The Fourth Dimension
Questions and Answers XIX
7 April 19021, Dornach
QUESTION: It has hem said that the three dimensions of space differ in structure. Where does this difference lie?
This statement was never formulated like that [Note 125] — "The three dimensions of space differ in structure." You are probably referring to the following thought.
First we have mathematical space, which we imagine — if indeed we imagine it with any precision at all — as consisting of three perpendicular dimensions or directions, which we define by means of a coordinate system on three perpendicular axes. When we consider this space from the usual mathematical perspective, we treat the three dimensions as if they were exactly the same. We make so little distinction between the dimensions of up and down, right and left, and forward and backward that we even can believe them to be interchangeable. In terms of merely mathematical space, it ultimately makes no difference whether we say that the plane of the \(y\)-axis, which is perpendicular to the plane formed by the \(x\)- and \(z\)-axes (which are also perpendicular to each other), is ''horizontal'' or "vertical." We are equally unconcerned about the boundedness of this type of space, which does not mean that we ordinarily get so far as to imagine it as limitless. We simply do not worry about its limits. We assume that from any point on the x-axis, for example, we can continue to move along the axis indefinitely, without ever reaching the end.
During the nineteenth century, metageometry presented many ideas contrary to this Euclidean concept of space. [Note 126] Let me simply remind you, for example, how Riemann distinguished between the "limitlessness" of space and the "infinity" of space. [Note 127] From the perspective of purely conceptual thinking, too, there is no need to assume that limitlessness and infinity are identical. Take the outer surface of a sphere, for instance. When you draw on such a surface, you never encounter any spatial limit that prevents you from continuing your drawing. Eventually, of course, you will intersect your previous drawing, but as long as you remain on the sphere's surface, you will never encounter a boundary that forces you to stop. Thus, you can say that a sphere's surface is limitless with regard to your ability to draw on it. This does not mean, however, that anyone claims that such a surface is infinite. In this way, on a purely conceptual level, we can distinguish between limitlessness and infinity.
Under specific mathematical conditions, this distinction also can be extended to space as a whole. If we imagine that we never will be hindered from extending an \(x\)- or \(y\)-axis by continuing to add segments to it, this property of space speaks for its limitlessness but not for its infinity. The fact that I can continue adding segments indefinitely does not mean that space is necessarily infinite. It might be simply limitless. We must distinguish between these two concepts. If space is limitless but not infinite, we can assume that it is inherently curved and returns to its starting point in some way, just as a spherical surface does. Certain ideas in modern metageometry depend on such assumptions. It is not easy to raise objections to these assumptions, because we cannot conclude that space is infinite from our experience of it. It equally well could be curved and finite.
I cannot carry this train of thought to its conclusion, of course, without explaining almost all of recent metageometry. Treatises by Riemann, Gauss, and others are readily available, however, and will provide you with plenty of food for thought if you are interested in mathematical ideas of this sort. [Note 128] These are the purely mathematical arguments against the fixed, neutral space of Euclidean geometry. All of the arguments I have mentioned so far are based purely on the concept of limitlessness. Your question, however, is rooted elsewhere, in the idea that space — the space of our calculations and the space we encounter in analytical geometry, for example, when we are dealing with a coordinate system of three perpendicular axes — is an abstraction. And what is an abstraction? This question must be answered first.
It is important to know whether we are restricted to an abstract idea of space. Is abstract space the only space we can talk about? To put it better, if this abstract concept of space is the only one we are justified in speaking of, only one objection is possible, and this one objection has been raised adequately by Riemann's geometry or other forms of metageometry. [Note 129]
Kant's definitions of space, for example, rest soundly on a very abstract concept of space. His concept is initially unconcerned with limitlessness or infinity. In the course of the nineteenth century, this concept of space was shattered — also internally, with regard to its conceptual content — by mathematics. [Note 130] It is impossible to imagine applying Kant's definitions to a space that is limitless but not infinite. Much of what Kant presents later in his Critique of Pure Reason — his theory of paralogisms, for example — would begin to totter if we were forced to substitute the concept of a limitless, curved space. [Note 131]
I know that this concept of curved space poses problems for our ordinary way of imagining things. But from the purely mathematical or geometric perspective, the only possible argument against the assumption that space is curved is that it forces us to move into a realm of pure abstraction that is initially quite remote from reality. Looking at the situation more closely, we discover that a curious circular argument exists in the derivations of modern metageometry, namely, that we arrive at them by taking as our starting point the ideas of Euclidean geometry, which is unconcerned with any limitations of space. We then move on to certain derivative ideas, such as those that apply to the surface of a sphere. On the basis of these derivatives and the forms that result, we can undertake certain transpositions and then make reinterpretations of space. Everything we say, however, presupposes Euclidean coordinate geometry. Under this presupposition, we get a specific rate of curvature. We arrive at the derivations. All this calculation presupposes Euclidean geometry. Here we come to a turning point, however. We use ideas such as the rate of curvature, which we developed only with the help of Euclidean geometry, to arrive at another idea that can lead to a new view and an interpretation of what we have gained from the curved forms. [Note 132] Essentially, we are functioning in a realm remote from reality by deriving abstractions from abstractions. This activity is justified only when an empirical reality forces us to align ourselves with the results of such abstractions. Thus the question is, Where does abstract space correspond to our experience? Space as such, as Euclid imagined it, is an abstraction. [Note 133] Where does its perceptible, empirical aspect lie?
