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.What passes over is merely the arithmetical expression of measure.If mechanical energy passes over into warmth the real occurrence is as follows: a certain quantity of this revelation which reveals itself in warmth is stimulated in a spiritual being by a certain quantity of mechanical energy.(This is so in a healthy fashion with Mayer.It was only Helmholtz who botched up this matter.)llNeither sound, nor warmth, nor light, nor electricity are vibrations, just as little as a horse is a sum of gallop-paces.Sound, for instance, is an essential entity, and the effect of this real quality in its passage through the air is vibration.For man as a sensing being, the vibration is motivation to imitate the essential entity in himself; this constitutes the perception of sound.It is similar with others: light, etc.llLight is that by which it is perceived.(See my introduction to Goethe's Theory of Color.) The vibration is the revelation of light in the ether.The refraction of light is the result of the effect of a certain force-direction upon the light-direction.Newton's color rings (circles), phenomena of interference, are results of light-radiation (effect of light in the ether), and of the effects of other forces found in the path of light (weakening effects, gradually weakening affects of other forces).The same goes for phenomena of polarisation.One should not seek the polarisation figures in the structure of the essence of the light but in the structure of the medium which places itself in the path of light.The speed of transmission is the result of a kind of friction of the light against the medium.llLight is not to be considered as a function of electricity, but the latter is to be considered as a kind of corporeal carrier of light.Electrically charged matter: certain accumulations of force retain those accumulations of force which manifest as electricity.llMathematics is the abstracted sum of the forces effective in space.If one says, “Mathematical propositions are valid a priori”, this comes from the fact that man exists within the same lines of force as the other beings, and that he can disassociate himself from everything that does not belong to the scheme of space, etc.lMathematics and OccultismBy Rudolf Steiner.IT is well known that the inscription over the door of Plato's school was intended to exclude anybody who was unacquainted with the science of Mathematics, from participating in the teachings of the Master.Whatever we may think of the historical truth of this tradition, it is based upon the correct understanding of the place that Plato assigned to mathematics within the domain of human knowledge.Plato intended to awaken the perceptions of his disciples by training them to move in the realm of purely spiritual being according to his “Doctrine of Ideas.” His point of view was that Man can know nothing of the “True World” so long as his thought is permeated by what his senses transmit.He demanded that thought should be emancipated from sensation.Man moves in the World of Ideas when he thinks, only after he has purged his thought of all that sensuous perception can present.The paramount question for Plato was, “How does Man emancipate himself from all sense-perception?” He considered this to be an all-important question for the education of the spiritual life.Of course, it is only with difficulty that Man can emancipate himself from material perceptions, as a simple experiment on one's own self will prove.Even when the man who lives in this every-day world does withdraw into himself and does not allow any material impressions of the senses to work upon him, the residues of sensuous perception still linger, in his mind.As to the man who is as yet undeveloped, when he rejects the impressions which he has received from the physical world of the senses, he simply faces nothingness — the absolute annihilation of consciousness.Hence certain philosophers affirm that there exists no thought free from sense-perception.They say, “Let a man withdraw himself ever so much within the realm of pure thought, he would only be dealing with the shadowy reflections of his sense-perceptions.” This statement holds good, however, only for the undeveloped man.When he acquires for himself the faculty of developing organs which can perceive spiritual truths (just as Nature has built for him organs of sense), then his thought ceases to remain empty when it rids itself of the contents of sense-perception.It was precisely such a mind emancipated from sense-perception and yet spiritually full, which Plato demanded from those who would understand his Doctrine of Ideas.In demanding this, however, he demanded no more than was always required of their disciples, by those who aspired to make them true initiates of the Higher Knowledge.Until Man experiences within himself to its full extent what Plato here implies, he cannot have any conception of what true Wisdom is.Now Plato looked upon mathematical science as a means of training for life in the World of Ideas emancipated from sense-perception.The mathematical images hover over the border-line between the material and the purely spiritual World.Let us think about the “circle”; we do not think of any special material circle which perhaps has been drawn on paper, but we think of any and every circle which may be represented or met with in Nature.So it is in the case of all mathematical pictures.They relate to the sense-perceptible, but they are not exhaustively contained in it.They hover over innumerable, manifold sense-perceptible forms.When I think mathematically, I do indeed think about something my senses can perceive; but at the same time I do not think in terms of sense-perception.It is not the material circle which teaches me the laws of the circle; it is the ideal circle existing only in my mind and of which the concrete form is a mere representation.I could learn the identical truths from any other sensible image.The essential property of mathematical perception is this: that a single sense-perceptible form leads me beyond itself; it can only be for me a representation of a comprehensive spiritual fact.Here again, however, there is the possibility that in this sphere I may bring through to sense-perception what is spiritual.From the mathematical figure I can learn to know supersensible facts by way of the sense-world.This was the all-important point for Plato.We must visualise the idea in a purely spiritual manner if we would really know it in its true aspect.We can train ourselves to this if we only avail ourselves of the first steps in mathematical knowledge for this purpose, and understand clearly what it is that we really gain from a mathematical figure.“Learn to emancipate thyself from the senses by mathematics, then mayest thou hope to rise to the comprehension of ideas independently of the senses”: this was what Plato strove to impress upon his disciples.The Gnostics desired something similar.They said, “Gnosis is Mathesis.” They did not mean by this that the essence of the world can be based on mathematical ideas, but only that the first stages in the spiritual education of Man are constituted by what is supersensible in mathematical thought.When a man reaches the stage of being able to think of other properties of the world independently of sense-perception in the same way as he is able to think mathematically of geometrical forms and arithmetical relations of numbers, then he is fairly on the path to spiritual knowledge [ Pobierz całość w formacie PDF ]
