The Foundations of Mathematics in Light of Anthroposophy

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1 The Foundations of Mathematics in Light of Anthroposophy Ernst Bindel teacher at the Free Waldorf School Stuttgart, Germany Published by the Astronomical-Mathematical Section of the School of Spiritual Science The Goetheanum, Dornach, Switzerland With 45 figures and 8 images Originally published: Waldorfschul-Spielzeug und Verlag, Stuttgart 1928 Translated by Charles Gunn cgunn3@gmail.com April 8, 2010

2 Translator s Remarks The current version is restricted to the first two sections of the book, entitled Introduction in the original. The rest of the translation has been completed but there are still some mathematical expressions are not yet formatted and figures have not been brought over to the translated version. I intend to extend the released translation as time and energy allow me to polish off the rough edges remaining. The book presented here should be compared to a book by the same author entitled Arithmetik: Menschenkundliche Begründung und pädagogische Bedeutung and published in 1967 by Ch. Mellinger, Stuttgart. The latter book, published 40 years after this work, contains some of the same material almost word-for-word, including the introductory sections and the treatment of the conditional equation. It lacks, however, the bulk of the historical, speculative, anthroposophical sections which in my opinion, give the older work its peculiar depth and fascination. Note that there are some terms in German which caused some problems for the translation. I mention them here although not all occur in the part of the translation currently available. These include: Arithmetik What we (Americans) mean with arithmetic is subsumed by Rechnen in german. And the German Arithmetik might be better translated as higher arithmetic in English, since it connotes the study of number in its widest sense. However, since I prefer the german usage, I have chosen to translate Arithmetik as arithmetic, and Rechnen with calculation. Goniometrie The root here refers to angle-measurement, but there is no convenient word in common english usage. It s related to trigonometry but isn t exactly the same. I have chose to use goniometry although the word doesn t appear in the standard english math literature. Bestimmungsgleichung This refers to a linear equation in one unknown, the kind that arises from most word problems encountered in beginning algebra. I have translated it according to the dictionary as conditional equation, although I have never encountered the latter phrase in the english mathematical literature. Falkensee, Germany, April, 2010 Charles Gunn 1

3 Preface After the publication last year of his book Calculation in the Light of Anthroposophy through the Mathematical-Astronomical Section of the Goetheanum, Ernst Bindel now has come forth with a larger work, with an essential historical-systematic content. The author characterizes the work himself in his forward as the outcome of a kind of treasure hunt. Only the display of these treasures can make the human being once more long to search in the ground, from which they have been raised into the light. It will also give him the trust that just as here the historical unfolding of mathematics is shown to be a mirror of the development of the human spirit he will have the necessary forces to create new forms out of his newly-awakened spiritual abilities. The first tender shoots of these spiritual seeds are beginning to show themselves in scientific fields, fructified by Rudolf Steiner s Anthroposophy. The Mathematical-Astronomical Section of the School of Spiritual Science sees it as one of its most beautiful tasks, to further nurture the seeds which grow in its domain. Hence this work from Ernst Bindel also appears as one of its publications. Dornach, March, 1928 For the Section: Elizabeth Vreede, Ph. D. 2

4 Forward The seed of this book can be identified in a lecture that I gave at Easter 1927, with the title Old and New Ways in Arithmetic and in Calculation, at the fifth public Teachers Conference of the Free Waldorf School, Stuttgart. With this book I have tried to remove the distrust and alienation vis-a-vis mathematics which are justifiably present in the widest circles of society, through a liberation of treasures, which lie submerged within contemporary mathematics. I have restricted myself to the arithmetic part of mathematics, and specifically to that part which, according to Rudolf Steiner s spiritilluminated directives, should be made accessible to every human being in his youth up to the 18 th year. The book is therefore primarily written for those who are acquainted through their education with the foundations of mathematics. As however, one has to reckon with the loss of this knowledge in most cases, I have tried to present all the calculations and operations which appear in the examples, so understandably that really only a dim memory of the erstwhile knowledge is assumed. Only in the two sections on differential calculus and its relationhip to the transcendental numbers (sections 15 and 16) was that impossible; anyone however who has read all the previous sections will, with some good will, manage to overcome the difficulties presented by these two sections. One would err, if one believed on the basis of this book, to be able to relearn mathematics. It restricts itself to a discussion of symptomatic questions, a discussion that is however as widely-framed as possible. I must admit to the hope, that a substantial number of readers will, after reading this book, undertake a study of mathematics based on ordinary textbooks, without having to fear they will be overcome with antipathy against mathematics. Valuable stimulations for my work were provided by Miss Elisabeth Vreede, the leader of the Mathematical-Astronomical Section of the School for Spiritual Science in Dornach, and by my collegue Dr. Walter Johannes Stein. I take this opportunity here to express my heartfelt thanks to them. Stuttgart, March, 1928 Ernst Bindel Teacher at the Free Waldorf School 3

