The Foundations of Mathematics in Light of Anthroposophy

Transcription

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 13, 2010

2 Contents Translator s Remarks 2 Preface 4 Forward 5 I Introduction 6 1 The Pedagogical Situation 6 2 On the Nature of the Human Being 8 II Algebra 11 3 Formula and Identity 11 4 Conditional equations 17 5 Physiology of arithmetic 25 6 Arithmetic as Experience of the Forces of Death 28 7 The Irrational and the Transcendental 33 1

3 Translator s Remarks The original book consists of an introductory section and two major parts, Algebra and Analysis. The translation you are reading is restricted to the introduction, and the first 5 (out of 10) sections of the algebra part. 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. It will also probably arouse in many readers, as it has in me, moments of exasperation, when one feels that the author goes too far in his efforts to find deeper meaning in innocent mathematical phenomena. I have learned to treasure also this capacity to provoke me, for out of such challenges I often find myself learning the most. He himself may have aged out of some of his youthful exuberance, and that may explain the difference between the earlier and the later (1967) work. Another aspect of this work which I find worth mentioning is that despite its title it has very little to do with geometry. Bindel himself is better known for his writings on music and mathematics, so that his predilection for the numerical-algebraic is obvious. But it is somewhat disconcerting to realize that the book contains almost no specifically geometric content. What geometry is present is presented almost exclusively to draw out of it some algebraic or arithmetic truth. It is probable that Bindel intended it this way. In Section Section 6 he writes: In fact, geometry yearns always to transform itself into arithmetic. For the modern human being is always working towards elevating the unconscious weaving of life forces into consciousness. Pure geometry cannot really be held in the human consciousness; it can only unfold itself within organic processes. Wherever it enters into consciousness, it immediately is penetrated with formative forces. So the reader who hopes to find here specific treatment of geometry or its place in the curriculum will be disappointed. Although I consider myself more a geometer than an algebraist, I have made my peace with this provocation, and no longer find any grounds to complain; for what Bindel does offer here is so full of treasures that complaint would be blasphemy. Note that there are some terms in German which caused some problems for the translation. 2

4 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 as 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. Perhaps simply angle measurement would be be better? 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. Additionally, German typesetting includes a form of emphasizing text by s t r e t c h i n g i t o u t adding space between the letters of the emphasized words. I ve mimicked this formatting, within the document typesetting system TEX, which I ve used to produce this translation. I am grateful for any corrections or suggestions regarding the translation. As you can see, it is a work in progress and can still be amended. Falkensee, Germany, April, 2010 Charles Gunn 3

5 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. 4

6 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 5

7 Part I Introduction Motto: The highest life is mathematics Novalis, Fragments Section 1 The Pedagogical Situation Mathematics instruction, especially in arithmetic, presents significant d i f f i c u l t i e s 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. 6

8 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 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. 7

9 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 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 Waldorfschul-Spielzeug und Verlag, Stuttgart, 1927 [needs updated] 2 [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, 8

10 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 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 9

11 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 selfmovement are for 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, 10

12 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 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 f r o m a b o v e t o b e l o w, 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. Part II Algebra Section 3 Formula and Identity What is it that impresses itself most strongly upon the experience of the human being in the era of the consciousness soul? those formative forces of the ego working upon the physical body. Here the invisible artist active in the human being confronted the most uncooperative material. Here one met also the possibility for the strongest possible experience of form; in fact, there was the possibility for a caricature of this experience. Such a development occurred in the work of the philosopher Immanuel Kant. Under the overwhelming influence of the formative forces active in his organism he tried, with unlimited one-sidedness, to trace back the formative activity of the ego to his threefold bodily organization and thereby lost the essential role of his own ego. It became for him the undifferentiated synthetic unity of perception. Instead of the real formative activity he found his three modes of perception : space, time, and categories. Alongside this he offered an evaluation of that domain of knowledge, in which the formative forces revealed a universal effectiveness mathematics; in it he saw a prototype for all specific branches of natural science. As a result of his work, the formative ego that small form within the great cosmic form, the formula within the Forma was seen not just as fulfilling thinking, but rather going beyond to actually Latin? subjugate thinking. More and more there came to the foreground, what can be characterized 11

