The Rudolf Steiner Archive

Episodic Reflections on Space, Time, and Movement

[ 1 ] I thought there would be at most a dozen people here today, and I wanted—as is indeed the intention—to say something that is, strictly speaking, quite unrelated to our usual considerations, but which may be important for some who can immerse themselves in the subject, in order to assess certain matters that are currently relevant with regard to specific conceptions of space, time, and motion.

[ 2 ] In fact, there are theoretical physicists today who believe that a profound revolution is taking place with regard to the simplest conceptions of the world. Among these simple conceptions of the world, which underlie theoretical physics, we would like to consider just a little bit today what relates to time, space, and motion. This will serve as the foundation for a more in-depth examination to be undertaken in the near future, which can lead us deeper into what is currently being sought in fundamental physical investigations.

[ 3 ] You have surely all heard by now that what is known as the theory of relativity in modern physics is currently gaining ground. The theory of relativity—and there are various nuances to it—is currently espoused by countless theoretical physicists. It is expected to bring about a complete reversal of all the concepts that physicists, when engaging in elementary theoretical considerations, have hitherto accepted as correct—concepts that essentially trace back to Newton. Now, today’s theoretical physicists believe that all these Newtonian concepts—which were accepted as completely irrefutable even during our student days—must undergo a radical overhaul; indeed, that, in a sense, the entire theoretical foundation of physics, as it has been and still is believed, is actually false. Well, why I must relate the discussion I intend to present to this newly emerging theory of relativity will become clear later.

[ 4 ] So that what I have to say does not remain entirely incomprehensible, I would like to start with very simple, elementary concepts in order to demonstrate to you right away, through these very concepts, what kind of mental image one can associate with the concept of time. Let us, as I said, start with very elementary things. Let’s assume, for the sake of argument, that some object—which I’ll call \(a\)—such as a rolling ball or something similar, is moving in a direction indicated by this line; that is, \(a\) is moving along the straight line in the direction of \(b\):

[ 5 ] Now, as you all know, the distance—the length of the path—that such a moving object travels in one second is called its speed. So let’s assume that \(a\) travels to this point, \(a_1\), in one second; then, in physics, this distance from \(a\) to \(a_1\) would be called the speed and denoted by \(c\). And if we further assume that the moving object continues through the following seconds, then—if it were moving uniformly (and we want to consider only such a case)—it would be at \(a_2\) at the end of the second second, where \(aa_1 = a_1a_2\), that is, moving at the same velocity \(c\), the object travels from \(a_1\) to \(a_2\) in the second second, from \(a_2\) to \(a_3\) in the third second, from \(a_3\) to \(a_4\) in the fourth second, and so on. Now let us assume that we observe this motion for a certain period of time and that our moving object travels a certain distance—let us assume as far as \(a_5\)

[ 6 ] then, when this moving object has rolled from \(a\) to \(a_5\), the portion of space—which we are considering here in one dimension—is called the path; so that \(a\) to \(a_5\) is the path it has traveled; \(c\) is the speed; the path is denoted by \(s\); and we say: the moving object \(a\) has traveled the path \(s\) at a speed of \(c\) in a certain time—here, five seconds. This elapsed time is denoted by \(t\).

[ 7 ] There is a specific relationship between distance, time, and speed. The simplest relationship that has been found is this: \(s\)—the distance—is five times the distance from \( \) to \(a_1\), that is, the distance \(a\) to \(a_1\) multiplied by \(5\), which is \(5\) seconds; that is the time; so we must multiply what we’ve called the speed—this segment \(aa_1\)—by \(5\), and then we get the distance \(s = c \cdot t\) (distance = speed \(\cdot\) time). So there are three terms in this formula: \(s\), \(c\), \(t\).

[ 8 ] Now, as you know, an infinite amount has been written about time by a number of philosophers, mathematicians, and even theoretical mechanics. Although people believe they have a mental image or a concept of time, anyone who were to explain and reflect on what they mean by time would very soon realize that they do not, in fact, have a proper understanding of this concept of time, which is one of the most fundamental concepts used in mechanics. In order to study the concept of time at all, let us stick to this formula, which, after all, initially translates the concept of time into uniform, rectilinear motion. But even though this formula appears in every physics textbook, it is nevertheless surrounded in physics by a great deal—I won’t say ambiguities, but rather a lack of clarity, a reluctance to delve deeper into the matter. And this stems in particular from the fact that in our schools, instruction regarding something we all learn does not teach us certain distinctions that are, however, important if one wishes to arrive at more precise concepts in a certain direction. After all, in our schools we learn to speak of four types of arithmetic: addition, subtraction, multiplication, and division. But when it comes to division, I don’t think we’re often made aware that this common arithmetic operation actually involves two completely different things. I’d like to show you this in a very simple way.