5 Motto: The highest life is mathematics Novalis, Fragments Section 1 The pedagogical situation Mathematics instruction, especially in arithmetic, presents significant difficulties for today s teacher. On the one hand, the curriculum has in the course of the recent centuries and even decades has become more and more abstract and dried out, and on the other hand, effective pedagogical instincts among teachers have largely died out. And so it can happen, that otherwise successful teachers will fail when faced with arithmetic. Whatever direction the teacher looks, he is faced with dangerous shoals. In regard to arithmetic, almost everything is problematic. You only need to consider the introduction of fractions, or negative numbers, or even multiplication itself, in order to confront a series of problems demanding resolution right at the beginning. And if you continue further into arithmetic, you continually meet new obstacles there. Another major problem is that nowhere else are there so many weakly endowed and handicapped students as in arithmetic. Indeed the teacher consoles himself over his failures with the thought that in this area it all depends on inborn gifts. This is a dangerous thought, because it cripples the efforts of the teacher and the student to achieve something. It is simply accepted as fact that some, in fact many, people take no inner interest in arithmetic. Can we understand in a fundamental way how this unfortunate situation has come about? More and more people have the tendency to separate themselves, so to speak, from the currents of world activity, to forget the relationships which connect the things of this world to the depths of their being. These relationships exist now just as they did before, but the human being has lost the conscious awareness of them; in fact, he even denies the possibility of such relationships. Things surround him, but since they appear separate from him, their being can hardly form a connection to his. The situation is particularly true for mathematics. It has become the bearer of a world view, which thinks of all visible processes as happening in such a way, that the inner human being need not be taken into account at all. And so it is not surprising, that the majority of us feel helpless when confronted by mathematics, and only a few, possessing a gift puzzling even to themselves, can still claim access to it. But even these few no longer recognize an inner relationship of mathematics to the depths of their being; essentially it has become a stranger to them also. What should a teacher do, confronted with such a situation? He does not have many choices, except to build his teaching upon general pedagogical principles on principles, that is, that are not actually derived from the substance of arithmetic. He attempts, for example, to connect with the experiences of his students, to bring about a change from tension to relaxation in the lesson, to make a place for visual representations, and similar strategies. But all this doesn t bring any change in the student in connection to arithmetic; it may be 4

6 somewhat more understandable but it doesn t bring it any closer in a human sense. In arithmetic, just as in any domain, the only subject matter that can be pedogogically fruitful, is that which presents itself from the inner being of the topic being studied. One must thus direct one s chief attention at the question of what is spiritually revealed in arithmetic. Such an essential investigation would then flow, as if by itself, into the educational theory. For pedagogy is not some special discipline of the human spirit, but always arises as the end result of a particular type of observation of the world. General pedagogical principles attain their requisite weight only when they are bound with a full insight into the structure of the specific content under consideration. But that is precisely the tragedy of the recent past, that it was no longer able to find an answer to the question of the nature of things. The human being ceased to understand his own nature, and thereby he also lost touch with the nature of the entities surrounding him. Under these conditions, it would not have helped, if someone had posed a question regarding the nature of arithmetic. It was only the appearance of anthroposophically oriented spiritual science, founded by Rudolf Steiner, that created an atmosphere where such questions might once again be asked. It gave back to the human being a knowledge of his own nature, and thereby simultaneously opened the door to understanding the world again. For, in a manner of speaking, everything is inside the human being, he carries all the archetypes of the universe within, as an image. He only needs to become aware of himself, in order to find access to everything else. This is indicated by the deep connection of the word science with the word conscience, Latin scire with conscientia. This linguistic context is also significant for arithmetic. In fact, the word scire is related to the word putare, which meant originally to reckon or calculate and only secondarily has taken on the ordinary meaning of opinion or belief. Next to the word conscientia one has to set the word computus. This word indicated, from ancient times through the Middle Ages, the complete body of calculation-practice; even today the words computer (English), conto (Italian) and compter (French) point to this connection. The people of ancient times connected this word with a high worth, with their own human worth. The church father Origenes wrote: If you remove number from all things, everything falls apart. Rob the age of calculation, and everything will be gripped in ignorance. You cannot distinguish man from the animals, if he hasn t learned how to calculate. Ancient man felt himself reminded of his higher nature, when he said the word computus. With time this picture has changed completely. The word conto repulses our finer-feeling contemporaries. They feel as if they are confronted with something inhuman in the word. Tragically, the human being has renounced something which makes up a fundamental part of his being. One of the most pressing tasks of the present day is to lead this part back to its original position. For the domain of calculation, an attempt has been made to fulfill this task in the author s book Calculation in the Light of Anthroposophy 1. The goal of the present book is 1 Waldorfschul-Spielzeug und Verlag, Stuttgart, 1927 [needs updated] 5

