Sunday, April 19, 2015

Where mathematics comes from. The Boundaries of Natural Science, lecture 3

The Boundaries of Natural Science. Lecture 3 of 8.
Rudolf Steiner, Dornach, Switzerland, September 29, 1920:

We have seen that one arrives at two limits when one seeks either to penetrate more deeply into natural phenomena or, proceeding from the state of normal consciousness, to penetrate more deeply into one's own being in order to uncover the essential nature of consciousness. Yesterday we showed already what happens at the one limit to knowledge. We have seen that man awakes to full consciousness in coming into contact with an external, physical world of sense. Man would remain a more-or-less drowsy being, a being with a sleepy soul, if he could not awake in confronting external nature. And what has happened in the spiritual evolution of humanity, in man's gradual acquisition of knowledge about external nature, is actually nothing other than what happens every morning when we awake out of sleep or dream-consciousness by confronting an external world. This latter is a kind of moment of awakening, and in the course of the evolution of humanity we have to do with a gradual awakening, a kind of long, drawn-out moment of awakening.
Now, we have seen that at this frontier a certain inertia on the part of the soul very easily comes into play, so that when we come up against the extended world of phenomena we do not proceed in the manner of Goethean phenomenology by halting at this frontier and ordering the phenomena according to the representations, concepts, and ideas we have already gained, describing them in a systematic, rational manner, and so forth. Instead, we roll on a bit farther beyond the phenomena with our concepts and ideas and thereby create a world — for example, a world of metaphysical atoms, molecules, and so forth. This world, when it is so constituted, is merely a fabrication of the mind, a world into which there enters a creeping doubt, so that we have to unravel again the theoretical web we have spun. And we have seen that it is possible to guard against such a violation of this frontier of our knowledge through phenomenalism, through working purely with the phenomena themselves. We have also had to show that at this point in our striving for knowledge something emerges that commends itself to our use as an immediate necessity: mathematics and that part of mechanics that can be comprehended without any empirical observation, i.e., the entire compass of so-called analytical mechanics.
If we call to mind everything comprehended by mathematics and analytical mechanics, we have before us the system of concepts that allows us to enter into phenomena with the utmost certainty. And yet, as I began to indicate yesterday, one should not deceive oneself, for the whole manner in which we call forth the notions of mathematics and analytical mechanics, this process within our souls, is entirely different from that employed when we experiment with or observe sensory data and then seek to comprehend them, when we try to gather knowledge from sensory experience. In order to arrive at the fullest clarity regarding these matters one must bring all one's mental energy to bear, for in this realm full clarity can be attained only with the greatest mental exertion.
What is the difference between accumulating knowledge from sensory experience in a Baconian manner and the more inward mode of apprehension we find in mathematics and analytical mechanics? One can sharply differentiate the latter from those modes of apprehension that are not inward in this way by formulating clearly the concepts of the parallelogram of motion and the parallelogram of forces. One theorem of analytical mechanics states that two angular vectors proceeding from one point result in a third vector. To say, however, that a vector of a specific force here [see diagram: a] and a vector of a specific force here [b] result in a third force, which can also be determined according to the parallelogram — that is another notion altogether.

Diagram 1

The parallelogram of motion lies strictly within the province of analytical mechanics, for it is internally consistent and demands no external proof. In this it is like the Rule of Pythagoras or any other geometrical axiom — but the existence of the parallelogram of forces can be determined only by experience, by experimentation. In this case, we bring something into that which we work through inwardly: the force that can be given only empirically from without. Here we no longer have a pure, analytical mechanics but an “empirical mechanics.” One can thus differentiate sharply between that which is still actually mathematical — as we still conceive mathematics today — and that which leads over into conventional empiricism.
Now one stands before this phenomenon of mathematics as such. We comprehend mathematical truths. We proceed from mathematical phenomena to certain axioms. We weave the fabric of mathematics out of these axioms and then stand before an architectonic whole apprehended by the mind's eye [im inneren Anschauen]. If we are able by means of energetic thinking to differentiate sharply this inner apprehension from anything that can be experienced outwardly, we must see in this fabric of mathematics something that arises through an activity of soul entirely different from that which underlies our experience of the outer objects of sensation. Whether or not we arrive at a satisfactory comprehension of the world depends to a tremendous extent on our being able to make this clear distinction out of inner experience. We thus must ask: Where does mathematics originate? Nowadays this question is still not pursued rigorously enough. One does not ask: how is this inner activity of the soul that we need in mathematics, in the wonderful architecture of mathematics — how is this inner activity of the soul different from that whereby we grasp external nature through the senses? One does not pose this question and seek an answer with sufficient rigor, because it is the tragedy of the materialistic worldview that, while on the one hand it presses for sensory experience, on the other hand it is driven unawares into an abstract intellectualism, into a realm of abstraction where one is isolated from any true comprehension of the phenomena of the material world.