[ 9 ] Let’s suppose we have an ordinary apple and divide it. We can divide it into five, ten parts, and so on; then, once we’ve divided it, we get a certain fraction of the apple. If we want to distribute the parts, what we’re distributing is simply a piece of the apple. We’re actually performing division here. I’ll write it as a fraction, because that’s the same as division. I can say: An apple is divided—let’s say into ten parts—and as a result, we get one-tenth of an apple. Now take a look at what I’ve written on the board:

$$\frac{1 apple (object)}{10 (number)} = \frac{1}{10} apple (object)$$

[ 10 ] In the numerator or dividend, we have a quality, something tangible; in the divisor or denominator, we have nothing tangible, but merely a number; \(10\) is merely a number here; and in the quotient, we again have something tangible: one-tenth of an apple.

[ 11 ] This doesn't change if we divide twenty apples instead of one. Suppose we divide \(20\) apples by \(10\); then, instead of one-tenth of an apple, we get \(2\) apples:

$$\frac{20 apples}{10} = 2 apples$$

[ 12 ] The \(20\) apples are, in turn, a concrete entity; on the bottom is simply the number, and as the quotient we again get a concrete entity. This is a division.

[ 13 ] But division can have a completely different meaning. I can have \(20\) apples as the dividend in the numerator, but in the denominator—or divisor—let’s say \(2\) apples; then I have a concrete quantity in both the numerator and denominator. What do I get as a result? I don’t get a concrete object as a result, but rather I find out how many times \(2\) apples fit into \(20\) apples—I get \(10\), which means I get a number:

$$\frac{20 apple (object)}{2 apple (object)} = 10 (number)$$

[ 14 ] Once again, I am dealing with a division, but this one now has a completely different meaning than the division in the first case. In the first case, I divide one thing and get another thing in return; in the second case, I don’t divide at all, but rather set out to investigate how often one thing is contained within another, and from that I arrive at a number.

[ 15 ] We can therefore say: Division is not always the same thing as dividing; rather, there are two types of division that are strictly distinct from one another. When teaching, one should therefore always make it clear that there are two types of division. In the first, the task is to investigate what results when one divides a concrete object; in the second, the task is to investigate how many times a concrete object is contained within another concrete object of the same kind—they must be of the same kind, because one cannot, of course, ask how many times \(2\) apples are contained in \(20\) pears—and then we obtain a number.

[ 16 ] This must be taken into account when studying the formula \(s = c \cdot t\).

[ 17 ] Now, this formula can also be written differently. I don't always have to find \(s\); I can also find \(c\) or \(t\), in which case the formula changes. If I’m looking for \(c\), I can find it by dividing \(s\) by \(t\). By dividing the total distance by \(t\), I get the distance traveled in \(5\) seconds divided by \(5\), which is the speed \(c\): $$c = \frac{s}{t}$$

[ 18 ] But you can also find \(t\): the time. Let’s assume you divide \(s\) by \(c\). If you ask: How many times does a one-second interval fit into the entire distance? The answer is five times. That gives you the time:

$$t = \frac{s}{c}$$

[ 19 ] Let’s take a closer look at these formulas. First, let’s take the second one and compare: \(s\)—that’s the distance here—the length from \(a\) to \(a_5\); we have that in the numerator. Here in the denominator, we have \(c\). What is \(c\)? Well, that’s the distance traveled in one second. Distances are like this: \(s\) is a distance, \(c\) is a distance. What kind of division does this resemble? Well, it resembles this form: \(20\) apples : \(2\) apples = \(10\). Here (in the numerator) you have apples, and here (in the denominator) you have apples; here (in the numerator of \(\frac{s}{c}\)) you have distance, and here (in the denominator) you have distance. What must be on the left side? Just a number. That means, in our physical considerations, \(t\) turns out to be nothing other than a number. For if I consider \(s\) and \(c\) as distance—that is, as tangible entities (since both are distance or a segment of a path)—then, by the very nature of the division, time \(t\) can only appear as a number. Just as the number \(10\) (\(20\) apples : \(2\) apples = \(10\)) is a number and nothing less or more, so in this division \(t\), time, can also be nothing other than a number.

