proceeds in the spirit of mathematics, even in those regions where mathematics itself ends. In its most immediate sense mathematics is only concerned with the quantitative. Its realm ends where the qualitative begins.
Thus it is also a question of researching in the strictest sense within the field of the qualitative. In this sense Goethe particularly strongly objected to an overestimation of mathematics. He did not wish to see the qualitative confined to a purely mathematical kind of treatment. Rather, he wished to think everywhere in the spirit of mathematics, everywhere according to the manner and model of mathematics. Goethe says: “even there, where we require no calculation, we should proceed as though we were accountable to the most stringent geometer. For on account of its deliberation and purity the mathematical method immediately exposes every jump in an assertion. Its proofs are nothing more than elaborate explanations of connections that were already implicit in their basic parts, and in their entire sequence demonstrate the whole to be correct and irrefutable.” Goethe seeks to grasp the qualitative in plant formations using the exactness and clarity of the mathematical manner of thinking. Just as one puts particular values into mathematical equations in order to understand the manifoldness of specific cases under a general formula, so Goethe sought the Archetypal or Primal Plant (Urpflanze), which is all-embracing in the realms of the qualitative and spiritual reality. Goethe wrote to Herder in 1787 concerning this: “Further I must confide to you that I am very close to the secret of plant generation and organisation and it is the simplest thing imaginable. […] The Primal Plant will be the most wonderful creation in the world, for which nature herself shall envy me. With this model and its key it will be possible to invent plants into infinity, that consequently must exist, that is to say, even if they do not actually exist, could nevertheless.” Thus Goethe sought the totally formless Primal Plant and strove to derive plant formations from it, just as a mathematician derives specific forms of lines and planes from an equation. – Goethe’s manner of thinking in this domain approached the manner of thinking of occultism. One can see this when one becomes more closely acquainted with Goethe’s work.
Thus it all depends on the human being rising, through the type of self-discipline indicated above, to sense-free perception. The doors to mysticism and occultism are only unlocked in this manner. One of the paths leading to a purification from life in the sense world is through a schooling carried out in the spirit of mathematics. And just as the mathematician first stands firmly in life, when because of his training he is able to build bridges and tunnels, that is say, he is able to quantitatively master reality, so they too are only able to master and understand the qualitative realm who have first understood it in the ethereal heights of sense-free perception. This is what the occultist does. And just as the mathematician uses mathematical laws to fashion iron structures into machines, so the occultist shapes the life and soul in the world using the laws of these realms that he has grasped in a mathematical way. The mathematician returns to the world with his mathematical laws; the occultist no less with his laws. And just as little as the non-mathematician can understand how a mathematician works on a machine, so just as little can the non-occultist understand the plans by which the occultist works upon the qualitative forms of life and soul.
Appendix: “The Theosophical Congress in Amsterdam” (1904) 
The Congress of the Federation of the European Sections of the Theosophical Society took place in Amsterdam from the 19th to the 21st June 1904. […] In the section ‘Philosophy’ Dr. Rudolf Steiner spoke on “Mathematics and Occultism”. He started from the fact that Plato demanded a preliminary mathematical training from his students, that the Gnostics had designated their higher wisdom as mathesis, and that the Pythagoreans had viewed number and form to be the foundation of existence. He explained that they all did not have abstract mathematics in mind, but rather the intuitive seeing (das intuitive Schauen) of the occultist. The latter grasps laws in the higher worlds with the aid of spiritual perception, which is represented in the spiritual sphere as music is in our ordinary sense world. Just as through oscillations air is able to stimulate musical sensations that can be expressed using numbers, so if the occultist is properly prepared by means of the secrets of numbers he is able to perceive spiritual music in the higher worlds. In an especially lofty development of the human being this may be heightened to a sensation of the music of the spheres. The music of the spheres is not a figment of the imagination, but constitutes a genuine experience for the occultist. The human being permits the hidden phenomena of the world to work upon himself by integrating mathesis into his own being, by the penetration of his astral and mental body with an intimate sense that is expressed in numerical relationships.
In modern times the occult sense withdrew from the sciences. Since the time of Copernicus and Galileo science has been concerned with the conquest of the physical world. However, it belongs to the eternal plan of human development that physical science should also find access to the spiritual world. In the epoch of physical research mathematics has become enriched by Newton’s and Leibniz’s analyses of the infinite, by differential and integral calculus. Whoever does not merely understand this in an abstract manner but attempts to inwardly experience what a differential really represents, impresses a sense-free intuition upon himself. For in the differential the sensible intuition of space itself has become overcome in a symbol; the cognition of the human being becomes purely mental for a moment. This is manifest to the clairvoyant insofar as the thought form of the differential is outwardly open, in contrast to the thought forms that the human being receives through sensible intuition. The latter are outwardly closed. Thus, by means of the analysis of the infinite a path becomes opened up through which the higher sense of the human being becomes outwardly open. The occultist knows what sort of process occurs with the charka (lotus blossom) situated between the eyebrows when he develops the spirit of the differential within himself. If the mathematician is in addition a selfless person then he may place whatever he has achieved in this way onto the universal altar of human brotherhood. And an important source for occultism therefore comes into being out of the apparently driest science. […]”.