What kind of capacity is it, then, that we acquire when we engage in mathematics? We want to address ourselves to this question. In order to answer this question we must, I believe, have reached a complete understanding of one thing in particular: we must take fully seriously the concept of becoming as it applies to human life as well. We must begin by acquiring the discipline that modern science can teach us. We must school ourselves in this way and then, taking the strict methodology, the scientific discipline we have learned from modern natural science, transcend it, so that we use the same exacting approach to rise into higher regions, thereby extending this methodology to the investigation of entirely different realms as well. For this reason I believe — and I want this to be expressly stated — that nobody can attain true knowledge of the spirit who has not acquired scientific discipline, who has not learned to investigate and think in the laboratories according to the modern scientific method. Those who pursue spiritual science [Geisteswissenschaft] have less cause to undervalue modern science than anyone. On the contrary, they know how to value it at its full worth. And many people — if I may here insert a personal remark — were extremely upset with me when, before publishing anything pertaining to spiritual science as such, I wrote a great deal about the problems of natural science in a way that appeared necessary to me. So you see it is necessary on the one hand for us to cultivate a scientific habit of mind, so that this can accompany us when we cross the frontiers of natural science. In addition, it is the quality of this scientific method and its results that we must take very seriously indeed.
You see, if we consider the simple phenomenon of warmth that appears when we rub two bodies together, it would be utterly unscientific to say, regarding this isolated phenomenon, that the warmth had been created ex nihilo or simply existed. Rather, we seek the conditions under which this warmth was previously latent and now appears by means of the bodies. We proceed from the one phenomenon to the other and thus take seriously this process of becoming [das Werden]. We must do the same with the concepts that we consider in spiritual science. So we must first of all ask: Is that which manifests itself as the ability to perform mathematics present in man throughout his entire existence between birth and death? No, it is not always present. It awakens at a certain point in time. To be sure, we can, while still remaining empirical regarding the outer world, observe with great precision how there gradually arise out of the dark recesses of human consciousness faculties that manifest themselves as the ability to perform mathematics and something like mathematics that we have yet to discuss. If one can observe this emergence in time precisely and soberly, just as scientific research treats the phenomena of the melting or boiling point, one sees that this new faculty emerges at approximately that time of life when the child changes teeth. One must treat such a point in the development of human life with the same attitude with which physics, for example, teaches one to treat the melting or boiling point. One must acquire the ability to carry over into the complicated realm of human life the same strict inner discipline that one can acquire by observing simple physical phenomena according to the methods of modern science. If one does this, one sees that in the course of human development from birth, or rather from conception, up to the change of teeth, the soul faculties enabling one to perform mathematics manifest themselves gradually within the organism but that they are not yet fully present. Now we say that the warmth that manifests itself in a body under certain conditions was latent in that body beforehand, that it was at work within the inner structure of that body. In the same way we must be entirely clear that the capacity to perform mathematics, which becomes most evident at the change of teeth and reveals itself gradually in another sense, was also at work beforehand within the human organization. We thus arrive at an important and valuable insight into the nature of mathematics — mathematics taken, of course, in the very broadest sense. We begin to understand how that which is at our disposal after the change of teeth as a soul faculty worked previously within to organize us. Yes, within the child until approximately its seventh year there works an inner mathematics, an inner mathematics not abstract like our external one but full of active energy, a mathematics which, if I may use Plato's expression, not only can be inwardly envisioned [angeschaut] but is full of active life. Up to this point in time there exists within us something that “mathematicizes” us through and through.
When we ask at first entirely superficially what can be seen by looking empirically at this “latent mathematics” in the body of the young child, we are led to three things resembling inner senses. In the course of these lectures we shall come to see that one can indeed speak of senses within as well. Today I want only to indicate that we are led to something that develops an inward faculty of perception similar to the outward perception developed by the eyes and ears, except that the former remains unconscious within us during these first years. And if we look within, look into our own inner organization not like nebulous mystics but with all our powers of apprehension, we can find within three functions similar to those of the outward senses. We find inner senses that exercise a certain activity, a certain inner mathematics, just in those first several years. One encounters first of all what I would like to call the sense of life. This sense of life manifests itself in later years as a perception of our inner state as a whole. In a certain way we feel either well or unwell. We feel comfortable or uncomfortable: just as we have a faculty for perceiving outwardly with the eyes, so also do we have a faculty for perceiving inwardly. This faculty is directed toward the whole organism and is for that reason dark and dull; yet it is there all the same. We shall have more to say about this later. For the moment I want to anticipate this later discussion only by remarking that this sense of life is — if I may use a tautology — especially active in the vitality of the child up until the change of teeth.