[ 20 ] You can also use the division form \(1 apple : 10 = \frac{1}{10} apple\), in which case it is equivalent to the formula \(c = \frac{s}{t}\). On the other hand, if a physical quantity is divided by a physical quantity, what must the result be? A number like \(t = \frac{s}{c}\) here, where \(t\) is simply a number. This means that both formulas indicate that—as long as we stick to physics—we obtain nothing other than a number for time based on the nature of the division. Specifically, here (\(20\) apples: \(2\) apples = \(10\)), it is a number that refers to apples and shows how many times \(2\) apples are contained in \(20\) apples, and here, with time, \(\frac{s}{c} = t\) is a number that shows how many times the velocity is contained in space.

[ 21 ] Now, surely none of you would see the number itself as a concrete thing. If you give a boy or girl not \(3\) apples, but merely \(3\) as a number, they will not be satisfied. So the number itself cannot be seen as a concrete thing, but rather as a mere abstraction—something that, in a sense, merely indicates relationships in the external world.

[ 22 ] From this consideration, we can see that time itself slips through our fingers when viewed from a physical perspective; it shrinks down to a mere number for us. Just as we cannot philosophize about a number, we cannot philosophize about time either—that is to say, it has been reduced to a mental image of a number. That is why we cannot find time in things, no matter how long we search everywhere, because it appears merely as a number. Why is that? Well, I believe a boy or a girl does not need to be particularly old to give an answer that springs from a healthy sense of intuition when asked: “Which interests you more, the apples or the number?” Certainly, someone could speak sophistically and say, “I’m interested in the number, because I prefer \(8\) apples to \(6\)”; but that is only because \(8\) apples are more than \(6\). So the number is not at all what matters to him here; rather, it is the apples—it is the concrete thing.

[ 23 ] It follows, however, that we must stick to the concrete and not to numbers when we speak of time, space, and velocity. And if we now consider the physical reality, time is ruled out from the outset—that is, it is a number and not a physical entity. You will therefore be able to say to yourselves: We have \(s\), the space, the segment of space that our moving object traverses. If it continues to roll, it can still traverse much, much more space. Space is, of course, a tangible entity out there. But that is not what matters at first, for one can conceive of space as extending indefinitely. However, something else has a great deal to do with what matters to us, and that is \(c\). For the way \(a\) travels through space depends entirely on whether it travels, say, \(20\), \(25\), or \(50\) cm in a second, and so on; and in turn, how far it travels depends on how fast it moves. But how fast it moves is something it possesses internally; it is inherent to it. And the entire process depends entirely on what is inherent to the moving object. So it all comes down to the speed of the moving object; this belongs to the moving object as such and is an intrinsic quality of the moving object. And when we look at the world—insofar as we consider it in terms of mechanical processes—then, when we speak of reality, we must speak of the intrinsic speed of bodies, atoms, or molecules. And the entire process compels us to speak of intrinsic speed as belonging to things, just as the red color belongs to the rose.

[ 24 ] So the fundamental concept is velocity; that is what matters. It follows that we must not adhere to the formula that has \(c\) here, \(c = \frac{s}{t}\), and must not believe that space and time represent anything particularly real; rather, what is real in things is speed, not time. Time, in turn, is only an abstraction derived from the concept of velocity, because things have different velocities. If we look at the different velocities and seek to reduce them to a common denominator, we arrive at the concept of time. This is an abstraction, just as the generic term “apple” is an abstraction, and only the particular, concrete apple is real. So when we examine the mechanically real nature of things, we must focus on speed and must not believe that we can place the concept of time in the foreground. This is the great mistake made throughout physics: failing to recognize that we must start with the speed that is inherent in things, which belongs to them just as life belongs to living bodies.

[ 25 ] So take note, my dear friends: it is not time, but speed that must underlie mechanics. You might say that making these distinctions is mere speculation. But they are not mere speculations; rather, these things are of fundamental importance for understanding certain aspects of reality, and I want to point out something to you right away that shows just how fundamentally important they are.

