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.In sending out its radiation, it is transmuted.It changes into helium, for example; so it becomes something quite different from what it was before.We have to do no longer with stable and enduring matter but with a complete metamorphosis of phenomena.Taking my start from these facts, I now want to unfold a point of view which may become for you an essential way, not only into these phenomena but into those of Nature generally.The Physics of the 19th century chiefly suffered from the fact that the inner activity, with which man sought to follow up the phenomena of Nature, was not sufficiently mobile in the human being himself.Above all, it was not able really to enter the facts of the outer world.In the realm of light, colours could be seen arising, but man had not enough inner activity to receive the world of colour into his forming of ideas, into his very thinking.Unable any longer to think the colours, scientists replaced the colours, which they could not think, by what they could, — namely by what was purely geometrical and kinematical — calculable waves in an unknown ether.This “ether” however, as you must see, proved a tricky fellow.Whenever you are on the point of catching it, it evades you.It will not answer the roll-call.In these experiments for instance, revealing all these different kinds of rays, the flowing electricity has become manifest to some extent, as a form of phenomenon in the outer world, — but the “ether” refuses to turn up.In fact it was not given to the 19th-century thinking to penetrate into the phenomena.But this is just what Physics will require from now on.We have to enter the phenomena themselves with human thinking.Now to this end certain ways will have to be opened up — most of all for the realm of Physics.You see, the objective powers of the World, if I may put it so, — those that come to the human being rather than from him — have been obliging human thought to become rather more mobile (albeit, in a certain sense, from the wrong angle).What men regarded as most certain and secure, that they could most rely on, was that they could explain the phenomena so beautifully by means of arithmetic and geometry — by the arrangement of lines, surfaces and bodily forms in space.But the phenomena in these Hittorf tubes are compelling us to go more into the facts.Mere calculations begin to fail us here, if we still try to apply them in the same abstract way as in the old wave-theory.Let me say something of the direction from which it first began, that we were somehow compelled to bring more movement into our geometrical and arithmetical thinking.Geometry, you know, was a very ancient science.The regularities and laws in line and triangle and quadrilateral etc., — the way of thinking all these forms in pure Geometry — was a thing handed down from ancient time.This way of thinking was now applied to the external phenomena presented by Nature.Meanwhile however, for the thinkers of the 19th century, the Geometry itself began to grow uncertain.It happened in this way.Put yourselves back into your school days: you will remember how you were taught (and our good friends, the Waldorf teachers, will teach it too, needless to say; they cannot but do so), — you were undoubtedly taught that the three angles of a triangle () together make a straight angle — an angle of 180°.Of course you know this.Now then we have to give our pupils some kind of proof, some demonstration of the fact.We do it by drawing a parallel to the base of the triangle through the vertex.We then say: the angle α, which we have here, shews itself here again as α'.α and α' are alternate angles and therefore equal.I can transfer this angle over here, then.Likewise this angle ß, over here; again it remains the same.Figure XaThe angle γ stays where it is.If then I have γ = γ', α = α' and ß = ß', while α' + ß' + γ' taken together give an angle of 180° as they obviously do, α + ß + γ will do the same.Thus I can prove it so that you actually see it.A clearer or more graphic proof can scarcely be imagined.However, what we are taking for granted is that this upper line A'B' is truly parallel to the lower line AB, — for this alone enables me to carry out the proof.Now in the whole of Euclid's Geometry there is no way of proving that two lines are really parallel, i.e.that they only meet at an infinite distance, or do not meet at all.They only look parallel so long as I hold fast to a space that is merely conceived in thought.I have no guarantee that it is so in any real space.I need only assume that the two lines meet, in reality, short of an infinite distance; then my whole proof, that the three angles together make 180°, breaks down.For I should then discover: whilst in the space which I myself construct in thought — the space of ordinary Geometry — the three angles of a triangle add up to 180° exactly, it is no longer so when I envisage another and perhaps more real space.The sum of the angles will no longer be 180°, but may be larger.That is to say, besides the ordinary geometry handed down to us from Euclid other geometries are possible, for