7 to solve the problem for a large part of the domain of arithmetic 2. The path to a solution proceeds in this way, that the teacher and educator first orients himself on the indications given by Rudolf Steiner concerning the human being, and from there attempts once more to approach arithmetic. In order to do this, you only need the courage to address what we may term human questions at arithmetic; the being of arithmetic will then give answers, as if out of our own selves. The teacher should ask questions such as: How is arithmetic differentiated in the body, soul, and spirit of the human being? Where may one search for number in the human being? What organic consequences does calculation have in the human being? What occurs in a person, when he solves an equation? It s clear that the answers to such questions will also turn out to be pedagogically fruitful in their own right. The teacher will once more be able to connect his students with arithmetic in a human way, even if he has to hold his deeper knowledge, and his conscience, in the background. If he does this then lack of ability, due to absence of interest, will also recede. For each student brings himself as his chief asset to this sort of mathematical inquiry; just by being human he contains all he needs. He is interested, because as a human being he himself is within it, because he senses, that he is meeting a piece of himself in mathematics, in this mathematics. It may well be that the sort of gift which makes possible the confident exercise of mathematics still remains in the possession of relatively few; at least antipathy against mathematics will disappear and be replaced by love and enthusiasm. Section 2 On the Nature of the Human Being At a certain point in human cultural development, the above-mentioned process took hold, resulting in the separation of mathematics from the depths of the human being. Rudolf Steiner characterized this moment in time in a series of lectures entitled The Appearance of Natural Science in World History and its Subsequent Development, delivered in Dornach from December 24, 1922 to January 6, In these lectures he explained, for example, how the human being in ancient times experienced the three directions of space with his whole body in connection with the earth, and how, with Descartes, a time then came, in which the human being no longer experienced his body as the crossing point of the three dimensions of space. There arose instead the abstract schema of space, constituted out of three mutually perpendicular but otherwise arbitrary directions, meeting at an arbitrary point in space. Dr. Steiner went on to point out that the same thing happened with other mathematical categories, as happened here with the experience of space; they were pushed out from the direct human experience into abstraction. In this way mathematics ceased to have something to do with the whole human being. Against this rather negative picture Dr. Steiner then placed a more positive one, in order to [Translator s note]: arithmetic as the study of number in its widest sense. 3 Rudolf Steiner, Der Entstehungsmoment der Naturwissenschaft in die Weltgeschichte, GA 326, Dornach, 6

8 make the inner meaning of this spiritual development more understandable. At the beginning of the 15 th century, humanity entered into the age of what he called the consciousness soul. The souls of the human beings began to have the urge, to bring into consciousness everything that manifests itself in sense-perceptible appearances, in order to engrave it, as it were, into their own body. More and more the human being renounced a dreamy, emotional experience of the world, and instead begin to strive for a wide-awake, thinking penetration into external appearances. This strengthening of consciousness in the direction of the sense world brought along with it the above-mentioned loss of the encompassing feeling of connection that reached deep into the things of the world. The human being separated himself from things, in order to confront them as physical sense objects. All this took place for the sake of a great goal, the intensification of the ego consciousness and the development of the forces of personality. The age of the consciousness soul is that of the ego-consciousness soul. Rudolf Steiner was able to share with us particularly deep and intimate statements about this ego. He knew how to represent it as the most valuable part of the human being, as his core, his most inner nature. It had been planted in the human body by divine powers, in the far reaches of time, in order that it could penetrate this body, step by step, in a mighty process of development. In order to follow this process, one must first create an exact picture of the constitution of the human being. One no longer knows this fully, but only as permitted by means of the physical senses and the understanding based on these senses. But that is, so to speak, only a third of the story. For as Dr. Steiner explicitly described in his book Theosophy, the human being has a threefold organization. That finds expression, among other ways, in the fact that the human being is surrounded both internally and externally by three domains of Nature. In this connection, the human body of mere material is connected to the domain of lifeless and unconscious minerals. The physiology of the human body thereby extends down into the world of physics, and so this body is called the physical body. But this body pulses with life, and reveals itself in this way to be related to the world of plants. The lifelessness and the unconsciousness of the minerals is raised in the world of plants to a condition of vitality and a dull sleeping consciousness. To the extent that the human being develops this vegetative side of his organization, he is a life body, an etheric body. Plants however have no movement of their own, they are rooted firmly in the earth, and form a unity with the earth. Once, however, the organism forms a boundary with respect to the earth, then you move from the plant kingdom up into the animal kingdom. In contrast to the plant kingdom, the animal has internal organs, and has wrested a soul and an inner nature away from life. The capacity for self-locomotion in the animal depends on the separation of the animal body from the earth, and this corresponds on the side of the soul, to a content that drives the animal an inner world of drives and sensations, to which the animal is given up in a dream-like consciousness. The animal wakes from the dull sleep of the plant, to a dreamy twilight consciousness. To the extent that the human body expresses itself in this way, it is a body of sensations. Now what the animals with their capacity for self-movement are for 7