13 as the universal counter-image of the individual formative force, the m a t h e m a t i c a l f o r m u l a. The formative activity of the human ego, directed at the body, projects itself into the objective-spiritual as the mathematical formula. We ll now attempt to penetrate to its true nature. If you feel your way energetically into any such formula, you cannot help but sense that it is arranged f r o m a b o v e t o b e l o w. You really move inwardly inside a formula from above to below, you experience a kind of mental fall. It s the same direction we also have to sense the formative path of the individuality in the bodily sheaths. What the ego does in the course of a lifetime, as it sinks from one body to the next, beginning with the astral and ending with the physical, is achieved spiritually in a mathematica formula. Every genuine mathematical formula lives in the vertical. That it happens to be written horizontally, isn t the reality of the matter. Let s take a familiar example. We have the expression a 2 + 2ab + b 2, that lets itself be rewritten as (a + b) 2. That produces the formula, well-known in mathematics as the binomial formula: a 2 + 2ab + b 2 (a + b) 2 Every formula represents a solution to the challenge of giving a new form to a mathematical expression. What comes out as the result, is the same as what was there to begin with, but in a different form. There is something i d e n t i c a l there, that is brought into a process of transformation. We can therefore speak of a mathematical formula as a formed identity. There are also u n f o r m e d i d e n t i t i e s, for example, when we say a equals a. The word equals doesn t really fit here; we ll return to this later. The archetypal phenomenon of all identity is however the immortal ego being that carries itself through repeated earth lives, remaining itself and yet in each life appearing differently, due to the spiritualized activity that it carries out on the enclosing bodily sheaths. This ego comprehends itself as a pure, unformed identity at a particular moment; if you d like to have a familiar image for this, you could think of the words of the Jahweh God coming out of the burning bush: I am the I am! In a mathematical formula, on the other hand, the ego comprehends itself in the flow of time, giving form to its bodies and thereby transforming itself. All forming is essentially a bringing-together, a contracting activity; in the German language, for example, the form-giver of language, the poet, is called a Dichter, which comes from the root meaning dense or contracted. Of the mathematicians of the modern age, Liebniz was perhaps the one who had the most spiritual access to mathematics. One finds with him a characteristic use of the word Form. He understood by a form, a sum of similarly constructed expressions and wrote this as a single term, under which he placed a double point, itself a symbol of contraction. So he let ạ. stand for the sum a + b + c +..., a.. 2 for the sum a 2 + b 2 + c , and ab.. for the sum ab + ac + bc +... With help of Leibnizian forms one 12

14 can, for example, write the binomial theorem in a very short, condensed form, as follows: (a + b + c + d) 2 a a3 b.. + 6a 2 b a 2 bc abcd Leibniz s notation has not been widely adopted; nevertheless it shows with extraordinary clarity, that for which the form inwardly stands. On the other hand, there is in our contemporary notation another mathematical symbol that shows you that you are entering into form. A stronger formation within the whole expression is made visible by the presence of parentheses. Even just from an external point of view, the parentheses are like a pair of hands, drawing out something which it prepares to form or model like a malleable material. Wherever parentheses appear, there is an intensification of the formative activity. The ego also increasingly brackets 4 the bodily sheaths over the course of repeated earth lives, and even in the course of a single life, too. Wherever, on the other hand, parentheses are dissolved in mathematics, then the opposite process manifests: form is released, relaxed; it is a question of a loss of form. Significantly, in arithmetic it is very rare that one encounters more than a three-fold bracketing; one works only with round, square, and curly brackets. From a purely abstract perspective, more would of course be possible. Things, however, have a certain reality of their own. Every mathematical formula unfolds its life between the poles of form and loss of form. It often happens, that you relax a mathematical formula, just in order to be able to put it under even greater tension. The original form of the formula didn t really correspond, in this case, to the contents. For example: (a + b) 2 (a b) 2 (a 2 + 2ab + b 2 ) (a 2 2ab + b 2 ) Dissolution a 2 + 2ab + b 2 a 2 + 2ab b 2 (a 2 a 2 ) + (2ab + 2ab) + (b 2 b 2 ) Formation 4ab Here we see, that the original form was split apart, in order to be able to afterwards compress it even more. Essentially, the initial release of form was really already a kind of forming. The whole process has something of a spiritual nature, that in terms of the body, corresponds to a jump: an apparent separation from the earth, in order that, through a controlled fall, you 4 [Translator s note]the original German uses umklammern here, which means both to parenthesize and to enclose. 13