Another inner sense that we must consider when we look within in this way is that which I would like to call the sense of movement. We must form a clear conception of this sense of movement. When we move our limbs, we are aware of this not only by viewing ourselves externally but also by means of an internal perception. Also when we walk: we are conscious that we are walking not only in that we see objects pass and our view of the external world changes but also in that we have an internal perception of the movements of the limbs, of changes within ourselves as we move. Normally we remain unaware of the inner experiences and perception that run parallel to the outer because of the strength of the external impressions, much as a dim light is “extinguished” by a bright one.
And a third inward-looking faculty is the sense of balance. The sense of balance is what enables us to locate ourselves within the world, to avoid falling, to perceive in a certain way how we can bring ourselves into harmony with the forces in our environment. We perceive this process of bringing ourselves into harmony with our environment inwardly. We thus can truly say that we bear within ourselves these three inner senses: the sense of life, the sense of movement, and the sense of balance. They are especially active in childhood up to the change of teeth. Around this time of the change of teeth their activity begins to wane — but observe, to take but one example, the sense of balance: observe how at birth the child has as yet nothing enabling it to attain the position of balance it needs in later life. Consider how the child gradually gains control of itself, how it learns at first to crawl on all fours, how it gradually achieves through its sense of balance the ability to stand and to walk, how it finally is able to maintain its own balance.
If one considers the entire process of development from conception to the change of teeth, one sees therein the powerful activity of these three inner senses. And if one can attain a certain insight into what is happening there, one sees that there is at work in the sense of balance and the sense of movement nothing other than a living “mathematicizing” [ein lebendiges Mathematisieren]. In order for it to come to life, the sense of life is there to vitalize it. We thus see a kind of latent realm of mathematics active within man. This activity does not entirely cease at the change of teeth, but it does become at that time considerably less pronounced for the remainder of life. That which is inwardly active in the sense of balance, the sense of movement, and the sense of life becomes free. This latent mathematics becomes free, just as latent heat can become liberated heat. And we see how that which initially was woven through the organism as an element of soul becomes free. We see how this mathematics emerges as abstraction from a condition in which it was originally a concrete force shaping the human organism. And because as human beings we are suspended in the web of existence according to temporal and spatial relationships, we take this mathematics that has become free out into the world and seek to comprehend the external world by means of something that worked within us up until the change of teeth. You see, it is not a denial but rather an extension of natural science that results when one considers rightly what ought to live within spiritual science as attitude and will.
We thus carry what originates within ourselves beyond the frontier of sense perception. We observe man within a process of becoming. We do not simply observe mathematics on the one hand and sensory experience on the other but rather the emergence of mathematics within the developing human being. And now we come to that which truly leads over into spiritual science itself. You see, that which we call forth out of our own inner life, this “mathematicizing,” becomes in the end an abstraction. Yet our experience of it need not remain an abstraction. In our time there is, to be sure, little opportunity for us to experience mathematics in a true light. Yet at a certain point in the development of Western civilization there does come to light something of this sense of a special spirit in mathematics. This comes to light at the point where Novalis, the poet Novalis, who underwent a good mathematical training in his studies, writes about mathematics in his Fragments. He calls mathematics a grand poem, a wonderful, grand poem.
One really must have experienced at some time what it is that leads from an abstract understanding of the geometrical forms to a sense of wonder at the harmony that underlies this inner “mathematicizing.” One really must have had the opportunity to get beyond the cold, sober performance of mathematics, which many people even hate. One must have struggled through as Novalis had in order to stand in awe of the inner harmony and — if I may use an expression you have heard often in a completely different context — the “melody” [Melos] of mathematics.
Then something new enters into one's experience of mathematics. There enters into mathematics, which otherwise remains purely intellectual and, metaphorically speaking, interests only the head, something that engages the entire man. This something manifests itself in such youthful spirits as Novalis in the feeling: that which you behold as mathematical harmony, that which you weave through all the phenomena of the universe, is actually the same loom that wove you during the first years of growth as a child here on Earth. This is to feel concretely man's connection with the cosmos. And when one works one's way through to such an inner experience, which many hold to be mere fantasy because they have not actually attained it themselves, one has some idea what the spiritual scientist [Geistesforscher] experiences when he rises to a more extensive grasp of this “mathematicizing” by undergoing an inner development of which I have yet to speak and which you will find fully depicted in my book Knowledge of the Higher Worlds and Its Attainment. For then the capacity of soul manifesting itself as this inner mathematics passes over into something far more comprehensive. It becomes something that remains just as exact as mathematical thought yet does not proceed solely from the intellect but from the whole man.