9 the earth, finds expression in the cosmos away from the earth, as the heavenly bodies, which in sublime self-movement follow, figuratively, their cosmic drives. And so ancient observers were right to call the ring of stars, along which this movement took place, the Zodiac. Thus it is reasonable that the body that we share with the animals, the body of sensation, can also be termed a star body, an astral body. We see that the ego of the human being is surrounded by three bodily sheaths an astral, an etheric, and a physical one, and it is the destiny of this ego, to work through this sheathnature, in order to wrest forth that which cannot be found in stone, plant or animal: ego consciousness. The ego s work is easiest in the astral body, because this already has a certain amount of consciousness within itself. The dull sleep consciousness of the etheric body already presents a bigger challenge to the penetration by the ego. But the most arduous effort must be undertaken by the ego within the physical body, since this has the deepest unconsciousness to illuminate. But precisely through the work which has to be done here, the personality rises to ever fuller consciousness. The ego awakens here to itself with clear and distinct contours. And because of the fact, that the penetration of the human ego into the physical body lies at the end of the path sketched out here, it is here that the ego - in a certain sense - concludes its bodily tasks. The moment in world development, where the ego began to work through this last stage, was the transition from the Middle Ages to modern times. With this, the age of the consciousness soul arrives. And it was just at this moment, that mathematics first retreated into the human head, after having been previously more connected with the whole human being. [Mention here the root for Zodiac means animal.] One can gain the definite impression that the time has returned, when the human being should apply the consciousness and ego strength which he has wrested from the physical world in the past centuries, to the higher regions of existence. He therefore stands before the daunting task, to orient himself once more to the more intimate relationships between himself and the world, without thereby dampening his consciousness in an atavistic way. With the light of consciousness he should once more grasp the spiritual situation which is currently forgotten. In this way we have also to get to the point, that we connect mathematics consciously with the depths of our nature, that we transform mathematics from a tool for penetrating into the physical-sensible world into an aid for understanding the activity of divine powers. Novalis had glimpsed mathematics from such an elevated position, when he spoke boldly the words: The life of the gods is mathematics. All our observations on the work of the ego upon the three sheaths in the past and in future would lack a foundation, if we count on just one earthly life for the human being. The validity of our observations assumes the acknowledgment of repeated earth lives, the rhythmically recurring incarnations of the human ego. If one admits the possibility of this venerable doctrine, which has with time been forgotten, one can observe otherwise unseen hues and shadings in individual contemporary biographies. You can then see how the human being, following lawful rhythms, unfolds talents that he has earned in previous incarnations, participating in the general trends of human development. The curriculum Rudolf Steiner developed for the founding of the first Waldorf school in Stuttgart, takes the greatest possible pains to consider these rhythms of the individual human life, particularly those active in 8

10 chidhood. You then become aware, how the eternal individuality gradually finds its way to form the three bodily sheaths. If you want to grasp this process in its essential direction, you need to imagine it as a path from above to below, from the heights of heaven to the depths of the earth. Already before birth, the individual follows such a path: she comes from the world of the stars, passes through the etheric sphere and lands on the physical earth, in order to unite with that which comes from the hereditary stream of the parents. To the visible birth of the physical body there follows the supersensible birth of the etheric body around the time of the change of teeth, and likewise the birth of the astral body at the time of sexual maturity, and from that point on, the ego forges its path down through the astral body, the etheric body, and the physical body; it pushes deeper and deeper into the bodily organization. The mathematician must also begin from such observations, if he wishes to once more bring his science to the human being. The picture that mathematics thereby presents will be sketched in the rest of this book. 9

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