which the sum of the three angles of a triangle is by no means 180°.Nineteenth century thinking went a long way in this direction, especially since Lobachevsky, and from this starting-point the question could not but arise: Are then the processes of the real world — the world we see and examine with our senses — ever to be taken hold of in a fully valid way with geometrical ideas derived from a space of our own conceiving? We must admit: the space which we conceive in thought is only thought.Nice as it is to cherish the idea that what takes place outside us partly accords with what we figure-out about it, there is no guarantee that it really is so.There is no guarantee that what is going on in the outer world does really work in such a way that we can fully grasp it with the Euclidean Geometry which we ourselves think out.Might it not be — the facts alone can tell — might it not be that the processes outside are governed by quite another geometry, and it is only we who by our own way of thinking first translate this into Euclidean geometry and all the formulae thereof?In a word, if we only go by the resources of Natural Science as it is today, we have at first no means whatever of deciding, how our own geometrical or kinematical ideas are related to what appears to us in outer Nature.We calculate Nature's phenomena in the realm of Physics — we calculate and draw them in geometrical figures.Yet, are we only drawing on the surface after all, or are we penetrating to what is real in Nature when we do so? What is there to tell? If people once begin to reflect deeply enough in modern Science — above all in Physics — they will then see that they are getting no further.They will only emerge from the blind alley if they first take the trouble to find out what is the origin of all our phoronomical — arithmetical, geometrical and kinematical — ideas.What is the origin of these, up to and including our ideas of movement purely as movement, but not including the forces? Whence do we get these ideas? We may commonly believe that we get them on the same basis as the ideas we gain when we go into the outer facts of Nature and work upon them with our reason.We see with our eyes and hear with our ears.All that our senses thus perceive, — we work upon it with our intellect in a more primitive way to begin with, without calculating, or drawing it geometrically, or analyzing the forms of movement.We have quite other categories of thought to go on when our intellect is thus at work on the phenomena seen by the senses [ Pobierz całość w formacie PDF ]
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.In sending out its radiation, it is transmuted.It changes into helium, for example; so it becomes something quite different from what it was before.We have to do no longer with stable and enduring matter but with a complete metamorphosis of phenomena.Taking my start from these facts, I now want to unfold a point of view which may become for you an essential way, not only into these phenomena but into those of Nature generally.The Physics of the 19th century chiefly suffered from the fact that the inner activity, with which man sought to follow up the phenomena of Nature, was not sufficiently mobile in the human being himself.Above all, it was not able really to enter the facts of the outer world.In the realm of light, colours could be seen arising, but man had not enough inner activity to receive the world of colour into his forming of ideas, into his very thinking.Unable any longer to think the colours, scientists replaced the colours, which they could not think, by what they could, — namely by what was purely geometrical and kinematical — calculable waves in an unknown ether.This “ether” however, as you must see, proved a tricky fellow.Whenever you are on the point of catching it, it evades you.It will not answer the roll-call.In these experiments for instance, revealing all these different kinds of rays, the flowing electricity has become manifest to some extent, as a form of phenomenon in the outer world, — but the “ether” refuses to turn up.In fact it was not given to the 19th-century thinking to penetrate into the phenomena.But this is just what Physics will require from now on.We have to enter the phenomena themselves with human thinking.Now to this end certain ways will have to be opened up — most of all for the realm of Physics.You see, the objective powers of the World, if I may put it so, — those that come to the human being rather than from him — have been obliging human thought to become rather more mobile (albeit, in a certain sense, from the wrong angle).What men regarded as most certain and secure, that they could most rely on, was that they could explain the phenomena so beautifully by means of arithmetic and geometry — by the arrangement of lines, surfaces and bodily forms in space.But the phenomena in these Hittorf tubes are compelling us to go more into the facts.Mere calculations begin to fail us here, if we still try to apply them in the same abstract way as in the old wave-theory.Let me say something of the direction from which it first began, that we were