We must take our human experience of space as our starting point. We actually perceive only one dimension of space — namely, the dimension of depth — as a result of our own active experience. This active perception of depth is based on a process in our consciousness that we very frequently overlook. This active perception, however, is very different from the idea of a plane, of extension in two dimensions. When we look out into the world with both eyes, these two dimensions are not the result of our own soul activity. They are there as givens, while the third dimension comes about as a result of activity that usually does not become conscious. We need to work at recognizing depths, at knowing how distant an object is from us. We do not work out the extent of a plane, — direct perception provides us with that knowledge. We do, however, use both eyes to work out the dimension of depth. The way we experience depth lies very close to the boundary between the conscious and the unconscious. But when we learn to pay attention to such processes, we know that the never fully conscious activity of estimating depth — it is at most semiconscious or one-third conscious — more closely approximates a rational activity, an active soul process, than does seeing objects only in a plane.
In this way, we actively acquire one dimension of three-dimensional space on behalf of our objective consciousness. And we are forced to say that our upright position contributes a quality to the dimension of depth — that is, forward and backward — that makes it non-interchangeable with any other dimension. The fact that we stand there actively experiencing this dimension makes it non-interchangeable with any other dimension. For the individual human being, the dimension of depth is not interchangeable with the other dimensions. It is also true that our perception of two-dimensionality — that is, of up and down and right and left, even when these two dimensions are in front of us — is associated with different parts of the brain. This perception is inherent in the sensory process of seeing, while the third dimension arises for us in parts of the brain located very close to the centers associated with rational activity. Thus, we see that even in terms of our experience, the third dimension arises in a way that is very different from the other two dimensions.
When we rise to the level of imagination, however, we leave our experience of the third dimension behind and see in two dimensions. At this level, we must work to experience right and left, just as experiencing forward and backward in our ordinary consciousness requires work of which we are not fully aware. And, finally, when we rise to the level of inspiration, the same is true of the dimension of above and below. [Note 134] As far as our ordinary nerve-related perception is concerned, we must work to experience the third dimension. When we exclude the ordinary activity of this system, however, and turn directly to the rhythmic system, we experience the second dimension. In a certain respect, this is what happens when we rise to the level of Imagination. I have not expressed this very precisely, but it will do for now. And we experience the first dimension when we rise to the level of Inspiration — that is, to the third member of our human organization.
What we encounter in abstract space proves to be exactly what it appears to be, because all of our mathematical accomplishments come from within ourselves. The mathematical consequence, threefold space, is something we derive from ourselves. When we move down through the levels of suprasensible perception, the result is not abstract space with three equivalent directions, but rather three different values for the three different dimensions of forward and backward, right and left, and above and below. These dimensions are not interchangeable. [Note 135]
We can then conclude that we also need not imagine the three dimensions as having the same intensity, which is essentially how we imagine the \(x\)-, \(y\)-, and \(z\)-axes in Euclidean space. If we want to abide by the equations of analytical geometry, we must see the \(x\)-, \(y\)-, and \(z\)-axes as equivalent in intensity. If we make the \(x\)-axis larger, stretching it with a certain intensity as if it were elastic, the \(y\)- and \(z\)-axes must grow with the same intensity. In other words, when I apply a certain intensity to expanding one dimension, the force of expansion must be the same for all three axes, that is, all three dimensions of Euclidean space. That is why I would like to call this type of space "fixed space."