15 can once again connect with it again. The physical confidence that comes from practicing jumping, corresponds to the spiritual confidence, that you can gain through transformation of arithmetic expressions. If you attempt to come even closer to the connection between forming and bracketing, there appears a wonderful lawfulness. We return to our observations to the binomial formula already considered. There we have the bracketless, that is unformed, or more precisely, less-formed expression in front of us: a 2 + 2ab + b 2 One can, without changing its meaning, rewrite it as follows: (a 2 ) + (2ab) + (b 2 ) It s conventional not to write it this way, but simply without parentheses. This represents a sum, whose members are powers and products; in this expression the powers and the products are, so to speak, overpowered by the sum. It s exactly the opposite in the transformed expression: (a + b) 2 Seen as a whole, this is a power, within which a sum appears; the sum is vanquished by the power. When I write it as follows: a 2 + 2ab + b 2 (a + b) 2 then the power conquers the sum, in other words: a lower arithmetic operation is suppressed by a higher one. For we have to keep in mind, with respect to the varieties of operations, that there is an underlying priority there, whose highest member is logarithmizing and whose lowest is addition. Logarithm Root Power Quotient Product Difference Sum This sequence is deeply rooted in the human being, as we will try to show later in this book. What goes on when we insert parentheses? Quite simply, something lower is overcome by something higher. Viewed from an inward perspective, however, this is just the process 14

16 of formation. Formlessness is equivalent to something lower overcoming something higher. One could argue, that this relationship is satisfied only by the relationship chosen by us an example. But that is not the case. The same thing can be confirmed in every case. Take for example another example: a c + b c (a + b) c In this formula the addition is taken hold of by division; once again a higher overcomes a lower, and once again parentheses express this symbolicallly. One more example: in the expression log a + b the sum rules over the logarithm; hence no brackets. The expression log (a + b) has a somewhat different meaning; the logarithm has overcome the addition and hence there are brackets. The relationship between form and bracket can in fact be used as a criterion to decide when to insert parentheses, and when not. Take the expression log a 3. There are no brackets, so the only meaning can be that log a is to be raised to the third power; the power as a lower being rules over the logarithm as a higher. If you want to have the opposite meaning, then you have to write brackets, and you then have the expression log(a 3 ). How helpless are most students when confronted by the question of brackets! Here a unifying thread, coming from deeper levels, can be placed in their hands, with which they can find their way through the labyrinth. In an example such as these brackets, we recognize how from time to time the secrets of the human being stand behind the apparent externalities of arithmetic. You d naturally hope that this were always so. However, the contemporary notation for arithmetic is not always grounded on an inner reality. We indicated as much, when we noted that formulae are usually incorrectly written horizontally with the familiar equals (=) sign. This is not entirely justified. One ought to apply a symbol for identity here, and differentiate this depending on whether one is dealing with a pure identity or a formed one. Already during Leibniz s time, people paid more attention to such details. Leibniz himself, who was really the spirit who first expressed in a thorough way all of mathematics in a symbolic form, drew a distinction between the equality sign and the identity sign. He used the symbol to represent the Symbol sign of equality, whereas you can find in an article found after his death, entitled Mathesis should be universalis, today s symbol for infinity used as a sign of identity. Today the identity sign flatter. is only rarely distinguished from the equality sign, and then one finds the symbol of three parallel strokes used to represent identity, as if to intensify the two strokes of the equality sign. If we accept this identity sign, then in conformance to what we said concerning the direction of a formula, we have to write our formula so: a 2 + 2ab + b 2 (a + b) 2 And how must a pure, unformed identity be written to correspond to spiritual reality? If you would write a = a, then the symbol would wrongly indicate the presence of an equation. 15

17 a You also cannot apply the pattern of a formula: a ; since in a pure identity there is no formative progress, but rather a motionless self-containment. You d come closest to the fact of the matter, if you write the identical quantity inside of a ring: a The ring, closed on itself, as a symbol of the ego being, the personality peaceful resting in itself think of the ring of the Nibelung. Putting a ring around a quantity was very common in the older mathematics. Such an identity was to be read: a is. By a formula, instead of this resting-in-itself, one is concerned with with a becoming and a dissolving: a 2 + 2ab + b 2 and (a + b) 2. It is a fascinating study, to investigate in the history of mathematics which is at the same time a history of human consciousness where exactly the individual symbols first appeared, and exactly why this one and not that one was chosen. You discover the astounding fact, that nearly all the numerical signs and mathematical symbols that serve us today, came into being since the beginning of the modern age. Before that one finds mathematical symbols only by certain outstanding individuals, and even then their occurrence forms a chaotic quest, a fluctuating usage. In general, before then, one did without symbols, one described and delineated everything with words. This phase of arithmetic is called therefore rhetorical arithmetic; this was dominated by a mood in respect to everything mathematical, which is still today to be found in the word account (as in give an account). Only since Leibniz and Euler are we in command of a fully developed mathematical symbollanguage. This also expresses the fact that human culture had entered the age of the consciousness soul. The human ego was able to create the sharply defined symbols in use today, as it itself was pushed down into the sharply defined physical body, and had to engender there intense formative forces. Indeed, you can experience the mathematical signs and symbols as the physical body of mathematics; in contrast, the linguistic-rhetorical expression of mathematics represents the etheric body of mathematics. The physical body of mathematics could only be built up once the ego had begun to direct its formative forces to the physical body of the human being. Among the consequences of the formative forces appeared now, that the human body lost the springing, swelling nature, that before had been part of its being. This body crystallized itself with increasing sharpness, and was in turn illuminated through and through by the human ego; the part of the human bodily form concerned only with the group withdrew more and more. This was particularly pronounced for the human head, that now became in much greater measure an expression of the human individuality. In this connection, you can see at this time, for example in painting, an increased occurrence of the portrait and particularly of the self-portrait. The quest for consciousness drove artists like Dürer and Rembrandt to observe themselves in the act of creating form. A similar process lies in the appearance of mathematical symbols. Mathematics was pushed out of the living body of the language, and rigidified into its current symbolism. As long as this science still was contained within the language, it had preserved an impersonal and generic quality. For example, the various European mathematicians didn t use their native tongues, but rather wrote in Latin. As the age of the consciousness soul burst forth, this situation changed. For example, in 1521 in Nuremburg there appeared, in German, the calculation book by Heinrich Schreiber, known as Grammateus. In this book one finds 16