[ 26 ] In the various discussions about the theory of relativity, people were primarily concerned with coming to terms with the concepts of time and speed. Now I would like to use two speculations to show you the way certain people think and how they formulate their thoughts when they talk about time and speed. To do so, I must introduce you to a curious character, Mr. Lumen, who plays a certain role in the theory of relativity. What kind of curious gentleman is this? Well, you see, he is, I would say, an “imaginary acquaintance” that Flammarion made. This Mr. Lumen has a very peculiar ability, which we can illustrate roughly as follows.

[ 27 ] As you all know from your physics classes, light travels at a certain speed; it covers 300,000 km per second. c—that is, everything that, according to our understanding, is intrinsically mechanical about light—has a speed of 300,000 km per second. Let’s assume, for example, that this is the Earth, and that a beam of light emanates into space from the objects and events taking place on Earth (this was schematically illustrated on the blackboard), and it is said that because the light goes out, we can see things. Now let’s assume the following. We are now having this somewhat abstruse math and physics lesson, and, let’s say, from three to four o’clock there was an eurythmy class. Light from all of this travels out into space, and one can observe from the outside what is happening here. And since the light travels at a speed of 300,000 km per second, what happened here this afternoon between three and four o’clock also radiated out into space at a speed of 300,000 km per second, so that if you imagine an observer 300,000 km away, that observer would not see what is happening here on Earth until one second later.

[ 28 ] Now Flammarion assumes that this Mr. Lumen is hurtling out into space even faster than light—namely, at a speed of 400,000 km per second. What will be the consequence of this? He will constantly overtake the light, for after the light has traveled for one second, he is already 100,000 km farther away; and as he races out like this and looks back, he must come upon the manifestations of the light, where he sees what has happened here now and between three and four o’clock. But since he not only catches up with the light but overtakes it, it must follow that he does not perceive the eurythmy lesson first and then our lesson, but rather the reverse—first the end and then what came before. It is a strange spectacle that this Mr. Lumen experiences. He sees everything in such a way that he sees the end first and then the beginning, for he is, after all, overtaking the light.

[ 29 ] As I said, such mental images played a certain role, particularly in the discussions about the theory of relativity. I’d like to present another mental image to you that also played a certain role—one that the naturalist Baer came up with. He said to himself: One could imagine that a person lives out their life not in about 70 or 80 years, but in 70 or 80 seconds. His pulse would simply have to beat so much faster that a single second would contain a year. This would mean that a person would not even be like a mayfly, but rather like a 70-second creature, if only his pulse beat fast enough. What would be the consequence? Such a person would experience immense things in 70 seconds. If, for example, he were to look at a plant that has remained true to its nature, he would never come to the conclusion that a plant grows out of the earth; rather, he would come to the conclusion that plants are eternal entities. Thus, such a person would have a completely different perspective on the world, simply because the speed of their life would have to be imagined as increased to the same extent as the speed of their pulse rate compared to the rest of us. Or, says Baer, let us imagine that a person lived not 80 seconds or 80 years, but 80,000 years, and that their pulse beat that much more slowly; then the whole world would be different once again. For example, while the sun moves across the sky at a certain speed for us, it would then race across the sky like a fiery wind; one would not distinguish the individual sun, but it would race around like a reddish wheel. Plants would shoot up in an instant and with breakneck speed wither away again, and so on.

[ 30 ] Baer presented this as a possible idea to show how one’s worldview depends on the subjective constitution of the organism. As you can see, everything—absolutely everything—is thrown into question.

