The Fourth Dimension
Questions and Answers II
28 June 1906, Nürnberg
QUESTION: Since [Note 6] time had a beginning, it is obvious to assume that space also has limits. What is the reality of the situation?
That's a very difficult question, because the faculties needed to understand the answer cannot be developed by most people of today. For now, you will have to simply take the answer at face value, but a time will come when it will be understood completely. The physical world's space with its three dimensions, as we human beings conceive of it, is a very illusory concept. We usually think that space either must reach to infinity or have boundaries where it is somehow boarded up and comes to an end. Kant put forward these two concepts of the infinity and the finiteness or limitedness of space and showed that there is something to be said for and against both of them. [Note 7]
We cannot judge the issue so simply, however. Since all matter exists in space and all matter is a condensed part of spirit, it becomes evident that we can achieve clarity on the question of space only by ascending from the ordinary physical world to the astral world. Our non-clairvoyant mathematicians already have sensed the existence of a strange and related phenomenon. When we imagine a straight line, it seems to reach to infinity in both directions in our ordinary space. But when we follow the same line in astral space, we see that it is curved. When we move along it in one direction, we eventually return from the other side, as if we were moving around the circumference of a circle. [Note 8]
As a circle becomes larger, the time needed to go around it grows longer. Ultimately, the circle becomes so huge that any given section seems almost like a straight line because there is so little difference between the circle's very slightly curved circumference and a straight line. On the physical plane, it is impossible to return from the other side as we would do on the astral plane. While the directions of space are straight in the physical world, space is curved in the astral world. When we enter the astral realm, we must deal with totally different spatial relationships. [Note 9] Consequently, we can say that space is not the illusory structure we think it to be but a self-contained sphere. And what appears to human beings as physical space is only an imprint or copy of self-contained space. [Note 10] Although we cannot say that space has limits where it is boarded up, we can say that space is self-contained, because we always return to our starting point.
This question-and-answer session took place during the lecture cycle The Apocalypse of St. John (GA 104).
Kant, Prolegomena to Any Future Metaphysics , "Cosmological Ideas", §50- 53, — and Critique of Pure Reason , "The Antinomies of Pure Reason, the First Conflict of Transcendental Ideas," §454ff. Kant shows that arguments can be presented both for and against the infinity of space. For him, the origin of this contradiction lies in the implicit assumption that space and its objects must be taken as absolute givens and as objective laws of things as they are ("von Dingen an sich"). If they were understood as what Kant says they are — namely, mere mental images (ways of looking at things, or phenomena) of things as they are — then the "conflict of ideas" dissolves.
Rudolf Steiner's statements here are based on the discovery that Euclidean geometry is embedded in projective geometry. A Euclidean straight line disappears into infinity in both directions, and the right and left directions are separated by infinity (the distant point). A projective straight line has no such limits — with regard to the sequence of its points, it is closed like a circle.
The text that has been preserved is insufficient to reconstruct whether Steiner attributes an actual geometric curve to astral space. In any case, a self-contained projective straight line is not curved. It is possible that Steiner simply wanted to point out the structural relationships on a projective straight line and how they behave on the circumference of a circle.
Here, too, Steiner presumably uses the term sphere only to draw attention to the self-contained character of astral space in the sense of a projective space. In the topological sense, neither the projective plane of a two-dimensional sphere nor the projective space of a three-dimensional sphere is equivalent.