Fixed space is an abstraction of real space, which is developed from within the human being, and the principle of equivalent intensity does not apply to real space. When we consider real space, we can no longer say that the intensity of expansion is the same for all three dimensions. Instead, it depends on human proportions, which are the result of spatial expansion intensities. For example, take they-axis, the up-down direction. We must imagine its expansion intensity as greater than that of the x-axis, which corresponds to the left-right direction. The formula that is an abstract expression of real space — we must be aware that this formula, too, is an abstraction — describes an ellipsoid with three axes.
Suprasensible perception dwells within the three very different expansion possibilities of this triaxial space. Our physical body provides direct experience of the three axes, and such experience tells us that this space also expresses the relationships among the effects of the heavenly bodies within it. Visualizing space in this way, we must also consider that everything we think of as existing in the three-dimensional universe cannot be accounted for if the expansion intensity of the \(x\)-, \(y\)-, and \(z\)-axes is the same, as is the case in Euclidean space. We must imagine the universe with a configuration of its own, corresponding to an ellipsoid with three axes. The configuration of certain stars suggests that this idea is correct. For example, we usually say that our Milky Way galaxy is lens-shaped, and so on. We cannot possibly imagine it as a sphere. We must find a different way of imagining it if we want to accommodate the facts of physics.
The way we treat space demonstrates how poorly modern thinking coincides with nature. In ancient times and cultures, the concept of fixed space did not occur to anyone. We cannot even say that the original Euclidean geometry incorporated a clear idea of fixed space with three equal expansion intensities and three perpendicular lines. It was only in fairly recent times, when abstraction became an essential attribute of our thinking and we began to apply calculations to Euclidean space, that the abstract concept of space emerged. [Note 136] The knowledge available to people in ancient times was very similar to what can be redeveloped now on the basis of suprasensible insights. As you see, concepts that we depend on heavily and take for granted today assume a high degree of importance only because they work in a sphere that is foreign to reality. The space we reckon with today is one such abstraction. It is far removed from anything real experience can teach us. We are often content with abstractions today. We harp on empiricism, but we refer very frequently to abstractions without even being aware of doing so. We believe that we are dealing with real things in the real world. You can see, however, how badly our ideas need correction in this respect.
Spiritual researchers do not simply ask if every idea they encounter is logical. Riemann's concept of space is thoroughly logical, though in a certain respect it depends on Euclidean space. It cannot be thought through to its conclusion, however, because we approach it by means of highly abstract thinking, and in this process our thinking is turned upside down because of one of the conclusions we draw. [Note 137] Spiritual researchers do not simply ask whether an idea is logical. They also ask whether it corresponds to reality. For them, that is the decisive factor in accepting or rejecting an idea. They accept an idea only if it corresponds to reality.
Correspondence to reality will apply as a criterion when we begin to deal appropriately with such ideas as the justification of the theory of relativity. In itself, this theory is as logical as it can possibly be, because it is understood purely in the domain of logical abstractions. Nothing can be more logical than the theory of relativity. The other question, however, is whether we can act on it. If you simply look at the analogies presented in support of this theory, you will discover that they are very foreign to reality. They are simply ideas being tossed around. The proponents of relativity theory tell us that these ideas are there only as symbols to help us visualize the issues. They are not merely symbols, however. Without them, the entire process would be left hanging in the air. [Note 138] This, then, is what I wanted to say in reference to your question. As you see, there is no easy answer to questions that touch on such domains.
Questions and answers (disputation) during the second anthroposophical conference at the Goetheanum in Dornach, April 3 to 10, 1921. Rudolf Steiner's lectures on "Anthroposophy and the Specialized Sciences" appeared, along with the question-and-answer sessions (disputations), in Die befruchtende Wirkung der Anthroposophie auf die Fachwissenschaften ("Anthroposophy's Positive Effect on the Specialized Sciences") (CA 76). Reports by Willy Stokar on this conference can be found in the periodical Dreigliederung des sozialen Organismus ('The Threefolding of the Social Organism"), vol. 2, (1920-1921), nos. 42 and 43. Eugen Kolisko's reports were published in Die Drei ("The Three"), vol. 1 (1921-1922), pp. 471-478. See also the invitation to this conference and the detailed program in Dreigliederung des sozialen Organismus, vol. 2 (1920-1921), no. 36.