[ 31 ] When considering the kind of thinking that underlies mental images such as Flammarion’s image of Mr. Lumen or Baer’s, one important point must be noted. Let us take Mr. Lumen once more. It is assumed that Mr. Lumen would be able to travel 400,000 km per second—that is, to outpace light and catch up with the subsequent light images. But now consider what you can actually accept as true if you delve more deeply into our Spiritual Science concepts. We can even set aside the coarser physical body entirely and turn directly to the etheric body. Yes, when we consider the etheric body, what is it? It is ether, light-ether; it is weaving light itself. Keep this in mind, for what follows from it? It follows, after all, that when we move through space, we can move at most at the speed characteristic of light. So if someone says that a person like Mr. Lumen is moving at a speed of 400,000 km per second, then we must ask—I’ll even set aside the physical body and simply assume that an etheric body could move on its own—how fast could it possibly move? Well, at most at a speed of 300,000 km per second—the speed of light. One cannot say that the etheric body outpaces light, for it is itself moving light. Thus, Mr. Lumen cannot be woven from anything that exists in space; in other words: He is an unreal mental image, a pure figment of the imagination. For the material or essential nature of things in the world has its speed immanent or inherent within it. It is inside it. It is its property. We cannot tear it out. We cannot even say, “We separate the thing’s speed from it”—rather, this is a property of the thing. We cannot speak of a property that lies separately outside the material world. Thus, we must also say regarding Baer’s mental images: The moment one realizes that the speed of the pulse belongs to the material nature of every human being, one also realizes that we cannot have any speed other than that of our pulse. We are human precisely because we have a certain pulse rate, and we cannot conceive of it arbitrarily, for we would cease to be human if, for example, our pulse were a thousand times faster than it actually is. Speed belongs to the physical reality.

[ 32 ] It is important to see how Spiritual Science leads to the essence of things, and where the kind of thinking that has developed right up to our time—without engaging with Spiritual Science—ultimately leads. It leads to the formation of mental images such as those of Mr. Lumen or the pulse rate accelerated a thousandfold, which are simply impossible or unreal. One operates with fantastical concepts if one does not realize that time is merely a number. Thus, so-called rational mechanics has led to entirely unreal concepts. Spiritual Science leads us to say: Yes, what exactly is this Mr. Lumen, who races at 400,000 km while he is at most 300,000 ... [gap in the postscript] ... He is nothing other than the famous gentleman who pulls himself up by his own forelock.

[ 33 ] From this perspective, then, Spiritual Science exists to bring human thinking—which has strayed into the realm of fantasy—back to reality, not to lead it away from reality. You see, while the Spiritual Science is accused of being fanciful, it is in fact intended to bring the fanciful mental images and concepts of physics back to reality. And it will be of extraordinary importance for sound thinking that, in the future, children are truly taught concepts such as the two types of division, so that they calculate not with all sorts of ambiguities but with definite concepts. One cannot arrive at mental images and concepts that have significance for reality except by truly confronting reality—that is, by thinking with Spiritual Science, for it is there that real, not fanciful, concepts become clear.

[ 34 ] Before the theory of relativity, physics was based on Newton’s mental image that space is a void, a sort of container—whether infinite or not, we will not examine that now—and that time flows like a uniform stream; things are contained within space, and events unfold in time; and depending on whether a thing takes this or that amount of time to traverse a certain distance, we attribute a certain velocity to it. This mental image is false because it fails to address the essential nature of space and time and thereby breaks down velocity—which is actually an intrinsic property—into two unreal concepts: space and time. Speed is truly the primary reality, whereas physics always regards speed as a function of space and time. What belongs to things, however, is their essence, and Spiritual Science shows that one must take certain paths in order not to arrive at fantasies about space and time—such as that of infinite space or that of time as a flowing stream—but rather to reach the true reality of speed. The entire field of mechanics, which we accepted in our youth as something immensely certain—as the most certain thing in science after mathematics—operates with very vague concepts because it does not know the nature of speed and does not regard it as fundamental.

[ 35 ] The impetus for the theory of relativity—developed by Minkowski, Einstein, Planck, Poincaré, the late mathematician and physicist, and so on—came precisely because they could no longer make sense of this childish Newtonian mental image of empty space, uniformly flowing time, and objects moving at a certain speed. Certain experiments gave rise to concepts that did not agree with what had been regarded as the most certain truth.

[ 36 ] Recently, I have developed a concept here in connection with Spiritual Science that may have come as a surprise to some. I have developed the idea that it is not at all true to believe that the most important thing in the head is substance, matter—because precisely where we suspect matter to be, there is a void, and from the point of view of Spiritual Science, we are all empty-headed. I used the analogy of the air bubbles in a bottle of seltzer water. It is the same there: where we believe we perceive something real and tangible, there is nothing. All around is the spiritually real, and within it there are holes everywhere; you can see them, just as with seltzer water you see only the bubbles—which are air—but you do not see the water. And if people believe that there is something where I bump into the table, that’s not true either, because there’s actually nothing there. I’m bumping into empty space, and because there’s nothing there, that’s why I can’t go any further.

