## The Fourth Dimension

GA 324a

### Questions and Answers XV

30 March 1920, Dornach

*QUESTION: How will anthroposophy affect the further evolution of chemistry?*

Assuming [Note 77] that we undertake the type of phenomenology described by Dr. Kolisko, this question is so all-encompassing that the answer can only be hinted at. First and foremost, we must realize that we would have to develop an appropriate phenomenology. Phenomenology is not simply an arbitrary assemblage of phenomena or experimental results. Real phenomenology is a systematization of phenomena, such as that attempted by Goethe in his theory of color. [Note 78] It derives the complicated from the simple, leading back to the foundations where the basic elements or phenomena appear.

Of course, I am quite aware that some truly intelligent people will argue that a sophisticated presentation of the connection between qualitative phenomena and archetypal phenomena is not comparable to the way in which complicated geometric relationships are mathematically derived from axioms. This is because geometric relationships are systematized on the basis of intrinsic structure. We experience the further development of mathematics from these axioms as an inherently necessary continuation of the mathematical process, while, on the other hand, we must depend on observing a physical state of affairs when we systematize phenomena and archetypal phenomena.

This argument, though it enjoys widespread support, is not valid and is simply the result of an incorrect epistemology, specifically, a confused mingling of the concept of experience with other concepts. This confusion results in part from failure to consider that human subjects shape their own experience. It is impossible to develop a concept of experience without imagining the connection of an object to a human subject. Suppose I confront a Goethean archetypal image. When I make it more complicated, the result is a derivative phenomenon, and I seem to depend on outer experience to support my conclusion. Is there any difference, in principle, between this subject-object relationship and what happens when I demonstrate mathematically that the sum of the three angles in a triangle is 180° or when I prove the Pythagorean theorem empirically? Is there really any difference?

In fact, there is no difference, as became evident from studies by very gifted nineteenth- and twentieth-century mathematicians who realized that mathematics ultimately also rests on experience in the sense in which the so-called empirical sciences use the term. These mathematicians developed non-Euclidean geometries that initially merely supplemented Euclidean geometry. [Note 79] Theoretically, the geometric thought that the three angles of a triangle add up to 380° is indeed possible, though admittedly we must presuppose that space has a different rate of curvature. [Note 80] Our ordinary space has regular Euclidean measurements/ dimensions and a rate of curvature of zero. Simply by imagining that space curves more, that is, that its rate of curvature is greater than I, we arrive at statements such as: The sum of the three angles of a triangle is greater than 180°. Interesting experiments have been conducted in this field, such as those of Oskar Simony, who has studied the subject in greater detail. [Note 81] Such efforts show that from a certain perspective, it is already necessary to say that conclusions we state in mathematical or geometric theorems need empirical verification as much as any phenomenological conclusions.

Question-and-answer session after

Eugen Kolisko'slecture on "Anthroposophy and Chemistry" during the conference on "Anthroposophy and the Specialized Sciences" held at the Goetheanum in Dornach from March 21 to April 7, 1920.Eugen Kolisko(1893–1939) was a physician and taught at the first Waldorf School in Stuttgart. To date, no transcript of his lecture has been discovered. See the brief report on the conference in the journalDreigliederung des sozialen Organismus('The Threefolding of the Social Organism"), vol. 1, 1919/1920, no. 45.Goethe,

Zur Farbenlehre("On Color Theory") [1810] andDer Versuch als Vermittler von Objekt und Subject('The Experiment as Mediator Between Object and Subject") [1823]. See Rudolf Steiner'sEinleitungen zu Goethes Naturwissenschaftlichen Schriften("Introduction to Goethe's Natural Scientific Works," GA 1), chapters X and XVI;Grundlinien einer Erkenntnistheorie der Goetheschen Weltanschauung("Outline of an Epistemology of the Goethean Worldview," GA 2), chapter 15; and the chapter inGoethe's Weltanschauung("Goethe's Worldview," GA 6) entitledDie Erscheinungen der Farbenwelt('The Phenomena of the World of Color").The discovery of non-Euclidean geometries showed that Euclidean geometry was not the only imaginable geometry. As a result, the question of which type of geometry applies to the space we experience became an epistemological problem for the sciences. For more on the impact of the discovery of non-Euclidean geometries, see also Rudolf Steiner's lectures of August 26, 1910 (GA 125); October 20, 1910 (GA 60); January 3, 1920 (GA 320),- March 27, 1920 (GA 73a),— January 1 and 7, 1921 (GA 323); and April 5, 1921 (GA 76). On the importance of the discovery of non-Euclidean geometry in the history of consciousness, see Ziegler [1987], On the history of this discovery, see Bonola/Liebmann [1919],- Klein [1926], chapter 4; and Reichardt [1976]. On the relationships of axioms, archetypal phenomena, and experience, see Ziegler [ 1992], chapters VII and VIII.

In an elliptical geometry such as Riemann's (Riemann [1867]), the rate of curvature of measurement is greater than 1, and the sum of the angles of a triangle is always greater than 180°. In

hyperbolic geometry, the rate of curvature of measurement is less than 1, and the sum of the angles of a triangle is always less than 180°. The relationship of spaces or manifolds with a constant curvature to non-Euclidean geometries was discovered byEugenio Beltrami(1835–1900) andBernhard Riemann(1826–1866). In contrast to Euclidean geometry (Pythagorean theorem), the measurement of such a space is determined by a function of the coordinates. In general, this function is no longer a sum of squares. On this subject, see Klein [1927], chapter 3C, and Scholz [1980], chapter III.See Simony [1888b], §5; [1883]; and [1886].