18 the beginnings of the symbolism which is still common today: the symbols + and are already used throughout. The individual celebrates his entrance into mathematics, and through these symbols creates for it a physical body of highly personalized form and masterly penetration. The contemporary mathematical symbols gaze at us, like highly individualized human portraits. If you investigate, on which basis this or that author chose this or that symbol, you re faced with the odd situation, that almost all of the symbols were introduced without any accompanying justification. And where such justification is attempted, you immediately have the feeling that it doesn t really penetrate to the depths of the innovation. From the depths of the body, formed by the ego, the mathematical symbols are brought to birth. In this way the identity sign consisting of the three strokes is extraordinarily striking. If it wasn t already there, then it would have to be invented again. It mirrors in itself, what the shaping ego itself is: t h e u n i t y o f t h e b o d i l y t r i n i t y! Section 4 Conditional equations A c o n d i t i o n a l e q u a t i o n 5 presents entirely dfferent relationships to the student than those present in an identity or a formula. In an identity, the ego weaves within itself; in a formula, it weaves in within the body. In both cases one is concerned with an internal affair of the human being, in no way requiring the existence of an external world. But in a conditional equation, as we shall see, the ego establishes a relationship to the external world. Hence, this type of mathematical object implies a process of cognition in which an inner world and an outer world are brought together. Rudolf Steiner described in detail the process of human knowledge in his philosophical writings 6, in a in a way that does justice to the spiritual realities. For the sake of our research into arithmetic, we cannot omit a condensed paraphrasing of his ideas here. All knowledge can be traced back to two factors, which we experience in the acts of perception and of thinking. Together these constitute what presents itself to us as reality. The word reality can lead us to a picture: through our thinking, we work the threads of our concepts into the carpet of sense perception which lies, as it were, before us. Without the activity of thinking, such a carpet would appear to have no patterning. Only because the human organization can grasp concepts out of the spiritual treasures of its pre-earthly existence, does a meaningful whole come into being. In addition to the various senses to which perception turns itself, the Sense 7 has also to be added. It is only when observation is illuminated by thinking, that the so-called object is created in the act of knowing. The object is not 5 [Translator s note] The original German Bestimmingsgleichung has no corresponding term in English. It refers to a type of mathematical problem in which several linear constraints (conditions) are present; the goal is to find the common solution to all the constraints by transforming them into a single linear equation and solving it by well-known operations. 6 (Fundamentals of a Theory of Knowledge of Goethe s Worldview, Truth and Science, The Philosophy of Spiritual Activity, etc 7 [Translator s note] The German Sinn used here can mean, as in English, both a human sense (sight, hearing, etc.) and also meaning. The capitalization used here denotes the latter possibility. 17

