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The Fourth Dimension
GA 324a

Questions and Answers XXII

29 December 1922, Dornach


As you will have gathered from the lecture, we must make a distinction between tactile space and visual space. This difference can stimulate us to move beyond considering mathematics on the one hand and the physical world on the other. As you may know from my lectures, [Note 157] it remains true that mathematics is a product of the human spirit or of the human being in general. And that as we move further into purely mathematical domains—that is, domains that are delineated in mathematical terms—we become less and less able to apprehend reality. [Note 158] You have all seen the difficulties that have arisen repeatedly in modern times when people have attempted to use mathematics to describe reality.

For example, if you consider the transition from an infinitely large sphere to a plane, you scarcely will be able to reconcile this cornerstone of projective geometry with our ordinary ideas of reality, which are based on empirical interaction with the world around us. [Note 159] Consequently, our task—and many people with the appropriate educational background would have to work very hard at it—is to attempt to use mathematical ideas to apprehend reality in very concrete domains. [Note 160] At this point, I would like simply to present the problem. It can be solved successfully only if mathematicians really begin to work seriously on it.

I have provided a theoretical explanation of tactile space. Now try to handle this space in a way that necessarily incorporates all of our earthly experience of touch—in fact, that is what we are dealing with. We must incorporate all of our tactile experience, including its inherent dimensionality, into our relationship to gravity. We are subject to gravity, and the various centripetal forces coming in different directions from the periphery make it possible to set up differential equations. With regard to tactile space, we must handle these equations in the same way that we handle equations for determined movements in analytical geometry and analytical mechanics. [Note 161] It then becomes possible to integrate these equations, which gives us specific integrals for what we experience in tactile space, whereas differentials always lead us out of reality.

Integrating these differentials results in the diagrams I told you about the day before yesterday. [Note 162] If you want to return to their reality, you must do it as I indicated in that lecture. You must work with the integral equations in the domain of real touch. It will become evident that with regard to touch, the vertical dimension has a certain differentiation, so that the variable x in this equation must be preceded by a plus or minus sign. This makes it possible to set up integrals for our experiences of tactile space. Let me formulate it like this:

$$\int f(x) \,dy$$

The result would be integrals for our experiences of tactile space.

Now let's move on and apply the same principle to visual space. Once again, we set up differential equations that we must handle in the same way that we handle equations for determined movements in analytical geometry and analytical mechanics. We will see that when we integrate, we get very similar integrals, but ones that must be thought of as negative (taking into account that the variable \(x\) was positive in the last instance). When we handle the integration in this way (I'll dispense with all the trimmings), we get a result that leads to other integrals:

$$\int f(x) \,dy$$

But when I subtract the two from each other, they almost cancel each other out and the result approaches zero. That is, when I integrate with regard to visual space, the result is integrals that cancel out those for tactile space. And the integrals for tactile space remind me very much—though they are more extensive—of all the formulas I need for circumstances and relationships that refer to analytical geometry or mechanics in general. The only difference is that gravitation must be included in the mechanical formulas.

I get integrals for visual space that seem applicable if I simply can find the right way to express the spatial aspect of vision in mathematical terms. It is always the case that we begin with a trivial instance and set up constructions about vision and fail to note that we must count on inevitable vertical movement when we consider visual space. We must accept that vision is always forced to work in the opposite direction from gravitation. [Note 163] Taking this fact into account, it becomes possible to relate the integrals to mechanics on the one hand and optics on the other hand. In this way, we formulate mechanics, optics, and so on in usable integrals that encompass the reality of a situation. It is not quite true, however, that the difference between the integrals is zero. In actual fact, it is a differential, and instead of writing zero, I must write:

$$dx = \int_+ - \int_-$$

If repeated searches for such integrals and the resulting differentials lead to differential equations corresponding to \(dx\), I then will see that when I take \(dx\) to be positive here and negative there, \(dx\) is an imaginary number in the mathematical sense.

If I integrate the resulting differential equation, however, the result is astounding. You can experience it for yourselves if you solve the problem correctly. This step leads to acoustics, to acoustical formulas. Thus, you really have used mathematics to apprehend an intrinsic reality. You have learned that we must write mechanics down below on the vertical and vision up above on the vertical—since light is equal to negative gravitation—while hearing, in reality, takes place horizontally. When you set up these calculations, you not only will observe discrepancies—mathematics on the one hand and physics on the other—as a result of the LaGrange equations. [Note 164] But you also will see that the work that can be done on this basis in the realm of mathematics and physics is just as productive as the work 1 pointed to earlier in the domain of phylogenetics. [Note 165] Along these lines—by working things out, not through merely descriptive considerations—we discover the differences between modern natural science and anthroposophy. We will have to demonstrate that our calculations are firmly rooted in concrete realities.

  1. Rudolf Steiner's additional comments during the lecture cycle Der Enstehungsmoment der Naturwissenschaft in der Weltgeschichte und ihre seitherige Entwickelung ('The Emergence of the Natural Sciences in World History and Their Subsequent Development"), GA 326. Comments on the discussion following a lecture by Ernst Blümel (1884–1952) on "Die vier Raumdimensionen im Lichte der Anthroposophie" ("The Four Dimensions of Space in the Light of Anthroposophy"). To date, no transcript of Blümel's lecture has been found.

  2. my lectures: The lectures given on December 26-28, 1922 (GA 326). On tactile and visual space, see Rudolf Steiner's lectures of March 17, 1921 (GA 324), and January 1, 1923 (GA 326).

  3. Rudolf Steiner points to the transition from a sphere to a plane or a circle to a straight line in many different places. See the parallel passages in this volume in the lecture of March 24, 1905, and in the questions and answers of September 2, 1906,; July 28, 1908; and November 25, 1912.

  4. For more about "apprehending reality" through projective geometry, see Rudolf Steiner's lectures of January 11, 1921 (published in Gegenwart ["The Present"], vol. 14, 1952, no. 2, pp. 49-67; planned for publication in GA 73a); April 5, 1921 (GA 76); and the question-and-answer session of April 12, 1922 (GA 324a and 82).

  5. Today inevitable movements are understood as movements possessing only one degree of movement, that is, movements that are so restricted that only one free parameter for movement exists. Presumably, however, what Steiner means here is the very general problem of movement subject to secondary conditions. The Newtonian formulation of mechanics proves unwieldy in calculating movements subject to secondary conditions. Furthermore, this formulation made it difficult to introduce standard, non-rectilinear coordinates for movement. The LaGrange equations, which are based on a principle of mechanical variation, offer elegant solutions to both problems.

  6. See Rudolf Steiner's lecture of December 27, 1922 (GA 326).

  7. On negative gravitation, see Rudolf Steiner's lectures of January 7 and 8, 1921 (GA 323).

  8. LaGrange equations: Joseph-Louis LaGrange (1736–1813), mathematician, physicist, and astronomer in Turin, Berlin, and Paris. The derivation, discussion, and application of the equations later named after LaGrange constitute the majority of his book Mécanique Analitique (Paris, 1788). On the LaGrange equations, see Note 159.

  9. Phylogatetics: See Rudolf Steiner's lecture of December 28, 1922 (GA 326).