somehow compelled to bring more movement into our geometrical and arithmetical thinking.Geometry, you know, was a very ancient science.The regularities and laws in line and triangle and quadrilateral etc., — the way of thinking all these forms in pure Geometry — was a thing handed down from ancient time.This way of thinking was now applied to the external phenomena presented by Nature.Meanwhile however, for the thinkers of the 19th century, the Geometry itself began to grow uncertain.It happened in this way.Put yourselves back into your school days: you will remember how you were taught (and our good friends, the Waldorf teachers, will teach it too, needless to say; they cannot but do so), — you were undoubtedly taught that the three angles of a triangle () together make a straight angle — an angle of 180°.Of course you know this.Now then we have to give our pupils some kind of proof, some demonstration of the fact.We do it by drawing a parallel to the base of the triangle through the vertex.We then say: the angle α, which we have here, shews itself here again as α'.α and α' are alternate angles and therefore equal.I can transfer this angle over here, then.Likewise this angle ß, over here; again it remains the same.Figure XaThe angle γ stays where it is.If then I have γ = γ', α = α' and ß = ß', while α' + ß' + γ' taken together give an angle of 180° as they obviously do, α + ß + γ will do the same.Thus I can prove it so that you actually see it.A clearer or more graphic proof can scarcely be imagined.However, what we are taking for granted is that this upper line A'B' is truly parallel to the lower line AB, — for this alone enables me to carry out the proof.Now in the whole of Euclid's Geometry there is no way of proving that two lines are really parallel, i.e.that they only meet at an infinite distance, or do not meet at all.They only look parallel so long as I hold fast to a space that is merely conceived in thought.I have no guarantee that it is so in any real space.I need only assume that the two lines meet, in reality, short of an infinite distance; then my whole proof, that the three angles together make 180°, breaks down.For I should then discover: whilst in the space which I myself construct in thought — the space of ordinary Geometry — the three angles of a triangle add up to 180° exactly, it is no longer so when I envisage another and perhaps more real space.The sum of the angles will no longer be 180°, but may be larger.That is to say, besides the ordinary geometry handed down to us from Euclid other geometries are possible, for which the sum of the three angles of a triangle is by no means 180°.Nineteenth century thinking went a long way in this direction, especially since Lobachevsky, and from this starting-point the question could not but arise: Are then the processes of the real world — the world we see and examine with our senses — ever to be taken hold of in a fully valid way with geometrical ideas derived from a space of our own conceiving? We must admit: the space which we conceive in thought is only thought.Nice as it is to cherish the idea that what takes place outside us partly accords with what we figure-out about it, there is no guarantee that it really is so.There is no guarantee that what is going on in the outer world does really work in such a way that we can fully grasp it with the Euclidean Geometry which we ourselves think out.Might it not be — the facts alone can tell — might it not be that the processes outside are governed by quite another geometry, and it is only we who by our own way of thinking first translate this into Euclidean geometry and all the formulae thereof?In a word, if we only go by the resources of Natural Science as it is today, we have at first no means whatever of deciding, how our own geometrical or kinematical ideas are related to what appears to us in outer Nature.We calculate Nature's phenomena in the realm of Physics — we calculate and draw them in geometrical figures.Yet, are we only drawing on the surface after all, or are we penetrating to what is real in Nature when we do so? What is there to tell? If people once begin to reflect deeply enough in modern Science — above all in Physics — they will then see that they are getting no further.They will only emerge from the blind alley if they first take the trouble to find out what is the origin of all our phoronomical — arithmetical, geometrical and kinematical — ideas.What is the origin of these, up to and including our ideas of movement purely as movement, but not including the forces? Whence do we get these ideas? We may commonly believe that we get them on the same basis as the ideas we gain when we go into the outer facts of Nature and work upon them with our reason.We see with our eyes and hear with our ears.All that our senses thus perceive, — we work upon it with our intellect in a more primitive way to begin with, without calculating, or drawing it geometrically, or analyzing the forms of movement.We have quite other categories of thought to go on when our intellect is thus at work on the phenomena seen by the senses [ Pobierz całość w formacie PDF ]