[ 37 ] We arrived at this conclusion quite systematically based on premises from Spiritual Science. By a different route, certain discerning and sensible physicists have now been led to a similar view, because certain phenomena in nature are simply incompatible with the concepts of Newtonian mechanics, which are considered so certain. And these phenomena include, for example, the processes involving cathode rays—with which you are no doubt familiar—which, as you know, can be observed in certain evacuated glass tubes. Here we are dealing with something that, as a moving entity, possesses velocity—with electrons, figuratively speaking, with flowing electricity. And through observation, through the experiments that physicists conducted by observing cathode rays in the tubes—which are flowing electricity—they arrived at some very peculiar mental images. And I would like to read one such mental image to you. It can be found in a lecture by Poincaré on “The New Mechanics.” There, he builds on the mental images arising from the cathode-ray experiment, because this experiment, in particular, does not agree with Newton’s concept of velocity. And after a rather convoluted train of thought, he finds himself compelled to make the following concession:... [gap in the transcript] ..., and the physicist feels compelled to say the following:

[ 38 ] “Matter has now become entirely passive. It no longer possesses, in the true sense of the word, the property of resisting the forces that seek to alter its motion. When a cannonball moves at great speed and thereby becomes the bearer of a living force, a tremendous energy that spreads death and destruction, it is no longer the iron molecules that constitute the seat of this energy; rather, this seat is to be found in the ether that surrounds the molecules. One could almost say that there is no longer any matter; there are only holes in the ether.”—Well, what more could you want, my dear friends?—“And insofar as these holes seem to play an active role, it consists in the fact that these holes cannot change their location without influencing the surrounding ether, which exerts a reaction against such changes.”

[ 39 ] Matter consists of holes in the ether! Based on current experimental evidence, physics is therefore compelled to admit this. Building on such findings, another physicist, Planck, made a statement that is most remarkable, namely the statement that says: We saw in the 1840s that Helmholtz approached a certain problem in this way—it was not Helmholtz, but Julius Robert Mayer, though we will not get into this important question of priority now—just as one who harnesses a horse not by the tail, but by the head. People had always said before that one must study the distribution of forces in space in a certain way. Helmholtz turned the matter on its head; he said that one must study the universe in such a way that only the universe as a whole can be a perpetual motion machine, whereas an individual process within the universe can never be a perpetual motion machine. People before him had, in fact, tried to explain the worldview entirely without a perpetual motion machine. But now Planck says that a process of this very kind must occur with regard to the ether. There are countless theories about the ether, ranging from the earlier mental image—when the ether was imagined as rarefied matter—to the ideas of Lord Kelvin or J. J. Thompson, who envisioned the ether as a rigid fluid—though, of course, one should not think of a fluid like water—with all intermediate stages represented. And now Planck, as a physicist, says: Physics will only be put on a sound footing when we start from the following principle: No mental image of the ether can provide a sound foundation for physics if it attributes material properties to the ether. - This is the statement made by one of the most eminent physicists of our time. This means, then, that if the ether is to serve as a sound foundation for physics, it may be attributed only spiritual properties. And it follows from this that today’s physicists are compelled to conceive of matter as holes surrounded by the ether—which, however, must be conceived in such a way that it has no material properties, but only spiritual ones. In short: holes surrounded by spiritual ether—that is what must be taken as the basis in order to arrive at a sound physics. This is being prepared today; it already exists.

[ 40 ] Now one might ask: If the physicist says that matter consists of holes and that the ether can have only spiritual properties, where, then, is there any possibility of justifying a materialistic worldview? So one is almost compelled to say: There is no longer any matter; there are only holes in the spiritual ether, and matter cannot change its location without exerting an influence on the surrounding ether—a reaction in the spiritual ether. This is what physics has come to.

[ 41 ] However, one will need a rigorous logic and must not shy away from addressing questions such as how the concept of speed can truly be understood if it is not to contradict what the experiment reveals.

[ 42 ] Consider these things as evidence that the Spiritual Science—so often denounced as unscientific—is, at its very foundations, infinitely more scientific than what is considered science today, for it tackles things—I would say—with the sharpest logic. And that is what we must seek above all else: a precise grasp of concepts, a clear understanding of what otherwise appears to us as vagueness in the world.