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.What passes over is merely the arithmetical expression of measure.If mechanical energy passes over into warmth the real occurrence is as follows: a certain quantity of this revelation which reveals itself in warmth is stimulated in a spiritual being by a certain quantity of mechanical energy.(This is so in a healthy fashion with Mayer.It was only Helmholtz who botched up this matter.)llNeither sound, nor warmth, nor light, nor electricity are vibrations, just as little as a horse is a sum of gallop-paces.Sound, for instance, is an essential entity, and the effect of this real quality in its passage through the air is vibration.For man as a sensing being, the vibration is motivation to imitate the essential entity in himself; this constitutes the perception of sound.It is similar with others: light, etc.llLight is that by which it is perceived.(See my introduction to Goethe's Theory of Color.) The vibration is the revelation of light in the ether.The refraction of light is the result of the effect of a certain force-direction upon the light-direction.Newton's color rings (circles), phenomena of interference, are results of light-radiation (effect of light in the ether), and of the effects of other forces found in the path of light (weakening effects, gradually weakening affects of other forces).The same goes for phenomena of polarisation.One should not seek the polarisation figures in the structure of the essence of the light but in the structure of the medium which places itself in the path of light.The speed of transmission is the result of a kind of friction of the light against the medium.llLight is not to be considered as a function of electricity, but the latter is to be considered as a kind of corporeal carrier of light.Electrically charged matter: certain accumulations of force retain those accumulations of force which manifest as electricity.llMathematics is the abstracted sum of the forces effective in space.If one says, “Mathematical propositions are valid a priori”, this comes from the fact that man exists within the same lines of force as the other beings, and that he can disassociate himself from everything that does not belong to the scheme of space, etc.lMathematics and OccultismBy Rudolf Steiner.IT is well known that the inscription over the door of Plato's school was intended to exclude anybody who was unacquainted with the science of Mathematics, from participating in the teachings of the Master.Whatever we may think of the historical truth of this tradition, it is based upon the correct understanding of the place that Plato assigned to mathematics within the domain of human knowledge.Plato intended to awaken the perceptions of his disciples by training them to move in the realm of purely spiritual being according to his “Doctrine of Ideas.” His point of view was that Man can know nothing of the “True World” so long as his thought is permeated by what his senses transmit.He demanded that thought should be emancipated from sensation.Man moves in the World of Ideas when he thinks, only after he has purged his thought of all that sensuous perception can present.The paramount question for Plato was, “How does Man emancipate himself from all sense-perception?” He considered this to be an all-important question for the education of the spiritual life.Of course, it is only with difficulty that Man can emancipate himself from material perceptions, as a simple experiment on one's own self will prove.Even when the man who lives in this every-day world does withdraw into himself and does not allow any material impressions of the senses to work upon him, the residues of sensuous perception still linger, in his mind.As to the man who is as yet undeveloped, when he rejects the impressions which he has received from the physical world of the senses, he simply faces nothingness — the absolute annihilation of consciousness.Hence certain philosophers affirm that there exists no thought free from sense-perception.They say, “Let a man withdraw himself ever so much within the realm of pure thought, he would only be dealing with the shadowy reflections of his sense-perceptions.” This statement holds good, however, only for the undeveloped man.When he acquires for himself the faculty of developing organs which can perceive spiritual truths (just as Nature has built for him organs of sense), then his thought ceases to remain empty when it rids itself of the contents of sense-perception.It was precisely such a mind emancipated from sense-perception and yet spiritually full, which Plato demanded from those who would understand his Doctrine of Ideas.In demanding this, however, he demanded no more than was always required of their disciples, by those who aspired to make them true initiates of the Higher Knowledge.Until Man experiences within himself to its full extent what Plato here implies, he cannot have any conception of what true Wisdom is.Now Plato looked upon mathematical science as a means of training for life in the World of Ideas emancipated from sense-perception.The mathematical images hover over the border-line between the material and the purely spiritual World.Let us think about the “circle”; we do not think of any special material circle which perhaps has been drawn on paper, but we think of any and every circle which may be represented or met with in Nature.So it is in the case of all mathematical pictures.They relate to the sense-perceptible, but they are not exhaustively contained in it.They hover over innumerable, manifold sense-perceptible forms.When I think mathematically, I do indeed think about something my senses can perceive; but at the same time I do not think in terms of sense-perception.It is not the material circle which teaches me the laws of the circle; it is the ideal circle existing only in my mind and of which the concrete form is a mere representation.I could learn the identical truths from any other sensible image.The essential property of mathematical perception is this: that a single sense-perceptible form leads me beyond itself; it can only be for me a representation of a comprehensive spiritual fact.Here again, however, there is the possibility that in this sphere I may bring through to sense-perception what is spiritual.From the mathematical figure I can learn to know supersensible facts by way of the sense-world.This was the all-important point for Plato.We must visualise the idea in a purely spiritual manner if we would really know it in its true aspect.We can train ourselves to this if we only avail ourselves of the first steps in mathematical knowledge for this purpose, and understand clearly what it is that we really gain from a mathematical figure.“Learn to emancipate thyself from the senses by mathematics, then mayest thou hope to rise to the comprehension of ideas independently of the senses”: this was what Plato strove to impress upon his disciples.The Gnostics desired something similar.They said, “Gnosis is Mathesis.” They did not mean by this that the essence of the world can be based on mathematical ideas, but only that the first stages in the spiritual education of Man are constituted by what is supersensible in mathematical thought.When a man reaches the stage of being able to think of other properties of the world independently of sense-perception in the same way as he is able to think mathematically of geometrical forms and arithmetical relations of numbers, then he is fairly on the path to spiritual knowledge [ Pobierz całość w formacie PDF ]