Metageometry is an almost obsolete term encompassing various types of non- Euclidean geometry. In the second half of the nineteenth century, these non- Euclidean geometries included projective geometry, hyperbolic and elliptical geometry, the geometry of general curved spaces (Riemann's geometry), and the geometry of higher-dimensional spaces.
Riemann: See Note 1, Lecture 1 (March 24, 1905).
Gauss: See Note 1, Lecture 1 (March 24, 1905).
Riemann’s metageometry" probably means either so-called elliptical geometry, which was first discovered and described by Riemann and is closely related to the geometry of a spherical surface, or the general theory — also based on Riemann's work — of curved spaces (manifolds with a Riemannian metric), of which elliptical geometry is only a special instance (space with a constant positive curve).
Kant did not distinguish between the mathematical or geometric view of the concept of space and the laws of perceived space. He interpreted the latter as necessary, subject-based prerequisites of sense perception. "Space is a necessary idea a priori and underlies all external views." (Critique of Pure Reason = CPR, B 38). 'The apodictic certainty of all geometric theorems is based on this necessity a priori, and the possibility of their construction a priori" (CPR, A 24). Thus, "Geometry is a science that determines the properties of space synthetically and yet a priori" (CPR, B 40). "For example, space has only three dimensions,- such statements, however, cannot constitute, and cannot be concluded on the basis of, empirical judgments" (CPR, B 41).
"How can the mind encompass an outer view that precedes the objects themselves and in which the concept of the latter can be determined a priori? Apparently only to the extent that it is affected by objects only in the subject, as the latter's formal constitution...that is, only as the form of the outer sense altogether." (CPR, B 41) Thus, "Space is nothing other than simply the form of all manifestations of outer senses, that is, the subjective condition of sensory nature, which alone makes our outer perception possible" (CPR, B 42).
Thus for Kant, the laws of perceived space coincide with geometric principles that can be thought. In Kant's time, ideas about non-Euclidean measurement and spaces with more than three dimensions had not yet appeared in mathematics. In particular, Kant lacked the clear distinction between topological and metric properties that dates back only to Riemann, so he saw no difference between the topological attribute of limitlessness and the metric attributes (that is, those pertaining to measured relationships) of infinity. Thus, in his explanations of the "antinomies of pure reason," where he proclaims the insolubility of certain problems that cannot be interpreted from his perspective, Kant says, 'The same is true of the dual answer to the question of the size of the cosmos, because if it is infinite and boundless, it is too big for all possible empirical concepts. If it is finite and limited, you are right to ask, What determines the limit?" (CPR, B 515). Kant's concept of space, which clings to three-dimensional Euclidean geometry, could no longer be reconciled with the various concepts of space that developed as mathematics continued to evolve. One of the first to point this out clearly from the perspectives of physics and physiology was Hermann von Helmholtz (1821–1894). On this subject, see Helmholtz's speech Die Thatsachen in tier Wahrnehmung ("Facts in Perception") .
Kant's discussion of the paralogisms (deceptive or faulty conclusions) and antinomies of pure reason constitute the major portion of the second volume, Transcendental Dialectics of The Critique of Pure Reason . Kant intended his critique of the paralogisms of pure reason as a critique of the claims of the rational psychology of his day (including the problems of unchangeability, preexistence of the soul, etc.) rather than as a discussion of classic paralogisms.
"A logical paralogism is the formal falsehood of a rational conclusion, regardless of its content. A transcendental paralogism, however, has a transcendental reason for coming to a formally false conclusion. In this way, a faulty conclusion of this sort has its reasons in the nature of human reason itself and carries an inevitable if not insoluble illusion with it" (CPR, B 399). As he does later in his discussion of the antinomies of pure reason, Kant also attempts here in his discussion of paralogisms to demonstrate that they "dissolve" only when his own view is applied, namely, that we can know only the manifestations of "things as such" and that while our reason can order these manifestations according to regulative principles (such as the perceived forms of space and time), no direct insight into the constitution of things as such is possible. The problem of space plays only a peripheral role in Kant's discussion of the paralogisms of pure reason, namely, in the fourth paralogism about the soul's relationship "to possible objects in space" (CPR, B 402). In contrast, Kant's view of space is of fundamental importance in his discussion of the system of cosmological ideas in the section on the antinomies of pure reason.
