The Fourth Dimension
Questions and Answers VII
25 November 1912, Munich
QUESTION: Has spiritual science achieved anything with regard to the fourth and higher dimensions?
It [Note 21] is not easy to make the answer to your question understandable. We human beings start from what we know from the physical, sense-perceptible world, where space has three dimensions. At least on a theoretical level, mathematicians formulate ideas about a fourth dimension and higher dimensions by analytically expanding their ideas on three-dimensional space through variables. At least in the context of mathematical thought, therefore, it is possible to speak of higher manifolds. [Note 22] For those familiar with these issues — that is, for those who put heart and soul into the question and also have the necessary mathematical knowledge— many things come to light. Let me mention Simony in Vienna as an example. [Note 23]
Initially, higher dimensions exist only in ideas. Actually seeing them begins when we enter the spiritual world, where we are immediately forced to come to grips with more than three dimensions. There, any image presented to us — that is, anything that still possesses intrinsic characteristics of three-dimensionality — is nothing more than a reflection of our own soul processes. In the higher worlds, very different spatial relationships prevail, if indeed we still want to call them spatial relationships. The same is true with regard to time. There are always many people who argue, How can we be sure that all your claims are not based on hallucinations? Such people need to consider the situation with regard to time, because they disregard the fact that the field of spiritual science works with phenomena that are totally different from hallucinations. Your question provides an opportunity to supplement what 1 said in the lecture, because it is never possible to say everything, and today's lecture was very long. Let me still point out the changes that take place with regard to time and space when we enter the spiritual world.
The return of the images that we have banished to Hades, as it were, makes sense only when approached in terms of higher dimensions. There, however, this is just as natural and self-evident as three-dimensionality is in the sense-perceptible world. That is why ordinary geometry is a poor match for the beings and events of the spiritual world. On behalf of mathematicians, it must be said that their speculations about the fourth dimension acquire real value when we enter the spiritual world. Usually, however, their conclusions about higher-dimensional space are only generalizations based on Euclidean three-dimensional perceived space rather than on reality, to which their conclusions do not fully correspond. We would need still better mathematics in order to perform calculations regarding the beings and events that spiritual researchers investigate.
And yet the answer to your question is "yes." Correlations to a suprasensible world, and also mathematical ideas about infinity, become a reality, especially certain subjects from the fringes of mathematics. Here is an example that I myself experienced many years ago. I know that I had a sudden flash of insight into an extremely important attribute of astral space when 1 was studying modern synthetic projective geometry and analytical mechanics at the university. [Note 24]
There is a relationship here to the concept that, on a straight line extending to infinity, the infinitely distant point on the left is identical to the infinitely distant point on the right. That a straight line, with regard to the arrangements of its points, is really a circle,- if we do not get winded and continue in a straight line long enough, we return from the other side. [Note 25] We may understand this, but we should not draw conclusions from it, since conclusions lead nowhere in spiritual research. Instead, allowing phenomena to work on us leads to knowledge of the supra-sensible world.
It is important not to overestimate mathematics when dealing with the supra-sensible world. Mathematics is useful only on a purely formal level. It cannot possibly grasp the reality of the situation. Like spiritual science, however, mathematics can be understood by means of forces inherent in the soul itself and is equally true for everyone. That is what mathematics and spiritual science have in common.
This question and answer session took place after a public lecture on "Wahrheiten der Geistesforschung," which was published in the periodical Mensch und Welt: Blätter für Anthroposophie, vol. 20, 1968, no. 5, pp.167—177. It has not yet been published in the complete edition (GA) of Rudolf Steiner's works.
Here Rudolf Steiner refers again to Bernhard Riemann's studies, mentioned several times in the lectures. See Note 1, Lecture I.
0skar Simony (1852-1915). See Rudolf Steiner's lecture of March 24, 1905 (Lecture 1), and Note 14, Lecture I.
See Rudolf Steiner, Autobiography (GA 28).
See the answers to the preceding questions and the accompanying notes.