19 something which exists ready-made outside of us, which comes in, as it were, through the senses, but the object arises only through the act of t h i n k i n g within ourselves. Rudolf Steiner then continues to describe this process further, how the concept and the object live actually only in the p r e s e n t. A concept is something totally vital and flowing, and it can be grasped no more than water can be. Only when it has been frozen to ice, can one grasp it. But at that moment that it comes into being, the still flowing object-knowledge solidifies into a solid, graspable image, which is familiar to everyone as a so-called m e n t a l p i c t u r e. In this moment, at which the sense perception and the concept join together to constitute reality, another sort of knowing p r e s e n t s i t s e l f to the human being. This is what can be remembered, and represents a sort of preserved knowledge. Just as the solidification of water into ice can be traced back to the wintery forces of death spread throughout the natural world, so is the hardening of concept into mental picture a kind of death process. It is brought about through the same forces with which the consciousness soul has to deal, the formative forces. The same process, which expresses itself in a mathematical formula, can be found in the process of knowledge in the hardening and contraction of the flowing, malleable concept. The more that modern human being has lost himself in this hardening, the more impossible has it became for him to appreciate the original pictorial nature of the concept. No post-mortem analysis of our knowledge will allow us to find it again. What is that we find there instead? The mental picture into which the living concept has died! Knowledge of reality, and with it the spirit, truly slips past us. What appears so substantial as mental picture to subsequent observation, has already been been left behind by the spirit, in which alone reality can be found. This has consequences for our every-day cognition. In every act of sense perception is also present the cache of mental pictures we have previously accrued. What we described at the beginning as the pure correspondence of perception and concept, actually occurs only for a very young child, which is void of all mental pictures, and whose whole bodily organization presents as it were one big sense organ. The adult always mixes in a good portion of his accumulated cache of mental pictures into each act of cognition, and the result is, that for him the act of cognition no longer has the freshness, which it does for the child. We experience ourselves as old and abandoned, due to the weight of the mental pictures attached to our cognition, and the only way to revitalize that is that we make ourselves inwardly empty with respect to new sense impressions. Otherwise it is as if reality flashes forth only at isolated points of our fleeting cognitive panorama. To gain a concrete insight into the underlying difference between concept and mental picture, simply bring to consciousness the difference that exists in a spoken sentence, between the fluctuating parts such as prepositions, conjunctions, articles, connectives, punctuation, and not least the voice modulation and the fixed parts, appearing primarily in the nouns. The former generally evade the consciousness which lies behind mental pictures, while the latter demand exactly such a consciousness, and the high wisdom of language shines through the fact, that one calls those words, which are most strongly supported by the fixity of mental 18

20 pictures, head words 8. Now, fortified with this brief excursion into the theory of knowledge, we can ask the question: which strings of the process of cognition are sounded when we solve a conditional equation? The conditional equation we have to be quite clear about this isn t the beginning, it is itself the result of a problem which has been posed with words, and it is with this that we have to begin. We wish to begin with the following problem: A businessman wants to produce 25 pounds of roasted coffee out of two sorts of raw coffee. One kind cost $2.50 a pound, the other costs $1.75 a pound. He wants the mixture to cost $2.40 a pound. How many pounds of the more expensive raw coffee should he take, if by roasting coffee loses 16 2/3% of its weight? We are confronted here by a series of sentences, sentences in which fluctuation and fixity, activity and inactivity are interwoven, just as it is in every act of cognition in our adult life. Concepts and mental pictures are mixed together here. The next step on the path lying before us, is the conditional equation. In order to extract it, the formative forces of the ego direct themselves at the problem, and strive to contract every conceptual element into a mental picture. All the conceptual elements, which by their nature resist this process, will be abruptly tossed out by the forming activity, so that all punctuation, prepositions, etc., disappear, and all that is left behind is a kind of scaffolding, or skeleton, which is just the conditional equation. Besides those conceptual elements, which defy the conversion into mental pictures, there are also some present in the problem, which o n l y i n i t i a l l y cannot be channeled into mental pictures. As concepts they are already present, although in a hidden form, but they have not yet been made into mental pictures. Mastering the problem consists of bringing even these hidden concepts into the sphere of mental picture, so that at the end, it is like all mental pictures a known thing. Concepts are recognized, mental pictures are known 9. Thus, the recognized but not yet known concepts in the problem (those which have not yet become mental pictures), are the u n k n o w n s. In our problem there is only once such unknown, the number of pounds of the better sort of coffee. We note, that each problem, which contains unknown quantities in it, objectifies in a universal way that which occurs in us individually, when we create a mental picture, when we cripple, so to speak, a concept into a mere mental picture. We ought to be interested to see how the solution is carried out, because at the same time it reveals an open secret of our body-soul-spirit being. Before we pursue this, however, we present a short overview of how in the history of mathematics that which today we call an unknown has been previously interpreted in other times and places. In the first document of calculation in history, the papyrus of the Egyptian 8 [Translator s note] The original German for nouns is wörter des hauptes, literally, words of the head 9 [Translator s note] In German, this is a play on the two words e r kannt [recognized] and b e kannt [known] 19