On this path of constant inner work — an inner work far more demanding than that performed in the laboratory or observatory or any other scientific institution — one comes to know what it is that underlies mathematics, that underlies this simple faculty of the human soul which can be expanded into something far more comprehensive. In this higher experience of mathematics one comes to know Inspiration. One comes to understand the differences between what lives in us as mathematics and what lives in us as outer-directed empiricism. In this outer-directed empiricism we have sense impressions that give content to our empty concepts. In Inspiration we have something inwardly spiritual, the activity of which manifests itself already in mathematics, if we know how to grasp mathematics properly — something spiritual which in our early years lives and weaves within us. This activity continues. In doing mathematics we experience this in part. We come to realize that the faculty for performing mathematics rests upon Inspiration, and we can come to experience Inspiration itself by evolving into spiritual scientists. Our representations and concepts then receive their content in a way other than through external experience. We can inspire ourselves with the spiritual force that works within us during childhood. For what works within us during our childhood is spirit. The spirit, however, resides in the human body and must be perceived there through the body, within man. It can be viewed in its pure, free form if one acquires through the faculty of Inspiration the capacity not only to think in mathematical concepts but to view that which exists as a real force in that it organizes us through and through up until the seventh year. And that which manifests itself partially in mathematics and reveals itself as a much more expansive realm through Inspiration can be inwardly viewed, if one employs certain spiritual scientific methods about which — as I have said — I plan yet to speak. One thereby gains not merely new results to add to those acquired through the old powers of cognition but rather an entirely new mode of apprehension. One acquires a new “Inspirative” cognition.
The course of human evolution has been such that these powers of Inspirative cognition have receded with the passage of time, after having been present earlier to a very high degree. One must come to understand how Inspiration arises within the inner being of man — that same Inspiration that survives in the West only in the diluted, intellectual experience of mathematics. The experience can be expanded, however, if only one comprehends fully the inner nature of that realm; only then does one begin to understand what lived in that earlier consciousness transmitted to us actually only from the East, from the Vedanta and the other Eastern philosophies that remain so cryptic to the Western mind. For what was it that actually lived within these Eastern philosophies? lt was something that arose through soul faculties of a mathematical nature. It was an Inspiration. It was not merely mathematics but rather something attained within the soul in a way similar to that in which one performs mathematics. Thus I would say that the mathematical atmosphere emanating from the Vedanta and similar ancient worldviews is something that can be understood from the perspective one attains in rising again to enter the realm of Inspiration. If one can raise to vivid inner life that which works unconsciously in mathematics and the mathematical sciences and can carry it over into another realm, one discovers the same mathematical element that Goethe viewed. Goethe modestly confessed that he did not have proficiency in mathematics in any conventional sense. Goethe has written on his relationship to mathematics in a very interesting series of essays, which you can find in his scientific writings under the heading “Relationship to Mathematics.” Extraordinarily interesting! For despite Goethe's modest confession that he had not acquired a proficiency in the handling of actual mathematical concepts and theories, he does require one thing: he calls for a phenomenalism such as he employed in his own scientific studies. He demands that within the secondary phenomena confronting us in the phenomenal world we seek the archetypal phenomenon [Urphänomen]. But just what kind of activity is this? He demands that we trace external phenomena back to the archetypal phenomenon, in just the same way that the mathematician traces the outward apprehension [äusseres Anschauen] of complex structures back to the axiom. Goethe's archetypal phenomena are empirical axioms, axioms that can be experienced.
Goethe thus demands, in a truly mathematical spirit, that one inwardly permeate phenomena with mathematics. He writes that we must see the archetypal phenomena in such a way that we are able at all times to justify our procedures according to the rigorous requirements of the mathematician. Thus what Goethe seeks is a modified, transformed mathematics, one that suffuses phenomena. He demands this as a scientific activity.
Goethe was able, therefore, to suffuse with light the one pole that otherwise remains so dark if we postulate only the concept of matter. We shall see how Goethe approached this pole; we moderns must, however, approach the other, the pole of consciousness. We must investigate in the same way how soul faculties manifest their activity in the human being, how they proceed from man's inner nature to manifest their activity externally. We shall have to investigate this. It shall become clear that we must complement the method of investigating the external world offered by Goethean phenomenology with a method of comprehending the realm of human consciousness. It must be a mode of comprehension justifiable in the sense in which Goethe's can be justified to the mathematician — a method such as I tried to employ in a modest way in my book The Philosophy of Freedom.
At the pole of matter we thus encounter the results yielded by Goethean phenomenology, and at the pole of consciousness those attained by pursuing the method that I sought to establish in a modest way in my Philosophy of Freedom.
Tomorrow we will want to pursue this further.

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