Of course, three-dimensional Euclidean space was the historical point of departure and, initially, the foundation on which non-Euclidean concepts were developed in projective geometry and the geometries of curved and higher-dimensional space. To this extent, these new forms of space were derivative in nature; although they were not special instances of Euclidean space, they expanded the concept of space on the basis of fundamental Euclidean concepts. Steiner's reference to circular logic has to do with the fact that we achieve only an apparent generalization of the view of space as long as the relevant concepts depend essentially on a Euclidean point of departure.
The further evolution of mathematics has shown that we can dispense with the Euclidean foundation, that the laws of space can be developed step by step without presupposing the development of any specifically Euclidean concepts. We begin with a topological manifold that is defined as coordinate-free, supplement it with metrical and, if needed, differential geometric structures, arriving at Euclidean geometry as a special instance of a three-dimensional metric manifold. Seen systematically, there is no longer any circular logic involved in this process. When Steiner answered this question, these issues had not been clarified finally, even among mathematicians. See also Rudolf Steiner's handwritten notes and the corresponding footnotes in no. 114/115 of Beiträge Zur Rudolf Steiner Gesamtausgabe ("Articles on Rudolf Steiner's Complete Works"), Dornach, 1995, p. 49. In any case, with regard to the structure of real space, mathematical concepts, which indicate only which spatial forms are possible, are indeed abstract and remote from reality in this sense, as long as their correspondence with reality has not been established.
The concept of space that dates back to Euclid (ca. 320–260 B.C.) can be found in his comprehensive, thirteen-volume work Elements, especially in book XI and, to a lesser extent, in book I. This view of space focuses on the fundamentals of stereometry, that is, calculating the volumes of three-dimensional objects.
On the relationship of Imagination, Inspiration, and Intuition to the dimensions of space, see Rudolf Steiner's lectures of August 19 and 26, 1923 (GA 227, pp. 39-41 and 161-163). See also his lectures of May 17, 1905 (GA 324a),- September 16, 1907 (GA 101, pp. 189ff.); January 15, 1921 (GA 323, pp. 274-283); April 8, 1922 (GA 82); June 24, 1922 (GA 213); and the question- and-answer session of April 12, 1922 (GA 82 and 324a).
See also Rudolf Steiner's lectures of April 9 and 10, 1920 (GA 201); March 17, 1921 (GA 324); December 26 and 27, 1922, and January 1, 1923 (GA 326). In the section on Goethe's concept of space in Einleitungen zu Goethes Naturwissenschaftlichen Schriften ("Introductions to Goethe's Scientific Works," GA 1, pp. 288-295), Steiner also develops the idea that the three dimensions are not interchangeable, but from a totally different perspective.
Essentially, Euclid's three-dimensional geometry is still stereometry, that is, the study of the geometric properties of three-dimensional objects. Right angles and the concept of the perpendicular play an important role in Euclidean geometry, but Euclid placed no particular emphasis on the cube or on the related system of three perpendicular axes.
The implicit introduction of such axes as a reference system for the algebraic treatment of curves dates back to Pierre de Fermat (1601–1665) and René Descartes (1596–1650). Both of these mathematicians, however, often used oblique-angled axes, and in their work the coordinate system did not yet play a role as an independent structure that could be dissociated from the geometric object being discussed. Until the end of the eighteenth century, the same was true of developments in analytical geometry based on the work of these pioneers. The systematic application of two perpendicular or oblique directions as a reference system for coordinates and the discussion of algebraic curves occurs first in a treatise by Isaac Newton (1643–1727) entitled Enumeratio Linearum Tertii Ordinis (1676). Newton was also the first to use negative coordinates systematically and to draw curves in all four quadrants of the coordinate system. The analytical geometry of three-dimensional space and the corresponding use of a system of three perpendicular axes dates back to systematic studies of surfaces conducted by Leonhard Euler (1707–1783). Analytical geometry in the modern sense was definitively formulated in the late eighteenth and early nineteenth centuries by Gaspard Monge (1746–1818) and his pupil François Lacroix (1765–1843), who was one of the nineteenth century's most successful authors of mathematical textbooks. Previously, coordinate systems had been used primarily in connection with specific geometrical figures, but in the new analytical geometry, a preexisting coordinate system provided a framework for the study of geometric figures, their internal proportions, and their interrelationships. See the standard work on this subject by Boyer ,
See the discussion of this problem in Note 8 above.
See the question-and-answer session of March 7, 1920, and the corresponding notes, particularly Note 3.