Translated from the German by David W. Wood
 Rudolf Steiner’s 1904 essay “Mathematik und Okkultismus” first appeared in the “Transactions of the First Annual Congress of the Federation of European Sections of the Theosophical Society”, edited by Johan van Manen, Amsterdam, 1906. This essay and many of the editorial notes can be found in Rudolf Steiner, Philosophie und Anthroposophie, Gesammelte Aufsätze – GA 35 [GA = Gesamtausgabe, the Complete Edition of Steiner’s works in German], Dornach 1984, pp. 7-18. The text is a write-up of an address that Steiner originally delivered in Amsterdam on the 21st June 1904 at the Congress of the Federation of European Sections of the Theosophical Society. This translation by David W. Wood was first published in September 1997 in the journal Archetype, issue 3, pp. 46-53. It appears here in a revised form and enlarged with an appendix containing Steiner’s 1904 report of his talk.
 According to tradition the inscription was: “Let no one ignorant of geometry enter here.” On the significance of geometry and mathematics for Plato, see among others The Republic, translated by Desmond Lee, Part 8, Book 7, (New York: Penguin Books, 1980), pp. 331-335; the Seventh Letter, in: Plato, Timaeus, Critias, Cleitophon, Menexenus, Epistles (Cambridge MA: Harvard University Press, 1999), pp. 477-573; and Plato, Protagoras and Meno (New York: Penguin Books, 1956).
 See I. Kant, Metaphysical Foundations of Natural Science (1776), edited by Michael Friedman, Preface: “ I assert, however, that in any special doctrine of nature there can be only as much proper science as there is mathematics therein.” (Cambridge: Cambridge University Press, 2004), p. 6.
 Karl Friedrich Gauss (1777-1855) discovered that the axiom system of classical Euclidean geometry is only one of many possible; Bernhard Riemann (1826-1866).
 Oskar Simony (1852-1915) was a professor of mathematics at the College of Agriculture in Vienna and personally known to Rudolf Steiner. Some of Simony’s central works include: “Gemeinverständliche, leicht kontrollierbare Lösung der Aufgabe: ‘In ein ringförmig geschlossenes Band einen Knoten zu machen’ und verwandter merkwürdiger Probleme” [Generally understandable and easily controllable solution to the problem: ‘To make a knot in a ring-shaped closed ribbon,’ and other curious related problems] 3rd enlarged edition Vienna 1881. See Steiner’s remarks on Simony in Rudolf Steiner, Towards Imagination GA Bibl. No. 169, Anthroposophic Press 1990, and Rudolf Steiner, The Fourth Dimension (Anthroposophic Press, 2001); (Die vierte Dimension, Mathematik und Wirklichkeit GA Bibl.No. 324a, Dornach 1995).
 Friedrich Jakob Kurt Geissler was born in 1859 and wrote among others, Die Grundsätze und das Wesen des Unendlichen in der Mathematik und Philosophie [The Principles and Essence of Infinity in Mathematics and Philosophy] (1902), Grundgedanken der Übereuklidischen Geometrie [Foundational Thoughts of Super-Euclidean Geometry] (1904).
 The expression “the birth of the higher manas out of kama-manas” is the birth of the higher mind (manas) out of the lower sensory or desire (kama) mind. Rudolf Steiner is here employing traditional theosophical Sanskrit terminology. He later he replaced these with anthroposophical terms.
 Cf. footnote 7 above. The arupa realm or plane signifies the formless (arupa) plane, and designates the spiritual world, that is the world of reason, the world of true intuition. This is in contrast to the lower realm of rupa or the world of (physical) forms.
 See Goethe’s essay, “Über Mathematik und deren Missbrauch” [On Mathematics and its Misuse]: “I hear that I am accused of being an adversary, an enemy of mathematics in general. However no one values mathematics higher than myself, for it accomplishes exactly what is wholly denied to myself to achieve. Concerning this I would gladly clarify my position and choose to this end a single means – the words and lectures of other significant and notable men. What concerns the mathematical sciences must by no means impress us through their nature or multitude. They owe their certainty primarily to the simplicity of their objects.” Goethes Naturwissenschaftliche Schriften, edited and commentated by Rudolf Steiner in Kurschner’s Deutsche National-Litteratur 1883-1897 (five volumes, reprinted Dornach 1975), GA 1b, pp. 45f.
 J.W. Goethe, “Der Versuch als Vermittler zwischen Objekt und Subjekt” (The Experiment as Mediator between Subject and Object), 1792/1823, ibid, p.19.
 Letter to Herder: 17th May 1787, in Italienische Reise (Italian Journey).